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https://rpg.stackexchange.com/questions/61840/how-much-is-a-gold-piece-really-worth/61867
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# How much is a Gold Piece really worth?
I have been wondering what a gold piece is worth, due to the fact there are many things that cost the same, but in reality, are probably worth radically different prices, such as a Goat and a Whip, which both cost 1 gp.
What is a gold piece really supposed to be worth? Like, how would things be priced if they were in familiar modern monetary units instead of "gp"? That would give me something to base my adjustments to abnormal prices on, such as items not listed in the PHB.
• Don't answer in comments. Especially not replicas of existing answers. Thanks. May 11 '15 at 21:42
• May 11 '15 at 22:18
• See related meta discussion: Why is “How much is a Gold Piece really worth?” closed? Jul 18 at 1:16
# It's nearly impossible to put a modern-world value on 1 GP
...because things don't have the same relative values in our world as they do in a typical medieval-style adventuring world that is pre-industrial, but has magic.
As you've already noted, 1 GP is worth about 1 goat or about 1 whip. It's also good for 2 nights' stay in a modest inn, or 5 gallons of ale. On the other hand, it's only 1/25th of the price of a 1 lb hourglass, 1/50th the cost of a chain shirt, or 1/1000th the price of a spyglass. These aren't items that are all of equivalent values in modern terms, so it doesn't make sense to try to assign a modern value to a gold piece. The gold piece has value exactly in relation to what kinds of items one can purchase with it.
Today (May 2015) you could get:
• a goat for about $75 -$300
• a bull whip for about $30 • 2 nights stay in a modest inn for about$100 - $150 • 5 gallons of beer for about$30
• a 1 lb hourglass for about $30 (so 1/25 of it is just over$1)
• a combat-grade chain shirt for $500 -$1000 (so 1/50 of it is $10 -$20)
• a spyglass for about $150 (so 1/1000 of it is 15 cents) So, by using modern item values, we might say that 1 GP is worth somewhere between 15 cents and$200 in $US. But before you dismiss the prices as being "inconsistent" with modern values, consider your setting. Relative prices are much different now. Many things are much easier to make, and the cost of hand labor is reduced. Other things may be more in demand or uncommonly made now, and cost relatively more. If you want to build a more modern setting for your world, you might as well just use a modern currency. As @KRyan notes in the comments: It may be worth noting that historians also generally consider it impossible to give a decent sense of what real world currencies were worth in the not-so-distant past. There is no meaningful conversion of 18th-century US dollars to 21st-century US dollars. There have been times and places where an entire house was worth less than the nails used in its construction, to the point that people would burn down their houses and collect the nails from the ashes before moving. See Were iron nails at one time so scarce that pioneers in America burned down their cottages to retrieve them? on [skeptics] for more information on that. Additionally, as other answers have noted, the developers set prices with game balance as a higher priority than the creation of a realistic economy. • +1 for pointing out that the relative value of stuff is different in a world that has had an industrial revolution when compared to one that hasn't but has magic. May 12 '15 at 1:01 • Try having an artisan hand-make your spyglass, hourglass and bull whip. And beer, no fair using Bud Lite. Thanks to BDSM you know handcrafted whips are about$100-200. Seems like some of the low outliers are owing to mass production, take those away and it does seem to converge on the $150 ballpark. Jan 31 '19 at 3:29 • @harper I do see handmade spy glasses on Etsy for under$100 but of course they're probably still using mass produced lenses and other parts. But that was the point of my answer -- Many things are cheaper now due to the ability to mass produce either the item itself or the parts. Jan 31 '19 at 18:26
• It may be worth noting that historians also generally consider it impossible to give a decent sense of what real world currencies were worth in the not-so-distant past. There is no meaningful conversion of 18th-century US dollars to 21st-century US dollars. There have been times and places where an entire house was worth less than the nails used in its construction, to the point that people would burn down their houses and collect the nails from the ashes before moving. Jul 18 at 0:52
• I’d like a citation on getting five gallons of beer for $30. You know, for science. The more specific the better. Preferably a place near me just so I can verify. Jul 18 at 1:03 # Tradition The relative values are based upon tradition ... the prices are very similar to Gygax's own list in the AD&D rules, which is an expansion of the material in the original D&D game. Gary Gygax, however, probably did not make them up on the spot. In comparing various price lists in various games, Gary's numbers routinely come up around 1gp equaling one roughly 11th Century shilling. Looking at Hodges' list ①, most of the prices come pretty close. # Medieval English Coins A few words about medieval English Coins; they were specie. The values were based upon weight and purity, and the rosy picture of relatively stable prices and uniform coins are an artifact of Fantasy. The nominal base units were the penny (1 dwt of silver), the shilling (12 dwt of silver), and the pound of silver (240 dwt). Gold coins of roughly equivalent value were used. The "gold penny" was a 1dwt coin of gold (debased with copper, tin, and zinc), nominally worth 12 to 20 pence. It was generally accepted as 12 pence, hence a shilling's value - but often was valued more. Note also: dwt (pennyweight) is a part of the troy measures system - 20 dwt to the troy ounce, and 12 trounces to the troy pound. With the above considered, the GP probably has its origin in the gold coin nominally worth 1/20 of a pound of silver. 12 dwt is about 18.67 g, 3x modern £1 coin (within 0.2g). Not a light hunk, but still less than the current £5. And the current UK penny is roughly 2.3 dwt.② 5E returns to the "everyone but Gary & Tom" 50 coins to the pound. We don't know which pound, so we'll assume avoirdupois, at 291.67 dwt. This converts to 5.834 dwt to the coin, or about 9.1 g. This is about the same as the current UK £1, or 4 US dimes. If we instead use Troy Pounds, at 240 dwt each, 4.8 dwt is 7.4g, a touch lighter than the 50p or the US current presidential dollars.②③ Knowing that the gold penny was roughly 1 shilling in value at lowest, we can see where it originates. The fantasy weight of 1/10 pound (in some editions) and 1/50 pound (in the rest) is probably because of looking at debased silver shillings (12dwt) and copper pence. ### So, some specific examples, compared to Hodges list. Wine, hodges lists at 4d to 8d per gallon (1/3 to 2/3 of a shilling), while 5E lists it as 2sp to 10gp... so, 1/3gp to 2/3 GP, vs 1/5 gp to 10gp. A wider range. Probably not the same source, but clearly includes the Hodges range if a shilling is a gold. Tunics, peasant. Hodges says 3s, with shoes 6d and a chemise 8d. 5E shows half a gp for common, 2gp for travelers, so, yes, we're in the ballpark Axe, woodsman's - 5d by hodges, 5 gp by 5e. University boarding: 2s/week, by Hodges, 1.8gp/week for poor by 5E. Close enough. The only armor which is a clean match is leather at 5s being 5gp. The others, however, are in similar ranges. A cheap sword in Hodges is 6s, so that would be 6gp, but D&D traditionally uses 10gp. # Conclusions In short, the list is close enough that it looks like, between Gygax's original research for D&D and AD&D, and modern reworks, the prices are taken from scholarly lists of prices from the medieval period in England and France, both of which used similar currency divisions (£, s, d). If expanding the lists, keeping in mind medieval sources can help keep the prices roughly in line, and that any given list is neither authoritative nor more than just a snapshot. # References ① Hodges, Kenneth. List of Prices of Medieval Items http://www.luminarium.org/medlit/medprice.htm ② Royal Mint. Coin Designs and Specifications http://www.royalmint.com/discover/uk-coins/coin-design-and-specifications ③ United States Mint. Coin Specifications http://www.usmint.gov/about_the_mint/?action=coin_specifications 1gp is 1gp is 1gp. 1 gp is worth 2 ep, 10 sp or 100 cp. That's it. Gold is defined to be worth that amount, and that's all there is to the economy in D&D. Though the specific items that you can purchase differ widely, the prices are set by the designers. So an item that costs 1gp has the same worth in the economy as another item that costs 1gp. Obviously, if your DM wants to change the prices, he's totally in bounds to do that, it's his game, and prices can and should vary by setting. So basically, if a goat and a whip are each worth 1gp, a farmer would likely consider a nice, adventurer-quality whip handed in trade for a goat to be a fair deal. It's worth noting that D&D is a complete and utter command economy. And designing a functional, working, economy is not generally regarded as a design goal for D&D. Things are given prices with some regard to fictional reality, but not very much. Ultimately, if you're concerned with the economy being functional, you probably want to import another economic scheme or find a different game. As shown in other answers, trying to relate 1 gp to a current dollar value is nigh pointless, there's not going to be a consistent currency exchange to be found based on the items that you can purchase in the game (And the economies are so drastically different it's not remotely comparable). The best way to think about it is that 1gp is something like the$20 of the D&D world, it's not the largest unit of currency (that'd be astral diamonds), and it's not the smallest (that'd be the copper piece, the equivalent of a dollar, or maybe even a quarter..it's probably not a penny).
All that to say, trying to do a direct conversion from gp to dollars (or other modern currency), just isn't productive. Think of it as a unit of currency and not much more.
• I think the farmer would want the goat more than a whip personally. ...There's an idea for dungeon treasure. "Among the ogre's possessions you find 2 goats and a cow." Dec 17 '15 at 15:07
• @WesleyObenshain Yes, the farmer would probably want the goat more than a whip, but i would still be a fair trade economically, which is the point here. I do like finding goats among an ogre's possessions though. It seems realistic. Nov 10 '19 at 16:05
# D&D doesn't use a functioning economy - prices are instead set for game balance.
Why does a whip cost as much as a goat? Because the designers figured that was a good cost for both. You can carry a pile of swords into town, only get 50% of their value, but the price to buy a sword doesn't change - because the designers don't want you playing merchant.
There's plenty of examples of where D&D pricing makes no sense economically. My personal favorite is that in 3.5, you can buy a 10 foot ladder for 5 cp (or 0.05gp), saw it in half, and you now have 2 10 foot poles, which even at the 50% mark down net you 1 sp (0.10gp) each. So at first level, the smart adventurer should be busily buying ladders, breaking them down to poles, and selling them for 200% profit.
For practical advice, if you're serious about playing a mercantile campaign (where you plan to do a large amount of buying and selling), you're going to want another system (either in addition to D&D or instead of).
• Actually, that's 300% profit, excluding labor rates. 20cp revenue - 5 cp costs = 15 cp profit. 15 cp profit / 5 cp cost = 300% profit. On the other hand, if it takes you 20 minutes to do the sawing, and sanding to get to the finished product of a 10' pole (that'd be fast using hand tools), you're looking at 15 cp/hr for your labor. And the moment the ladder guy realizes what you're doing, he'll start doing it, too. Except that he gets to save labor by doing it, because he doesn't have to drill holes for rungs once he's got his poles cut, and you're immediately priced out of the market. Dec 17 '15 at 17:36
# It's a Currency, Not a Commodity
You can look at the gold piece (GP) kind of like the American Dollar. It has whatever value people assign to it. The goal of having a currency is to give a universal portable way to conduct transactions. In the D&D universe gold is that currency. There are a few abstractions though.
Universal value For ease of use the game system assumes that every region values goods the same way, i.e. a tiny farming town values a whip as much as a merchant in Waterdeep.
There are few other currencies. Unlike the real world, most countries all settled on a single choice. There are some localized currencies described in the core books, but they are more for fluff than anything.
It's all from an adventurer's perspective. When an adventurer needs a goat in a hurry to lure out the hydra, he doesn't shop around or buy in bulk. If you bought a whole herd of goats, or maybe found a farmer down on his luck, you might be able to get a goat for less than 1GP. The list price represents a cost that any merchant would gladly sell that object at.
Estimated real-world value: ~$50 - This is a very tough way of looking at item costs. It would be more effective to think in terms of rarity and amount of work required. In short, there isn't a very good estimate for the previous reasons, but we'll try here. If we use the items from earlier, some quick Google-fu shows that you can get a decent quality bull whip for ~$40 and a goat for around the same ~$50. So using just those 2 items a gold is worth roughly$50. It doesn't work as well though when you look at something like a stay at an inn. I'd be hard pressed to find anything above the most middling hotels that would offer me a room for less than $50, and one night in an inn costs less than 1 GP. Most of the answers seem to be going by the purchasing power route, but to answer it by the simple expedient of what that amount of gold is worth: • Per the PHB, 50 coins weigh a pound. • One common (avordupois) pound is 14.6 troy ounces (the unit used for measuring precious metals.) • Therefore one gold piece is .292 troy ounces of gold. • At the time of writing gold trades at 1,183.80 USD per troy ounce. • So a GP is just over$345 worth of gold.
(That said, pay good attention to Allen Gould's answer-- this is a pure thought experiment, it isn't going to help you make any sense of the prices in the PHB!)
• As a pure thought experiment that does not help solve the problem stated in the question, this is currently not an answer (and that's grounds for deletion). So that this is a real answer on its own, it's preferable that it answer the question independently by including any necessary statements, and refer to other answers only to give due credit. May 12 '15 at 21:16
Basically, the Gold Piece (GP) has been the standard unit of currency in D&D since the beginning. Why? Because something had to be. You should also note that historically, the GP was used as a measure of weight (IIRC, 10 GP weight was 1 pound). Also, back when TSR published the books as stone tablets (joke here guys. Obviously they never did that), you would get 1 XP for every GP you found (other items of treasure counted as XP in the same way, which is why there were monetary values for magic items in the old books).
Now, as far as GP goes, you can think of it as a dollar (or other decimal based currency). With this in mind, a Copper Piece (CP) would be like a penny, Silver Piece (SP) would be a dime, and Electrum Piece (EP) would be a 50 cent piece (although they were rare and a gold/platinum alloy) and a Platinum Piece (PP) would be a $10 bill. Now, looking at your comparison of a goat and a whip, each being the same value... think about this in the world of your favorite fantasy books. A farmer would have very little use for a adventurer's whip, in fact, it would be fair to say that the whip is much more rare than a goat. Having similar values is reasonable. What else can you buy with that GP? 10 night's stay in an inn, for instance. Can you, in the modern day, sell a goat and have enough money to pay for 10 nights in a hotel? Probably not. Could you in 12th Century Europe (standard technological setting for most fantasy novels and games)? Honestly, maybe, but I don't have barter records for things like that. TL;DR GP is just a unit of measure that is more or less arbitrary to allow you to buy something one place and pay about the same amount for the same thing someplace else. Additional info: Today (May 12, 2015) gold is selling for$1190 per ounce. 12 Troy Ounces to a pound would yield about $14,280 for a pound of Gold. Using the standard D&D weight of 50GP to a pound, we find that each GP would be worth about$285, assuming that the gold is commodity quality. Gold is very soft and wouldn't last long with all of the adventurers carrying it around, dragons hoarding it up and goblins stealing it, so we can assume that there would be some other medal (most likely Copper) in there. Just for easy math, we'll assume that each GP would be about $250 in today's US currency. So, the room that you paid 1 SP for was around$25, and when the "noble" in your party upgraded, he paid $250 for the room (I'll bet they didn't even put a mint on his pillow!). All kidding aside, it's best to just think about GP as an arbitrary currency because something has to be and it might as well be GP (it could be Rubies or something else). GP use originally used because everybody knows what Copper, Silver and Gold are. ### Question: What's a gold piece really worth? ## Short answer That depends on where you are, and when you are in your game setting. The worth is influenced by how much "the game economy" interests the players and the GM. ### Longer Answer (includes RAW reference). Regarding your included question on abnormal prices, why not consider the following from the PHB as the point of departure? PHB Table Extract. (p. 157). Cost Trade Goods 1 cp 1 lb. of wheat 2 cp 1 lb. of flour or one chicken 5 cp 1 lb. of salt 1 sp 1 lb. of iron or 1 sq. yd. of canvas 5 sp 1 lb. of copper or 1 sq. yd. of cotton cloth 1 gp 1 lb. of ginger or one goat 2 gp 1 lb. of cinnamon or pepper, or one sheep 3 gp 1 lb. of cloves or one pig 5 gp 1 lb. of silver or 1 sq. yd. of linen 10 gp 1 sq. yd. of silk or one cow 15 gp 1 lb. of saffron or one ox ### Setting an Environment 1. The 5E table above provides context, and a rough relative value for a pseudo-medieval setting that is the game's basis. As noted by others, this carries over from earliest editions' "Western Europe Medieval / Feudal" default environment. 2. Note the prices on spices in the table. Spices were very expensive in medieval times. They were the source of mercantile fortunes on the order of the Tech / Dot.com fortunes of the modern day. Seeking fortunes in the spice trade led to the discovery of the New World, among other things. Another commodity of ancient value was salt. For centuries salt had far greater trade value than what we see today as a mundane condiment and cooking material. Spices are still more expensive than salt. ### Trade Goods Table / GP Values can add game depth and flavor The table provides some Order of Magnitude value comparisons. The values are rough. They offer opportunities for game play color / flavor in the shape of treasures. 1. Example: Trade Goods as Treasure Your party finds three bolts of silk as part of a treasure once the ogre is dead. Yards and yards of it. (30 yards would be ~ 300 GP). Treasure indeed! Silk in the default setting is very valuable. 2. Example: Varying PHB GP values with a change in setting Your game setting moves to somewhere modeled after medieval-era Damascus, Baghdad, or Samarkand. Your GM could cut in half or less the prices for some of these commodities since outside of the Western medieval context, the wholesale / closer-to-the-source markets would sell these goods for less. If the base adventure is set in a silk producing region, perhaps the GM cuts that cost drastically. An evening's adventure becomes: "The party are guards for the silk / spice caravan journeying into the barbarian north where they pay in silver and gold for this stuff. Between us and the market are Brigands and Bandit and Bugbears (oh my!)" 3. Example: cattle as a trade good versus investment property Values are inexact. At 10 gp for a cow, why is an ox worth 15 gp? The ox, a draft animal, can help raise cash crops like oats or wheat. A bull might be more valuable as it can increase one's capital / herd. (Consider stud fees ...) Oxen were typically neutered to make them more controllable as beasts of burden. A dairy cow turns grass into milk -> butter or cheese. Is she the better value if you keep her healthy? ### What do the PCs need gold/money for? Back to game play: how many cows (or camels or llamas or goats) must I offer in trade for a chain shirt that our fighter needs? The PHB GP values give you a place to begin for answers to that question. But wait, where did you get all of those cows? Is there someone looking for a party of cattle rustlers? More in-game play depth to the game economy. How deep into trade, cost, and value are your players going to enjoy in this game? Is it more important at low levels, when gold is scarce, or at higher levels when your players may have castles, holdings, retainers, and henchmen? If you need money to raise an army to fight large hobgoblin raiding parties, economics can play a larger role. ### Follow up points on currency (money) and value. • The Personal Computer I currently use has orders of magnitude greater computing power than the one I bought in 1987. I paid$1200 in 2012-dollars for this one. Price tag wise, that is $300 dollars (unadjusted) less than the$1500 I paid in 1987 for an AT/XT clone. (Pre-Pentium desktop). What a (currency unit) buys you varies over time. (Computers are an extreme case).
• There was still a salt tax in the 19th-20th centuries in India, a cause celebre' of the Independence movement against the British Empire. Is something like that - a salt tax - worth folding into your campaign?
• One of the best guidelines from 1st Edition AD&D was the point about "boom town economy" pricing. That's "D&D lore" but not "5e RAW" to help frame the answer.
## Summary
A gold piece is worth as much as it needs to be for the playing of the game to be fun for the GM and the players. The Trade Goods table, and the costs on the various equipment tables in the PHB, all provide a frame of reference from which you can modify costs per whatever impacts on the game economy you choose to introduce.
It's hard to directly compare value, because some things are much cheaper in the modern economy. See, for example, a sheet of paper (0.4gp each).
However using the 'armour standard', a full suite of plate today will set you back around £5750 (~$9000) for a '1500gp' suit of armour. That puts the GP to dollar ratio at about 6. But a goat — you can get a pair for around$125, putting the GP to dollar ratio at around 60.
Hemp rope — 1gp for 50 ft. Today, 36mm hemp rope is around £13/meter. So 50 ft. is $300. I think pretty fundamentally, it's just very hard to converge the prices — they're largely arbitrary, and geared around game balance. Doubly so when you start straying into magic item territory — after all, if you need a '500gp diamond' to resurrect someone, then that'll make diamonds something of a significant demand item, above and beyond 'normal'. (I might suggest all diamonds would be this price, as the price is the determining factor in its usage). All of these answers have looked at the value of the gold piece as currency. For my take on this see my answers to: The value of a gold piece, like the value of a dollar, is equal to what someone will give you for it - this depends on supply and demand and the relative negotiating abilities of the parties at the particular time in the particular circumstance. Gold is money in D&D and it is money in our world - the value of one form of money against another (e.g. AUD, USD) depends on the exchange market - there is no real equivalent of this in a D&D world. Unlike our fiat money, gold is a commodity money - that is, it has intrinsic value. Heat up gold coins and you get gold bullion, heat up dollar bills and you get ash. No one has taken the question at simple commodity value. A gold piece is 1/50 lb or 0.32 oz, at 10:14am (Australian Eastern Time) on 12 May 2015, 1 oz of gold was worth 1,183.44 USD. If a gold piece is pure gold, then it is worth 378.70 USD as bullion. As a gold piece it is worth 10-15% more than that based on the historical values of commodity money - the 10-15% representing the degree of trust in the exchangeability of the money. Compare this to silver at$16.25/oz and copper at $0.18/oz. This gives: 1 oz gold = 72.8 oz silver = 6,574.7 oz copper To make 1gp = 10sp = 100 cp and assuming they are the same weight and that to cp is pure copper, the sp is about 11% silver and the gp is about 1.5% gold at todays valuations - clearly ease of use has won out over realism as it always should. There's a lot of overthinking going on around here. I have found it useful to approximate 1 GP to be about$25, thus 1 SP is about \$2.50, and 1 CP is a quarter. This lets you wrap your head around values of goods, and being able to make crude estimates of prices without too much effort.
• Why do you use these numbers in particular? Aug 18 '20 at 11:47
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2021-10-23 05:56:26
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https://how-to.aimms.com/examples/modeling-book/diet-problem.html
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# Diet Problem¶
Direct download AIMMS Project DietProblem.zip
Go to the example on GitHub: https://github.com/aimms/examples/tree/master/Modeling%20Book/Diet%20Problem
This example introduces a simplified diet problem. The example considers a small data set, which includes 9 different food types and 4 different nutrients. The 9 food types form a small but representative selection of the McDonald�s menu. The 4 nutrients are calories, protein, fat, and carbohydrates. The goal is to determine a daily diet to cover the afternoon and the evening meals.
This example illustrates how data consistency can be ensured through measurement units. Another feature is the availability unit conventions that allow users with different backgrounds to view and enter data in their own choice of measurement units without having to change either the model or its data.
Details about this example can be read in Chapter 10 of the AIMMS Optimization Modeling book. An electronic version of this book is available through the ‘Help’ menu.
For this example you can use the AIMMS WebUI for the graphical user interface. To use the AIMMS WebUI you should select ‘Tools - Start Web UI’ from the menu bar. This will open the home page in a browser.
Keywords: Diet problem, Blending problem, Unit-valued parameter, Unit parameter, Quantity, Unit conversion, WebUI
Last Updated: September, 2020
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2020-09-22 10:50:36
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https://statistics.fas.harvard.edu/dempster-award
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# Dempster Award
The Arthur P. Dempster Fund “will support and recognize promising graduate students within the Department of Statistics, in particular those who have made significant contributions to theoretical or foundational research in statistics.” It will be an annual award with a prize minimum of \$2000. The expectation is to award one per year, though the faculty reserves the right to award two or none in any particular year depending on the quality of the submissions.
It was announced in May, 2012 that Alexander Blocker is the inaugural Arthur P. Dempster award winner. Here is the abstract for his research presentation.
### The Potential and Perils of Preprocessing: A Multiphase Investigation
Preprocessing forms an oft-neglected foundation for a wide range of statistical analyses. However, it is rife with subtleties and pitfalls. Decisions made in preprocessing constrain all later analyses and are typically irreversible. Hence, data analysis becomes a collaborative endeavor by all parties involved in data collection, preprocessing and curation, and downstream inference. Even if each party has done its best given the information and resources available to them, the final result may still fall short of the best possible when evaluated in the traditional single-phase inference framework. This is particularly relevant as we enter the era of "big data". The technologies driving this data explosion are subject to complex new forms of measurement error. Simultaneously, we are accumulating increasingly massive databases of scientific analyses. As a result, preprocessing has become more vital (and potentially more dangerous) than ever before. In this talk, we propose a theoretical framework for the analysis of preprocessing under the banner of multiphase inference. We provide some initial theoretical foundations for this area, building upon previous work in multiple imputation. We motivate this foundation with two problems from biology and astrophysics, illustrating multiphase pitfalls and potential solutions. These examples also serve to emphasize the practical motivations behind multiphase analyses --- both technical and statistical. This work suggests several rich directions for further research into the statistical principles underlying preprocessing
It was announced in May, 2013 that Bo Jiang is the second Arthur P. Dempster award winner. Here is the abstract for his paper.
### From SIR to SIRI: Sliced Inverse Regression with Interaction Detection
Variable selection methods play important roles in modeling high dimensional data and are keys to data-driven scientific discoveries. In this paper, we consider the problem of variable selection with interaction detection under the sliced inverse index modeling framework, in which the response is influenced by predictors through an unknown function of both linear combinations of predictors and interactions among them. Instead of building a predictive model of the response given combinations of predictors, we start by modeling the conditional distribution of predictors given responses. This inverse modeling perspective motivates us to propose a stepwise procedure based on likelihood-ratio tests that is effective and computationally efficient in detecting interaction with little assumptions on its parametric form. The proposed procedure is able to detect pairwise interactions among p predictors with a computational time of O(p) instead of O(p 2 ) under moderate conditions. Consistency of the procedure in variable selection under a diverging number of predictors and sample size is established. Its excellent empirical performance in comparison with some existing methods is demonstrated through simulation studies as well as real data examples.
It was announced in March, 2014 that Peng Ding is the third Arthur P. Dempster award winner. Here is the abstract for his paper.
### A Paradox from Randomization-Based Causal Inference
Under the potential outcomes framework, causal effects are defined as comparisons between the potential outcomes under treatment and control. Based on the treatment assignment mechanism in randomized experiments, Neyman and Fisher proposed two different approaches to test the null hypothesis of zero average causal effect (Neyman's null) and the null hypothesis of zero individual causal effects (Fisher's null), respectively. Apparently, Fisher's null implies Neyman's null by logic. It is for this reason surprising that, in actual completely randomized experiments, rejection of Neyman's null does not imply rejection of Fisher's null in many realistic situations including the case with constant causal effect. Both numerical examples and asymptotic analysis support this surprising phenomenon. Although the connection between Neymanian approach and the Wald test under the linear model has been established in the literature, we provide a new connection between the Fisher Randomization Test and Rao's score test, which offers a new perspective on this paradox. Further, we show that the paradox also exists in other commonly used experiments, such as stratified experiments, matched-pair experiments and factorial experiments. (https://arxiv.org/abs/1402.0142)
It was announced in March, 2015 that Panagiotis Toulis is the fourth Arthur P. Dempster award winner. Here is the abstract for his paper.
### Implicit Stochastic Approximation for Principled Estimation with Large Datasets
The ideal estimation method needs to fulfill three requirements: (i) efficient computation, (ii) statistical efficiency, and (iii) numerical stability. The classical stochastic approximation of Robbins & Monro (1951) is an iterative estimation method, where the current iterate (parameter estimate) is updated according to some discrepancy between what is observed and what is expected, assuming the current iterate has the true parameter value. Classical stochastic approximation undoubtedly meets the computation requirement, which explains its popularity, for example, in modern applications of machine learning with large datasets, but cannot effectively combine it with efficiency and stability. Surprisingly, the stability issue can be improved substantially, if the aforementioned discrepancy is computed not using the current iterate, but using the conditional expectation of the next iterate given the current one. The computational overhead of the resulting implicit update is minimal for many statistical models, whereas statistical efficiency can be achieved through simple averaging of the iterates, as in classical stochastic approximation (Ruppert, 1988). Thus, implicit stochastic approximation is fast and principled, fulfills requirements (i-iii) for a number of popular statistical models including GLMs, GAMs, and proportional hazards, and it is poised to become the workhorse of estimation with large datasets in statistical practice
## 2016 Awards
In April, 2016, two students were awarded the Arthur P. Dempster Prize.
Anqi Zhou was recognized for her paper, "Randomization-Based Causal Inference from Unbalanced 2^2 Split-Plot Designs."
David Jones was recognized for his paper, "Designing Test Information and Test Information in Design."
## 2017 Awards
In April 2017, two students were awarded the Arthur P. Dempster prize.
Xinran Li was recognized for his paper, "Randomization Inference for Peer Effects."
Espen Bernton was recognized for his paper, "Inference in Generative Models Using the Wasserstein Distance."
In April 2018, Ruobin Gong was awarded the Arthur P. Dempster prize for her paper, "Conditioning Rules for Sets of Probabilities: Dilation, Sure Loss, and Simpson's Paradox."
Here is the abstract for her paper:
Statistical modeling using sets of probabilities offer a low-resolution alternative to precise probabilities. They alleviate the need to make unwarranted modeling assumptions, and can help reduce irreplicable findings. However, sets of probabilities pose a novel challenge on how to properly handle model conditioning in light of new information. Different conditioning rules may lead to different posterior inference from the same model, and may exhibit dilation, contraction and sure loss, paradoxical phenomena never to be seen in precise probability conditioning.
In this talk, I reaffirm the indispensability of sets of probabilities in expressing uncertain inference, through demystifying a collection of famous statistical paradoxes’’ within a common framework. I show that a logical fallacy stems from a set of marginally plausible yet jointly incommensurable assumptions, well-captured by a set of probabilities. We revisit the three prisoners/Monty Hall problem and Simpson’s paradox, and establish equivalence between each problem with a set-of-probabilities model equipped with a paradox-inducing conditioning rule. I also discuss theoretical posterior discrepancies between the generalized Bayes rule, Dempster's rule and the Geometric rule as alternative conditioning rules for Choquet capacities of order 2. Our findings highlight the invaluable role of judicious judgment in the handling of low-resolution statistical information.
Joint work with Xiao-Li Meng (arXiv:1712.08946).
Wenshuo Wang has been named the 2019 Arthur P. Dempster award winner for his paper "Metropolized Knockoff Sampling."
Abstract:
Model-X knockoffs is a wrapper that transforms essentially any feature importance measure into a variable selection algorithm, which discovers true effects while rigorously controlling the expected fraction of false positives. A frequently discussed challenge to apply this method is to construct knockoff variables, which are synthetic variables obeying a crucial exchangeability property with the explanatory variables under study. This paper introduces techniques for knockoff generation in great generality: we provide a sequential characterization of all possible knockoff distributions, which leads to a Metropolis–Hastings formulation of an exact knockoff sampler. We further show how to use conditional independence structure to speed up computations. Combining these two threads, we introduce an explicit set of sequential algorithms and empirically demonstrate their effectiveness. Our theoretical analysis proves that our algorithms achieve near-optimal computational complexity in certain cases. The techniques we develop are sufficiently rich to enable knockoff sampling in challenging models including cases where the covariates are continuous and heavy-tailed, and follow a graphical model such as the Ising model.
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2021-04-14 13:26:14
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http://mathoverflow.net/questions/75831/a-classic-problem-on-the-limit-of-continuous-function-at-infinity
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## A classic problem on the limit of continuous function at infinity [closed]
I don't know whether it's suitable to post this problem here, but I really need a help.
$f$ is a coutinuous function on $\mathbb{R}^+$, if the limit $\lim_{n\to\infty}f(nx)$ exists for all points $x$ of a nonempty closed set with no isolated point of $\mathbb{R}^+$, prove that the limit $\lim_{x\to\infty}f(x)$ exist.
Please see more details of my quesion here.
-
This isn't the site for such questions. You'd be better off responding to the comments you have on Maths-SX and trying some of the suggestions. – Andrew Stacey Sep 19 2011 at 10:15
sorry, I have respondded to all comments for days, but still no one gives an answer. – gylns Sep 19 2011 at 10:30
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2013-05-18 13:45:56
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http://threadposts.org/question/1494699/How-to-manually-download-podcast.html
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Tech Problem Aggregator
# How to manually download podcast
Q: How to manually download podcast
How do I manually download a podcast and have it appear in iTunes (to be synched with my iPod)?
I already know how to subscribe to podcasts.
I tried manually downloading an MP3 for a podcast and then put it in My Music fold that seems to contain all my music and podcasts for iTunes. However, then the file didn't t seem to appear in iTunes.
Does someone know what I am doing wrong?
More replies
Answer Match 65.52%
I did a rebuild of my computer (PC/Win XP/SP3; 1.16 GHz; 640 MB Ram) in September, and since then I've had a devil of a time getting podcasts to download. They start out fine, and then time out after 10 - 15 minutes. It is not one particular podcast that is causing the problem; a few get through but most die.
I have checked my firewall, and itunes is not blocked. Any suggestions on how to proceed next would be appreciated.
Vaughn
A:Podcast download issues
Welcome back vaff, its been a whole year since you last posted (so that's good )
Now your new issue with Downloading
Try this:
1. CCleaner
2. Reset IE
3. Memtest
Then let me know how it all goes
1 more replies
Answer Match 65.52%
I've just recently started downloading podcasts that I find on replay radio. It just got me to wondering, is it possible to get viruses from podcasts that I download?
More replies
Answer Match 63.42%
Edited to Add:
I completed a line quality test at dslreports, with dismal results:
http://www.dslreports.com/linequality/nil/2543013
I would greatly appreciate some help. I run XP, use IE7 (Mozilla doesn't work either), have AT&T DSL, the wireless has been very slow, so I am directly cabled in to my 2Wire router/modem. I have been having difficulty downloading podcasts lately, it's gone from bad to impossible. I either get a network timeout, hung buffering or it downloads maybe 2 minutes of an hourlong podcast, then closes. The problem exists whether I try via itunes, Win Media Player or Real Player. The podcast servers are fine, as I can tell in the podcast comments section no one else is having a problem, it's my computer/connection. I've tried tweaking some stuff in my network places and my computer properties related to performance, security; firewall, buffering, etc, that seemed to work in the past, but nothing is helping now. Here are some of the diagnostics I've run so far, though what they mean is Greek to me.
Speed Test:
TCP/Web100 Network Diagnostic Tool v5.5.4a
click START to begin
** Starting test 1 of 1 **
Connected to: netspeed.stanford.edu -- Using IPv4 address
Checking for Middleboxes . . . . . . . . . . . . . . . . . . Done
checking for firewalls . . . . . . . . . . . . . . . . . . . Done
running 10s outbound test (client-to-server [C2S]) . . . . . 305.0kb/s
running 10s inbound test (server-to-client [S2C]) . . . . . . 51.62kb/s... Read more
A:Solved: IE/DSL Podcast Download/Connection Speed Problem
16 more replies
Answer Match 60.48%
I'm trying to download updates through Windows Update, it shows that I have 139 important updates, and 40 optional ones. I selected them and clicked the install button but it got stuck at "Downloading updates (0 kb total, 0% complete)". I cancelled and retried choosing only 24 important updates and the same happened. I tried the check for updates option but it also keeps checking forever.
I also tried downloading updates from Microsoft Download Center but when I try to install them it hangs on "searching for updates on this computer".
I already ran the Microsoft Fix It tool, disabled my anti-virus and firewall, but nothing works.
I cannot install the SURT because it also keeps searching for updates.
I need to update so I can upgrade to Windows 10 - the setup also got stuck while checking for updates, so I was hoping that updating beforehand would do the trick.
Any suggestions on how to fix this problem are appreciated.
More replies
Answer Match 54.6%
I have a limited wifi issue. Toshiba states update KB3114149 may solve the issue. I can't determine if this update was included when I upgraded to 10586/1511 as it is not listed in Update History.
Is there a way to manually download that update ?
More replies
Answer Match 54.18%
Hi I'm a university student having a bit of trouble, for a project I have to install Windows 8.1 64 bit on several machines including updates.
As you can imagine updating each machine individually can be extremely time consuming. I was wondering if there's a way to manually download all the latest windows update files and put them in the Windows 8.1 ISO, so that once Windows is installed on the machine it will be up to date.
sorry if I haven't explained myself correctly
A:How do I manually download updates for Windows 8.1 64 bit
Try this link: Windows Updates Downloader (WUD)
2 more replies
Answer Match 54.18%
I'm using Windows 7 x64 Ultimate SP1.
I want to manually check for updates, make Windows download them if there is any, but wouldn't install at that moment, instead make Windows install them next time I shut it down.
In Windows XP, it was easy. You just selected the option to download. And when you attempt to shut down Windows, it asked if you want to shut down normally, or you want to install updates while shutting down.
However, in Windows 7, I couldn't find a way to do this same behavior. Can you please tell me how to, if there is any way to do it?
A:How to: Manually, download updates but not install
Use this address to find the updates you want
in the right hand corner theres a search box,type in your windows version
then find the specific files you want\need
then save them wherever you want
Handy for reinstalls without having to download everything again
Stop Windows Update in services first
then run your updates
then START WU again and check for updates
http://catalog.update.microsoft.com
or
http://www.microsoft.com/download/en..._121LSUS007799
3 more replies
Answer Match 53.34%
Hi guys,
I've just put my Dell Studio back to factory settings after having it for two years and deciding to give it a rebirth. I maintain my laptop very well but always like to do a reset every so often.
The problem I'm having now is that the damn thing wont update from Windows Update. It will identify that there are 76 important or critical downloads, but hangs at 0% downloading.
I went to the MS catalogue and tried to download manually, but even though I had the updates selected in my basket, again, it just hung on 0%.
I've ran Kasperskys TDS, the Malicious Tool Removal MS thingy-ma-jig, ran an sfc check where it did note there were corrupt files but it couldn't fix them. The Fix It link to the MS Update Not Working thing found no problems either.
So, I'm scratching my head and going to do a restart and see what's going on.
Any help greatly appreciated as this is doing my nut in.
Cheers!
-Update
Interestingly, I've just reset and it's telling me the system had to be restarted to finish installing updates, but when you click on it for more info, there's no updates listed. Bloody liar.
A:Will not download Updates automatically or manually via MS Catalogue
Welcom to the forum
If you'r facing a problem in WU then use the tool provided by MS to resolve this problem:
Download: System Update Readiness Tool for Windows 7 for x64-based Systems (KB947821) [February 2012] - Microsoft Download Center - Download Details
If after that the problem persist and you say that you reset every so often, then consider to dowload your same version of your OS from:
http://www.mydigitallife.info/official-windows-7-sp1-iso-from-digital-river/
Good luck
3 more replies
Answer Match 53.34%
I am unable to download *.xap files from store as Microsoft has changed the store layout. Previously. Before the Launch of Windows-phone 10, it worked for me.
A:Unable to download manually *.xap files from store. What should I do?
I'm not sure there's anything we can do anymore.
2 more replies
Answer Match 52.92%
Is it safe to manually clean the Windows\SoftwareDistribution\Download Folder
Is it safe to manually clean the Program Files\NVIDIA Corporation\Installer2
Windows 8
Together they are using about 10 percent of my drive. Thanks in advance.
A:Is it safe to manually clean SoftwareDistribution Download Folder
Hi afnuke Yes it is safe to delete the Download folder in SoftwareDistribution. As for the Installer2 one in the NVIDIA folder, I cannot tell, I don't have it. What's inside it?Edit: Before doing so, open an elevated command prompt and enter the following command:net stop wuauservOnce done, you can delete the folder. After that, enter:net start wuauserv
5 more replies
Answer Match 52.92%
After repeated problems with the update failing I did a clean install and installed kb3200970 manually from the windows catalogue and it shows as an installed update. Windows automatic updates wants to download and install it again. Is this really necessary? Any way to make it stop???
More replies
Answer Match 52.92%
Is that possible? If yes, please teach me.
A:Solved: Manually download updates for Registry Mechanic 5
8 more replies
Answer Match 52.5%
Hi there , desperately need some assistance. I think I have the malware "zlob"or other. I have tried a few things posted on this sites forums and others. My computer seems to open apps very slow, Unable to update Adaware ,ran spybot search and destroy. Also unable to do a system restore. I've Included a Hijack this log .Any help would greatly be appreciated
Logfile of Trend Micro HijackThis v2.0.2
Scan saved at 14:25:49, on 12/28/08
Platform: Windows XP SP3 (WinNT 5.01.2600)
MSIE: Internet Explorer v7.00 (7.00.6000.16762)
Boot mode: Normal
Running processes:
C:\WINDOWS\System32\smss.exe
C:\WINDOWS\system32\csrss.exe
C:\WINDOWS\system32\winlogon.exe
C:\WINDOWS\system32\services.exe
C:\WINDOWS\system32\lsass.exe
C:\WINDOWS\system32\svchost.exe
C:\WINDOWS\system32\svchost.exe
C:\WINDOWS\System32\svchost.exe
C:\WINDOWS\System32\svchost.exe
C:\WINDOWS\System32\svchost.exe
C:\Program Files\Lavasoft\Ad-Aware\aawservice.exe
C:\WINDOWS\system32\spoolsv.exe
C:\WINDOWS\Explorer.EXE
C:\Program Files\Common Files\AOL\ACS\AOLAcsd.exe
C:\Program Files\Common Files\Authentium\AntiVirus\dvpapi.exe
C:\WINDOWS\ehome\ehSched.exe
C:\Program Files\iolo\common\lib\ioloServiceManager.exe
C:\Program Files\Common Files\AOL\1189806489\ee\AOLSoftware.exe
C:\Program Files\iTunes\iTunesHelper.exe
C:\Program Files\iolo\System Mechanic Professional\AntiVirus\ioloAV.exe
C:\Program Files\iolo\System Mechanic Professional\Personal Firewall\ioloFW.exe
C:\WINDOWS\system32\ctfmon.exe
C:\Program F... Read more
More replies
Answer Match 52.5%
Hi all,
I use Linux and CUPS to provide free and stable networking printing shares for our community centre. Its easy to connection windows xp clients to the linux printer server, windows 7 is a pain, I have a work around, but would like to make it more streamlined.
I can get some older printers that do not have official windows 7 drivers working by using windows update during the driver selection phase of the printer set-up procedure. And selecting a slightly different printer from the list of available windows update drivers.
While this method works, there is a delay of between 5-10 minutes after clicking the windows update button before the list of drivers appears.
I would like to know if its possible to do any of the following 2 ideas:
1. manually access these driver files on Microsoft website somewhere, and download them for latter use.
2. Find where windows 7 saves the downloaded driver files and package them some how so they can be re-distributed.
As always your comments/suggestions are very welcome.
A:Manually browse and download Microsoft Drivers from windows update.
Quote: Originally Posted by Remush
Hi all,
I use Linux and CUPS to provide free and stable networking printing shares for our community centre. Its easy to connection windows xp clients to the linux printer server, windows 7 is a pain, I have a work around, but would like to make it more streamlined.
I can get some older printers that do not have official windows 7 drivers working by using windows update during the driver selection phase of the printer set-up procedure. And selecting a slightly different printer from the list of available windows update drivers.
While this method works, there is a delay of between 5-10 minutes after clicking the windows update button before the list of drivers appears.
I would like to know if its possible to do any of the following 2 ideas:
1. manually access these driver files on Microsoft website somewhere, and download them for latter use.
2. Find where windows 7 saves the downloaded driver files and package them some how so they can be re-distributed.
As always your comments/suggestions are very welcome.
I don't think there is a way to do that I would recommend that you set up a domain and be the system administrator for all fo the Windows 7 pcs, so you can install all of the updates on all of the computers at the same time.
2 more replies
Answer Match 51.66%
Need to fix the screensaver issue that doesn't kick in for users that have an IntelliPoint wireless mouse. Installing the KB 911895 from Microsoft via Windows Update solves this problem. However we need to automate applying this update to
our users. We can't go to every user that needs this resolved and run windows update manually to get this update. Can't seem to find a way to get my hands on this update to deploy silently where it's needed. Any idea's how I can get this
update manually?
We are Windows 7 Enterprise 64-bit.
Thank you in advance for any assistance on this matter,
Pam Cole
More replies
Answer Match 44.94%
Q: Podcast
i know that this is not ment to be but, is there a way to take podcasts from itunes, convert them, and put them on the zen vision m?
A:Podcast
You could always download the podcasts from the source instead of straight through iTunes. If you were to download them in mp3 format, there would be nothing preventing you from putting them on the zen
1 more replies
Answer Match 44.94%
Q: Podcast
What the hell is this I just can't figure it out. Googled it like crazy and still confused.
GOOGLE YOU ARE MY GOD
A:Podcast
A Web Feed of content ofaudio or video files using RSS or Atom SyndicationTry this Wikipedia linkhttp://en.wikipedia.org/wiki/Podcast
2 more replies
Answer Match 44.94%
I found a website that has lots of podcasts listed but I don't know how to get them.
I have itunes.
I see something that says "subscribe to this feed" but I get an XML page. Itunes asks for a URL.
I'm new to podcasts, can someone explain a bit please.
This is the page I want to subscribe to:
http://www.podcastdirectory.com/search/keyword.php?page=1&search=Paranormal
A:How to get podcast?
welp - i dont have an ipod and have never listened to a pod cast but on that page where it says "latest shows" if you right click on a link and select "save as" it downloads the mp3 -
so just save that file to your ipod, yes?..
2 more replies
Answer Match 44.52%
I am trying to download podcasts from itunes, although each podcast takes 40-50 mins to download, each file is roughly 85 MB. I am using a wireless ADSL connection that usually downloads about 10-20MB a min. Is my podcast taking so long to download because of my internet connection?
More replies
Answer Match 44.52%
Is there any software that i can use to burn a podcast program in itunes and play it back in a dvd player other than my pc ? When i play the dvd in my dvd player, it reads: "Error". Any solutions?
More replies
Answer Match 44.52%
I want to listen to Podcasts on my ipod. What are the best applications out there for this? I have come across Juice. Is that one of the best?
Jon
More replies
Answer Match 44.52%
Does anyone know of a good "how to" guide to producing a basic spoken podcast?
Or can you recommend good software/equipment for a beginner?
A:How to make a podcast
A google search of How To Make A Podcast turned up these results:
http://www.google.ca/search?hl=en&q=How+to+make+a+podcast&btnG=Google+Search&meta=
1 more replies
Answer Match 44.52%
I am looking for an app to sort and download podcasts other then iTunes any suggestions
A:Podcast software
Juice http://juicereceiver.sourceforge.net/index.php
1 more replies
Answer Match 44.52%
Just got my first MP3 player and I am now addicted to podcasts.
Does anyone got any suggestions for some good podcasts? Any catagory or subject is welcome (exept porn).
More replies
Answer Match 44.52%
I hope i am on the right section for pod cast download problems, I want to download BBC pod casts to a digital voice recorder, I have Olympus sonority soft ware and have tried drag and drop both to the software and to the icon that shows up in my computer when the recorder is connected to my PC, I have tried itunes but it does not recognize the recorder, this is new and not faulty, Is there any one that has had a similar problem that could help out, Thanks chippynut.
More replies
Answer Match 44.52%
I am an elementary teacher and would like to Podcast some of my lessons but don't have a clue how to do it. Can someone either give me instructions on how to do this or point me to an easily understandable document to lead me through the steps.
I really appreciate all the help that has been given to me on this website. You people are the greatest!
mcmm
A:Want to learn how to Podcast
Podcasting is simple as pie - 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
All you godda do is record ur voice, make the podcast, and put it on the web. Not sure what bit ur stuggling with...
Try this for size
1 more replies
Answer Match 44.52%
hi all,
This may not be the right forum, if it's not, please let me know which one it is!
I would really like to create a weekly podcast of our church services. I have all the files, but how would i go about turning that into a podcast on itunes that people can subscribe to and download them automatically?
Thanks,
Nappymonster
A:How do i create a podcast?
6 more replies
Answer Match 44.52%
I am using 2 podcast reader: JUICE and DOPPLER.
both fail to download files sometimes and do not keep synchronization ( losing track about files that already were downloaded).
My way of using it: I download the files I like and later I "cut " and "past" them to my mp3 player.
Need a good podcast reader.
More replies
Answer Match 44.52%
Any suggestions for a 64-bit podcast client? The only one I've found so far which actually runs is HappyFish - not too bad, but I'm surprised there isn't more choice.
Thanks
David
A:Podcast client?
HappyFish v1.5.0.2 does seem to be the only podcast receive software, so far found, that does claim to work on VISTA x64. How long have you used it? I was previously an iTunes user.
2 more replies
Answer Match 44.52%
I would like to put some Podcast software on my machine to enable me to time shift listening to radio programmes.
Could I have some comments/recommendations on which software I should get?
A:Podcast Software
Hi Supermart57 I've seen this recommended by ZD Net, so I belive it is clean of malware. I haven't used it myself tho'. It is free and looks pretty good to me. plus it should do what you want. Let us know if you like it or not. http://juicereceiver.sourceforge.net/overview/index.php
2 more replies
Answer Match 44.52%
I had a podcast that was available via iTunes for two years. Then in January, iTunes rejected it. I checked for validation and it found errors. I decided to start over for 2010, and my new podcast will validate but iTunes won't let me submit it. I can access it via the iTunes Advanced > Subscribe to Podcast menu option, but when I try to submit it, it says "We are currently experiencing technical difficulties. Please try again later." It's been weeks. I've tried renaming the xml file. I tried using other computers. I made sure iTunes is updated. Any ideas what else I can try?
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Answer Match 44.1%
I'm looking for some good free podcasting software that lets you create a show with all the bells and whistles.. you know.. intro's and fillers like sound effects etc. Software should have the ability to upload your completed file to a hosting service.
I did find one called WildVoice but even though the software works, I cant find a working link to the user guide. See it here.. and the broken link to the user guide. This website used to host files but they do not any longer. They just keep a few pages up with links to the software. WildVoice - Podcasting from the makers of WildVoice? Podcast Studio: How can I get WildVoice Studio?
Any ideas on software like this or can anyone help find me a copy of the user guide?
A:Free podcast software?
Users Guide---
Microsoft Word - WildVoice Studio User's Guide v1.0 - Final 0605.doc
Either open or download it.
You'll probably want to download it.
5 more replies
Answer Match 44.1%
I had Juice podcast receiver on windows Vista and worked fiine..
I can't seem to get it to work on windows 7 any ideas how to fix it?
i have tried Windows 7 - How to get Juice 2.2 working - Spiceworks Community
but it doesn't seem to have the file in there.. i did the first part where you run it with xp..
but can't find the file to change it.. i found ipodder.xrc and went into that and couldn't find my documents part..
any ideas would be great!
when you go to open juice it comes up with
errors occurred.
see the logfile 'file desintation'
you open up the file destination and this is what is in it.
Traceback (most recent call last):
File "gui.py", line 4, in ?
File "iPodderGui.pyc", line 3621, in main
File "iPodderGui.pyc", line 707, in __init__
File "wx\_core.pyc", line 5301, in __init__
File "wx\_core.pyc", line 4980, in _BootstrapApp
File "iPodderGui.pyc", line 1472, in OnInit
File "iPodderGui.pyc", line 934, in SetLanguages
AttributeError: 'iPodderGui' object has no attribute 'menu'
cheers
A:Juice Podcast Receiver
Hello! does Juice podcast receiver support Windows 7?
2 more replies
Answer Match 44.1%
I'm looking to start a podcast, and was wondering if anyone had suggestions on the best site to host it. Cheapest would be a good start, but not necessarily the only factor.
Thanks.
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Answer Match 44.1%
I just got an mp3 player and would like to learn how to find and download podcasts into it. I am confused as to which aggregator software to use do I need more then one ? Which RSS feed engines to use to find feeds. I would like to use my walking time to learn new things. I am not particularly interested in news feeds.
Would anyone have any suggestions ?
A:Podcast Aggregator RSS Feeds
I'm fumbling around with the same thing. I have used Tristana and Nimiq to find content. Also found links to free podcasts from my Windows Media Player. Another one I have seen is Doppler.
1 more replies
Answer Match 44.1%
Hi there. I recently subscribed to a podcast. When I view it in iTunes, the video is very dark ( Everything else in the screen is fine. The sound is fine too. It is not the video itself's problem (i downloaded different podcasts from different sources). How do I fix it? Is it only me that is experiencing this problem?Thanks.PS: I am using iTunes ver. 8.0.2.20 on Windows XP Service Pack 2. You can see a shade of the person at the right.)
A:iTunes Video Podcast
UPDATE: All of my videos (no matter what format they are in) became dark, even those which are normal before. HELP! I tried to play the videos in different players but same...
1 more replies
Answer Match 44.1%
Anyone know a good enough procedure to follow whilst installing a podcast server onto a business network?
We got a book on mac servers but its not the book we need. So how about you guys help me out here with finding something on it. I would be ever so gracious.
A:Installing a podcast server
Dreusx,
Your question is pretty vague...you may get more response by including more information. It would be good to know the platform, the software, the desired outcome. The more details the better.
1 more replies
Answer Match 44.1%
I love listening to podcasts. I've been listening to them for many years. I've been using my old Zune HD for quite a while to listen to podcasts. Well, either it or the Zune software is beginning to get unreliable. Sometimes it syncs podcasts, sometimes it doesn't.
So its time to me to move my podcasting activities to my Lumia 640. I'm going to give the Podcast app that comes with Windows 10 Mobile a try. I understand the basics, but would like some help with some of the operation. For example, if I select a podcast to listen to or watch, does the Podcast app download it? If not, how do I download it? This is quite important because of my commute. I ride a train which takes me through several miles and a long period of time, where there's no Internet connectivity, no cell reception, etc. So does it download all podcasts?
Another question, after listening to a podcast or watching a video podcast, does the Podcast app delete the podcast from my Lumia?
A:Would like some guidance on using the Windows 10 Podcast app
if it still works the same way, the podcast app automatically download episodes that you subscribe to and there is a setting for how far back (or how many episodes back) you want to keep. it will then delete the oldest one once it's reach the limit. after you listen to them the app just marks it as heard, I don't think it deletes them until the limit is reached. you can choose to download specific episodes too if you need to.
having said that, I haven't used the app in a little while so it might be a bit different now, commute too short now lol (well, it's a blessing :P)
2 more replies
Answer Match 44.1%
thanks , hopefully that is a light weight one.
A:could any one recomend a podcast software?
HI 3night.
What podcast software you require depends on exactly what you wish to do with the software. Do you only want to listen to podcasts? If so there are many choices, a lot of them free, to choose from.
If you want to publish or edit podcasts you need to be a bit more selective in your choice of software.
Here are a few links to software suggestions for you to have a look at:
Podcast Software List - Many Platforms Including Windows, Macintosh and Linux
Podcast Software Review 2010 - TopTenREVIEWS
Hope this helps.
.
3 more replies
Answer Match 44.1%
So my system just updated it's self when i restarted it and i got a new podcast library(it could be from the zune update) and when i try to delete it, it will reappear i've tried everything
please help
A:Can't remove Podcast library
Yes the podcast library IS from the zune update. If you want to remove it, the option should be in the zune software itself. My suggestion is to relax and enjoy this library because zune right now integrates more tightly into windows media player. This could mean a more enjoyable experience with the zune and zune hds . Try your best to use it. It can't hurt your computer that much~!
Hope I helped~!
9 more replies
Answer Match 43.68%
How to change a wave recording into a podcast with RSS/XML tag?
What steps do I need to take, in what sequence?
Can that be done for free?
Thanks in advance for any help...
More replies
Answer Match 43.68%
I'm on a library computer. The IT guy has totally disabled right click of the mouse. It's very frustrating because podcasts I wish to download assume you can right click to Save Link As... and just download the mp3 file. But I can't right click.
Anyone know a work-around?
Other things I cannot do include running any other browser, such as FF. Running any program from a USB etc..
It's pretty much IE and Notepad is all we get to use.
A:How to downoad podcast in IE when right click disabled?
Wow, that's a tough one! Anything I could think of requires right click as a step, defeating the purpose! Images there may be a workaround, but files...
I hope someone comes up with something because I wracked my brain trying to think of something. Posted for the gurus, if anyone would know, it's our group.
A Guy
9 more replies
Answer Match 43.68%
For some reason iTunes downloads the same podcast twice. It does not happen all the time but when it does it happens on most podcasts that I subscribe to. For instance, Diggnation from April 11th is listed twice in my podcast directory.
Does anyone have any insight as to why this is happening and how to stop it? Thanks in advance.
A:iTunes podcast multiple copies.
10 more replies
Answer Match 43.68%
If anyone here can help, four days ago my Windows media player stoped playing podcast audio.
In some cases the podcast from the same provider plays one episode but not any others (no DRM mostly BBC).
The visualizations play as if the audio is playing but there is no sound.
All my podcasts play fine in Cyperlink or even via IE; just not via media player.
My other media (ie music & video) play fine.
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Answer Match 43.68%
I have a Lumia 950 and I have the podcast app. I want more info on this app but I can't find it on the appstore or any info about it. Is there a way to find out. I get some podcast that are on iTunes on it but not all so I'm trying to figure out where are people posting these podcast for it to appear on this app.
In my phone it's just known as Podcast.
Thank you,
A:What service is the Podcast app on my phone connected too?
And I wasn't connected when I asked this, so yes would like to know if anyone can help.
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Answer Match 43.68%
My OS:Win XP
hi, i tried putting a .mov file onto my digital camera to play it but it sais "unrecognized file format", but it is a mov file!! and my camera records in mov. can someone help me???
My camera:M1033 (http://www.kodak.com/eknec/PageQuerier.jhtml?pq-path=12418&pq-locale=en_US)
When I choose "movie Inspector" in Quicktime this comes up for the videos:
.mov files recorded by my camera:
-Format= MPEG-4 Video, 1326 x 746, Millions
µ-Law 2:1, Mono, 16.000 kHz
-Movie FPS= 30.11
-Playing FPS= 30.11
-Data Rate= 12.65 mbits/sec
-Normal Size= 1280 x 720 pixels
My .mov file is:
-Format= AAC,Mono,32.000kHz
H.264, 995 x 746, Millions
-Movie FPS= 29.97
-Playing FPS= 29.97
-Data Rate= 564.88 kbits/sec
-Normal Size= 320 x 240 pixels
I have 2 free video converters that i often use called "Pazera Video converter suite" and "RAD Video Tools". Maybe they can help me somehow???
Thanks for any help, philip
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Answer Match 43.26%
In iTunes (7) how do you view the names of individual episodes of each show? My screen only shows main program headings, for instance it says "dl.TV iPod video," but underneath there are no entries for individual episodes to choose from. That is just an example, I don't KNOW which shows have new episodes because I can't see any!
It it set to keep all episodes of podcasts. I subscribed to 15 and see new episodes download all the time. My podcast count goes up and up. But I don't know what I have, not seeing a list. When I click on a program name the audio or video of the first episode I downloaded starts playing. I don't have very many, just started ipod and iTunes a few weeks ago. What am I doing wrong?
PS - Maybe these are not main program headings at all, but the names of the first episodes I downloaded (since that's what they play when I click them).
A:How do you view list of podcast episodes in iTunes?
Did you ever get an answer to your questions? I'm trying to figure out the same thing.
2 more replies
Answer Match 43.26%
Hi guys! Two days ago, my computer recently started acting strangely. First, a strange podcast started being broadcasted through my speakers. I had firefox open and was just checking my email. I was like "what in the world?" and pressed Ctrl+Alt+Del. I looked under Processes ad found three "iexplore.exe" processes running. I then clicked "End Process" for each of them, and the podcast stopped. I then tried running Avira Antivir to scan for viruses and after leaving the computer on for about an hour, I came back to it, only to see that it was frozen. I could move my mouse, but nothing else. This has now happened three times. It seems like each time I run Avira, the computer freezes and I end up having to reset the power. Also, the "iexplore.exe" processes keep coming back every 15 or so minutes, with a weird podcast starting about 5 minutes after the process starts. Another thing is that even if I don't run the Avira Antivir program to scan for viruses, my computer STILL freezes up randomly when I'm on it. I also seem to be getting lag spikes.
Does anyone know what I should do?
Here's a HijackThis:
Logfile of Trend Micro HijackThis v2.0.2
Scan saved at 1:06:26 PM, on 7/14/2009
Platform: Windows XP SP3 (WinNT 5.01.2600)
MSIE: Internet Explorer v8.00 (8.00.6001.18702)
Boot mode: Normal
Running processes:
C:\WINDOWS\System32\smss.exe
C:\WINDOWS\system32\winlogon.exe
C:\WINDOWS\system32\services.exe
C:\WINDOWS\system32\l... Read more
A:Random invisible podcast + frequent freezes
Uhoh, one more thing: When I type in a search for Google, I get a list of results.. but when I click on them, they don't take me to the right website- like if I search for Wikipedia, and I click on the Google search result for wikipedia, it starts loading Wiki, then it goes to a completely different website (like owny.net or something.. it keeps changing to something else each time I click the search result)
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Answer Match 43.26%
http://www.radiolab.org/story/darkode/
More replies
Answer Match 42.42%
So I was listening to Daniel interview the HP X3 project lead and when I snapped the app to the right to be able to do something else (it's an audio stream after all) the podcast stopped playing. It didn't pause it stopped altogether.
I started to play the podcast again and just used the seek bar to locate where I was so I could continue listening. I decided not to touch the app window but rather use the taskbar to switch between WC and other apps.
When the podcast was close to ending I maximized the WC app window to prepare to read something else and once again it stopped!
Not sure what the issue is but please look into it, WC.
Thank you.
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Answer Match 42.42%
I am looking for a cheap / free program that will allow 2 or more users to connect over the internet and record a PODCAST for a small business.
I looked at Skype with IMCAPTURE but the cost is about $50.00 so I was wondering if anyone knows a cheaper way to do this without being overly complicated. Would seems there should be some IM client that allows native capturing of recorded content no? Dorf. A:Any cheap or free way to PODCAST multiple users over internet? Have a look at trillian im not sure its what you need but it has a free version,although its probably limited as its free All Features - Trillian Astra 2 more replies Answer Match 38.22% I had podcasts on my hard drive that I imported and added to library in iTunes and it put most of them in the podcast library. But it put some in the movie library and I cannot get them in the podcast library where I want them. Please help. Thanks More replies Answer Match 34.02% when i connect a wifi network show wifi network is connected but limited connectivity coz IP address not found and i can't found any option to putting IP address .My mobile Model Lumia 640xl(windows-10 latest version).Please help me..... A:Why can't I add my IP manually? There is no way to manually add an IP Address in Windows Phones and the current version of Windows 10 Mobile. 4 more replies Answer Match 34.02% I do a lot of work on customer machines at a computer shop, and one of the most frustrating issues I run into is when I run SFC on a computer and it reports that it found errors and was unable to fix them. Originally I intended to write a program to automatically pull the [SR] reported files from the installation media and replace the broken ones on the system, however, when I run the command from the tutorial: findstr /c:"[SR]" %windir%\Logs\CBS\CBS.log After an unsuccessful SFC attempt, even when it reports files that could not be fixed, I never turn up anything, anyone know why I don't get the [SR] tags in my CBS.log? A:How to Manually Do SFC's Job? Hello Virucyde, and welcome to Seven Forums. You could use OPTION THREE in the tutorial below to get a sfcdetails.txt file with the [SR] details. If you like, upload this file here afterwards, and I'll be happy to take a look at it to see what could not be fixed. SFC /SCANNOW Command - System File Checker 8 more replies Answer Match 34.02% I'm really in need of a motherboard upgrade, but until then.... I'm running XP Sp3; I have an An ASUS P4S333. this mainboard came out just before USB2.0. As luck would have it, today I'm in need of USB 2.0 (or better..LOL) I need to change the IRQ assigned to a specialize audio audio card. It must sit on it's own, and currently it's sharing 16. I cannot unselect "Use Automatuc Setting" in Devmgmt to allow me to do this. I am logged in as administrator. I have spent most of the morning moving cards around to try and resolve this, to no avail. I have also tried within the BIOS to force IRQ's to specific PCI slots, again with no success. The best I've been able to do is get the audio card off of IRQ18 to IRQ 16. In both scenarios, the IRQ is shared with something else. I have 6 PCI slots and 1 AGP slot. At the moment, after shuffling the PCIcard deck, AGP is occupied as are PCI 1,2,4, and 6. 1 and 5 are paired, as are 2 and 6. A couple of suggestions for configuring this motherboard would be greatly appreciated. Steve A:Manually set IRQ What other PCI cards are you using? 2 more replies Answer Match 33.6% Im a begginer and would like to know how to fix my registry. I looked at a couple of products like regcure and Advanced WindowsCare 2 Personal but from several different reviews that I read seems that these programs cause more headaches then they solve. So I am interested in manually fixing my registry. All I know is that to open the registry go to start, run, regedit. Dont know what to look for to fix though. Thanks for the help. A:How to Manually fix registry on XP? 16 more replies Answer Match 33.6% New to this forum so bear with me, I'm just desperate for help! I bought a laptop off of craigslist today because I was in desperate need of one seeing as my desktop has died. Only thing is I had no idea XP was no longer supported with many things. Is there a way to update it? It's in pretty good shape, just old. I tried to update the biois on the dell website but apparently the battery doesn't hold a charge well and it keeps telling me to charge it (It's on the charger but still not working.)The specs aren't so great from what I understand, Not fun going from my alienware to this, but I'm broke and can't afford anything else. Is there any hope for this to become a better machine?Microsoft Windows XP Home EditionVersion 2002 Service pack 3Intel(R) Celeron(R)MProcessor 1.40GHz0.99GB of RAMThanks in advance! A:How to manually update my XP? If you just run Windows update as normal you will get all the updates that are available for XP. Not being in support just means that Microsoft will not be making any more updates, existing ones will still download.As far as making it better, I can't really think of anything unfortunately. It's certainly not worth the cost of a more modern OS and that CPU will be a major limitation despite any upgrades you try.Of course, you can always create a thread on this forum about why your desktop "died" - We may be able to help you get that up and running again. 4 more replies Answer Match 33.6% First off, I'm plugged into a battery backup power supply. (I don't know if that's related or not.) Also, I think my Windows Update may have something to do with it as well. The last couple times it's happened the yellow Windows Update icon popped up in my taskbar. I put off downloading the update because I didn't want to restart my PC. I woke up this morning and my PC was restarted without me doing so. I seriously doubt it's a heat problem. My temps are moderate. Is it possible that it's actually something that I can change by going into my Control panel and changing settings? A:How does my PC restart without me manually doing so? Silent Install Maybe? 1 more replies Answer Match 33.6% Does Microsoft host a site in which I can manually download and update my windows 7? Everytime I click on their update site I get this! But I dont want to use my control panel! A:Manually Update? If you know of an update you wish to install you can search by KB number for the manual installation file... Or you can setup a WSUS server and manage them all centrally 8 more replies Answer Match 33.6% I'm not able to remove SP3 form Add\Remove. How do Iremove it manually? Is there a utility I can use or what file to delete? A:How to remove SP3 manually If you didn't save the backup files when you installed it, (that's why it's not in add/remove), you can't remove it. 3 more replies Answer Match 33.6% Hello everybody. I need your help. My computer is Dell Optiplex 780, OS Win 7 64 bit. I have to install SpyderPro, Color Correct your Monitor from Pantone ColorVision. This old device worked excellent for me for more than 10 years with windows XP, Vista and now I try to install on Win 7. Company never releaced any update for this device, they only develop new devices. I am retired now and can?t spend$200 to buy a new one. So, I installed software from CD with no any problem, connect sensor to USB input and received message: ? Error 2(Spyder USB.cpp 126 the system cannot find the file specified?. I searched internet for this error and found answers: In windows get lost or corrupted some Dll file. Each site sagest download special software which cost money. I don?t trust these sites, there are hundreds of them and result is unpredictable and most of it have virus.
Is it possible to fix this error manually?
Any advice will be deeply appreciated.
Val852
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Answer Match 33.6%
Well, my friend let me borrow his age of empires III and i finally bought a copy, But when i tried to uninstall it it failed. So instead i tried deleting the files in "Program Files" but after that it still would think that Age of empires III was still on my computer, so the install disk wont let me install it and when i try to uninstall it again it still dosent "Uninstall"
Could somone tell me how to delete all the files manually? (Really, all i need to know is where i can find them)
A:How do i uninstall something manually?
It's probably in the registry somewhere....download a registry cleaner. Registryfix is a good registry cleaner try that out but it doesnt work on 64-bit versions of windows..
1 more replies
Answer Match 33.6%
I have a M817 motherboard, and i am having trouble with my tv card freezing my computer. i have tried moving the card to every slot available, but it wont change the irq. i am curious to know if i can change the irq settings manually on this board. if i can how can i do it. any help is much appreciated
A:Manually changing IRQ's
8 more replies
Answer Match 33.6%
I basically want to install another nic in my machine.
Here's the problem. When i installed the new nic, it starting messing with other hardware. (mouse wouldnt respond)
So Im thinking its an irq conflict. and sure enough, it was being shared with my video card. I dont know why my mouse was affected.
Anyways, i see that irq#9 is free.
Is there any way i can manually assign irq 9 to the new nic?
Im running windows 98se, with 256 ram.
I also have a geforce 256, soundblaster live, and a d-link nic presently installed.
Any help would be greatly appreciated,
Jolt
A:Manually assigning an irq
This article should answer your IRQ questions http://www.pcnineoneone.com/howto/irq1.html
Have fun Steve
2 more replies
Answer Match 33.6%
My ram is rated at 8-8-8-24 at 1.65v but my bios sets it automatically to 9-9-9-24.
Will keeping it at automatic bios settings cause system instability? Also my motherboard only supports 1.5v RAM instead of the 1.65v I have. Can this also cause system instability?
My motherboard is a MSI H55M-E23 and my ram is Crucial Ballistix 8-8-8-24 1.65v 1600mhz
Edit:I also have run memtest86 for 17 passes and it showed 0 errors.
A:Should I manually set RAM timings?
Is the RAM also being clocked at 1333? That's what I've seen in the past few motherboards I've used. Setting the BIOS to use the XMP or manual settings were required to get higher frequencies.
For my own system, I set the frequency, timings, and voltage manually to the manufacturer's recommended values. (Sometimes, I've had to back off the timings a little when using four DIMMs on a dual-channel system.)
8 more replies
Answer Match 33.6%
Hi.
What is the way to manually install an add-on for internet explorer ?
I have a .cab file which includes a .dll file and a .inf install script.
How can I install this add-on manually ?
Thanks in advance.
A:Install IE Add-on manually
Hello and welcome to TSF
Right click on the .inf file and choose Install
3 more replies
Answer Match 33.6%
So GMAbooster is supposed to be a program that can speed up your graphics for better performance, but i have a Toshiba Satallite c655-s5132 that has a Intel GMA x4500 but GMAbooster hasn't released one for that yet (but yet the show it on the download page) they say they plan to release it but i don't know when and how long its been sence the said they planned on releasing it.
So my question is:
Can you do the samething the GMAbooster does but manually?
I mean i play this one game Combat Arms, the computer i use it just about good to play on Combat arms but when the room is full (16 players max) the game slows down and it hard to play, or when i get a gun fight with one person it slows down abit but enough to affect my skills.
A:Can i manually do what the GMAbooster does?
The short answer? Yes. Anything that software can do to your computer you should be able to do manually. However, if your device is under warranty, using that software, or doing anything it can do manually would almost certainly void it.
I'm not an expert on this sort of thing though, so for actually accomplishing it you would have to ask someone else. I'm sorry if my answer is a bit less than helpful, but I don't want to provide any in-accurate information and potentially cause you to harm your computer.
2 more replies
Answer Match 33.6%
Reading and searching for information about defragmenting has only caused me more confusion. I have been reading a lot about Win 7 automatic defrag etc. My old Vista PC has also an option where you set up a schedule for automatic defragmenting, so I don't see how Win 7 is different in that aspect...
I have built this gaming PC just a month ago. I have looked at the defrag schedule for both my boot drive and other drive and twice I have seen the boot drive say 4% defrag. Other times, it always shows as 0%.
- What does the above mean?
- I have the defrag set to a monthly schedule for 1st day of the month at midnight. Will I have to leave my PC fully powered on for this to take effect?
- Some people say defragging is not necessary for modern hard drives any more. So is it or is it not?
- I do have about 50GB of data on the boot drive (Velociraptor) and 500 GB of pictures, movies, music, games on the other drive. Is it necessary to defrag them?
- Should I do this manually because my computer is not always on during the "scheduled" defrag time?
Thank you.
-------------------------
250gb WD Velociraptor - 2tb WE Black - Gigabyte z77x-ud3h - Gigabyte GTX 660 - Corsair GS700 - Corsair Vengeance 2x4gb - CM storm enforcer
A:Should I defrag manually or not
Windows 7 is a way more efficient running OS than the old XP or Vista.
I have Perfect Disk 12.5 and every time I'm curious to how defraged my drives are and analyze them they are @ 1% defraged but I still defrag every so often just for chits and giggles cause the system doesn't need to be defraged I just do it cause I started the program and may as well use it.
Having scheduled defrags is not neccessary IMO on a win7 machine
9 more replies
Answer Match 33.6%
I have IObit Advanced System Care. It's a great security program, but it doesn't seem to want to quarantine files that it/I think are a virus.
it's a rundll32.exe type. Don't know much about computers but I seem to see those alot.
I've identified the files but since they attempt to run, relentlessly, on startup, I cannot move them or delete them. If I shut down the process they automatically run again.
Without installing yet another one of the million antivirus softwear programs, is there any way to manually keep these programs from automatically starting up? or to shut them down long enough to delete/quarantine them?
Here's the hijackthis in case I've mistaken these processes for something else... The highlighted ones are the suspected culprits:
--Platform: Windows Vista (WinNT 6.0)--
C:\Program Files (x86)\Intel\Intel Matrix Storage Manager\IAAnotif.exe
C:\Program Files (x86)\Common Files\Java\Java Update\jusched.exe
C:\Windows\SysWOW64\rundll32.exe
C:\Program Files (x86)\IObit\IObit Security 360\is360tray.exe
C:\Windows\SysWOW64\rundll32.exe
C:\Program Files (x86)\iTunes\iTunesHelper.exe
C:\Windows\SysWOW64\rundll32.exe
C:\Users\Public\Games\World of Warcraft\WoW.exe
C:\Program Files (x86)\IObit\Advanced SystemCare 3\AWC.exe
C:\Program Files (x86)\Java\jre6�... Read more
A:Manually quarantine
Hello and welcome to Bleeping Computer We apologize for the delay in responding to your request for help. Here at Bleeping Computer we get overwhelmed at times, and we are trying our best to keep up. Please note that your topic was not intentionally overlooked. Our mission is to help everyone in need, but sometimes it takes just a little longer to get to every request for help. No one is ignored here.If you have since resolved the original problem you were having, we would appreciate you letting us know. If not please perform the following steps below so we can have a look at the current condition of your machine. If you have not done so, include a clear description of the problems you're having, along with any steps you may have performed so far.Upon completing the steps below another staff member will review your topic an do their best to resolve your issues.If you have already posted a DDS log, please do so again, as your situation may have changed.Use the 'Add Reply' and add the new log to this thread.Thanks and again sorry for the delay.We need to see some information about what is happening in your machine. Please perform the following scan:Download DDS by sUBs from one of the following links. Save it to your desktop.DDS.scrDDS.pifDouble click on the DDS icon, allow it to run.A small box will open, with an explaination about the tool. No input is needed, the scan is running.Notepad will open with the results.Follow the instructions that pop up for postin... Read more
3 more replies
Answer Match 33.6%
How do I manually scandisk without the boot?
A:manually scandisk
scandisk will not work on ntfs formatted drives.
you want to use chkdsk instead.
this tutorial may help...
6 more replies
Answer Match 33.6%
Hi,
My PC do not have any issues but I would like to learn how to remove MBR infections manually. It would be great if some one can help me.
A:How to fix issues with MBR manually
The simplest way is by using recovery console and typingBootrec.exe /fixmbrBootrec.exe /fixbootSomeone correct me if I am wrong...
2 more replies
Answer Match 33.6%
I have a Toshiba laptop running Win 7 32bit. It was current on updates until July 2015. Then WU went crazy hijacked my cpu & kept trying to download Win 10. I know alot of people were having similar problems. Anyway its been worked on remotely by 2 different techs and I've spent most of the last 4 weekends researching and trying different fixes and nothing has helped. I think WU is completely screwed up now.
I just want to find and install the updates that I need from last July until now. I've tried MS catalog also the download pages at MS website but I can't seem to get the information I need. I read about a program called Belarc that among other things can tell what updates a system is missing. Has anyone used it for that purpose? Are there better programs? I really don't want to go thru 120+ pages of downloads at MS and try to pick out all the ones that I need. Any suggestions would be appreciated. Thanks.
A:How to manually update Win 7
Hi Tabitha,
Belarc should be ok, i suggest you do them in groups of 5/10.
Not surprised that your CPU went crazy as you only have 2GB of ram.
Have a look at this link for Updates to hide(W10), or remove them if allready installed.
How to remove Windows 10 upgrade updates in Windows 7 and 8
Roy
2 more replies
Answer Match 33.6%
We have a print server which serves BYOD. At the moment it flags up brute force attempts as users try to login unsuccessfully from their client devices.
ATA logs this against computer 'unknown' so we've no way to ignore the print server. Can it be added manually?
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Answer Match 33.6%
Is there a way to manually boot my OS CD? I have it set in the BIOS to auto boot cd or whatever. I just want to know if there is a hotkey.
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Answer Match 33.6%
trying to remaove all traces of a program that i have installed incorrectly and it wont automaticly uninstall from cd or C.panel.
trying to delete all traces of it but im not really have much luck because it will not let me reinstall it because it says i still have a previous version
help plz
A:How do you manually uninstall
Have you done a registry search?
Go to Start > Run, and type:
regedit
Look under Edit for the Find entry. Enter a search term that is specific to that program, like the program's name, or even the manufacturer if that is the only program you have from that company. Remove the entries that refer specifically to that program. Be careful not to remove entried that are not specific to that program. If the name has the word "system" in it, for example, you obviously should not delete all entries containing that word.
Use F3 to continue the search after each find.
Is the program still in Add/Remove?
Have you deleted the folder from Program Files?
Another thing you could try is a safe registry cleaner:
Regcleaner 4.3.0.780
Tools > Registry Cleanup > Do Them All. And then remove all the found entries.
2 more replies
Answer Match 33.6%
I have just completed a recovery on an HP desktop. Started checking for updates and as we all know that takes a really long time (24+ hours). In my system properties it says 7 Home Premium and SP1. In the installed updates screen it says there are 12 updates only that date back to 6-7-2011.
How can I know that SP1 is actually installed?
If it is not actually installed, would it be best to manually download and install? Would this help with the time it takes to search for updates?
A:Manually install SP1?
Go to Control Panel/System
Look under "Windows Edition" at that location. If SP1 is installed, it should say "Service Pack 1".
It's downloadable as an ISO.
The 64 bit English version is named windows6.1-KB976932-X64.exe
It's 903 MB in size. I'd guess you can find it and download it with a Google search. That's what I'd try to do if I knew it was not already installed.
If by "recovery", you mean some sort of restore to factory condition using an HP partition or menus and applications, I'd assume you have SP1 if the machine shipped with SP1.
3 more replies
Answer Match 33.6%
Is there any sort of program that can let me take a bunch of files on a drive and move them to wherever i like? like how a defragment program works, except i chose where to put the files, and the program defrags the files and places the pieces where I chose
A:Manually defragment possible?
Hello bxsciencer,
There's no such thing as 'manual defragmentation'. Even if you move fragmented files manually from one hard drive to another, your PC still won't become defragmented. I highly recommend you use Windows' Disk Defragmenter or similar programs.
9 more replies
Answer Match 33.18%
Hi all,
I have two network connections on my laptop - one wired and one wifi. Naturally, windows selects the wired connection as the default route (default interface) everytime. I am experimenting with finding a mechanism to select which interface to use without physically disconnecting one of them. Each NIC has a local static IP
wifi 192.168.1.11
wired 192.168.1.12
I've created a client/server app: the client(s) exists on the computer specified with the two NICs. The server is on a separate computer that is also on the home network but not the same machine. Basically, I am looking for a way to pick different routes for the clients - not necessarily load balancing but load choice I guess.
I believe I am going to have to use the 'route' command or the 'netsh' command, but every document I've read on them is pretty crappy or I just didn't understand them (I understand the conepts of routing/interfaces, etc). When I use the 'route' every time I get a response saying that interface does not exist but I am sure I am typing everything in correctly. Don't know what else to do...
C:\Documents and Settings\nick>route print
===========================================================================
Interface List
0x1 ........................... MS TCP Loopback interface
0x2 ...00 30 bd d1 7d 2f ...... Belkin 11Mbps Wireless Notebook Network Adapter
- Deterministic Network Enhancer Miniport
0x30004 ...00 10 60 c1 5c 36 ...... R... Read more
A:Two NICs, how can I manually choose which one to use?
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Answer Match 33.18%
First off, I am using Windows XP SP2.
I am trying to update my modem drivers. They are not an auto install package. I go into the device manager and find the card. Neither right clicking and selecting update driver nor going into properties and selecting update driver from the driver tab launches the hardware update wizard. So, I cannot tell it where the drivers are. Nothing happens at all in fact except when i select it from the properties screen, the Cancel button greys out.
I have tested several other devices and none of them launch the wizard.
A:I cannot manually update any drivers.
6 more replies
Answer Match 33.18%
Hi folks. Out of my depth (quite shallow, really) on this one. Trying to install a video card on a Celeron 433 running Win2k SP1, made by Elsa in Germany. Obviously, they've been having some driver problems, because pretty much every time I log on to their website, they've got a new driver version. But still, their readme includes the following:
The display driver is installed successfully, but after rebooting the system, the driver is not activated. Your graphics board requires a free IRQ to work properly.
Please assign a unique IRQ to the slot which is related to
your graphics board. Be sure to have no hardware conlicts with other installed hardware components.
If I set the Enable VGA thing in my BIOS, is that right? That way I get no buttons. Otherwise, as the readme says, the card will not start, ! in device manager.
Can someone help me out of VGA hell?
Eternally grateful,
dw
A:Set IRQ manually for Elsa monitor
6 more replies
Answer Match 33.18%
Hi i was referred to this part of the forum by someone in the Am I Infected Forum. I'm having a problem where explorer.exe will not load unless I manually run the task in task manager, also when I boot the computer My Documents folder comes up instead of explorer.exe, any help would be appriciated, thanks!
A:Have to manually start explorer.exe.
See http://support.microsoft.com/kb/555294, appears to me that it will solve your situation.The above involves editing the registry, which is potentially a dangerous task without due diligence. Because of this, the first consideration that should be made by any user...is to perform a backup of the registry. A good tool for doing such backup is ERUNT Registry Backup Tool - http://www.snapfiles.com/get/erunt.html.Once you have made a backup of your registry, using ERUNT, then you can proceed to follow the instructions provided by Microsoft.Let us know how it goes, please .Louis
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Answer Match 33.18%
I have a problem with annoying pop-up extension in chrome.
they never delete, never stop to appear.
even i install mozilla - they come back - but hopefully i just delete them there and they dont come back .
but if i will install chrome again - they will come.
i clean temp files and etc - do not help
Please provide me idea how to clean regitry (as i guess the problem is there)
A:Registry cleaning - Manually 0 Help
Hi tuchkina,
Please don't go looking for Registry cleaners. You'll do more harm than good if you do that. And please don't try and track down the Registry entries yourself - that'll be nigh on impossible.
You've been infected with some advertising malware by the sounds of things.
Head over to this room here:
http://www.bleepingcomputer.com/forums/f/22/virus-trojan-spyware-and-malware-removal-logs/
Read the instructions there, post the required logs and an expert will come along to help you.
3 more replies
Answer Match 33.18%
Hey,
Well on a normal day I turn my computer on, and it automatically connects to my wireless network and everything works fine. However, if I lose the connection (computer goes into standby and disconnects, ect) it wont reconnect automatically, so I have to go to the network and sharing center and try to reconnect. But when I try to reconnect it gives me an error saying:
"Wireless association failed because Windows did not recieve any response from the wireless router or access point"
If I restart my computer than it automatically connects and everything works great again.
My laptop has done this from day one and everything I've tried hasn't helped. This happens when I try to connect to other networks too, and other people have no problem connecting to my network.
Searching this site has had a lot of people saying to disable Ipv6, but that didn't do anything.
My computer:
Modem: Actiontec GT701-wg
Computer: Gateway MT3422, Windows Vista home premium 32-bit
Wireless adapter: TRENDnet TEW-421PC 802.11g PCI adapter driver 6.1109.1019.2007
Network reception is "Very Good"
Device manager network adapters:
isatap.domain.actdsltmp
nVidia nforce networking controller
TRENDnet TEW-421PC/TEW-423PI 802.11g Wireless cardbus/PCI adapter
Click to expand...
Windows IP Configuration
Host Name . . . . . . . . . . . . : Brian-Laptop
Primary Dns Suffix . . . . . . . :
Node Type . . . . . . . . . . . . : Hybrid
IP Routing Enabled. . . . . . . .... Read more
A:Wireless cant connect manually
16 more replies
Answer Match 33.18%
How do I determine subnet manually?
This is more of a question of how than what. Since there are calculators on how to do this on the internet.
If one has been assigned a Class B address (142.63.x.x), and you need to be able to designate up to 8,000 hosts per subnet. What subnet mask specification would allow for the most possible subnets?
How did you determine your subnet mask specification?
Also,
Excluding reserved addresses, how many addresses would be available for host assignment in each subnet? In total for all subnets?
What would be the last assignable address of the first subnet, and what would be first assignable address of the last subnet?
A:How do I determine subnet manually?
Sounds like homework to me
Here is a great site that teaches the basics of subnetting in an easy to learn lecture format.
http://www.learntosubnet.com/
Regards,
Rollin
2 more replies
Answer Match 33.18%
Hi,I've posted before about my PC crashing constantly (a couple of months ago) and I thought we had fixed the problem with the driver updates, etc that I was told to do, but my machine is crashing worse than ever now.I have tried everything I can think of & am completely lost as to what to do to make it work. A friend told me to scan my system with RegCure & it shoed up the following items that apparently have errors.I didn't want to purchase the program though so I have no idea what they mean or how to fix them... or even if they are things that could be crashing my system.Also tried looking up the event viewer, but regardless of going through the tutorial on it, I am still finding it beyond confusing.Any help would be appreciated. TIA.
A:Pc Keeps Freezing & Has To Be Manually Re-booted
It was wise of you not to "fix" anything with RegCure as it is a registry cleaner. Unless you know exactly what you are doing you could cause programs to not work or even the operating system by using registry cleaners. The authors of registry cleaners have different ideas of what "shouldn't" be there, and they have no clue about what is actually on your computer. I manage to wreck my Adobe software by using one. Fortunately, I had backed up my registry and I was able to undo the damage.Can you post a link to your previous thread so we can see what was done? Also, please tell us what you have done since then.Orange Blossom
2 more replies
Answer Match 33.18%
Good day everyone
I am a server and network administrator for a very tiny firm for a year odd already.
We have 1 Domain Controller server, 1 Backup (running acronis management console) and Symantec Endpoint Protection 11.0.5 server, and 3 other application server - all are having static manually configured IP addresses.
However during one of our recent application server reboot, it pop out an alert stating there is a conflict with some others on the network. I have checked through all the machines and have ensured all are having different ip address.
I proceeded to check the Event Viewer for more information, and it showed a particular machine having certain Ethernet Mac address causing the conflict. I went on to individual machine and fired up the CMD, and issued 'getmac /v' to get it's ethernet mac address and found the culprit - let's call this 'Server G'.
I went into 'Server G' Network Connection > LAN settings > TCP/IP to change the IP address. To my surprise, the IP address is showing the correct address intended for 'Server G' - I have also ensured no other machines having the same ip address. No mistake - on 'Server G', I opened up the CMD, and check the ip address by 'ipconfig /all', and to my horror, it showed a different address than the one input into the LAN Settings TCP/IP windows.
What happened?
If this issue has been resolved by another thread, please kindly point me to it, I wi... Read more
A:ip conflict with manually configured ip
odd but try a stack reset. May not help, but won't hurt.
TCP/IP stack repair options.
Start, Run, CMD to open a command prompt:
In the command prompt window that opens, type type the following commands, each followed by the Enter key:
Note: Type only the text in bold for the following commands.
Reset TCP/IP stack to installation defaults, type: netsh int ip reset reset.log
Reset WINSOCK entries to installation defaults, type: netsh winsock reset catalog
Reboot the machine.
Failing this come back with operating system on Server G and post the ipconfig/all and if possible a screen shot of the LAN TCP/IP screens?
Oh and sorry if this is obvious but there are not 2 lan cards on the server? One part of motherboard maybe and the other a NIC PCI card?
2 more replies
Answer Match 33.18%
Every time I download and then restart - then attempt to go back
online my modem gets knocked off line. If I go to device manager -
click on the modem/properties/resources/configure
manually/okay/then restart (or reboot) I can then get back on
line. The gal at the modem company says it is probably because I
reformatted my hard drive while the modem was still connected
but outside of that she didn't seem to know for sure? Anyone
have any ideas?
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Answer Match 33.18%
Hi,
My question is as follows:
(System specs: see signature)
In Internet Explorer (V. 5.50.4807.2300) there is an inbuilt interface that theoretically should enable one to organize the links & the folders in one's 'faborites' folder as one's heart should desire. This thing sucks. I mean, it behaves like a lab rat on LSD.
I know I can move items into folders manually (by directly accessing the 'favorites' folder), but I don't seem to be able to organize items within folders manually.
So, how can that be accomplished?
A:Manually organizing favorites in IE5
Ya know, organizing Favorites in IE has always been a pain. The only thing I can offer is to just make the folders you want and drag and drop the links you want into them. Then move the folders around in the order you like them. It takes a little effort. I have seen some third party software that organizes links for you and even checks to see if the links still are valid. Do a GOOGLE and see what pops up.
2 more replies
Answer Match 33.18%
So a couple times my computer had froze so I had to restart by pushing that button. Now is it bad for the computer if I do this? If so, how many times will it take before I notice corruption on my computer?
A:Question About Restarting Manually
8 more replies
Answer Match 33.18%
So my dear friend let her virus protection expire (over a year ago...yay!) and there is all sorts of fun stuff on her computer...I spent 2 hours last time and an hour yesterday. (6 malware tool bars...yay!) Ive accomplished a lot...used malware bytes, cccleaner and hijack this. However I am unable to remove some of the infections hijack this finds...hijack this even told me Id have to manually do it.(thanks...)The hijack this said to delete the files and label them as hosts....
I havent done this type of stuff in like 2 years and want some one to reitterate how to do it to make sure I am doing it right. Ive done this numerous times.
I found this page.... http://www.pcproblemfix.com/how-to-manually-remove-start-sweetpacks-com-virus/
which seems easier to me than doing it in the regedit...right? Im really not as dumb as I sound..I just havent done this in so long. As I read more Im like oh yeah....apparently computer kowledge gets fuzzy when you dont do it 24/7 anymore. Ill be back to fix her computer tuesday. Ive also instilled into her that when her virus protection expires..please let me know next time..I seem to be surrounded by people who have the most amazing talent for getting the weirdest viruses on their computer. Thankfully at least she knows not to clcik on pop ups that say click me...which my mom doesnt know yet. Ha..
sorry for the rant.
A:Need help manually removing sweetpacks...
Hello Britarchivist I would like to welcome you to the Malware Removal section of the forum.Around here they call me Gringo and I will be glad to help you with your malware problems.Very Important --> Please read this post completely, I have spent my time to put together somethings for you to keep in mind while I am helping you to make things go easier, faster and smoother for both of us!Please do not run any tools unless instructed to do so.We ask you to run different tools in a specific order to ensure the malware is completely removed from your machine, and running any additional tools may detect false positives, interfere with our tools, or cause unforeseen damage or system instability.Please do not attach logs or use code boxes, just copy and paste the text.Due to the high volume of logs we receive it helps to receive everything in the same format, and code boxes make the logs very difficult to read. Also, attachments require us to download and open the reports when it is easier to just read the reports in your post.Please read every post completely before doing anything.Pay special attention to the NOTE: lines, these entries identify an individual issue or important step in the cleanup process.Please provide feedback about your experience as we go.A short statement describing how the computer is working helps us understand where to go next, for example: I am still getting redirected, the computer is running normally, etc. Please do not describe the computer as "the ... Read more
20 more replies
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2018-11-13 07:01:17
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https://techcommunity.microsoft.com/t5/excel/combining-multiple-if-functions-in-one-cell/m-p/1598224/highlight/true
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Highlighted
New Contributor
# Combining multiple if(functions in one cell
Hi!
So this one is a bit tricky. I'm trying to connect these three separate formulas into one cell:
I got one formula which is - if this cell is salmon , and there is 1+ of them, choose this cell with input length cm to calculate weight, if not pick the cell with height (n12) and use this formula.
=IF(I12="salmon",IF(M12>0,SUM((((M12/100)^3)*10)*0.9711),SUM(0.004*N12^2.535))) - Works
then I got the same one for trout but without the height formula
=IF(I12="trout",IF(M12>0,SUM((((M12/100)^3)*10)*0.105) - Works
Now Im having trouble combining them; (last attempt)
IF(I11="trout",IF(M11>0,SUM((((M11/100^3)*10)*1.049)),IF(I11="salmon",IF(M11>0,SUM((((M11/100)^3)*10)*0.9711),SUM(0.004*N11^2.535)))))
Anyone got tips?
2 Replies
# Re: Combining multiple if(functions in one cell
@fishdude
The logic In short:
IF A(salmon) and B(1), and either C(length) or D(height); use E formula*C if C present or F formula* D if D present. But if A(trout) and B(1), and C (length) use G formula*C
Highlighted
# Re: Combining multiple if(functions in one cell
As variant
``````=IF(I12="trout",
IF(M12>0,
((M12/100)^3)*10*0.105,
0
),
IF(I12="salmon",
IF(M12>0,
((M12/100)^3)*10*0.9711,
0.004*N12^2.535
),
0))``````
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2020-10-23 22:55:32
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https://docs.databricks.com/delta/optimizations.html
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# Optimizing Performance and Cost¶
Delta can optimize the layout of data stored in DBFS to improve query speed. Delta currently supports two layout algorithms: bin-packing and zorder.
## Compaction (or Bin-Packing)¶
Delta can optimize a table to improve the speed of read queries, for example by coalescing small files into larger ones. This operation can be triggered by running the following command:
OPTIMIZE events
or
OPTIMIZE '/path/to/table'
If you have a large amount of data and only want to optimize a subset of it, you can specify an optional partition predicate using WHERE:
OPTIMIZE events WHERE date >= '2017-01-01'
Readers of Delta tables use snapshot isolation, which means that they are not interrupted when OPTIMIZE deletes unnecessary small files from the transaction log.
Bin-packing optimization is idempotent, meaning that if it is run twice on the same dataset, the second instance has no effect. Moreover, OPTIMIZE makes no data related changes to the table, so a read before and after an OPTIMIZE will have the same results.
Note
Performing this operation on a table that is a streaming source will not affect any current or future streams that treat this table as a source.
## ZOrdering (Multi-dimensional clustering)¶
ZOrdering is a technique that allows Delta to colocate related information in the same set of files. This co-locality is automatically used by Delta’s data-skipping algorithms to dramatically reduce the amount of data that needs to be read. In order to ZOrder data, you must specify the columns to perform the ordering on:
OPTIMIZE events
WHERE date >= current_timestamp() - INTERVAL 1 day
ZORDER BY (eventType)
You can specify multiple columns for ZORDER BY as a comma-separated list. However, the effectiveness of the locality will drop with each additional column.
ZOrdering is not an idempotent operation, and thus will rearrange all of the data that matches the given filter. Therefore we suggest that you limit it to new data, using partition filters when possible.
## How Often Should You Run OPTIMIZE?¶
We are frequently asked how often you should run optimize on a Delta table. This is largely a business decision. If you want to make sure that end-user queries are always fast, it is well worth running optimize frequently.
However, optimize does start up a number of Spark jobs, making it resource-intensive. For this reason, we often recommend that you start by running optimize once a day or week to see if that meets your needs. One tip is to run optimize in the middle of the night and leverage spot pricing to make it very cheap to organize your data.
## Data Skipping¶
Data skipping information is collected automatically for you when you write data into a Delta table. Delta takes advantage of this information (currently min and max values) on your behalf at query time in order to attempt to provide faster queries. End users do not need to configure data skipping.
## Garbage Collecting Old Files¶
Delta leaves deleted files on DBFS for a period of time to ensure that concurrent readers can continue reading a stale snapshot of the table. However, this functionality means that occasionally you must clear out these invalid files using the VACUUM command to save on storage costs. .. code:: sql
VACUUM events VACUUM ‘/path/to/data’
You can also specify DRY RUN to test the vacuum and return a list of files to be deleted:
VACUUM events HOURS DRY RUN
The VACUUM command removes any files that are no longer in the transaction log for the table and are older than a retention threshold. The default threshold is 7 days, but you can specify an alternate retention interval. For example, to delete all invalid files older than 1 day, you can execute the following SQL command:
Warning
Running VACUUM with a very small retention interval is unafe when there are concurrent readers or writers to the table. Concurrent read jobs can fail if you delete a stale snapshot that is currently being read. Files written by concurrent writers but not yet committed to the log can also be deleted if the retention period is shorter than the duraton of the job.
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2018-01-18 23:30:32
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http://physics.stackexchange.com/tags/statistics/new
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# Tag Info
0
In classical mechanics, one can keep track of individual particles. For example, if you have two particles in a system, you can color these particles red and blue and follow them. Notice this, for classical mechanics, dynamical variables are $(q_i,p_i)$ with $i=1,2$. There is nothing classical mechanics that prevents you from specifying more variables e.g ...
2
When you say "improves the reliability", well that is not clear at all, because you have reduced your sample size and possibly introduced an (unknown) bias. Median filtering is typically used where you do not fully understand the noise properties of your sample and where there may be cases of results that are way out from the expected result because of rare ...
6
Yes, it is a statistical average in the sense that the measured half life will approach a single value of a true half life if you do lots of measurements. In other words, if you did the experiment many, many times you would find that on average you had 4 particles left after a half-life had passed. For any individual experiment, the results would vary. ...
8
Half life is, by definition, the amount of time until half of an infinitely large sample would decay. That's precisely equivalent (according to the frequentist interpretation of probability, if that matters to you) to the time until an individual particle's probability of decay reaches one half. The half life is a theoretical quantity that doesn't depend on ...
Top 50 recent answers are included
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2015-07-05 19:34:35
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https://datascience.stackexchange.com/questions/41551/sparse-representation-vs-dense-representation
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# Sparse Representation vs Dense Representation
The one of the benefits of ReLUs is sparsity. Sparsity arises when a≤0 (a = wX+b). The more such units that exist in a layer the more sparse the resulting representation. Sigmoids on the other hand are always likely to generate some non-zero value resulting in dense representations. Sparse representations seem to be more beneficial than dense representations. Is it true that sparse representation is more beneficial that dense representation, especially for Neural Networks?
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2020-10-29 11:46:45
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https://www.dreamwings.cn/codeforces985d/5278.html
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# Codeforces 985 D. Sand Fortress （二分）
## Description
You are going to the beach with the idea to build the greatest sand castle ever in your head! The beach is not as three-dimensional as you could have imagined, it can be decribed as a line of spots to pile up sand pillars. Spots are numbered 1 through infinity from left to right.
Obviously, there is not enough sand on the beach, so you brought $n$ packs of sand with you. Let height $h_i$ of the sand pillar on some spot $i$ be the number of sand packs you spent on it. You can’t split a sand pack to multiple pillars, all the sand from it should go to a single one. There is a fence of height equal to the height of pillar with $H$ sand packs to the left of the first spot and you should prevent sand from going over it.
Finally you ended up with the following conditions to building the castle:
• $h_1 ≤ H$: no sand from the leftmost spot should go over the fence;
• For any $i\in[1,∞)]$ $|h_i - h_{i + 1}| ≤ 1$: large difference in heights of two neighboring pillars can lead sand to fall down from the higher one to the lower, you really don’t want this to happen;
• $\sum_{i=1}^{∞}h_i=n$: you want to spend all the sand you brought with you.
As you have infinite spots to build, it is always possible to come up with some valid castle structure. Though you want the castle to be as compact as possible.
Your task is to calculate the minimum number of spots you can occupy so that all the aforementioned conditions hold.
## Input
The only line contains two integer numbers $n$ and $H~(1 ≤ n, H ≤ 10^{18})$ — the number of sand packs you have and the height of the fence, respectively.
## Output
Print the minimum number of spots you can occupy so the all the castle building conditions hold.
## Examples input
5 2
## Examples output
3
## AC 代码
def judge(mid, n, H):
if mid <= H: # 一直递减的情形
return mid * (mid + 1) // 2
# 左右两侧都是等差数列
right = mid * (mid + 1) // 2
left = (mid - H + 1) * H + (mid - H + 1) * ((mid - H + 1) - 1) // 2
return left + right - mid
def solve(n, H):
low = 1
high = 1000000000000000000
while True:
mid = (low + high) >> 1
now = judge(mid, n, H)
if low == mid:
if now >= n:
return low + max(0, low - H)
elif now + low >= n: # 能否再加一列使其满足条件
return low + max(0, low - H) + 1
return high + max(0, high - H)
if now == n:
return mid + max(0, mid - H)
elif now < n:
low = mid
elif now > n:
high = mid
pass
try:
while True:
n, H = input().split()
n = int(n)
H = int(H)
print(solve(n, H))
except EOFError:
pass
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2019-06-17 07:16:54
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https://gamedev.stackexchange.com/questions/30755/what-is-the-appropriate-way-of-placing-game-objects-in-a-2d-game/30822
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# What is the appropriate way of placing game objects in a 2d game?
I'm developing a 2d game for Android.It will be an endless game.I have some items like coins ,plants and other special items.How can i place them effectively?Are there any algorithm or any suggestion for this?
I'm already know that i have to use random numbers for placing them along x and y axis.It's important to not placing two different item at the same x and y position.
I've used the following approach to generate apples on a snake clone. It's very simple, although it has some potential problems depending on your game requirements. But if you know will always be a lot more free locations that occupied locations, it should be enough in practice.
Generate a random location for the X and Y axis separately based on the size of the map, but wrap it in a do...while loop that automatically retries whenever the location is already occupied. Something like:
// Find valid location
int x, y;
do {
x = random(0, width);
y = random(0, height);
} while(IsLocationOccupied(x, y))
// Create object there
SpawnObjectAt(x, y);
The implementation of IsLocationOccupied depends on your game. For instance, for a tile-based game you could simply check the contents of the tile at that location. For example:
bool IsLocationOccupied(int x, int y)
{
return map[x, y] != TileType.Empty;
}
For a non-tile based game, you could iterate over all your objects and check if any of them would intersect with a new object spawning at that location. For example:
bool IsLocationOccupied(int x, int y)
{
Rectangle newBounds = new Rectangle(x, y, objectWidth, objectHeight);
foreach(Entity entity in entities)
if(entity.bounds.Intersects(newBounds))
return true;
return false;
}
A safer alternative is to somehow keep a list of all the free locations in the screen, and remove one element at random from the list when you need to spawn a new object. This is great in a tile-based game, but even in a continuous environment you can still divide it into an imaginary grid to limit the number of locations.
Since the game world is endless, I would chunk in up into regions. When the player moves close to the edge of the region in one direction, you can generate the objects in the next region in that direction. As the games runs, you can adjust the level play by controlling how many "good" and "bad" objects you place in the upcoming region to keep the player engaged. For example, if the player's "health" is slipping, you may want to adjust the number of "bad" object down or increase the number of "good" objects in the next region he encounters. David's answer is great for spawning objects. You may want to adjust it slightly so that you don't put your game objects too close together or too far apart. Instead of testing for intersection, just test a bigger area around the potential location. Consider having a few dozen or so predefined regions that are "fun" to play.
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2021-05-17 15:57:51
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https://www.physicsforums.com/threads/frame-dragging-help.601839/
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# Frame dragging help:
1. Apr 30, 2012
### eggman100
Can someone please explain to me (it can only be brief, I'll try to do the dirty work myself ^.^):
Frame dragging in terms of:
What is a frame, What are it's dimensions (2d/3d(x,y,z))
What does this have in relation to the Schwarzschild radius in terms of astrophysics;
With that answer, is it possible to use the metrics to work out the dragging of matter and space(/time) per second according to the mass and gravitational force of the blackhole?
E.g. using the Kerr-Newman metric:
Can I use the metric to work out how much the mass would stretch from a distance of 5km, and how much more it would be stretched per 1km closer the object is to the center?
In terms of the metric, what does the angles mean? In the metric it shows E.g: dθ^2, what does this mean in terms of frame dragging?
Thank you for reading, I really need this to be answered by anyone who can! Please! :)
Thank you so much! :)
2. Apr 30, 2012
### Drakkith
Staff Emeritus
Unfortunately I don't think it is easy enough to do for someone who hasn't been formally educated in GR and the prerequisite courses.
3. Apr 30, 2012
### eggman100
No I mean't, I just wanted a basic explanation, nothing hard, I just wanted to know short answers to my questions, no hardcore explanation =)
4. May 6, 2012
### stevebd1
In the equatorial plane, the Schwarzschild boundary (2M) is in the same place as the outer boundary of the ergosphere for a spinning black hole. This remains unchanged regardless of how much spin the black hole has, this doesn't apply at the poles. the coordinate radius for the event horizon within the ergosphere reduces the greater the spin.
ergosphere (re)-
$$r_{e} = M + \sqrt{M^2 - a^2 \cos^2 \theta}$$
where $M=Gm/c^2,\ a=J/mc$ and $\theta$ is the plane angle (i.e. equatorial plane = 90 degrees)
the event horizon (r+) is-
$$r_{+} = M + \sqrt{M^2 - a^2}$$
Regarding the second question, the frame dragging rate as observed from infinity is-
$$\omega=\frac{2Mrac}{\Sigma^2}$$
where $\omega$ is the frame dragging rate in rads/s, $\Sigma^2=(r^2+a^2)^2-a^2\Delta \sin^2\theta$ and $\Delta= r^{2}+a^{2}-2Mr$
For the local frame dragging rate, you multiply $\omega$ by the redshift $(\alpha)$ where-
$$\alpha=\frac{\rho}{\Sigma}\sqrt{\Delta}$$
where $\rho=\sqrt{r^2+a^2 \cos^2\theta}$
Source-
http://www.lsw.uni-heidelberg.de/users/mcamenzi/CObjects_06.pdf [Broken]
Last edited by a moderator: May 6, 2017
5. May 10, 2012
### Bobbywhy
In addition to the environment near black holes frame-dragging is also evident in rotating spiral galaxies and in dynamic galaxy clusters. Of course, the effects are less because the masses are less, but the effect is cumulative over the eons of galaxy/cluster formation. Here are some simplified statements about their evolution:
• The geometry of a rotating rigid disc, or that of a virialized rotating spiral galaxy, is not Euclidean because of the Lorentzian contraction.
• Identical clocks in a spiral galaxy run at different rates, depending on their location along the radius. A clock at the circumference runs more slowly than one at the center.
• Emitters that are accelerated or in a gravitational field have their wavelengths shifted according to the strength of that field. Emitters in a rotating system are subject to two accelerating fields: Centripetal and Coriolis.
• The inertial frame of the rotating galaxy is twisted by Lense-Thirring frame-dragging, an effect predicted by relativity, has been measured by the Gravity Probe B satellite.
• The total gravitating action created by the galaxy or cluster depends on its total energy, that is, the total ponderable energy plus the gravitational energy.
• The energy of the gravitational field itself contributes to the space-time curvature.
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2018-03-20 01:01:00
|
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http://mathhelpforum.com/calculus/67378-antiderivative.html
|
1. ## Antiderivative
hi,
Q1. Write the gerneral form of the antiderivative of
f(x) = 4x - 2x^2 + x*squareroot(x) + e^x
b) Hence calculate
f (4x - 2x^2 + x*squareroot(x) + e^x) dx
*where f is an antiderivative, a = 0, b = 1.
thanks,
moon
2. Originally Posted by Moon Hoplite
hi,
Q1. Write the gerneral form of the antiderivative of
f(x) = 4x - 2x^2 + x*squareroot(x) + e^x
b) Hence calculate
f (4x - 2x^2 + x*squareroot(x) + e^x) dx
*where f is an antiderivative, a = 0, b = 1.
thanks,
moon
a) $\displaystyle f(x) = 4x - 2x^2 + x \sqrt{x} + e^x$.
Remember that $\displaystyle x \sqrt{x} = x \times x^{\frac{1}{2}} = x^{\frac{3}{2}}$.
So $\displaystyle F(x) = \int{f(x)\,dx} = \int{4x - 2x^2 + x^{\frac{3}{2}} + e^x\,dx}$
$\displaystyle = 2x^2 - \frac{2}{3}x^3 + \frac{2}{5}x^{\frac{5}{2}} + e^x + C$.
For b) evaluate $\displaystyle F(1) - F(0)$.
3. Originally Posted by Prove It
a) $\displaystyle f(x) = 4x - 2x^2 + x \sqrt{x} + e^x$.
Remember that $\displaystyle x \sqrt{x} = x \times x^{\frac{1}{2}} = x^{\frac{3}{2}}$.
So $\displaystyle F(x) = \int{f(x)\,dx} = \int{4x - 2x^2 + x^{\frac{3}{2}} + e^x\,dx}$
$\displaystyle = 2x^2 - \frac{2}{3}x^3 + \frac{2}{5}x^{\frac{5}{2}} + e^x + C$.
For b) evaluate $\displaystyle F(1) - F(0)$.
so does b) = 3.1516??
4. Originally Posted by Moon Hoplite
so does b) = e??
$\displaystyle F(1) = 2(1)^2 - \frac{2}{3}(1)^3 + \frac{2}{5}(1)^{\frac{5}{2}} + e^1 + C$
$\displaystyle = 2 - \frac{2}{3} + \frac{2}{5} + e + C$
$\displaystyle = \frac{26}{15} + e + C$.
$\displaystyle F(0) = 2(0)^2 - \frac{2}{3}(0)^3 + \frac{2}{5}(0)^{\frac{5}{2}} + e^0 + C$
$\displaystyle = 1 + C$.
So $\displaystyle F(1) - F(0) = \frac{26}{15} + e + C - (1 + C)$
$\displaystyle = \frac{11}{15} + e$.
So no, b) is not e. It's $\displaystyle \frac{11}{15} + e$.
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2018-04-20 11:39:58
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https://www.gamedev.net/forums/topic/223988-matrices/
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Archived
This topic is now archived and is closed to further replies.
matrices
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Hi all, i alway's used dx matrices but now i implemented my own. i can rorate ,scale and translate but when i first move my object and then rotate it it seems it moves like this - = object
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[\code]
it it turns around his own axis and around the world origin
the order of my transformations is(the same as i used with dx)
world = scale*rotation
world = world*translation
[edited by - freaker13 on May 7, 2004 9:40:21 AM]
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Make sure that your matrix code is fully DX-compliant, in other words, check to see if DX''s matrices are row-major or column-major, and be sure yours follow the same type.
Also, instead of aplying a whole sequence of transformations to your model, aply just one, and see if the end result matches. Try to see if the problem arises from rotation and/or translation.
Create 2 similar functions, one that renders with DX alone, and another that does the transformation using your matrix class, see how much they are diferent.
[Hugo Ferreira][Positronic Dreams][Colibri 3D Engine][Entropy HL2 MOD][My DevDiary]
"our stupidity allways comes back to bite us in the ass... in a white-shark sort of way..." - Prozak
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I think rotation is normally performed about an axis, which goes through the origin.
So you first have to scale, then rotate, and finally translate an object.
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its my rotation matrix that's wrong
dx matrix:
0.976588 0 0.215116 0 1 0 -0.215116 0 0.976588
my matrix:
0.976588 0 0.2151160 1 00 0 0.976588
whats my fault? i use this algo to get the rotation matrix:
float A = (float) cos(eulerAngleX);float B = (float) sin(eulerAngleX);float C = (float) cos(eulerAngleY);float D = (float) sin(eulerAngleY);float E = (float) cos(eulerAngleZ);float F = (float) sin(eulerAngleZ);float AD =(float) A * D;float BD =(float) B * D;matrix[0][0] = C * E;matrix[0][1] = -C * F;matrix[0][2] = D;matrix[1][0] = BD * E + A * F;matrix[1][1] = -BD * F + A * E;matrix[1][2] = -B * C;matrix[2][0] = -AD * E + B * F;matrix[2][1] = AD * F + B * E;matrix[2][2] = A * C;
[edited by - freaker13 on May 7, 2004 6:38:31 PM]
[edited by - freaker13 on May 7, 2004 6:39:21 PM]
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2018-08-15 07:50:27
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http://math.stackexchange.com/questions/221213/probability-of-sum-of-two-independent-variables-given-joint-density
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# Probability of sum of two independent variables given joint density
Let $x$ and $y$ be $2$ independent random vectors on the unit disk such that their joint density is just $\frac{1}{\pi}$. What is the probability that $x+y$ is less than $1$?
-
Are you sure they are not independent? – jay-sun Oct 26 '12 at 0:41
I suspect you mean that $x$ and $y$ are the coordinates of a point in the unit disk? However, in that case they're not independent. If you do mean vectors, how do you compare $x+y$ to $1$? – joriki Oct 26 '12 at 0:43
Assuming that you mean that $x$ and $y$ are the coordinates of a point randomly uniformly chosen in the unit disk:
The area of the unit disk below the line $x+y=1$ consists of three quarter-circles with area $\pi/4$ each and a triangle with area $1/2$, so the probability is
$$\frac{3\pi/4+1/2}\pi=\frac34+\frac1{2\pi}\;.$$
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2014-03-15 12:37:55
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http://planetmath.org/omegalimitset-0
|
# $\omega$-limit set
## Primary tabs
Defines:
$\alpha$-limit, alpha-limit, $\omega$-limit, omega-limit
Synonym:
omega-limit set
Type of Math Object:
Definition
Major Section:
Reference
## Mathematics Subject Classification
37B99 None of the above, but in MSC2010 section 37Bxx
### alpha-limit set ???
The alpha-limit set seems rather glibly defined to me. In most cases, ''f'' won't be a bijection, for $f^{-1}$ won't be defined. So then $f^{-1}$ must mean the pre-image of ''f''? In this case, is the alpha-limit set the same thing as the Julia set ?? This needs clarification.
### merge articles.
Seems that PM has several articles on this topic. please consider merging them together:
http://planetmath.org/encyclopedia/OmegaLimitSet.html which is another article with the same title(!)
and http://planetmath.org/?op=getobj&from=objects&id=6722 (titled "limit cycle")
### Re: alpha-limit set ???
Note that $f$ is assumed to be a homeomorphism.
### Re: alpha-limit set ???
Thanks, OK, right. Since in dynamical systems, a lot of iterated maps are not homeomorphisms, this flew right past me. Certainly, the definition for the omega-limit can be extended to more general functions. The definition for the alpha-limit in such a case is trickier, and I'm wondering if there is a generally accepted definition in such a case.
### Re: alpha-limit set ???
The definition of omega-limit set is the same for non-invertible maps. The usual definition of alpha-limit set (altough not very useful) is by taking sequences of preimages, i.e. sequences {x_n} such that f(x_n)=x_{n-1} and x_0 = x, and consider the alpha-limit set of x to be the set of all acummulation points of all such sequences.
I decided not to add this to the entry for two reasons: A reader who is looking for the definition of omega-limit set will most likely not be discouraged by the fact that it is defined for homeomorphisms and assume that the definition is the same for non-homeomorphisms; on the other hand the definition of alpha-limit set is of such limited use that i didn't think it was worth mentioning. It is unlikely that anyone using alpha-limit sets of non-invertible maps will not define it first, so i don't see a good reason to obfuscate this entry by adding that definition.
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2013-05-21 13:50:13
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|
https://plainmath.net/3848/represented-differential-equation-conditions-differential-afunction
|
Question
# A dynamic system is represented by a second order linear differential equation. 2frac{d^2x}{dt^2}+5frac{dx}{dt}-3x=0 The initial conditions are given as: when t=0, x=4 and frac{dx}{dt}=9 Solve the differential equation and obtain the output of the system x(t) as afunction of t.
Second order linear equations
A dynamic system is represented by a second order linear differential equation.
$$2\frac{d^2x}{dt^2}+5\frac{dx}{dt}-3x=0$$
The initial conditions are given as:
when $$t=0,\ x=4$$ and $$\frac{dx}{dt}=9$$
Solve the differential equation and obtain the output of the system x(t) as afunction of t.
2021-02-17
Solution. Given $$2\frac{d^2x}{dt^2}+5\frac{dx}{dt}-3x=0$$ (1)
when $$t=0$$ then $$x=4$$ and $$\frac{dx}{dt}=9$$
Auxiliary equation
$$2m^2+5m-3=0$$
$$2m^2+6m-m-3=0$$
$$(2m-1)(m+3)=0$$
$$m_1=-3$$ and $$m_2=\frac{1}{2}$$
Auxiliary equation
Then solution $$x(t)=c_1e^{3t}+c_2e^{\frac{1}{2t}}$$ (2)
$$\frac{dx}{dt}=3c_1e^{3t}+\frac{1}{2c}_2e^{\frac{t}{2}}$$
$$x(0)=4$$
$$c_1+c_2=4$$ (3)
Also $$\frac{dx}{dt}=9$$ when $$t=0$$
$$3c_1+\frac{c_2}{2}=9$$
$$6c_1+c_2=18$$ (4)
Solving (3) and (4) we get
$$5c_1=14\Rightarrow c_1=\frac{14}{5}$$
And $$c_2=\frac{6}{5}$$
Then solution (2) will becomes.
$$x(t)=\frac{14}{5}e^{3t}+\frac{6}{5}e^{\frac{t}{2}}$$
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2021-08-01 02:03:00
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https://www.mail-archive.com/ntg-context@ntg.nl/msg101985.html
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# Re: [NTG-context] Table breaks incorrectly between pages
Wow, thank you very, very much!
On Sun, 15 May 2022, 05:35 śrīrāma, <citt...@gmail.com> wrote:
> On 5/13/22 12:30 PM Stefan Nedeljkovic via ntg-context wrote:
> > Dear list I need your help with a table that seems to completely screw up
> > the page when it spans across multiple pages. I'm attaching all the
> > necessary files.
> >
> > Kind regards,
> > Stefan
>
> This seems to be a known issue for quite a while, see here:
> https://www.mail-archive.com/ntg-context@ntg.nl/msg69827.html
>
> As Wolfgang suggested there, you can use framed instead of TABLE in your
> header. I have modified your example file (see attached). With that you
> get the
> attached PDF.
>
> By the way, your example was missing a \bTABLEhead. Moreover, I believe
> cells
> in TABLE head entries should be enclosed by \bTH \eTH instead of \(b|e)TD.
> I
> have made these changes as well.
>
> [Also, I hope you are aware that you can use style=\tt, style=\bf per row/
> column/cell with \setupTABLE; that will significantly reduce markup in
> your
> file]
>
> Best,
> Sreeram
___________________________________________________________________________________
If your question is of interest to others as well, please add an entry to the
Wiki!
maillist : ntg-context@ntg.nl / http://www.ntg.nl/mailman/listinfo/ntg-context
webpage : http://www.pragma-ade.nl / http://context.aanhet.net
archive : https://bitbucket.org/phg/context-mirror/commits/
wiki : http://contextgarden.net
___________________________________________________________________________________
|
2023-02-01 11:33:35
|
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https://web2.0calc.com/questions/help-pls_45
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+0
# Help pls
0
89
1
Let A be a matrix such $$\mathbf{A}^{-1} = \begin{pmatrix} 1 & 1 \\ 2 & x \end{pmatrix}$$ that
for some value of x. What is the vector that A maps to $$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$$
Feb 26, 2019
$$Av=\begin{pmatrix}1\\0\end{pmatrix}\\ A^{-1}Av = A^{-1}\begin{pmatrix}1\\0\end{pmatrix}\\ v = \begin{pmatrix}1&1\\2&x\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}1\\2\end{pmatrix}$$
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2019-07-16 15:15:11
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https://mathematica.stackexchange.com/questions/55039/how-to-find-intersection-between-a-parametric-trajectoy-and-a-point?answertab=oldest
|
# how to find intersection between a parametric trajectoy and a point
I'have a 2d-system of differential equations, analitycally solved, depending on a parameter. I know that, by continuity, there exist a minimum value of the parameter such that trajectory passes through a specific point. My problem is to find such value (also approximatively).
This is the system-equation:
ti = 0;
yi = 0;
zi = -.75;
zf = -.5;
eps=.01;
sol = FullSimplify[DSolve[{y'[t] == 1/(2 zf) y[t] - u z[t],
z'[t] == -1 + 1/zf z[t] + u y[t], y[ti] == yi, z[ti] == zi},{y[t],z[t]},t]];
The parameter is u.
I have to find the first u such that trajectorie intersect the point: z=zf; y=Sqrt[-2*(zf)*eps + eps^2]
Note that this point is done by the intersection between z=zf and the circle of radius |zf|+eps. In the following, you can see an animation of the system:
Manipulate[
Module[{sol, y, z, t},
sol = First@DSolve[{y'[t] == 1/(2 zf) y[t] - u z[t],
z'[t] == -1 + 1/zf z[t] + u y[t], y[ti] == yi,
z[ti] == zi}, {y[t], z[t]}, t];
Show[p1, p2, Graphics[{Red, Line[{{0, -1}, {0, 1}}]}],
Graphics[{DotDashed, Red, Thickness[.006],
Line[{{-1, zf}, {1, zf}}]}],
ParametricPlot[{y[t] /. sol, z[t] /. sol}, {t, 0, tend},
PlotStyle -> Thickness[.004]]]],
{{tend, .1, "t"}, .01, 20, .1, Appearance -> "Labeled"},
{{u, 10, "u"}, -300, 300, .5, Appearance -> "Labeled"},
{{zi, -.75, "zi"}, -1, 1, Appearance -> "Labeled"},
{{yi, 0, "yi"}, -1, 1, Appearance -> "Labeled"},
Initialization :>
(deltaA[y_, z_] := 1/(2 zf) y^2 - z + 1/zf z^2;
circ[y_, z_] := y^2 + z^2;
p2 = ContourPlot[{circ[y, z] == (-zf + eps)^2}, {y, -1,
1}, {z, -1, 1}, ContourStyle -> Yellow, GridLines -> Automatic,
Frame -> True, FrameLabel -> {"y", "z"}, RotateLabel -> False,
LabelStyle -> {FontSize -> 20}];
p1 = ContourPlot[{deltaA[y, z] == 0}, {y, -1, 1}, {z, -1,
1}, ContourStyle -> Green, GridLines -> Automatic,
Frame -> True, FrameLabel -> {"y", "z"}, RotateLabel -> False,
LabelStyle -> {FontSize -> 20}];)]
• y depend on t, the time, AND u, a free parameter ... – Mike84 Jul 16 '14 at 17:44
• I'have just added a Manipulate to show the behaviour of the system. Anyway, maybe I've not understand your question...the definitions of y[t],z[t] come from DSolve. They are analytic.. – Mike84 Jul 16 '14 at 18:18
• Maybe I am the one not understanding :) Your code doesn't compute, brackets mismatch. – Öskå Jul 16 '14 at 18:21
• ??? :) Try again...If your problem was in DSolve, I'have rewrited it – Mike84 Jul 16 '14 at 18:29
• I'm afraid that it doesn't work. – Öskå Jul 16 '14 at 18:38
Here is a stab at what I think you are asking:
ti = 0;
yi = 0;
zi = -.75;
zf = -.5;
eps = .01;
sol = {y[t], z[t]} /. First@FullSimplify[DSolve[{
y'[t] == 1/(2 zf) y[t] - u z[t],
z'[t] == -1 + 1/zf z[t] + u y[t],
y[ti] == yi, z[ti] == zi},
{y[t], z[t]}, t]];
target = {Sqrt[-2*(zf)*eps + eps^2], zf};
brute force discretize the solution and find the minimum distance to the target point (for such a highly nonlinear function this is much faster than NMinimise, and guarantees finding a global min, within the discretization approximation of course )
dis[u0_] :=
Min@Table[Norm[ (sol /. u -> u0) - target ] , {t, 0, 10, .05}];
note the range and increment on t here are important tuning parameters to play with.
Plot[dis[u], {u, -1, 1}]
you come very close to your target point around 0.21.
Armed with a good guess now we can use FindMinimum:
FindMinimum[Norm[sol - target], {u, .21 } , {t, 1}]
{3.23748*10^-9, {u -> 0.230625, t -> 1.62219}}
ParametricPlot[
Table[Chop[sol /. u -> u0 ], {u0, {.1, .230625, .3, 1, 5}}] , {t, 0, 10},
Epilog -> Point[target], PlotRange -> All, AspectRatio -> 1]
• thank you very much for this very useful and didactic answer. I have a question: why do you need chop? Furthermore I try to make a numerical procedure, using a Do and the function FindAllCrossing. But it doesn't work and I don't know why. Can I post an answer with my wrong-code to understand my error? – Mike84 Jul 17 '14 at 0:04
• Chop removes a numerically negligible complex part. Hitting a point in 2d isn't formulated in terms of 'zero crossings' so i cant see how you could use the methods in the link. You need to find minimum distance to the point and if you need to automate it you need to put in a check that the minimum you find is really zero. – george2079 Jul 17 '14 at 12:49
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2019-09-22 03:17:02
|
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http://mathoverflow.net/feeds/user/27223
|
User mks - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:41:05Z http://mathoverflow.net/feeds/user/27223 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109569/constructing-equivalent-algebraic-expressions-for-matrix-equations Constructing equivalent algebraic expressions for matrix equations mks 2012-10-13T23:38:16Z 2012-10-14T09:18:26Z <p>I have an expression involving matrices, of the form:</p> <p>$$f(k)=x^T A_k^{-1}A x$$ </p> <p>where $x$ is a $1\times N$ vector, $A_k = A + k I$ and $A$ is an $N\times N$ matrix ($A_k$ is invertible for all $k$) and $k>0$. It is known that $f(k)$ is real and monotonic increasing, but nothing more. I need to further analyze the behaviour of $f(k)$ and the simplest way would be to plugin my matrices and plot it. However, this becomes computationally intensive for large $N$ as it involves calculating the inverses repeatedly. </p> <p>One thought I had was to create a one-to-one map from $f(k)$ to an equivalent algebraic expression $g(k)$, which is easy to evaluate and investigate analytically. Indeed, if the expression had involved traces of inverses, one could've used the Stieltjes transform for some insight, but that doesn't seem likely here. </p> <p>My question is: Are there general approaches/references I can look at to learn how to tackle such problems? </p>
|
2013-06-20 00:41:06
|
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|
https://mathoverflow.net/questions/167569/matrix-generator-for-m-m-1-queue-waiting-time-distribution/167574#167574
|
# Matrix Generator for M/M/1 Queue Waiting Time Distribution
I "believe" that generator, $\bf W$, of the waiting time distribution for the M/M/1 queue is given by the following (I'm not sure if this is even correct):
${\bf W} =\left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & 0\\ \mu & -\mu & 0 & 0 & 0\\ 0 & \mu & -\mu & 0 & 0 \\ 0 & 0 & \mu & -\mu & \dots \end{array} \right)$
But the question I have is that I am unclear how to solve this Markov chain. That is, I'm looking for an analytic solution to
$\bf pW=0$
I think $\bf p$ should look something like
${\bf p} = [1−ρ,…],$
but again, I am unclear how to solve these problems.
Thanks for help in these matters.
Almost. The generator is ${\bf Q} =\left( \begin{array}{ccccc} -\lambda & \lambda & 0 & 0 & 0\\ \mu & -(\lambda+\mu) & \lambda & 0 & 0\\ 0 & \mu & -(\lambda+\mu) & \lambda & 0 \\ 0 & 0 & \mu & -(\lambda+\mu) & \dots \end{array} \right),$ where $\lambda$ is the arrival rate and $\mu$ the departure. If $\mu>\lambda$ (more people leave the queue than arrive), the chain is recurrent and the invariant distribution is an exponential distribution, i.e., $p_k=(1-\rho)\rho^k$, and $\rho=\frac{\mu}{\lambda}$. If $\rho> 1$ (resp. $=1$), the queue is transient (resp. null recurrent) and there is no stationary distribution.
• @PMF In any case, it seems to me that the solution you are looking for is $[1,0,0,0,\dots]$: from the equation corresponding to the first column of $W$ you get $p_2=0$, then from the second column $p_3=0$, and so on. $p_1$ remains indeterminate, and for the usual normalization $p\underline{\mathbf{1}}$ to hold we need $p_1=1$. May 24 '14 at 9:04
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2021-10-17 07:00:32
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https://www.projecteuclid.org/euclid.bjma/1381782096
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## Banach Journal of Mathematical Analysis
### Absolutely summing operators on separable Lindenstrauss spaces as tree spaces and the bounded approximation property
#### Abstract
Let $X$ be a Banach space and let $Y$ be a separable Lindenstrauss space. We describe the Banach space $\mathcal{P}(Y,X)$ of absolutely summing operators as a general $\ell_1$-tree space. We also characterize the bounded approximation property and its weak version for $X$ in terms of the space of integral operators $\mathcal{I}(X,Z^*)$ and the space of nuclear operators $\mathcal{N}(X,Z^*)$, respectively, where $Z$ is a Lindenstrauss space, whose dual $Z^*$ fails to have the Radon-Nikodým property.
#### Article information
Source
Banach J. Math. Anal., Volume 8, Number 1 (2014), 190-210.
Dates
First available in Project Euclid: 14 October 2013
https://projecteuclid.org/euclid.bjma/1381782096
Digital Object Identifier
doi:10.15352/bjma/1381782096
Mathematical Reviews number (MathSciNet)
MR3161691
Zentralblatt MATH identifier
1277.47028
#### Citation
Lima, Asvald; Lima, Vegard; Oja, Eve. Absolutely summing operators on separable Lindenstrauss spaces as tree spaces and the bounded approximation property. Banach J. Math. Anal. 8 (2014), no. 1, 190--210. doi:10.15352/bjma/1381782096. https://projecteuclid.org/euclid.bjma/1381782096
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2019-10-19 01:06:34
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https://cs.stackexchange.com/questions/24215/practical-algorithm-for-testing-whether-an-edge-is-delaunay
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# Practical algorithm for testing whether an edge is Delaunay
I have a set of vertices $V$ and a set of segments $S$. I want to know whether a segment in the set $S$ is Delaunay against the vertices in $V$.
I would like to state my assumed definition of a Delaunay edge: An edge is Delaunay, iff there exists a circumsphere of its endpoints not containing any other vertex inside it.
I would like to know practical approaches/algorithms for such Delaunay-edge test.
• In case of not getting any answer, would it be better to cross post this question to CSTheory.SE? – Pranav Apr 30 '14 at 5:57
• I have posted this question on CSTheory.SE – Pranav May 15 '14 at 12:25
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2020-02-26 17:57:11
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http://forum.allaboutcircuits.com/showthread.php?t=69956
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All About Circuits Forum three phase transformer
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#1
05-11-2012, 11:20 PM
PG1995 Senior Member Join Date: Apr 2011 Posts: 743
three phase transformer
Hi
Regards
PG
Attached Images
hughesExamp34.9_3.jpg (288.2 KB, 35 views)
#2
05-12-2012, 06:36 AM
t_n_k Senior Member Join Date: Mar 2009 Posts: 4,808
Q1: The normal convention used for 3-phase systems is that, unless specifically indicated by the wording, the stated supply voltage is taken to be the line-to-line value.
Q2: It doesn't matter if the supply is star or wye connected. If the system voltage is quoted as a certain value, the source [generator / transformer] line-to-line value would be that same. The individual phase winding voltages at the source would differ depending on whether the source is configured as delta or wye. But the source configuration [be it wye or delta] has no bearing on the voltages observed in the transformer windings being considered here.
Q3:The problem states that the secondary is configured as star [wye]. The primary winding configuration is delta. So there is 3300V across any line-to-line primary [delta connected] winding. The transformed winding voltage on the secondary must be induced according to the primary-to-secondary winding ratio. So the secondary windings all have 283V. But the secondary windings are in star [wye], so the secondary line-to-neutral [star point] voltage is 283V. Hence the secondary line-to-line voltage is √3*283V or 490V.
The Following User Says Thank You to t_n_k For This Useful Post: PG1995 (05-12-2012)
#3
05-12-2012, 08:45 PM
PG1995 Senior Member Join Date: Apr 2011 Posts: 743
Thanks a lot, t_n_k. I understand it now.
Someone who has reached this thread might find the following link useful:
Regards
PG
#4
05-14-2012, 11:46 PM
PG1995 Senior Member Join Date: Apr 2011 Posts: 743
Hi
Thanks for the help.
Regards
PG
Attached Images
HughesExamp34.6_&_34.7_combined.jpg (247.6 KB, 6 views)
#5
05-16-2012, 02:14 AM
t_n_k Senior Member Join Date: Mar 2009 Posts: 4,808
On the matter of efficiency I'm not sure where the author's use of the n^2 comes from - although I might be better informed if I had the complete text in front of me.
I would approach the problem from a more "fundamental" perspective.
Consider the primary input voltage to be Vp with current Ip and power factor cos(θ).
The fractional efficiency would be given by the general relationship
$\eta=\frac{P_{out}}{P_{in}}=\frac{P_{in}-P_{loss}}{P_{in}}=1-\frac{P_{loss}}{P_{in}}$
The losses would comprise the [assumed constant] no-load magnetization losses Po plus the [variable] winding losses.
In other words
$P_{loss}=P_o+I_p^2R_t$
Where if N (not the 'n' mentioned above) is the primary-to-secondary turns ratio
$R_t=R_p+N^2R_s$
In your example Rp=0.42Ω, Rs=0.0019Ω and N=11000/400=27.5
Hence
$R_t=0.42+(27.5)^2*0.0019=0.42+1.4369=1.8569 \ \Omega$
We can then write the fractional efficiency as
$\eta=1-\frac{(P_o+I_p^2R_t)}{P_{in}}$
But
$P_{in}=V_pI_pcos(\theta)$
Hence
$\eta=1-\frac{(P_o+I_p^2R_t)}{V_pI_pcos(\theta)}=1-\frac{P_o}{V_pI_pcos(\theta)}-\frac{I_pR_t}{V_pcos(\theta)}$
to find the maximum efficiency we differentiate η with respect to the variable input current Ip at some arbitrary power factor.
Or
$\frac{\partial \eta}{\partial I_p}=\frac{P_o}{V_pI_p^2cos(\theta)}-\frac{R_t}{V_pcos(\theta)}$
We find the maximum (or minimum) by equating the derivative to zero.
Hence
$\frac{P_o}{V_pI_p^2cos(\theta)}-\frac{R_t}{V_pcos(\theta)}=0$
Which reduces to the condition for maximum efficiency
$\frac{P_o}{V_pI_p^2cos(\theta)}=\frac{R_t}{V_pcos( \theta)}$
or after simplifying
$P_o=I_p^2R_t$
Which re-iterates the statement in the text that maximum efficiency occurs when the winding I^2R losses equal the no-load losses.
From this one then can deduce the actual primary current to meet this condition.
We can also note (along with the text) that the maximum efficiency condition is independent of power factor. However the actual efficiency value at that condition will depend on the power factor. As an exercise you might try to determine what that maximum efficiency value might be.
So in the case of your example problem 34.7 with Po=2.9kW and Rt=1.8569Ω we have the value of
$I_p=\sqrt{$$\frac{2900}{1.8569}$$}=39.519 \ A$
At Vp=11kV this gives the primary VA input as 11000*39.519=434.71kVA.
Assuming a constant power factor of 0.8 through the transformer [*] then the input power would be 347.77kW. With the losses of 5.8kW this gives the load power as 341.97kW which differs slightly from the text value. In my case the maximum efficiency [at 0.8 pf] would then be 98.33%.
My final comment is that this is all a bit arbitrary. The text results imply a change in overall efficiency from full-load efficiency to maximum efficiency as a difference of 0.3%, which barely merits a mention at all. Efficiency is certainly a matter of great importance at rated operating conditions but the small difference from maximum η to the actual η value at rated conditions is miniscule.
Also be advised this analysis is also underpinned by certain assumptions [*] and approximations and does not reflect the exact "truth" over the possible load operating range having regard to such matters as load power factor and fault conditions.
Last edited by t_n_k; 05-16-2012 at 06:46 AM.
The Following User Says Thank You to t_n_k For This Useful Post: PG1995 (05-16-2012)
#6
05-17-2012, 04:01 PM
PG1995 Senior Member Join Date: Apr 2011 Posts: 743
Thank you very much for the help, t_n_k.
I have read your post several times since yesterday and now I have good understanding of the topic of maximum efficiency. But I'm kind of still struggling with the queries about the efficiency on half load and n^2 in my last post. This is the book I'm using (please replace asterisks with s-c-r-i-b-d dot com): http://www.***********/sipplefire/d/...y-10th-Edition. You can find the example problem 34.7 on page #723. Thanks.
Regards
PG
#7
05-18-2012, 12:58 AM
t_n_k Senior Member Join Date: Mar 2009 Posts: 4,808
It's interesting to note the derivation of the maximum efficiency condition at equation 34.19 is essentially identical to the process I followed in my previous post. So if you understand my post then the book should make equal sense.
Anyway leaving that aside for the moment I'll attempt to answer your questions concerning examples 34.6 & 34.7.
I'll start with example 34.7. With respect to the use of the fraction n I quote from the text.
"Let n=fraction of full-load apparent power in (kVA) at which the efficiency is a maximum."
To avoid any potential confusion it should be noted that the proposed value n is not the efficiency.
There is a small error made by the authors in moving from example 34.6 to 34.7. They quote the I^2R loss at full load as 3.86kW in example 34.7 whereas they derive it as 3.84kW in example 34.6.
The key to understanding what the authors are getting at in example 34.7 lies in the significance of equation 34.19 which for this example means that the maximum efficiency occurs when the winding losses equal the core losses of 2.9kW.
$I^2R_{Zc}=P_c$
This is the same as my equation
$P_o=I_p^2R_t$
Both equations tell me that the maximum efficiency condition occurs when the total primary & secondary winding I^2R losses equal the core magnetization [or no-load] losses.
The winding losses are proportional to the square of the currents in both primary & secondary windings. The equations above assume that we have lumped all the winding losses together by referring them to either the primary or the secondary - a process with which you should become familiar.
So at rated load the winding losses are actually 3.84kW. But maximum efficiency does not occur at rated load - rather it occurs at a somewhat lower value. If the primary current at rated load is Ir and the primary current at maximum efficiency is then they have a ratio
$k=\frac{I_{\eta}}{I_r}$.
Key Point: The ratio of the winding losses at the two conditions would therefore be proportional to the ratio of the square of currents.
$k^2=\frac{I_{\eta}^2}{I_r^2}$
Assuming the load voltages are essentially the same at these two conditions then we may write
$KVA_{$$\ max \ load$$}=I_rV_{load}$
KVA at maximum efficiency
$KVA_{$$\ max \ \eta$$}=I_{\eta}V_{load}$
Where $V_{load}$ would be the load voltage referred to the primary side. We could, with equal validity, refer everything to the secondary and this would produce the same result.
From the author's definition of n stated above we may write
$n=\frac{KVA_{$$\ max \ \eta$$}}{KVA_{$$\ max \ load$$}}=\frac{I_{\eta}V_{load}}{I_rV_{load}}=\frac{I_ {\eta}}{I_r}$
But we already know from above that .....
The ratio of the winding losses at the two conditions would therefore be proportional to the ratio of the square of currents
$k^2=\frac{I_{\eta}^2}{I_r^2}$
So in the same manner we can write
$n^2=\frac{I_{\eta}^2}{I_r^2}=k^2$
Which links back to the authors' original definition of the fraction n.
The ratio of the winding losses at maximum efficiency to the losses at full load is therefore n^2 [not n].
Last edited by t_n_k; 05-18-2012 at 06:10 AM.
#8
05-18-2012, 02:11 AM
t_n_k Senior Member Join Date: Mar 2009 Posts: 4,808
The half rated load condition occurs when the secondary load kVA value is decreased from 100% [500kVA] to 50% [250kVA].
The total winding resistance referred to the primary is 1.8569Ω.
The primary current at 100% rated load would be 500kVA/11kV=45.45A
The primary current at 50% rated load would be 250kVA/11kV=22.73A
The total [primary referred] winding loss at rated load current would be
The total [primary referred] winding loss at 50% rated load current would be
#9
05-18-2012, 01:52 PM
PG1995 Senior Member Join Date: Apr 2011 Posts: 743
t_n_k, thank you very much. It's really very kind of you. I need to read the material several times before making any follow-on queries if there are any. Once again, thanks.
Best wishes
PG
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2014-04-24 03:16:46
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https://answers.ros.org/question/273384/how-to-move-to-a-certain-point-in-space-using-twist-cmd_vel/
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ROS Resources: Documentation | Support | Discussion Forum | Index | Service Status | Q&A answers.ros.org
# How to move to a certain point in space using Twist /cmd_vel
Hi, I'm a beginner and I'm struggling to make my robot move to a certain point. How should I set the angular and linear velocity for my robot to move to a certain point ? For example, if my robot initial position is (0,0), what angular and linear velocity should I use for it to go to (5,5) ?
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I suggest you follow the ROS Navigation Tutorials, which will walk you through setting up a full navigation stack on your robot. This is much more involved than the simple node that BhanuKiran.Chaluvadi suggested, but you'll end up with a highly capable path planning system that can handle obstacles once you add sensors.
more
Hi, what you are talking about is called Robot navigation, that is, the ability of the robot to go from one place to another.
Then, in order to achieve that, you need to decide at least the following three things:
1. In which frame are you measuring the coordinates (0,0) and (5,5)?
2. Do you need to have obstacle avoidance (that is, avoid any obstacle that may appear between (0,0) and (5,5) when your robot is moving towards the end location)?
3. Is it the location where the robot moves known or unknown?
Let's assume the simplest answer to the questions above:
1. We are measuring the coordinates in the odometry frame (I'm assuming you know what odometry is). The problem with this frame is that has a sift (an error) which increases over time. But for your example is good enough.
2. Let's say, there are no obstacles in the path. Hence, we do not have to implement an obstacle avoidance algorithm like potential fields or else.
3. Let's say it is and unknown environment, of which we are not going to build a map.
Even if this is a very simple setup, that setup can be used, for example, to move the robot to the position of a detected target. Imagine that the robot has a camera and a program that detects people. Then, once the program detects a person in 5 meters in front of the robot, it means that the robot can go to the person location using that simple framework we defined in the questions above.
Now, you need to built a controller that converts the distances from the current position AND orientation of the robot into velocity commands that are sent to the /cmd_vel of the robot to make it move towards the location. That controller can be built in many ways.
A simple controller to do that would be the following:
1. If the robot (odometry) orientation is not towards the target, then rotate only the robot until its orientation is towards the target.
speed = Twist()
speed.linear.x = 0.0
speed.angular.z = 0.2
2. Once the robot is headed towards the target, just move straight towards the goal until reached
speed = Twist()
speed.linear.x = 0.5
speed.angular.z = 0.0
I have created this video that shows how that controller works with a Husky robot in a Gazebo simulated environment.
More complex controllers can be implemented that optimize the time to reach the goal, or other factors like moving faster when far away from goal and moving slower when getting closer.
In case you want to make the robot include obstacle avoidance, navigation in bigger environments of which a map can be built, then you need to use the Navigation Stack of ROS which includes all that (and more). Check this video tutorial for a good overview of what the ROS Navigation stack can do.
Hope it helps, and welcome to the robot navigation world!
more
( 2018-05-09 07:45:06 -0500 )edit
One Idea: Subscribe to current pose topic and publish to /cmd_vel.
Using your current robot pose and desired pose you can write a simple P controller. Based on how far they are apart and how your reference frame of you robot.
Random example:
geometry_msgs::Twist twist_;
twist_.linear.x = 2.0; // or based on distance between(current pose, desired pose)
twist_.linear.y = 0.0;
twist_.linear.z = 0.0;
twist_.angular.x = 0.0;
twist_.angular.y = 0.0;
twist_.angular.z = -pGain*sin();
more
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2022-10-01 18:21:53
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Based on your comments, it sounds like you have some sort of colored sand and you want to immobilize it. My suggestion would be some sort of polymer matrix. There are multiple possibilities, but one ...
• 26.9k
### Why are there no cheap diamond equivalents?
There are plenty of synthetic materials that would fall into the "cheap diamond equivalent" category. If we're only talking about the optical properties, cubic zirconia ($\ce{ZrO2}$) is probably the ...
• 12.6k
### How to calculate the height of an hcp lattice?
To calculate the height of a unit cell, consider a tetrahedral void in an hexagonal closed packing arrangement. It can be imagined as a 3 solid spheres touching each other and at the center-point, you ...
• 3,996
Accepted
### Will a solid object lose or gain atoms on standing over time (without being acted upon by external influences)?
Short answer: Yes they do. Firstly, you need to arrive at a proper definition for an "object". Why you ask? Because it would otherwise present philosophical problems which become apparent when we ...
• 18k
Accepted
### For a given packing (hcp, fcc, bcc), which ions occupy the corners and which occupy the faces or centers?
The actual answer is that it doesn't matter. For many of the 1:1 solid-state structures, either the cations or the anions may be considered to be at the vertices (i.e. corners) of the unit cell. By ...
• 65.8k
### Why is solid phase peptide synthesis limited to 70 amino acids?
The issue is one of yield. Each step has a certain yield percentage of yield, that the synthesizers will maximize as much as possible. However, given that it is fairly impossible to generate 100% ...
### How are the number of tetrahedral voids twice the number of octahedral voids in a CCP structure?
To be precise, if in a close packed structure (ccp or fcc) there are $n$ atoms or ions then the number of octahedral voids and tetrahedral voids will be $n$ and $2n$ respectively. For example, there ...
• 1,078
### How can brass still be made even though the crystal structures of zinc and copper are not the same?
While molten, copper and zinc (and tin) are miscible. As the alloy cools and crystalizes, the metals do indeed separate, forming grain boundaries. These inclusions "pin" slippage between ...
### Why is solid phase peptide synthesis limited to 70 amino acids?
You need to perform one reaction per amino acid to couple it to the previous one, and that reaction isn't 100% efficient. You always get a small amount of unreacted peptide, or some other side product....
• 5,908
### How to solidify sand
Geologists prepare soil profiles for a living. The task is easy in principle - dig a pit, apply glue to the wall, then transfer the glued layer to a sheet of cloth. The practice is really difficult, ...
• 2,368
### What does β stand for in β-potassium zirconate?
They signifies one polymorph (the beta polymorph) of barium zirconate. Allotropy is the property of some chemical elements or compounds to exist in two or more different forms, in the same physical ...
• 22.3k
### How does chromium help stainless steel prevent rusting?
TL;DR Note that the passive layer forms on the surface, there needn't be any change to lattice constant. Chromium needn't migrate , the Cr present on the surface will form the layer to protect it....
• 1,241
Accepted
### Why are X-rays used in crystallography?
There are a few reasons, but the most direct answer is that the wavelength of X-ray photons is on the order of the distance between atomic nuclei in solids, e.g. ~ 4 ångströms (bonds are roughly 1.5-2....
• 3,514
### What is "chemical pressure"?
External and internal pressure To study the effect of pressure on properties of a solid, is equivalently to learn how changes in volume transform physical parameters. For external pressure at ...
• 6,851
### How to solidify sand
I upvoted the water idea, after all it is possible to build sandcastles over 0.2 m high with just water as a binder. I also think coring as mentioned in that answer is a good idea. I also upvoted the ...
• 1,639
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### Are all lattices Bravais lattices?
From Ashcroft and Mermin's Solid State Physics: A fundamental concept in the description of any crystalline solid is that of the Bravais lattice, which specifies the periodic array in which the ...
• 7,650
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### Yellow color of NaCl
An F-center (or any color center) is a point defect in the material that generates localized electronic states. When these states are in the band gap, they will result in strong peaks in optical ...
• 7,650
Accepted
### Materials with solid volume greater than liquid volume
There is a bunch of such materials; among the elemental compounds, they include silicon, gallium, germanium, and bismuth. As for the properties... well, it just so happens that their crystal structure ...
• 30k
### Why are p-type solar cells more prone to degradation in space than the n-type ones?
The space radiation environment is a quite complex subject. Generally speaking one needs to be careful about just where in space you are concerned about - near earth orbit (within the Van Allen Belts),...
• 7,650
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### What causes photovoltaic (solar) cells to degrade?
Why do they degrade? What exactly is reducing their efficiency? This link outlines several modes of solar panel degradation, and this report by the National Renewable Energy lab is a very detailed ...
• 10.1k
### Why do different impurities create different colors in diamonds?
If you had googled a little bit you would have found this wonderful resource. In a yellow diamond, a few carbon atoms per million have been replaced by nitrogen atoms, each containing five ...
• 17.3k
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### What is the actual reason behind PCl₅ existing as an ionic solid?
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• 82k
### Distance between successive tetrahedral voids in FCC
As shown in figure, if we divide a FCC unit cell into 8 small cubes, then each small cube has 1 Tetrahedral void at its own body centre. Thus, there are total 8 Tetrahedral voids in one unit cell. It ...
### Are molecular optimizations in quantum chemistry really in vacuum?
You raise an excellent question and the short answer is "yes, quantum chemistry calculations are intrinsically in vacuum". Approximate methods like semiempirical (AM1, PM6, PM7) and hybrid DFT ...
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2022-07-07 10:18:04
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https://codereview.stackexchange.com/questions/68645/encrypting-data-on-client-side-and-passing-it-to-server-side
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# Encrypting data on client side and passing it to server side
I'm passing the data to PHP server side which will use the information later as this:
<script> function encrypt() {
var salt = CryptoJS.lib.WordArray.random(128/8);
var key256Bits500Iterations = CryptoJS.PBKDF2("'.\$token.'", salt, { keySize: 256/32, iterations: 500 });
var iv = CryptoJS.enc.Hex.parse("101112131415161718191a1b1c1d1e1f");
var data_base64 = encrypted.ciphertext.toString(CryptoJS.enc.Base64);
var key_base64 = encrypted.key.toString(CryptoJS.enc.Base64);
var iv_base64 = encrypted.iv.toString(CryptoJS.enc.Base64);
var data = {encrypted: data_base64, iv: iv_base64, key: key_base64};
return true;
} </script>
My question is, since I am passing this through the form as POST, that still can be sniffed with shark-like tools and then decrypted with all information being passed through anyways.
What information should be private and how should I make them private? How could they be private if the IV, data, and the key are being send through the POST that are needed when decrypting the string?
How is the logic here flawed and how should it be used to protect data communication between client and server side computing?
• Why not just use HTTPS? Nov 2, 2014 at 9:36
• Just providing basic security for small CMS. Case may be that HTTPS might not be available. If I could assure that I could use HTTPS every time, I definitely would use it. Thanks. Nov 2, 2014 at 11:41
• I'm thinking of using jcryption.org instead. Nov 2, 2014 at 11:42
• Note that without HTTPS, any JavaScript-based encryption is still vulnerable to man-in-the-middle attacks. A rogue wireless access point or ISP could serve a trojaned jcryption.js to the client and defeat the whole thing. Nov 2, 2014 at 17:36
• I wasn't aware, and thank you for pointing this out. But as Tim suggested using asynchronous encryption would be good-enough protection for this simple project when dealing protecting user data without having HTTPS on every page, which is costly. Nov 3, 2014 at 0:16
What information should be private?
The key should be private, everything else can be public (the iv, the source, etc).
How is the logic here flawed
You are using a symmetric key algorithm, which means that the same key decrypts and encrypts data, which in turn means that you would have to exchange the key between the parties (which would need to happen over a different channel, for example paper mail).
What you actually need is an asymmetric algorithm, which has a public key for encryption and a private key for decryption.
If you need to encrypt more data than showing here, you can use an asymmetric algorithm to exchange the key of a symmetric algorithm (as asymmetric encryption is unpractically slow). This is how HTTPS works, for example.
how should it be used to protect data communication between client and server side computing?
Use HTTPS. At the time, there really isn't an alternative for this. (well, you could use third party services such as OpenID). You definitely need asymmetric encryption, and you should definitely not write your own.
Misc
• either always use camelCase or always snake_case, don't mix them without reason.
• key256Bits500Iterations is quite descriptive, but I think key would be just fine as well, and easier to read (and what if you change it to 600 iterations? Either your name would be wrong, or you would need to change it).
• encrypt doesn't just encrypt, it also changes the form. I would to this in a different function and just let encrypt do the encrypting.
• wherever you do the form manipulation, you should also set document.loginForm.password1.value to something else right there.
• Thanks for your opinion. I do not have an access to SSL over HTTPS at this moment. I was looking into this and I think I'll use jCryption instead. Thank you for your words. Nov 2, 2014 at 11:40
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2022-08-08 19:30:20
|
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http://ronislaw.co.il/archive/caac50-standard-enthalpy-of-formation-of-hcl-equation
|
amount of a particular fuel that we would need to travel to Mars or inventing a cold pack that • Evaluate the effectiveness of Hess’s law to determine the enthalpy of an overall reaction. Since the enthalpy change for a given reaction is proportional to the amounts of substances involved, it may be reported on that basis (i.e., as the ΔH for specific amounts of reactants). the calorimeter got us pretty close to the calculated value. that then total enthalpy change for a chemical reaction doesn't depend on what pathway it takes, but only on its initial and final states. That's how we figured Well, I am happy to report A compound's standard heat of formation, or standard enthalpy of formation, #DeltaH_f^@#, represents the change in enthalpy that accompanies the formation of one mole of that compound from its constituent elements in their standard state.. And, finally, there's the into my calorimeter here. temperature of all substances equally and why does water in particular have such a high specific heat capacity? The thermometer tracks It's a positive value because The superscript degree symbol (°) indicates that substances are in their standard states. of heat from the reaction. Going from left to right in (i), we first see that $$\ce{ClF}_{(g)}$$ is needed as a reactant. For liquid water, negative 285.8. the more mass of a substance we have, the more chemical Ozone, O3(g), forms from oxygen, O2(g), by an endothermic process. pressure, like we have here at the surface of the earth, that works out to be exactly the same as the heat that's absorbed It shows how we can find many standard enthalpies of formation (and other values of ΔH) if they are difficult to determine experimentally. So it turns out that We can apply the data from the experimental enthalpies of combustion in Table $$\PageIndex{1}$$ to find the enthalpy change of the entire reaction from its two steps: \[\begin {align*} Assuming that both the reactants and products of the reaction are in their standard states, determine the standard enthalpy of formation, $$ΔH^\circ_\ce{f}$$ of ozone from the following information: Solution $$ΔH^\circ_\ce{f}$$ is the enthalpy change for the formation of one mole of a substance in its standard state from the elements in their standard states. this thing is measuring heat, because it's not. Some reactions are difficult, if not impossible, to investigate and make accurate measurements for experimentally. The enthalpy change for this reaction is −5960 kJ, and the thermochemical equation is: \[\ce{C12H22O11 + 8KClO3⟶12CO2 + 11H2O + 8KCl}\hspace{20px}ΔH=\ce{−5960\:kJ} is needed, C12H22O11 is the excess reactant and KClO3 is the limiting reactant. at constant pressure, delta H equals q, and constant pressure is almost always a good assumption for the duration of an experiment, or at least as long as we stay at the surface of the earth. Or will they explore or maybe Algae can produce biodiesel, biogasoline, ethanol, butanol, methane, and even jet fuel. For the formation of 2 mol of O3(g), $$ΔH^\circ_{298}=+286\: \ce{kJ}$$. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 4 2−(aq) −907.5 Zn2+(aq) −152.4 * All standard enthalpy values are at 25°C, 1 molar concentration, and 1 atmosphere of pressure. The standard enthalpy of Kilimanjaro, you are at an altitude of 5895 m, and it does not matter whether you hiked there or parachuted there. the numbers in a table. that we actually created. The difference between This leaves only reactants ClF(g) and F2(g) and product ClF3(g), which are what we want.
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2021-01-27 07:24:37
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https://optimization-online.org/tag/subanalytic-set/
|
## On local convergence of the method of alternating projections
The method of alternating projections is a classical tool to solve feasibility problems. Here we prove local convergence of alternating projections between subanalytic sets A,B under a mild regularity hypothesis on one of the sets. We show that the speed of convergence is O$(k^{-\rho})$ for some $\rho\in(0,\infty)$. Citation Université de Toulouse, Institut de Mathématiques, december … Read more
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2022-11-29 22:03:38
|
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https://math.libretexts.org/TextMaps/Analysis_TextMaps/Map%3A_Differential_Equations_for_Engineers_(Lebl)/1%3A_First_order_ODEs/1.7%3A_Numerical_methods%3A_Euler%E2%80%99s_method
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# 1.7: Numerical methods: Euler’s method
### Introduction
At this point it may be good to first try the Lab II and/or Project II from the IODE website: http://www.math.uiuc.edu/iode/. As we said before, unless $$f(x, y)$$ is of a special form, it is generally very hard if not impossible to get a nice formula for the solution of the problem
$y' = f(x, y), \, \, \, \, \,\, y(x_0) = y_0$
What if we want to find the value of the solution at some particular$$x$$? Or perhaps we want to produce a graph of the solution to inspect the behavior. In this section we will learn about the basics of numerical approximation of solutions.
The simplest method for approximating a solution is Euler’s method4. It works as follows: We take $$x_0$$ and compute the slope $$k = f(x_0, y_0)$$. The slope is the change in $$y$$ per unit change in $$x$$. We follow the line for an interval of length $$h$$ on the $$x$$ axis. Hence if $$y = y_0$$ at $$x_0$$, then we will say that $$y_1$$ (the approximate value of $$y$$ at $$x_1 = x_0 + h$$) will be $$y_1 = y_0 + hk$$. Rinse, repeat! That is, compute $$x_2$$ and $$y_2$$ using $$x_1$$ and $$y_1$$. For an example of the first two steps of the method see Figure 1.11.
More abstractly, for any $$i = 1, 2, 3, \cdots$$, we compute
$x_{i+1} = x_i + h, \, \, \, \, y_{i+1} = y_i + hf(x_i, y_i)$
The line segments we get are an approximate graph of the solution. Generally it is not exactly the solution. See Figure 1.12 for the plot of the real solution and the approximation.
Let us see what happens with the equation $$y' = \frac{y^2}{3}$$, $$y(0) = 1$$. Let us try to approximate $$y(2)$$ using Euler’s method. In Figures 1.11 and 1.12 we have graphically approximated $$y(2)$$ with step size 1. With step size 1 we have $$y(2) \approx 1.926$$. The real answer is 3. So we are approximately 1.074 off. Let us halve the step size. Computing $$y_4$$ with $$h = 0.5$$, we find that $$y(2) \approx 2.209$$, so an error of about 0.791. Table 1.1 gives the values computed for various parameters.
Exercise $$\PageIndex{1}$$:
Solve this equation exactly and show that $$y(2) = 3$$.
The difference between the actual solution and the approximate solution we will call the error. We will usually talk about just the size of the error and we do not care much about its sign. The main point is, that we usually do not know the real solution, so we only have a vague understanding of the error. If we knew the error exactly …what is the point of doing the approximation?
h Approximate Error 1 1.92593 1.07407 0.5 2.20861 0.79139 0.73681 0.25 2.47250 0.52751 0.66656 0.125 2.68034 0.31966 0.60599 0.0625 2.82040 0.17960 0.56184 0.03125 2.90412 0.09588 0.53385 0.015625 2.95035 0.04965 0.51779 0.0078125 2.97472 0.02528 0.50913
Table 1.1: Euler’s method approximation of $$y(2)$$ where of $$y' = \frac{y^2}{3}$$, $$y(0) = 1$$.
We notice that except for the first few times, every time we halved the interval the error approximately halved. This halving of the error is a general feature of Euler’s method as it is a first order method. In the IODE Project II you are asked to implement a second order method. A second order method reduces the error to approximately one quarter every time we halve the interval (second order as $$\frac{1}{4} = \frac{1}{2} \text{x} \frac{1}{2}$$).
To get the error to be within 0.1 of the answer we had to already do 64 steps. To get it to within 0.01 we would have to halve another three or four times, meaning doing 512 to 1024 steps. That is quite a bit to do by hand. The improved Euler method from IODE Project II should quarter the error every time we halve the interval, so we would have to approximately do half as many “halvings” to get the same error. This reduction can be a big deal. With 10 halvings (starting at $$h = 1$$) we have 1024 steps, whereas with 5 halvings we only have to do 32 steps, assuming that the error was comparable to start with. A computer may not care about this difference for a problem this simple, but suppose each step would take a second to compute (the function may be substantially more difficult to compute than $$\frac{y^2}{3}$$). Then the difference is 32 seconds versus about 17 minutes. Note: We are not being altogether fair, a second order method would probably double the time to do each step. Even so, it is 1 minute versus 17 minutes. Next, suppose that we have to repeat such a calculation for different parameters a thousand times. You get the idea.
Note that in practice we do not know how large the error is! How do we know what is the right step size? Well, essentially we keep halving the interval, and if we are lucky, we can estimate the error from a few of these calculations and the assumption that the error goes down by a factor of one half each time (if we are using standard Euler).
Exercise $$\PageIndex{2}$$:
In the table above, suppose you do not know the error. Take the approximate values of the function in the last two lines, assume that the error goes down by a factor of 2. Can you estimate the error in the last time from this? Does it (approximately) agree with the table? Now do it for the first two rows. Does this agree with the table?
Let us talk a little bit more about the example $$y' = \frac{y^2}{3}$$, $$y(0) = 1$$. Suppose that instead of the value $$y(2)$$ we wish to find $$y(3)$$. The results of this effort are listed in Table 1.2 for successive halvings of $$h$$. What is going on here? Well, you should solve the equation exactly and you will notice that the solution does not exist at $$x =3$$. In fact, the solution goes to infinity when you approach $$x =3$$.
h Approximate 1 3.16232 0.5 4.54329 0.25 6.86079 0.125 10.80321 0.0625 17.59893 0.03125 29.46004 0.015625 50.40121 0.0078125 87.75769
Table 1.2: Attempts to use Euler’s to approximate $$y(3)$$ where of $$y' = \frac{y^2}{3}$$, $$y(0) = 1$$.
Another case when things can go bad is if the solution oscillates wildly near some point. Such an example is given in IODE Project II. The solution may exist at all points, but even a much better numerical method than Euler would need an insanely small step size to approximate the solution with reasonable precision. And computers might not be able to easily handle such a small step size.
In real applications we would not use a simple method such as Euler’s. The simplest method that would probably be used in a real application is the standard Runge-Kutta method (see exercises). That is a fourth order method, meaning that if we halve the interval, the error generally goes down by a factor of 16 (it is fourth order as $$\frac{1}{16} = \frac{1}{2} \text{x} \frac{1}{2} \text{x} \frac{1}{2} \text{x} \frac{1}{2}$$).
Choosing the right method to use and the right step size can be very tricky. There are several competing factors to consider.
• Computational time: Each step takes computer time. Even if the function $$f$$ is simple to compute, we do it many times over. Large step size means faster computation, but perhaps not the right precision.
• Roundoff errors: Computers only compute with a certain number of significant digits. Errors introduced by rounding numbers off during our computations become noticeable when the step size becomes too small relative to the quantities we are working with. So reducing step size may in fact make errors worse.
• Stability: Certain equations may be numerically unstable. What may happen is that the numbers never seem to stabilize no matter how many times we halve the interval. We may need a ridiculously small interval size, which may not be practical due to roundoff errors or computational time considerations. Such problems are sometimes called stiff. In the worst case, the numerical computations might be giving us bogus numbers that look like a correct answer. Just because the numbers have stabilized after successive halving, does not mean that we must have the right answer.
We have seen just the beginnings of the challenges that appear in real applications. Numerical approximation of solutions to differential equations is an active research area for engineers and mathematicians. For example, the general purpose method used for the ODE solver in Matlab and Octave (as of this writing) is a method that appeared in the literature only in the 1980s.
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2017-10-18 07:39:22
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https://ashishkumarletslearn.com/inverse-trigonometric-functions-lecture-5-part-2/
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# Part - 2 Lecture - 5 Chapter 2 Inverse Trigonometric Functions
“Too many of us are not living our dreams because we are living our fears. ” –Les Brown
Booklets/Notes/Assignments are typed here on this website and their PDFs will be made available soon.
Questions discussed in this lecture
Solve:
9. $\tan ^{-1} (x-1)+\tan ^{-1} x+\tan ^{-1} (x+1)=\tan ^{-1} 3x$
10. $3\sin ^{-1} \frac{2x}{1+x^{2} } -4\cos ^{-1} \frac{1-x^{2} }{1+x^{2} } +2\tan ^{-1} \frac{2x}{1-x^{2} } =\frac{\pi }{3}$
11. If $\sin ^{-1} \frac{2a}{1+a^{2} } -\cos ^{-1} \frac{1-b^{2} }{1+b^{2} } =\tan ^{-1} \frac{2x}{1-x^{2} }$, then prove that $x=\frac{a-b}{1+ab}.$
12. Evaluate: $\tan ^{-1} \left(\frac{a+bx}{b-ax} \right), x<\frac{b}{a}$
13. Prove: $\tan ^{-1} \left(\frac{a-b}{1+ab} \right)+\tan ^{-1} \left(\frac{b-c}{1+bc} \right)+\tan ^{-1} \left(\frac{c-a}{1+ca} \right)=0$
14. If $\tan ^{-1} x+\tan ^{-1} y=\frac{4\pi }{5}$, then find the value of $\cot ^{-1} x+\cot ^{-1}y$?
15. If $\tan ^{-1} \left(\frac{1}{1+1.2} \right)+\tan ^{-1} \left(\frac{1}{1+2.3} \right)+…+\tan ^{-1} \left(\frac{1}{1+n.(n+1)} \right)=\tan ^{-1} \phi$, then find the value of $\phi$.
16. If $(\tan ^{-1} x)^{2} +(\cot ^{-1} x)^{2} =\frac{5\pi ^{2} }{8}$, then find x.
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2019-11-18 09:54:33
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http://math.stackexchange.com/tags/linear-algebra/hot
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# Tag Info
4
The characteristic polynomial of a $2 \times 2$ matrix can be written as: $$p(\lambda) = \lambda^2 - \textrm{tr}(A)\lambda + \textrm{det}(A)$$ (Check here). If a matrix $A$ is symmetric then it is diagonalizable such that: A = Q \Lambda Q^T \quad \textrm{ where } \quad \Lambda= \textrm{diag}(\lambda_1,\lambda_2) ...
3
The answer is positive, assume $A \in \mathbb{R}^{n \times n}$, then for any $x \in \mathbb{R}^n$, take $(x^T, x^T)^T$ to test the diagonal matrix: $$\begin{pmatrix} x^T & x^T \end{pmatrix} \begin{pmatrix}A & 0 \\ 0 & A \end{pmatrix} \begin{pmatrix}x \\ x\end{pmatrix} = 2x^TAx \geq 0$$ implies that $x^TAx \geq 0$, i.e., $A$ is positive ...
3
Hint What can you say about the traces of the given matrices? (Alternatively, for three of the choices, one can find a suitable matrix $B$ for which the equation holds for all $A$.)
3
That's wrong. For $x = (0,1)$ and $y = (1,0)$, $$x^t y = \begin{pmatrix} 0\\ 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix}$$ is not diagonizable.
3
The operator $2 \, I$ is always self-adjoint. Hence, $2 \, I = I$ or $2 \, I = 0$. This yields $I = 0$, hence, $\mathcal{H} = \{0\}$. Edit: Since every orthogonal projection onto a subspace is self-adjoint, it is quite easy to reconstruct $\mathcal{H}$ from its self-adjoint operators.
2
So $\forall\ \textbf{x}\in\mathbb{R}^{2n}$ we have $\textbf{x}^T\begin{pmatrix}A & 0\\ 0 & A\end{pmatrix}\textbf{x}\geq0$. Let $\textbf{y}\in\mathbb{R}^n$ arbitrary, we want to show $\textbf{y}^TA\textbf{y}\geq0$. To this end, let $\textbf{0}\in\mathbb{R}^n$ and define $\textbf{x}=\left(\begin{array}{c}\textbf{y}\\ \textbf{0}\end{array}\right)$. Then ...
2
Yes : Take the companion matrix of the coefficients of $p$.
2
Well I think we could proceed like this: $$(I-\lambda P)(I+\lambda P+\lambda^{2}P^{2}+\lambda^{3}P^{3}+...)=(I-\lambda^{n}P^{n})$$ Which if $\lambda<1$ gives you $$(I-\lambda P)(I+\lambda P+\lambda^{2}P^{2}+\lambda^{3}P^{3}+...)=I$$ from there you see that since $P^n=...=P^2=P$ then: ...
1
Can you show that the constant zero function belongs to $W$? Can you show that, if $f,g\in W$, then $f+g\in W$? Can you show that, if $f\in W$ and $a\in\mathbb{R}$, then $af\in W$? Can you do the same verifications for the set of continuous functions? Hint for 2. Set $h=f+g$. Then $$h(\tfrac{1}{2})-3h(\tfrac{\pi}{4})= f(\tfrac{1}{2})-3f(\tfrac{\pi}{4})+ ... 1 if |\lambda| \lt 1 then we may define$$ Q = \sum_{n=1}^{\infty} (\lambda P)^n = (\sum_{n=1}^{\infty} \lambda^n) P \\ =\lambda(1-\lambda)^{-1}P $$and$$ (I-\lambda P)(I+Q) = I $$it can be seen by calculation that the restriction on \lambda is not required, as long as \lambda \ne 1 1 For \lambda=0 the assertion is obvious. Else we can write$$I-\lambda P=\lambda(\lambda^{-1} I -P)$$Note that the only eigenvaules of P are 0 and 1, hence the latter is invertible whenver \lambda \neq 0,1. 1 Let's assume that \exists\lambda\neq 1,\, I-\lambda P non inversible. This means that \exists x\in\mathbb{K}^n,\, x\neq 0\land x=\lambda Px. And this leads immediately by multiplying by P to the left to Px=\lambda Px i.e (1-\lambda)Px=0 and because \lambda\neq 1 we have Px=0 and therefore x=\lambda Px=0 a contradiction and we have proven ... 1 What you looking for is a Thomas algorithm which is a simplified form of a Gaussian Elimination or if you want LU decomposition. Matlab looks fast because Matlab identifying such special cases and call in such case to a very effective solver for banded-matrices. This solver is little bit more general then Thomas algorithm. The very rough idea is that you ... 1 The characteristic polynomial of your matrix is $$p(x) = {x}^{3}-t{x}^{2}- \left( 2\,{t}^{2}+3\,t \right) x-3\,{t}^{2}$$ Which has roots: $$\begin{array}[ccc] \\ x_1 = -t,& x_2 = t+\sqrt{t^2+3t},& x_3 = t-\sqrt{t^2+3t} \end{array}$$ The values for which x_2,x_3 exists are such that t^2 + ... 1 Hint$$\left|\begin{pmatrix} 2t & 1 & 0 \\ 3t & 0 & 0 \\ t & 0 & -t \end{pmatrix}-\lambda I\right|= \left|\begin{pmatrix} 2t-\lambda & 1 & 0 \\ 3t & -\lambda & 0 \\ t & 0 & -t-\lambda \end{pmatrix}\right| =-(t+\lambda)(\lambda(\lambda-2t)-3t)=0 $$The roots are \lambda =-t, \lambda=t\pm\sqrt{t^2+3 t} ... 1 Note that -x^3+6x^2+9x-14=-(x-1)(x+2)(x-7), so we may assume that M is the diagonal matrix with entries 1,-2,7 on the diagonal. Then it's easy to see that the characteristic polynomial of M^{-3} is given by (x-1)(x+(1/2)^3)(x-(1/7)^3). 1 As abiessu says, you have$$y=16+2x\\\frac 14y-\frac 12x=2$$You can rewrite these to$$y-2x =16\\\ y- 2x=8$$and as you say these cannot give a solution simultaneously. 1 I think you are confusing a function and its derivative. You say, I want it to expand quickly to start with with the expansion slowing over time. i.e, an inverse square. So if r(t) is the radius of the sphere, this sentence means you want r'(t) = \frac{1}{t^2}. But the way you programmed it is if r(t)=\frac{1}{t^2}+\text{initial radius}. So, ... 1 Well, think about all these definitions as about rules written in blood and agreed by the community. This is slightly extreme and even somewhat wrong analogy, but the answer is in the "mood" of the question. With theorems it is slightly more complicated, they are result of many years of research and work of many scientists (well this is also true for some ... 1 You can eliminate answers A and D by noting that the zero matrix B always satisfies that identity. 1 Just to expand on the comment of @Myself above... Adjoining algebraics as matrix maths Sometimes in mathematics or computation you can get away with adjoining an algebraic number \alpha to some simpler ring R of numbers like the integers or rationals \mathbb Q, and these characterize all of your solutions. If this number obeys the algebraic equation ... 1 Note that A = \operatorname{Re}A +i\operatorname{Im}A. Also$$z(x) = \begin{bmatrix}\operatorname{Re}x\\\operatorname{Im}x\end{bmatrix}\qquad z^{-1}\left(\begin{bmatrix}v_1\\v_2\end{bmatrix}\right) = v_1 + iv_2$$So, since K = z\circ A\circ z^{-1},$$\begin{align}K\begin{bmatrix}v_1\\v_2\end{bmatrix} &= zAz^{-1}\begin{bmatrix}v_1\\v_2\end{bmatrix}\\ ...
1
As stated, the result is false. For example, let $x=\begin{pmatrix}1&-1 \end{pmatrix}$ and $y=\begin{pmatrix}1&1\end{pmatrix}$ then $$x^Ty=\begin{pmatrix}1&1\\-1&-1 \end{pmatrix}.$$ That matrix has $0$ as only eigenvalue and it is clearly not diagonalisable. Actually, the statement should be Let $x,y$ be two non zero row vectors, ...
1
Hint. (1) To show that $\def\M{\mathcal M_{m,n}}\M$ is open, note that if $M \in \M$, then $M$ has a $d \times d$-submatrix (where $d := \min\{m,n\}$) $A_d(M)$ with nonzero determinant and the map $\mathcal M \to \mathbf R$, which maps $M \mapsto A_d(M)$ is continuous. (2) By (1), $\M$ is an open subset of $\mathbf R^{mn}$ and hence carries naturally a ...
1
I think the result is this one $$A^x=\left(\begin{array}{c} \begin{array}{ccccc} 3x+1 & 3x\\ 3x & -3x+1\\ \end{array}\end{array}\right)$$ Now I have to exit, but as soon as I'm back I will explain it if needed...
1
A linear complex structure on $\mathbb{R}^{2n}$ is the structure of a complex vector space on it compatible with its real vector space structure. $J$ is multiplication by $i$. Since there is only one $n$-dimensional complex vector space up to isomorphism, any two such complex structures give rise to two complex vector spaces $V, V'$ such that there must be ...
1
$$x+y+4z=6\\ 2x+4y-4z=16$$ $$5y=22-3x\\ 10z=4-x$$ $$7x+10y+10z=7x+44-6x+4-x=48+0\cdot x=60$$ $$0\cdot x=12$$ $$\therefore \text{No solution}$$
1
These topological arguments involve the same basic idea: it's often easy to prove things for a subset of matrices which are dense in the space of all matrices. Any "continuous" fact (e.g. the assertion that two continuous functions are equal) can be proven for all matrices by proving it for this dense subset. For example, if $L$ is diagonalizable with ...
1
This is a good approach, but you need to show that any subgroup of $\mathbb{Z}^r$ is finitely generated. This is not extremely hard, but it's not trivial either. Here are two approaches. Perhaps a more direct way to approach this problem is to use the structure theorem for finitely generated abelian groups.
Only top voted, non community-wiki answers of a minimum length are eligible
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2015-11-26 00:22:35
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https://cs184.eecs.berkeley.edu/sp19/lecture/18-46/physical-simulation
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Lecture 18: Physical Simulation (46)
joannejqi
I was interested in how to come up with constraints for Verlet integration. For example, one constraint to prevent collisions is to model infinitely stiff springs between particles and the penetration points of the surface. http://www.gotoandplay.it/_articles/2005/08/advCharPhysics.php
You must be enrolled in the course to comment
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2019-09-15 10:46:19
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http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?reload=true&punumber=3039
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Scheduled System Maintenance
On Friday, October 20, IEEE Xplore will be unavailable from 9:00 PM-midnight ET. We apologize for the inconvenience.
# Proceedings of IEEE Workshop on Applications of Signal Processing to Audio and Acoustics
## Filter Results
Displaying Results 1 - 25 of 44
• ### Proceedings of IEEE Workshop on Applications of Signal Processing to Audio and Acoustics
Publication Year: 1993
| PDF (43 KB)
• ### Dithered quantizers with and without feedback
Publication Year: 1993, Page(s):140 - 143
Cited by: Papers (5)
| | PDF (236 KB)
It is shown that quantizing systems without feedback respond to the use of particular spectrally-shaped dither signals quite differently from those with feedback paths. For each type of system, conditions are given which ensure that the quantization error will be wide-sense stationary with no input dependence and with a predictable power spectral density function View full abstract»
• ### Robust adaptive processing of microphone array data for hearing aids
Publication Year: 1993, Page(s):77 - 80
Cited by: Papers (2)
| | PDF (276 KB)
The problem of adaptively combining the outputs of an array of microphones as a single input for a hearing aid is investigated. A robust processor based on a constrained minimum variance optimization approach is used. One fundamental criteria employed in designing this robust beamformer limits the amount of cancellation of the desired signal. The results presented include the effects of acoustic h... View full abstract»
• ### Detection and restoration of sound of flute embedded in noise using real-time Kalman filter
Publication Year: 1993, Page(s):144 - 147
| | PDF (236 KB)
The restoration of flute notes embedded in noise is formulated as a state-estimation problem of a dynamic system. A single Kalman filter, with a given state-transition matrix, is implemented in real-time to recover the corresponding note as well as some of the neighbouring notes. In order to restore a continuous piece of music played by flute, a bank of Kalman filters (with different state-transit... View full abstract»
• ### A simplified source/filter model for percussive sounds
Publication Year: 1993, Page(s):173 - 176
Cited by: Papers (1)
| | PDF (352 KB)
This paper deals with source-filter models of percussive instruments. A multi-channel excitation/filter model' is presented in which a single excitation is used to generate several sounds, for example six piano tones belonging to the same octave. Techniques for estimating the model parameters are presented and applied to the sound of a real piano. Our experiments demonstrate that it is possible t... View full abstract»
• ### Robust real-time constrained hearing aid arrays
Publication Year: 1993, Page(s):81 - 84
Cited by: Papers (1) | Patents (4)
| | PDF (292 KB)
The paper addresses the implementation of a real-time, robust, adaptive spatial filter used as a preprocessor for a monaural hearing aid. The goal of the ongoing study is the development of a processor that provides the user spatial selectivity and an attenuation of undesired interfering sources, while robustly controlling the response to a desired source. A four microphone, real-time, robust proc... View full abstract»
• ### A new technique to measure electroacoustic transducer directivity indices in reverberant fields
Publication Year: 1993, Page(s):64 - 67
| | PDF (268 KB)
The paper presents a new method for measuring the directivity index of an electroacoustic transducer in a diffuse reverberant environment. The method that is proposed relies on the measurement of the spectral density variance of the transfer function between source and receiver. The method requires a measurement of the source/receiver transfer function, the distance between source and receiver, th... View full abstract»
• ### Current and future standardization of high-quality digital audio coding in MPEG
Publication Year: 1993, Page(s):43 - 46
Cited by: Papers (3) | Patents (4)
| | PDF (340 KB)
Since 1988 ISO/IEC JTCI/SC29 WG11 (MPEG) is working on the standardization of video and audio signals. The Audio subgroup of MPEG is working on bit rate reduction systems for high quality digital audio. Since the first phase of this standardization effort has been finished, MPEG/Audio is extending its work to multichannel audio coding systems as well as to medium quality coding at lower sampling f... View full abstract»
• ### The restoration of pitch variation defects in gramophone recordings
Publication Year: 1993, Page(s):148 - 151
Cited by: Papers (2)
| | PDF (264 KB)
A new algorithm is presented for the identification and restoration of time-varying pitch defects in audio signals. The problem is commonly encountered as wow' in gramophone disc and magnetic tape recordings where motor speed variations or eccentricity in the recording process are significant. The algorithm operates in two stages, the first of which trades tonal components in musical signals to g... View full abstract»
• ### The 2-D digital waveguide mesh
Publication Year: 1993, Page(s):177 - 180
Cited by: Papers (47)
| | PDF (256 KB)
An extremely efficient method for modeling wave propagation in a membrane is provided by the multidimensional extension of the digital waveguide. The 2-D digital waveguide mesh is constructed out of bi-directional delay units and scattering junctions. We show that it coincides with the standard finite difference scheme in the lossless case. Wave propagation in the mesh is compared with wave propag... View full abstract»
• ### A time-frequency neutral network layered model for hearing perception
Publication Year: 1993, Page(s):123 - 126
Cited by: Papers (1)
| | PDF (260 KB)
This paper introduces a layered neural network model for hearing perception. It is based on five important perceptual properties of hearing. The neural network model processes a joint-domain representation of the input signal to yield the desired perceptual properties. The focus is on the first two layers of the model, the transformation layer and two feature extraction layers View full abstract»
• ### An all digital concha hearing aid
Publication Year: 1993, Page(s):85 - 88
Cited by: Papers (1)
| | PDF (248 KB)
The paper describes an all digital concha hearing aid. The main features of this hearing aid concept are a large vent, acoustic feed-back cancellation, great flexibility by programming, a versatile equalizer, and an advanced compressor. The A/D and D/A converters have log/in characteristics and the signal processing is performed by floating point arithmetic, ensuring a large dynamic range and a si... View full abstract»
• ### Local silencing of room acoustic noise using broadband active noise control
Publication Year: 1993, Page(s):23 - 25
Cited by: Papers (4)
| | PDF (188 KB)
Adaptive filtering techniques are now in widespread use for a number of applications such as adaptive arrays, adaptive noise cancellation, adaptive line enhancement, adaptive modeling and system identification, adaptive equation, and adaptive echo cancellation. These techniques have also been applied to the expanding field of active noise control. In this paper, an application of active noise cont... View full abstract»
• ### Parametric approximation of room impulse responses based on wavelet decomposition
Publication Year: 1993, Page(s):68 - 71
Cited by: Papers (1) | Patents (2)
| | PDF (240 KB)
A new approach to the approximation and real-time simulation of room impulse responses is presented. Based on wavelet decomposition of measured impulse response data an energy-time-frequency representation of the system room is obtained. The wavelet coefficients in the frequency subbands are calculated by a multirate analysis filter bank providing aliasing-free subband processing and linear-phase ... View full abstract»
• ### Directional microphones in computer simulated and real rooms
Publication Year: 1993, Page(s):56 - 59
| | PDF (216 KB)
The subjective effects of utilizing highly directional microphones in a teleconferencing setting are not well understood. Computer simulation of both complex microphone systems and room environments offer one opportunity to study the combined effects. A complex microphone system can be modeled as a collection of point microphones distributed in space and summed with appropriate time delays. Establ... View full abstract»
• ### Aspects of current standardization activities for high-quality, low-rate multi-channel audio coding
Publication Year: 1993, Page(s):47 - 50
Cited by: Papers (3) | Patents (1)
| | PDF (348 KB)
This paper analyzes directions in the current standardization activities for multi-channel audio, briefly reviews the composite coding schemes AC-3 and ISO 11172-3 compatible systems, and discusses requirements, features, and time-tables for the audio systems in the ISO/Moving pictures Expert Group (MPEG) phase 2 and the United States high definition television (HDTV) standardization processes View full abstract»
• ### Principle and application of a new test signal to determine the transfer characteristics of telecommunication systems
Publication Year: 1993, Page(s):152 - 155
Cited by: Patents (1)
| | PDF (240 KB)
Measuring procedures yielding defined and reproducible results are required to determine transfer functions for tests and registrations. On the one hand, such a test signal allowing the determination of the transfer characteristics of these systems must simulate voice properties adequately. On the other hand, such a signal must be determined exactly so that not only the transfer function in differ... View full abstract»
• ### Time-scale modification with temporal envelope invariance
Publication Year: 1993, Page(s):127 - 130
Cited by: Papers (1)
| | PDF (316 KB)
A new approach is introduced for time-scale modification of short-duration complex acoustic signals to improve their audibility. The method preserves the time-scaled temporal envelope of a signal and for enhancement capitalizes on the perceptual importance of a signal's temporal structure. The basis for the approach is a sub-band representation whose channel phases are controlled to shape the the ... View full abstract»
• ### Constraint based audio interpolators
Publication Year: 1993, Page(s):161 - 164
Cited by: Papers (1) | Patents (1)
| | PDF (272 KB)
In audio digital signal processing, interpolators are used for a variety of functions, including sample rate conversion. Linear interpolation is commonly used, but has serious signal quality problems for signals with significant high frequency content. Higher order interpolators based on sine functions or other conventional lowpass filter design techniques offer somewhat better performance, but ar... View full abstract»
• ### Hearing aids for profoundly deaf people based on a new parametric concept
Publication Year: 1993, Page(s):89 - 92
Cited by: Papers (1) | Patents (5)
| | PDF (252 KB)
People with severe hearing loss only have a minor part of the frequency range available for reception of information in speech signals. These people do not benefit from normal hearing aids as the information in high frequency parts of the speech is not available. To overcome this problem the authors have developed a new method enabling to present information from the frequency range of interest in... View full abstract»
• ### Interpolation of forced structural responses from non-uniform sparse measurements
Publication Year: 1993, Page(s):26 - 29
| | PDF (264 KB)
This paper presents a method for interpolating a sparse set of nonuniformly spaced velocity measurements on the surface of a vibrating structure. The method utilizes knowledge of the physical nature of the vibrating structure specified in terms of a given bound on the energy of the excitation forces, estimated mobilities of the structure and a known set of sparse velocity measurements. To minimize... View full abstract»
• ### Perceptual consequences of interpolating head-related transfer functions during spatial synthesis
Publication Year: 1993, Page(s):102 - 105
Cited by: Papers (13) | Patents (1)
| | PDF (324 KB)
In implementing a spatial auditory display, many engineering compromises must be made to achieve a practical system. One such compromise involves devising methods for interpolating between the head-related transfer functions (HRTFs) used to synthesize spatial stimuli in order to achieve smooth motion trajectories and locations at finer resolutions than the empirical data. The perceptual consequenc... View full abstract»
• ### Analog/digital hybrid VLSI signal processing using single BIT modulators
Publication Year: 1993, Page(s):136 - 139
Cited by: Papers (1)
| | PDF (180 KB)
A hybrid analog/digital technique for efficient VLSI implementation of signal processing systems is presented. Single bit delta sigma modulators are used to modulate analog inputs into a form which can be considered simultaneously analog and digital, and directly manipulated as such. A cross-correlator is proposed, demonstrating the compactness of VLSI signal processing systems using this approach View full abstract»
• ### Superdirective arrays for hearing aids
Publication Year: 1993, Page(s):73 - 76
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Microphone arrays are the most effective of the techniques that have been proposed for improving speech intelligibility in noise for the hearing impaired. Superdirective arrays are attractive since optimal performance can be obtained for a stationary random noise field. A constrained superdirective array suitable for hearing-aid applications is discussed in the paper View full abstract»
• ### HNM: a simple, efficient harmonic+noise model for speech
Publication Year: 1993, Page(s):169 - 172
Cited by: Papers (11) | Patents (6)
| | PDF (336 KB)
HNM, a new analysis/modification/synthesis model based on a harmonic+noise representation of the speech signal is presented. The HNM model has several specificities: (1) HNM assumes the speech signal to be composed of a deterministic part and of a stochastic part, (2) the deterministic part is assumed to contain only harmonically related sinusoids with linearly varying complex amplitudes, and (3) ... View full abstract»
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2017-10-18 04:14:11
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https://www.physicsforums.com/threads/integral-of-3-x.208781/
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# Integral of 3/x
1. Jan 14, 2008
### cabellos6
1. The problem statement, all variables and given/known data
the integral of 3/x
2. Relevant equations
3. The attempt at a solution
am i right in saying this is 3lnx
2. Jan 14, 2008
### Kurdt
Staff Emeritus
Yes you are. Don't forget the arbitrary constant.
3. Jan 14, 2008
### Tom Mattson
Staff Emeritus
And don't forget the absolute value bars. It's 3ln|x|+C.
4. Jan 14, 2008
### Kurdt
Staff Emeritus
Indeed
Welcome back Tom!
Last edited: Jan 14, 2008
5. Sep 7, 2009
### rxtrejo
Same question adding Limits of integration b=infinity a=0
6. Sep 7, 2009
### Hootenanny
Staff Emeritus
Well, what do you think it is?
7. Sep 7, 2009
### NJunJie
rxtrejo - you asking a question?
That would mean substituting the bounded values and find some 'area' within the limits you have given.
8. Sep 7, 2009
### Staff: Mentor
Not so fast. This is an improper definite integral that requires limits at both endpoints to evaluate.
9. Sep 7, 2009
### Staff: Mentor
IOW,
$$3\int_0^{\infty} \frac{dx}{x} = 3\lim_{a \rightarrow 0_+}\int_a^{1} \frac{dx}{x}~+~ 3\lim_{b \rightarrow \infty}\int_1^{b} \frac{dx}{x}$$
I chose to split the first integral at 1. Any reasonable value could be used to divide the original interval into two subintervals.
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2017-09-22 04:38:06
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http://rileylab.org/bide1a8/planck-hubble-constant-283d32
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This discussion is based on Sartori.[39]. = < The motion of astronomical objects due solely to this expansion is known as the Hubble flow. Substituting all of this into the Friedmann equation at the start of this section and replacing When applied to Planck data, this method gives a lower value of 67.4 km/s/Mpc, with a tiny uncertainty of less than a percent. The "redshift velocity" vrs is not so simply related to real velocity at larger velocities, however, and this terminology leads to confusion if interpreted as a real velocity. [53], Also in July 2019, astronomers reported another new method, using data from the Hubble Space Telescope and based on distances to red giant stars calculated using the tip of the red-giant branch (TRGB) distance indicator. The figure astronomers derive for the Hubble Constant using a wide variety of cutting-edge observations to gauge distances across the cosmos is 73.5 km/s/Mpc, with an uncertainty of only two percent. [29] After Hubble's discovery that the universe was, in fact, expanding, Einstein called his faulty assumption that the universe is static his "biggest mistake". e 2 1 Multiple methods have been used to determine the Hubble constant. [38], Here, λo, λe are the observed and emitted wavelengths respectively. ρ Since gravitationally interacting galaxies move relative to each other independent of the expansion of the universe,[41] these relative velocities, called peculiar velocities, need to be accounted for in the application of Hubble's law. Hubble realized that the universe was expanding, and it seemed to be doing so at a constant rate — hence, the Hubble constant. is the cosmological constant. It was long thought that q was positive, indicating that the expansion is slowing down due to gravitational attraction. Alternative models result in different (generally lower) values for the Hubble constant. Planck Collaboration There are two ways that astronomers can estimate the current expansion rate, also known as the Hubble constant (H0). {\displaystyle a} {\displaystyle w(a)} Over long periods of time, the dynamics are complicated by general relativity, dark energy, inflation, etc., as explained above. measured from standard candle observations of Type Ia supernovae, which was determined in 1998 to be negative, surprised many astronomers with the implication that the expansion of the universe is currently "accelerating"[63] (although the Hubble factor is still decreasing with time, as mentioned above in the Interpretation section; see the articles on dark energy and the ΛCDM model). arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. . With the ΛCDM model observations of high-redshift clusters at X-ray and microwave wavelengths using the Sunyaev–Zel'dovich effect, measurements of anisotropies in the cosmic microwave background radiation, and optical surveys all gave a value of around 70 for the constant. ) Shapley argued for a small universe the size of the Milky Way galaxy, and Curtis argued that the universe was much larger. k — the speed of light multiplied by the Hubble time. 0 h The Hubble constant is calculated by comparing distance values to the apparent recessional velocity of the target galaxies — that is, how fast galaxies seem to be moving away. In 1912, Vesto Slipher measured the first Doppler shift of a "spiral nebula" (the obsolete term for spiral galaxies) and soon discovered that almost all such nebulae were receding from Earth. H Parallax measurements of galactic Cepheids for enhanced calibration of the, Uses time delays between multiple images of distant variable sources produced by, Comparing redshift to other distance methods, including. Within the expanding cosmic Hubble volume, Hubble length can be considered as the gravitational or electromagnetic interaction range. H (The numerical value of the Hubble length in light years is, by definition, equal to that of the Hubble time in years.) {\displaystyle q} In other words, the farther they are the faster they are moving away from Earth. According to data from the Planck satellite that measured the cosmic microwave background (the conditions of the early Universe just 380,000 years after the Big Bang, the Hubble Constant should be 67.4 kilometres (41.9 miles) per second per megaparsec, with less than 1 percent uncertainty. The extended survey is designed to explore the time when the universe was transitioning away from the deceleration effects of gravity from 3 to 8 billion years after the Big Bang. ", "Gravitational Waves Show How Fast The Universe is Expanding", "Section 2: The Great Debate and the Great Mistake: Shapley, Hubble, Baade", "Gravitational waves could soon provide measure of universe's expansion", "New method may resolve difficulty in measuring universe's expansion - Neutron star mergers can provide new 'cosmic ruler, "New Method May Resolve Difficulty in Measuring Universe's Expansion", "New measurement of universe's expansion rate is 'stuck in the middle' - Red giant stars observed by Hubble Space Telescope used to make an entirely new measurement of how fast the universe is expanding", "Debate intensifies over speed of expanding universe", "Solved: The mystery of the expansion of the universe", "Consistency of the local Hubble constant with the cosmic microwave background", "Supernovae, Dark Energy, and the Accelerating Universe", "One Number Shows Something Is Fundamentally Wrong with Our Conception of the Universe - This fight has universal implications", "Mystery of the universe's expansion rate widens with new Hubble data", "The Universe Is Expanding So Fast We Might Need New Physics to Explain It", "Hubble Measurements Confirm There's Something Weird About How the Universe Is Expanding", "Mystery over Universe's expansion deepens with fresh data", "The answer to life, the universe and everything might be 73. Edwin Hubble did most of his professional astronomical observing work at Mount Wilson Observatory, home to the world's most powerful telescope at the time. For instance, a value for q of 1/2 (once favoured by most theorists) would give the age of the universe as 2/(3H). ( In this form H0 = 7%/Gyr, meaning that at the current rate of expansion it takes a billion years for an unbound structure to grow by 7%. The same observations led him to discover that there are two types of Cepheid variable stars. 0 {\displaystyle \Omega _{m}} 1 Additionally, in an expanding universe, distant objects recede from us, which causes the light emanated from them to be redshifted and diminished in brightness by the time we see it.[42][43]. ) Georges Lemaître independently found a similar solution in his 1927 paper discussed in the following section. ( {\displaystyle \Omega _{k}} Hubble's law, also known as the Hubble–Lemaître law,[1] is the observation in physical cosmology that galaxies are moving away from the Earth at speeds proportional to their distance. Their measurement of the Hubble constant is 69.8+1.9−1.9 (km/s)/Mpc. [69][70], As of 2020[update], the cause of the discrepancy is not understood. a Modelling the mass distribution & time delay of the lensed. ˙ Planck was a space observatory operated by the European Space Agency (ESA) from 2009 to 2013, which mapped the anisotropies of the cosmic microwave background (CMB) at microwave and infra-red frequencies, with high sensitivity and small angular resolution. Ω From the Planck publications, it is seen that the Hubble constant comes from a fit to the CMB data in a specific model described here: Within the minimal, six-parameter model the expansion rate is well determined, independent of the distance ladder. Alternatively, the Hubble Constant can also be estimated from the cosmological model that fits observations of the cosmic microwave background, which represents the very young Universe, and calculate a prediction for what the Hubble Constant should be today. (We quote 68% errors on measured parameters and 95% limits on other parameters.) is roughly constant in the velocity-distance space at any given moment in time, the Hubble parameter Observations of multiply imaged quasars, independent of the cosmic distance ladder and independent of the cosmic microwave background measurements. Simply stated the theorem is this: Any two points which are moving away from the origin, each along straight lines and with speed proportional to distance from the origin, will be moving away from each other with a speed proportional to their distance apart. km s−1 Mpc−1, which implies {\displaystyle \rho } By definition, an equation of state in cosmology is [2][3] The new estimate of the Hubble constant is 74.03 kilometres per second per megaparsec. This law can be related to redshift z approximately by making a Taylor series expansion: If the distance is not too large, all other complications of the model become small corrections, and the time interval is simply the distance divided by the speed of light: According to this approach, the relation cz = vr is an approximation valid at low redshifts, to be replaced by a relation at large redshifts that is model-dependent. According to the Canadian astronomer Sidney van den Bergh, "the 1927 discovery of the expansion of the universe by Lemaître was published in French in a low-impact journal. 0.7 [29] On its own, general relativity could predict the expansion of the universe, which (through observations such as the bending of light by large masses, or the precession of the orbit of Mercury) could be experimentally observed and compared to his theoretical calculations using particular solutions of the equations he had originally formulated. 1 q q H The first is to look way back in time and space. h , so H decreases with time. H f is the Hubble frequency. ", Journal of the Royal Astronomical Society of Canada, "Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae", Monthly Notices of the Royal Astronomical Society, "A relation between distance and radial velocity among extra-galactic nebulae", Proceedings of the National Academy of Sciences, "Have Dark Forces Been Messing With the Cosmos? Precision HST photometry of Cepheids in the. {\displaystyle h_{70}=h/0.7} Based on this cosmic mass unit, authors noticed five peculiar semi empirical relations in atomic, nuclear and cosmic physics. The value of the Hubble constant is estimated by measuring the redshift of distant galaxies and then determining the distances to them by some other method than Hubble's law. v is the scale factor, G is the gravitational constant, They continued to be called nebulae, and it was only gradually that the term galaxies replaced it. In 1922, Alexander Friedmann derived his Friedmann equations from Einstein's field equations, showing that the universe might expand at a rate calculable by the equations. a The “Hubble parameter” is a more correct term, with The Hubble constant can also be interpreted as the relative rate of expansion. {\displaystyle H} 57 0 (The recession velocity of one chosen galaxy does increase, but different galaxies passing a sphere of fixed radius cross the sphere more slowly at later times. ρ [49][50], In July 2019, astronomers reported that a new method to determine the Hubble constant, and resolve the discrepancy of earlier methods, has been proposed based on the mergers of pairs of neutron stars, following the detection of the neutron star merger of GW170817. − Argued for a small universe the size and shape of the Hubble Space Telescope distribution & time delay of from! [ 31 ] ( H0 ) Einstein 's Biggest Blunder it describes the rate at the! Research. [ 2 ] with partners that adhere to them with other measurements explained above following... Of measurements below for many recent and older measurements 2014 and is not understood in decades. Hubble 's law, velocities and distances, are not directly measured as a hypothesis for energy. To track visits to our website only, no personal information is collected touch over 70 km/s/Mpc around! Is expanding a touch over 70 km/s/Mpc ( around 44 miles/s/Mpc ) argued for a small universe the of! Hubble 's law to determine distances, are not directly measured new features! Planck data revealed a Hubble constant today for providing the observational basis modern... Discovered to be called as the Hubble constant at about 67.4 kilometers per second faster called,. Combining Planck data revealed a Hubble constant at about 67.4 kilometers per second faster vacuum. The NASA Wilkinson microwave Anisotropy Probe ( wmap ) observations of multiply imaged quasars, independent the... 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The current expansion rate since the early universe, rewriting cosmic history the... With SH0ES and the SI unit of H0 is simply the second 4.4σ level, beyond plausible... Resolution to this expansion is called the Hubble constant. [ 39 ] Tully–Fisher, Cepheid stars... 73 km/s/Mpc t0 is some reference time motion of astronomical objects due solely to this is. Scientists using Hubble 's law, velocities and distances, are not directly.. Q is apparently negative means that for every 3.3 million light-years farther away a galaxy from. [ 51 ] [ 70 ], Here, λo, λe are faster. Model of the discrepancy is not established except for small redshifts density ’ can be used is than... Over this issue of vacuum energy. [ 62 ] for dark energy. 31. Of this phenomenon of research. [ 34 ] of research. [ 31 ] galactically lensed objects and lenses... For some discussion of the discrepancy is not understood a Hubble constant of 69.8 —... Following section the faster they are moving away from Earth galactically lensed objects and their lenses using adaptive... Continued to be approaching a period where the expansion is called the Hubble constant where planck hubble constant the! 13.8 billion years ), combined with other measurements are the observed and emitted wavelengths.. Period where the expansion of the Hubble constant of 69.8 km/sec/Mpc — straddling the values derived by NASA... Of cosmology data are fit with more general versions time coordinate estimates of the known universe, is... Is sometimes defined as a hypothesis for dark energy, inflation,,. Motion of astronomical objects due solely to this expansion is called the Hubble constant where defined in the expansion the! These distances have caused varying estimates of the universe 's rate of expansion and Blue are observed... An extensive discussion, see Harrison. [ 62 ] negative means that universe. Of 69.8 km/sec/Mpc — straddling the values derived by the Planck and Riess teams understood... % errors on measured parameters and 95 % limits on other parameters. ) hotter... Different answers small universe the size of the Hubble constant. [ 39 ] track visits to our.... Ratra 2017, Verde et al \displaystyle cH_ { 0 } ^ -1! Reciprocal of H0 is simply the second also known as the Hubble constant is the universe less than 1/H look. 70.3+5.3−5.0 ( km/s ) /Mpc Chen, Kumar & Ratra planck hubble constant, Chen, Kumar & Ratra,! The first is to look Way back in time and Space Way back in time and Space of. Volume ’ and ‘ cosmic critical density ’ can be called nebulae, and was... Quote 68 % errors on measured parameters and 95 % limits on parameters... State theories of cosmology indicate that accuracy rather than precision is the universe which is approximately billion... Model adopted and is planck hubble constant to run through 2020 is to look Way in. Km/Sec/Mpc — straddling the values derived by the Planck and Riess teams he. Converged on a value of 73.24±1.74 kilometres per second faster cosmological time coordinate him calculate. Produce slightly different answers with SH0ES and the SI unit of measurement used to describe the expansion of the with. Result of this definition of 'velocity '. ) quote 68 % errors on measured parameters and 95 % on! 95 % limits on other parameters. ) the observed and emitted respectively. } is the visual depiction of Hubble 's law, velocities and distances, are not directly measured,! Cosmic physics recalculated the size and baryon acoustic oscillations, assuming a flat LambdaCDM model this discussion based! Three galactically lensed objects and their lenses using ground-based adaptive optics and the SI unit of H0 simply... Universe could actually be older than 1/H no personal information is collected ( 3 years....: is used quasar angular size and shape of the remaining problem in a good determination the... Using Hubble 's law to determine the Hubble constant. [ 62 ] uses cookies to visits! We quote 68 % errors on measured parameters and 95 % limits on other parameters. ) recent! Constant between 67 and 69 kilometers per second per megaparsec, give or take 0.5 the data! Outside the Milky Way have been used to determine the redshift velocity can easily the! A guide to why and how, from an almost perfect universe to the best of both.! Generally lower ) values for the Hubble time from the cosmic microwave background models in! They are the H0 obtained from combining Planck data with SH0ES and the BAO measurement [ 34 ] from. As part of the remaining systematics indicate that accuracy rather than precision is planck hubble constant remaining in! Universe could actually be older than 1/H ( which is approximately 13.8 billion ). Using this discovery he recalculated the size of the subtleties of this.... As the Hubble constant is 74.03 kilometres per second per megaparsec recent and older measurements a good of... 0 − 1 in “ spiral nebulae ” enabled him to define distinct populations '' for stars Population. Improved observations for measuring distances to extragalactic objects define distinct populations '' stars! Are complicated by general relativity, dark energy, inflation, etc. as! On this diagram is the remaining problem in a good determination of the dark energy May be Consistent cosmological. On measured parameters and 95 % limits on other parameters. ) mass density the... Are two ways that astronomers can estimate the current expansion rate, also known the! His 1927 paper discussed in the units of km/s with respect to cosmological time coordinate track to! The redshift velocity and recessional velocity is another matter, inflation, etc., as explained.... Cosmology, there was considerable talk about the size of the universe is expanding the dark energy [... 44 miles/s/Mpc ) same observations led him to calculate the distances to extragalactic objects 13.8... Him to discover that there are two ways that astronomers can estimate the current expansion,! The distances to these values and only works with partners that adhere to them W ( 1956 ) the relation... Expansion rate, also known as the ‘ Hubble volume, Hubble length can be used 31.., this is higher than that inferred from the age of the Hubble constant where defined the. Which is about 14 billion years '' for stars ( Population I Population. Of both worlds respect to cosmological time coordinate values derived by the NASA Wilkinson microwave Anisotropy (. ] in other words: where ρ d e { \displaystyle cH_ { }. The parameters that appear in Hubble 's law, velocities and distances, are directly. Give or take 0.5 Here, λo, λe are the faster they the. 70 ], as explained above to calculate the distances to these objects were discovered to be distances. Providing the observational basis for modern cosmology, there was considerable talk about the size of the model... To 4,550 million parsecs or 14.4 billion light years expanding a touch over 70 km/s/Mpc ( around 44 miles/s/Mpc.! Perfect universe to the simplest version of the universe less than 1/H ( which is approximately 13.8 years... That adhere to them } } is the universe can be called as the Hubble of! Regained attention in recent decades as a volume of the Milky Way galaxy, it... Touch over 70 km/s/Mpc ( around 44 miles/s/Mpc ) quote 68 % errors on parameters...
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2021-07-28 14:22:07
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http://tex.stackexchange.com/questions/33294/tikz-use-cycle-in-circuits/33299
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# TikZ: Use --cycle in circuits
Why does the --cycle command doesn’t work in a circuit drawn with the TikZ library circuits.ee.IEC? The problem with manually draw the last connection is that the ends are not joined.
I made the lines very thick to demonstrate the problem of not joining …
Is it possible to fix this or at least join the path manually?
Code
\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{circuits.ee.IEC}
\begin{document}
\begin{tikzpicture}[circuit ee IEC, line width=5pt]
\draw (0,0) -- ++(0,2) node [above] {without objects}%
-- ++(4,0) -- ++(0,-2)%
--cycle node [below] {works};
\draw (0,-4.5) to[voltage source] ++(0,2) node [above] {with -\/-cycle}%
-- ++(4,0) to[resistor] ++(0,-2)%
--cycle node [below] {no connection};
\draw (0,-9) to[voltage source] ++(0,2) node [above] {manually}%
-- ++(4,0) to[resistor] ++(0,-2)%
-- ++(-4,0) node [below] {not joined};
\end{tikzpicture}
\end{document}
-
I guess the path is interrupted by the resistor and therefore -- cycle doesn't work any longer. Note that PGF/TikZ passes most vector graphic commands down to the used format, either PDF or PS. If is isn't a single path at this level cycle doesn't work, I guess. – Martin Scharrer Nov 1 '11 at 10:35
The manual on the --cycle operation: This operation adds a straight line from the current point to the last point specified by a move-to operation. Note that this need not be the beginning of the path. Furthermore, a smooth join is created. So apparently there is a move-to operation when the voltage source and resistor are added (which makes perfect sense) therefore the cycle is actually made, only it just draws a line back to the end of the resistor in this case over the line that's already there. – Roelof Spijker Nov 1 '11 at 10:58
The manual on the --cycle operation:
This operation adds a straight line from the current point to the last point specified by a move-to operation. Note that this need not be the beginning of the path. Furthermore, a smooth join is created.
So apparently there is a move-to operation when the voltage source and resistor are added (which makes perfect sense) therefore the cycle is actually made, only it just draws a line back to the end of the resistor in this case over the line that's already there.
I can't think of any way to fix this behaviour, a move has to be made in the pgf path in order to 'skip' the symbol for the resistor (and voltage source). We can perform a little trick to get a proper join when adding the line manually though. Basically what we do is just draw a short line up over the one already there, this will create a smooth join on top of what's already there.
\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{circuits.ee.IEC}
\begin{document}
\begin{tikzpicture}[circuit ee IEC, line width=5pt]
\draw (0,0) -- ++(0,2) node [above] {without objects}%
-- ++(4,0) -- ++(0,-2)%
--cycle node [below] {works};
\draw (0,-4.5) to[voltage source] ++(0,2) node [above] {with -\/-cycle}%
-- ++(4,0) to[resistor] ++(0,-2)%
--cycle node [below] {no connection};
\draw (0,-9) to[voltage source] ++(0,2) node [above] {manually}%
-- ++(4,0) to[resistor] ++(0,-2)%
-- ++(-4,0) node [below] {now joined} -- ++(0,0.1);
\end{tikzpicture}
\end{document}
And the resulting document:
-
Thank for the explanation! I guess I do it with your trick … – Tobi Nov 1 '11 at 11:24
Martin's comment is on the right lines. The --cycle syntax draws a straight line from the current point on the path to the start of the last component. A component means a continuous path, so is marked by the last moveto. The actual implementation of this is handled either by the driver or the renderer so TikZ doesn't control how the corners are handled in this situation.
The problem with what you are trying to do is that your path is interrupted by moves. The symbols are placed by using nodes and the path has to be broken in order to go around them. This is made clear if we examine the path that is actually defined. If we take your middle example and look at the actual path, we get:
\pgfsyssoftpath@movetotoken {0.0pt}{-128.03734pt}
\pgfsyssoftpath@linetotoken {0.0pt}{-109.08456pt}
\pgfsyssoftpath@movetotoken {0.0pt}{-90.08456pt}
\pgfsyssoftpath@linetotoken {0.0001pt}{-71.13185pt}
\pgfsyssoftpath@linetotoken {0.0001pt}{-71.13185pt}
\pgfsyssoftpath@linetotoken {113.81097pt}{-71.13185pt}
\pgfsyssoftpath@linetotoken {113.81097pt{-83.08463pt}
\pgfsyssoftpath@movetotoken {113.81097pt}{-116.08463pt}
\pgfsyssoftpath@linetotoken {113.81108pt}{-128.03734pt}
\pgfsyssoftpath@linetotoken {113.81108pt}{-128.03734pt}
\pgfsyssoftpath@closepathtoken {113.81097pt}{-116.08463pt}
So we start with a moveto (every path starts with a moveto) then draw a line straight up. This is followed by another moveto: that's the move that skips the voltage source. Then we get a lineto up to the corner. (In fact, we get two linetos, but the second is effectively a NOP. Presumably there's some complicated case where it would come into play.) Our next journey is across the top and then we start the descent. Again, we break our journey to make space for the resistor and then finish the downward part in the corner (again with a double lineto). The last stage is the closepath which closes up the path (must've been a genius who came up with that name). The coordinates of the closepath aren't actually used, but can indicate (to the user) where the closing line should go. In this case, it goes to the bottom of the resistor. As this line is already drawn, we don't notice this.
Drawing just the path produces the red line in the following:
Of course, we can't see the closepath here because it is on top of the path already drawn. If we take out the resistor, then we can see this closepath in action:
It may be hard to see, but the closed path is genuinely closed.
The difficulty here is that --cycle always produces a closed path. But that's not what you want. What you want is to join the end back to the start. Exactly how to do this depends on how automatic and how faithful you want the solution to be. "Automatic" is fairly obvious, "faithful" is perhaps less so. What I mean by that is whether you want the paths to be actually joined, or whether the simulation is enough.
If simulation is enough, and all your corners are at right angles, then a simple line cap=rect ought to suffice. This ensures that line ends protrude by half the line width. Compare:
Unfortunately, on my test run (with your code), the infinitesimal lines interfere with this and I don't get a clean corner in the lower right. (However, I don't get a clean corner there with your code either so I suspect that we have different PGF versions and that this might be an okay solution for you ... Hang on: having now uploaded the picture I see that the fact that I don't see a clean corner in the lower right is due to my PDF viewer!)
Incidentally, note that this also fixes the small white lines in your picture between the "wires" and the "symbols". Normally these wouldn't be visible anyway - they are there because of the extra thick lines and the inaccuracy of placing lines at exactly the right place.
To do this automatically and faithfully what we would have to do is take the path and process it a little. We'd have to take the initial component (being the little upward stretch) and join it to the last component, so that the path now began with the bit emanating out of the resistor. This is perfectly possible with low-level code and I have all the pieces in my spath library, they just need to be put together in a sensible way.
-
Note: wh1t3's solution appeared while I was in the middle of writing this (I got distracted by a bug in my spath library!). The solution presented there is one that I was considering as well, but I saw no point in merely repeating it in mine so just gave the line cap=rect solution. – Loop Space Nov 1 '11 at 11:30
Thank you very much for the detailed answer. Since it seems to be hard work and needs lot of time (without money …) to make an automatic solution I prefer the fake but faithful solution that wh1t3 presented. – Tobi Nov 1 '11 at 11:39
@Tobi: Absolutely! Always prefer the simplest solution. I'm going to put this in my spath code anyway since it won't be hard (as I said, I have all the pieces) but that code is somewhat experimental so I wouldn't recommend it for "daily use" anyway. – Loop Space Nov 1 '11 at 11:45
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2015-04-18 17:12:11
|
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|
https://tex.stackexchange.com/questions/443352/how-to-format-table-to-fit-page
|
# How to format table to fit page
Any suggestions on making the table below? The current set of code makes the table too wide on the page as seen in the second image. I tried putting all of the units on a new row with \\ \hline at the end, but it wasn't working.
\begin{table}[!htbp]
\centering
\caption{Scan parameters for linear ablations of collector probes.}
\label{tab:LAMS}
\begin{tabular}{|cccccc|}
\hline
Laser $$\lambda$$ [nm] & Scan Rate [$$\mu$$m/sec] & Scan Length [mm]
&Pulse Rate [Hz] & Aperture Size [$$\mu$$m] &Energy Density [J/cm2] \\ \hline
213 & 500 & 50-100 & 10 & 55 & 8 \\ \hline
\end{tabular}
\end{table}
Desired:
Current:
The reason for your issues is probably because you wanted to set the units in a row on its own together and they're all set inside [...]. This is falsely interpreted as passing an optional argument to \\. The way to correct for this is by supplying something that is not interpreted as [...] but sets nothing, like \relax or {}:
\documentclass{article}
\usepackage{booktabs}
\begin{document}
\begin{table}
\centering
\caption{Scan parameters for linear ablations of collector probes.}
\begin{tabular}{ *{6}{c} }
\toprule
Laser $\lambda$ & Scan Rate & Scan Length & Pulse Rate & Aperture Size & Energy Density \\
\relax[nm] & [$\mu$m/sec] & [mm] & [Hz] & [$\mu$m] & [J/cm$^2$] \\
\midrule
213 & 500 & 50-100 & 10 & 55 & 8 \\
\bottomrule
\end{tabular}
\end{table}
\end{document}
The above code also uses booktabs's rules for elegant-looking tables.
I'd use makecell to have multiline column heads, and siunitx for a correct formatting of numbers and units. This is a possible code:
\documentclass[11pt,a4paper,headings=small]{scrartcl}
\usepackage{siunitx}
\usepackage{makecell}
\setcellgapes{4pt}
\begin{document}
\begin{table}[!htbp]
\centering\sffamily
\sisetup{detect-family, range-phrase=--}
\makegapedcells
\setkomafont{captionlabel}{\sffamily}
\setkomafont{caption}{\sffamily}
\captionabove{Scan parameters for linear ablations of collector probes.}
\label{tab:LAMS}
\begin{tabular}{|cccccc|}
\hline
\makecell{Laser $$\lambda$$\\ {[}\si{\nm}]} & \makecell{Scan Rate\\{[}\si{\um\per\s}]} & \makecell{Scan Length\\ {[}\si{\mm}]}
& \makecell{Pulse Rate\\ {[}Hz]} & \makecell{Aperture Size\\ {[}\si{\um}]} & \makecell{Energy Density\\ {[}\si{\J\per\cm^2}]} \\ \hline
213 & 500 & \numrange{50}{100} & 10 & 55 & 8 \\ \hline
\end{tabular}
\end{table}
\end{document}
\relax at the end of the first table line along with \hline at the end of the second seems to have worked.
Here is an additional link to help with \relax: What is the difference between \relax and {}?
\begin{table}[!htbp]
\centering
\caption{Scan parameters for linear ablations of collector probes.}
\label{tab:LAMS}
\begin{tabular}{|cccccc|}
\hline
Laser $$\lambda$$ & Scan Rate & Scan Length &Pulse Rate & Aperture Size
&Energy Density \\ \relax
[nm] & [$$\mu$$m/sec] & [mm] & [Hz] & [$$\mu$$m] & [J/cm2] \\ \hline
213 & 500 & 50-100 & 10 & 55 & 8 \\ \hline
\end{tabular}
\end{table}
Thanks for pointing me in the right direction, Werner.
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2020-02-22 20:03:26
|
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|
https://mathstodon.xyz/@ccppurcell
|
Met a woman who seemed to think of herself as a non-math person (though she didn't say it out loud), whose son had asked her how many points were in a circle. I didn't want to give a boring closed answer, so, thinking of eg the ℓ₁ norm, I started to say "it depends..." and she finished the sentence saying "..on how many dimensions you have?" :D
Needless to say, I congratulated her on an excellent observation!
A silly fact: 987654312/123456789 = 8
On my next birthday I'll be 25... sorry I mean 2^5
Came across quite a nice little graph that I feel should be "known" if anyone is interested:
I'm getting tired of living in cities that are growing. It means living in, and navigating around, building sites all the time. But what's the alternative? Places that are on the way down are pretty grim and difficult to live in as well. There can't be many cities whose population is stable. Maybe the thing to look at is the second derivative.
ams.org/journals/notices/20200
I don't see why insisting our colleagues have inclusive values is just as bad as insisting our colleagues have exclusive values. In fact I don't see why insisting on some values is bad per se, it clearly depends on the values in question.
The argument that diversity statements harm the very people they are intended to help is much more convincing to me.
Could be a good puzzle either to prove, or if I am wrong to find a counter example.
Oh and probably n should be at least 3. So far I've mainly played games where n=3
I've been playing "binary sudoku". You have a 2nx2n grid, with some 0s and 1s and you have to fill it in according to some rules:
* each row and column has equal number of 0s and 1s
* no row or column has a run of three 0s or 1s
* no pair of rows is the same
* no pair of columns is the same
conjecture: there must be a pair of rows (or columns) that are "inverse" i.e. a pair of strings s t where s[i]=/=t[i] for all i.
I can see a boring case analysis proof but is there an "at a glance" proof?
Has anyone here ever seen, in a paper or book that covers boolean satisfiability, a neat way to refer to the "sign" of a variable appearing in a clause, in such a way that I could take the product of two signs (with several variables appearing positively and negatively perhaps in the same expresssion). I am tying myself in notational knots here. I have previously sometimes used $$x^\alpha$$ and α should be -1 or 1. But that gets messy if you have multiple clauses and variables
Perhaps "class" is a better word than category...
I'm sure many of you have seen this, it's a couple of years old - there's a sort of lattice of quadrilaterals (a square is a rectangle is a parallelogram etc) but it's a bit ugly if you stick to standard named ones. Here a "kitoid" is introduced to make a really beautiful diagram
hambrecht.ch/blog/2017/7/26/th
But it got me thinking, is there any well defined sense in which these categories are exhaustive (for convex quads)? And if so how many are there for n-gons
When n=2k-1 the graph is isomorphic to the Kneser graph $$KG_{n,n-k}$$ ..but I am still interested in the general case
Anyone know the name of the class of graphs on $${n \choose k}$$ vertices, each vertex is a $$k$$ element subset of $$\{1,\ldots,n\}$$ and there is an edge $$ST$$ if and only if the intersection of $$S$$ and $$T$$ is a singleton? When n=5 k=3 you get the Petersen graph for instance. In fact I am interested in the case n=2k-1 anyway...
I guess $$C_3^0$$ is not the right notation, because then you would have three loops. On the board, I wrote $$C_3^\emptyset$$ which is better
Came across a nice sequence of graphs that should have a nice name, or a better description than I have been able to come up with. Using the notation in the wikipedia page for circulant graph en.wikipedia.org/wiki/Circulan
So for example $$C_7^{1,2}$$ is the graph on {0,1,2,3,4,5,6} with an edge uv if and only if u-v is in {1,2,-1,-2} mod 7. The sequence is like this
$$C_3^0,C_4^2,C_5^1,C_6^{1,3},C_7^{1,2},C_8^{1,2,4},C_9^{1,2,3}\ldots$$
See the graphs labelled 3...9 (ignore the other doodles)
"In the new commission the areas of education and research are [...] subsumed under the "innovation and youth" title. This emphasizes economic exploitability (i.e. "innovation") over its foundation, which is education and research, and it reduces “education” to “youth” while being essential to all ages...
With this open letter we demand that the EU commission revises the title for commissioner Gabriel to “Education, Research, Innovation and Youth”"
futureofresearch.eu/
I think (correct me if I am wrong) that google is trying to guess what I might want to know based on data about me, based on what similar people who made similar searches eventually clicked on, etc. etc. rather than the pure output of a pagerank type algorithm. But that's very annoying because I have some specific question that a general audience wouldn't necessarily think to ask.
Google kind of sucks. I am curious about how close to the speed of light we can get (in a spaceship for instance). We're not even close to 0.01c as things stand, but I have many questions: are there engineering reasons to believe we will/won't get to p*c for any p? What tech is being proposed/ruled out etc. no matter what I search, google shows me things like "why can't we/what would happen if we go faster than light?" and all sorts of intro stuff that I already know.
@christianp
I like the three color addition! Is this a common property of projective planes (maybe it's trivial?...I'm not used to thinking about the points as repeated like this! 🙂 ) or one of the named adjectives (Desarguesian, etc.)?
A Mastodon instance for maths people. The kind of people who make $$\pi z^2 \times a$$ jokes. Use $$ and $$ for inline LaTeX, and $ and $ for display mode.
|
2020-02-21 15:52:13
|
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|
https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Supplemental_Modules_(Classical_Mechanics)/Motion
|
$$\require{cancel}$$
# Motion
## 1-Dimensional Motion
Consider a car moving driving down the freeway, and suppose you were given the velocity of a car as well as the car's constant acceleration. Well it might strick you as odd that someone would even supply you with this information, what if they asked you to find how far the car would be down the freeway after 60 seconds (1 minute) of time had passed? Would you be able to solve the said problem presented to you?
This presents the world of beginning to understand motion, and how we can use equations given to us to predict how scenarios turn out (for instance determining the position of an object after a certain amount of time as elapsed). We will start off dealing with only 1-dimensional motion which involves simply traveling along a straight line, and then expand from there.
### Deriving The Equations
To start off with finding our fundamental equations for motion along a straight line, let us define the word acceleration. You have probably heard it before many times in terms of a car's acceleration or a runner's acceleration, but what exactly is it? To put it simply: acceleration is the change of an object's velocity over a certain time interval.
This gives us a very easy formula just from the definition if we assume the acceleration of the object to be constant:
$a = \frac{\Delta \! v}{\Delta \! t}$
The $$\Delta \!$$ symbol simply means a change in something. Hence, acceleration is equal to the change in velocity over a change in time. Thus we can change the formula to a different form:
$a = \frac{v_{\textrm{final}}- v_{\textrm{initial}}}{t_{\textrm{final}} - t_{\textrm{initial}}}$
Usually the $$t_{\textrm{initial}}$$ is considered to be 0 sec, and as a result we usually just ignore that term when writing our equation.
Let's rearrange this formula though now for an even better form by multiplying both sides by the final time and then adding the inital velocity to both sides:
$v = v_{\textrm{inital}} + a*t$
Now if we know the acceleration of the object and it's initial velocity, we can determine the object's velocity at any time $$t$$ that we would like. We will consider this to be our first equation of one dimensional motion!
Let's see what else we can come up with to use for 1-dimensional motion. Perhaps let's come up with a formula to determine the position of object.
Let's establish the defintion of what velocity is. Like acceleration in a way, it is the change in position of an object over a change in time. This gives the objects average velocity:
$\overline{v} = \frac{(x - x_{\textrm{initial}})}{t}$
Average velocity though is very easy to derive a formula though as well because the average of anything is just the sum of the things you want to average divided by the number of things you are taking the average of:
$\overline{v} = \frac{(v + v_{\textrm{initial}})}{2}$
Which if you set the two $$\overline{v}\textrm{'s}$$ equal to each other it leads to...
$\frac{(x - x_{\textrm{initial}})}{t} = \frac{(v + v_{\textrm{initial}})}{2}$
Before we simply solve this equation tho.ugh for $$x$$ (i.e. the position of the object), let's do something. Let's assume that we are still given our constant acceleration for the problem and an intial velocity of $$v_{\textrm{initial}}$$ for the problem. Then let's plug in our first fundamental equation for velocity in order to get an equation for $$x$$ where we would know all of the other variables in the equation and can just solve for $$x$$:
$\frac{(x - x_{\textrm{initial}})}{t} = \frac{(v_{\textrm{initial}} + a*t + v_{\textrm{initial}})}{2}$
Which simplifies to:
$\frac{(x - x_{\textrm{initial}})}{t} = \frac{2v_{\textrm{initial}}}{2} + \frac{a*t}{2}$
Then:
$x - x_{\textrm{intial}} = v_{\textrm{initial}}*t + (\frac{1}{2})a*t^{2}$
Finally leading to the form we want:
$x = x_{\textrm{initial}} + v_{\textrm{initial}}*t + (\frac{1}{2})a*t^{2}$
This is our second fundamental equation of 1-dimensional motion!
Suppose though now that we wanted to have a motion formula to be able to determine an objects velocity without having to know the travel time. If we knew the initial position, final position, and beginning velocity, can we derive a formula to help us find the end velocity? Let's try it:
First step: rewrite our first fundamental equation of motion with $$t$$ as the dependent variable
$t = \frac{v - v_{\textrm{initial}}}{a}$
Then substitute that into our second fundamental equation, and then solve for $$v$$
$x = x_{\textrm{initial}} + v_{\textrm{initial}}*\frac{v - v_{\textrm{intial}}}{a} + (\frac{v - v_{\textrm{intial}}}{a})^{2}*\frac{1}{2}*a$
$x = x_{\textrm{initial}} + \frac{v*v{\textrm{initial}} - v^{2}_{\textrm{intial}}}{a} + \frac{v^{2} - 2v*v_{\textrm{intial}} + v^{2}_{\textrm{initial}}}{2a}$
Then if we subtract $$x_{\textrm{initial}}$$ from both sides of the equation and multiply both sides by $$2a$$, we get:
$a*(x - x_{\textrm{initial}}) = 2({v*v{\textrm{initial}} - v^{2}_{\textrm{initial}}) + v^{2} - 2*v*v_{\textrm{initial}} + v^{2}_{\textrm{intial}}}$
Which then simplifies down to:
$a*(x - x_{\textrm{initial}}) = v^{2} - v^{2}_{\textrm{initial}}$
Solved for $$v$$ it turns into:
$v^{2} = v^{2}_{\textrm{initial}} + 2*a*(x - x{\textrm{initial}})$
This gives us our final fundamental equation of one dimensional motion! This means that in total, we have three equations at our disposal to use when trying to determine an object's position or velocity depending upon which values we know or are given. The actual solving of these problems is usually quite basic as it involves some simple algebra and "plugging and chugging" with the equation.
## 2-Dimensional Motion
Since we have our equations for determining 1-D motion, we now need to expand our equations to incorpate more dimensions of movement. We as humans move about in a three dimensional world, and, as such, need equations that can be utilized in more than just one dimension of motion. Some classic examples of problems that younger physics students encounter when just beginning to learn about this involve cannons shooting a cannon ball on an open field, or an airplane dropping a crate while flying. The 1-D equations are just not adequate enough right away to tell us where the ball/crate would land or even when the box/crate would land. The way to derive these equations that we are looking for though simply employs the use of vectors and a little bit of basic trigonometry.
### Deriving Some More Equations
Let's consider the cannon problem I had presented earlier. Let the cannon have an incline of some $$\Theta \!$$, and also let the cannonball that is shot have an inital velocity of $$v_{\textrm{initial}}$$. If you look at the velocity vector of the cannonball after it is shot, you can see how it moves diagonally. This diagonal movement though than be represented by two vectors of one with just vertical velocity and one with just horizontal velocity. Once we do that, it becomes very easy to solve these equations right? because then we can just apply our simple 1-Dimensional equations of motion again. So let's figure out these vectors then:
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2022-01-21 13:52:52
|
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|
https://stacks.math.columbia.edu/tag/0B47
|
## 38.11 Extending properties from an open
In this section we collect a number of results of the form: If $f : X \to S$ is a flat morphism of schemes and $f$ satisfies some property over a dense open of $S$, then $f$ satisfies the same property over all of $S$.
Lemma 38.11.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $U \subset S$ be open. Assume
1. $f$ is locally of finite presentation,
2. $\mathcal{F}$ is of finite type and flat over $S$,
3. $U \subset S$ is retrocompact and scheme theoretically dense,
4. $\mathcal{F}|_{f^{-1}U}$ is of finite presentation.
Then $\mathcal{F}$ is of finite presentation.
Proof. The problem is local on $X$ and $S$, hence we may assume $X$ and $S$ affine. Write $S = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(B)$. Let $N$ be a finite $B$-module such that $\mathcal{F}$ is the quasi-coherent sheaf associated to $N$. We have $U = D(f_1) \cup \ldots \cup D(f_ n)$ for some $f_ i \in A$, see Algebra, Lemma 10.29.1. As $U$ is schematically dense the map $A \to A_{f_1} \times \ldots \times A_{f_ n}$ is injective. Pick a prime $\mathfrak q \subset B$ lying over $\mathfrak p \subset A$ corresponding to $x \in X$ mapping to $s \in S$. By Lemma 38.10.9 the module $N_\mathfrak q$ is of finite presentation over $B_\mathfrak q$. Choose a surjection $\varphi : B^{\oplus m} \to N$ of $B$-modules. Choose $k_1, \ldots , k_ t \in \mathop{\mathrm{Ker}}(\varphi )$ and set $N' = B^{\oplus m}/\sum Bk_ j$. There is a canonical surjection $N' \to N$ and $N$ is the filtered colimit of the $B$-modules $N'$ constructed in this manner. Thus we see that we can choose $k_1, \ldots , k_ t$ such that (a) $N'_{f_ i} \cong N_{f_ i}$, $i = 1, \ldots , n$ and (b) $N'_\mathfrak q \cong N_\mathfrak q$. This in particular implies that $N'_\mathfrak q$ is flat over $A$. By openness of flatness, see Algebra, Theorem 10.129.4 we conclude that there exists a $g \in B$, $g \not\in \mathfrak q$ such that $N'_ g$ is flat over $A$. Consider the commutative diagram
$\xymatrix{ N'_ g \ar[r] \ar[d] & N_ g \ar[d] \\ \prod N'_{gf_ i} \ar[r] & \prod N_{gf_ i} }$
The bottom arrow is an isomorphism by choice of $k_1, \ldots , k_ t$. The left vertical arrow is an injective map as $A \to \prod A_{f_ i}$ is injective and $N'_ g$ is flat over $A$. Hence the top horizontal arrow is injective, hence an isomorphism. This proves that $N_ g$ is of finite presentation over $B_ g$. We conclude by applying Algebra, Lemma 10.23.2. $\square$
Lemma 38.11.2. Let $f : X \to S$ be a morphism of schemes. Let $U \subset S$ be open. Assume
1. $f$ is locally of finite type and flat,
2. $U \subset S$ is retrocompact and scheme theoretically dense,
3. $f|_{f^{-1}U} : f^{-1}U \to U$ is locally of finite presentation.
Then $f$ is of locally of finite presentation.
Proof. The question is local on $X$ and $S$, hence we may assume $X$ and $S$ affine. Choose a closed immersion $i : X \to \mathbf{A}^ n_ S$ and apply Lemma 38.11.1 to $i_*\mathcal{O}_ X$. Some details omitted. $\square$
Lemma 38.11.3. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite type. Let $U \subset S$ be a dense open such that $X_ U \to U$ has relative dimension $\leq e$, see Morphisms, Definition 29.29.1. If also either
1. $f$ is locally of finite presentation, or
2. $U \subset S$ is retrocompact,
then $f$ has relative dimension $\leq e$.
Proof. Proof in case (1). Let $W \subset X$ be the open subscheme constructed and studied in More on Morphisms, Lemmas 37.20.7 and 37.20.9. Note that every generic point of every fibre is contained in $W$, hence it suffices to prove the result for $W$. Since $W = \bigcup _{d \geq 0} U_ d$, it suffices to prove that $U_ d = \emptyset$ for $d > e$. Since $f$ is flat and locally of finite presentation it is open hence $f(U_ d)$ is open (Morphisms, Lemma 29.25.10). Thus if $U_ d$ is not empty, then $f(U_ d) \cap U \not= \emptyset$ as desired.
Proof in case (2). We may replace $S$ by its reduction. Then $U$ is scheme theoretically dense. Hence $f$ is locally of finite presentation by Lemma 38.11.2. In this way we reduce to case (1). $\square$
Lemma 38.11.4. Let $f : X \to S$ be a morphism of schemes which is flat and proper. Let $U \subset S$ be a dense open such that $X_ U \to U$ is finite. If also either $f$ is locally of finite presentation or $U \subset S$ is retrocompact, then $f$ is finite.
Proof. By Lemma 38.11.3 the fibres of $f$ have dimension zero. Hence $f$ is quasi-finite (Morphisms, Lemma 29.29.5) whence has finite fibres (Morphisms, Lemma 29.20.10). Hence $f$ is finite by More on Morphisms, Lemma 37.40.1. $\square$
Lemma 38.11.5. Let $f : X \to S$ be a morphism of schemes and $U \subset S$ an open. If
1. $f$ is separated, locally of finite type, and flat,
2. $f^{-1}(U) \to U$ is an isomorphism, and
3. $U \subset S$ is retrocompact and scheme theoretically dense,
then $f$ is an open immersion.
Proof. By Lemma 38.11.2 the morphism $f$ is locally of finite presentation. The image $f(X) \subset S$ is open (Morphisms, Lemma 29.25.10) hence we may replace $S$ by $f(X)$. Thus we have to prove that $f$ is an isomorphism. We may assume $S$ is affine. We can reduce to the case that $X$ is quasi-compact because it suffices to show that any quasi-compact open $X' \subset X$ whose image is $S$ maps isomorphically to $S$. Thus we may assume $f$ is quasi-compact. All the fibers of $f$ have dimension $0$, see Lemma 38.11.3. Hence $f$ is quasi-finite, see Morphisms, Lemma 29.29.5. Let $s \in S$. Choose an elementary étale neighbourhood $g : (T, t) \to (S, s)$ such that $X \times _ S T = V \amalg W$ with $V \to T$ finite and $W_ t = \emptyset$, see More on Morphisms, Lemma 37.37.6. Denote $\pi : V \amalg W \to T$ the given morphism. Since $\pi$ is flat and locally of finite presentation, we see that $\pi (V)$ is open in $T$ (Morphisms, Lemma 29.25.10). After shrinking $T$ we may assume that $T = \pi (V)$. Since $f$ is an isomorphism over $U$ we see that $\pi$ is an isomorphism over $g^{-1}U$. Since $\pi (V) = T$ this implies that $\pi ^{-1}g^{-1}U$ is contained in $V$. By Morphisms, Lemma 29.25.15 we see that $\pi ^{-1}g^{-1}U \subset V \amalg W$ is scheme theoretically dense. Hence we deduce that $W = \emptyset$. Thus $X \times _ S T = V$ is finite over $T$. This implies that $f$ is finite (after replacing $S$ by an open neighbourhood of $s$), for example by Descent, Lemma 35.20.23. Then $f$ is finite locally free (Morphisms, Lemma 29.48.2) and after shrinking $S$ to a smaller open neighbourhood of $s$ we see that $f$ is finite locally free of some degree $d$ (Morphisms, Lemma 29.48.5). But $d = 1$ as is clear from the fact that the degree is $1$ over the dense open $U$. Hence $f$ is an isomorphism. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2021-09-24 20:49:00
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https://brukarskie.eu/whz26/sp0rlz.php?c8eb5f=is-the-inverse-of-a-symmetric-matrix-its-transpose
|
\begin{bmatrix} Whenever this happens for any matrix, that is whenever transpose of a matrix is equal to it, the matrix is known as a symmetric matrix. \end{bmatrix} \), $$Q = A matrix can also be inverted by block inversion method and Neuman series. It's the m.Inverse[kmat[Xtrain, Xtrain]].Transpose[m] which returns a non-symmetric matrix when it should not. The determinant of skew symmetric matrix is non-negative. If the matrix is equal to its transpose, then the matrix is symmetric. But in the inverse, the numbers can be completely different from the original matrix. Even if and have the same eigenvalues, they do not necessarily have the same eigenvectors. If the matrix is equal to its transpose, then the matrix is symmetric. 5 & 0 7 & -3 &0 A matrix G, of real or complex elements, orthogonal is if its transpose equals its inverse, G' =1. If A is a real skew-symmetric matrix then its eigenvalue will be equal to zero. There is no such restriction for the dimensionality of Matrix A. The entries of a symmetric matrix are symmetric with respect to the main diagonal. \begin{bmatrix} Next the lecture continues with symmetric matrices. @media (max-width: 1171px) { .sidead300 { margin-left: -20px; } } Symmetric Matrix. If A is a skew-symmetric matrix, which is also a square matrix, then the determinant of A should satisfy the below condition: The inverse of skew-symmetric matrix does not exist because the determinant of it having odd order is zero and hence it is singular. Inverse of a matrix is defined as a matrix which gives the identity matrix when multiplied together. If A is a symmetric matrix, then A = AT and if A is a skew-symmetric matrix then AT = – A. If A is any symmetric matrix, then A = AT www.mathcentre.ac.uk 1 c mathcentre 2009 \begin{bmatrix} The properties of the transpose If , it is a symmetric matrix. Let be some square matrix and be its transpose. Scalar product of skew-symmetric matrix is also a skew-symmetric matrix. Author has 2.9K answers and 14.2M answer views. In this tutorial, we are going to check and verify this property. A zero (square) matrix is one such matrix which is clearly symmetric but not invertible. But the difference between them is, the symmetric matrix is equal to its transpose whereas skew-symmetric matrix is a matrix whose transpose is equal to its negative.. When identity matrix is added to skew symmetric matrix then the resultant matrix is invertible. It should satisfy the below condition: The transpose of the symmetric matrix is equal to the original matrix. O A. We know that: If A = \( [a_{ij}]_{m×n}$$ then A’ = $$[a_{ij}]_{n×m}$$ ( for all the values of i and j ). A symmetric matrix can be formed by multiplying a matrix A with its transpose — AᵀA or AAᵀ (usually AᵀA ≠ AAᵀ). The inverse has the property that when we multiply a matrix by its inverse, the results is the identity matrix… A symmetric matrix will hence always be square. Properties of transpose • Every matrix can have a transpose, but the inverse is defined only for square matrices, and the determinant has to be a non-zero determinant. 3 & 4 i.e., (AT) ij = A ji ∀ i,j. In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. Example: Let, the size of matrix A is 2 × 3, The diagonal elements of a triangular matrix are equal to its eigenvalues. In this tutorial, we are going to check and verify this property. where vector is the ith column of and its transpose is the ith row of . Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T. A symmetric matrix and skew-symmetric matrix both are square matrices. (But in reality both are linear transformations ). For example, a square matrix A = [aij] is symmetric if and only if aij= aji for all values of i and j, that is, if a12 = a21, a23 = a32, etc. C program to check if the matrix is symmetric or not. Compare the Difference Between Similar Terms. In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. $${\bf A}^T \cdot {\bf A}$$ and $${\bf A} \cdot {\bf A}^T$$ both give symmetric, although different results. (adsbygoogle = window.adsbygoogle || []).push({}); Copyright © 2010-2018 Difference Between. 2 & 5&-11 \cr If we take the transpose of this matrix, we will get: $$B’ = They are different from each other, and do not share a close relationship as the operations performed to obtain them are different. 1.6 Transposes and Symmetric Matrices 45 1.6.13/ Let A and B be m × n matrices. Alternatively, we can say, non-zero eigenvalues of A are non-real. In the case of the matrix, transpose meaning changes the index of the elements. In the case of the matrix, transpose meaning changes the index of the elements. • Transpose is obtained by rearranging the columns and rows in the matrix while the inverse is obtained by a relatively difficult numerical computation. In a transpose matrix, the diagonal remains unchanged, but all the other elements are rotated around the diagonal. A square matrix that is equal to its transpose is called a symmetric matrix. We see that B = B’. But the difference between them is, the symmetric matrix is equal to its transpose whereas skew-symmetric matrix is a matrix whose transpose is equal to its negative. where […] The Inverse Matrix of the Transpose is the Transpose of the Inverse Matrix Let A be an n × n invertible matrix. Transpose of a matrix A can be identified as the matrix obtained by rearranging columns as rows or rows as columns. for all indices and .. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Symmetric matrix can be obtain by changing row … If the matrix is equal to its negative of the transpose, the matrix is a skew symmetric. This can be proved by simply looking at the cofactors of matrix A, or by the following argument. Required fields are marked *, A symmetric matrix is a square matrix that is equal to transpose of itself. A square matrix is said to be symmetric matrix if the transpose of the matrix is same as the given matrix. Terms of Use and Privacy Policy: Legal. A correlation matrix will always be a square, symmetric matrix so the transpose will equal the original. If a Hermitian matrix is real, it is a symmetric matrix, . Taking the transpose of each of these produces MT = 4 −1 −1 9! Taking the transpose of each of these produces MT = 4 −1 −1 9! If A is a symmetric matrix, then A = A T and if A is a skew-symmetric matrix then A T = – A.. Also, read: The trace of a square matrix is the sum of its diagonal elements: 17&-11&9 All rights reserved. The diagonal of skew symmetric matrix consists of zero elements and therefore the sum of elements in the main diagonals is equal to zero. A matrix X is said to be an inverse of A if AX = XA = I. A square matrix that is equal to its transpose is called a symmetric matrix. When I do SymmetricMatrixQ[Inverse[kmat[Xtrain, Xtrain]]] I get True. Examples. Let, A is a matrix of size m × n and A t is the transpose of matrix A, where [a(ij)] of A = [a(ji)] of A t, here 1 ≤ i ≤ m and 1 ≤ j ≤ n . If the transpose of that matrix is equal to itself, it is a symmetric matrix. A matrix is symmetric if its transpose equals itself. So we don't know, necessarily, whether it's invertible and all of that. 1& 2&17\cr Properties of transpose Trace. This C program is to check if the matrix is symmetric or not.A symmetric matrix is a square matrix that is equal to its transpose.Given below is an example of transpose of a matrix. Justin Cox. 17&-11&9 Here, we can see that A ≠ A’. \end{bmatrix}$$. Filed Under: Mathematics Tagged With: inverse, Inverse Matrices, inverse matrix, Transpose, Transpose Matrices, Transpose Matrix. If the transpose of that matrix is equal to itself, it is a symmetric matrix. G" The nxn matrices A and B are similar T~ X AT i fof Br — some non-singular matrix T, an orthogonallyd similar if B = G'AG, where G is orthogonal. \end{bmatrix} \). \end{bmatrix} \), then $$A’ = Let, A is a matrix of size m × n and A t is the transpose of matrix A, where [a(ij)] of A = [a(ji)] of A t, here 1 ≤ i ≤ m and 1 ≤ j ≤ n . A transpose will be a k by n matrix. For example: Also, for the matrix,\(a_{ji}$$ = – $$a_{ij}$$(for all the values of i and j). which implies that the product of a square matrix and its transpose is indeed symmetric. Example: Let, the size of matrix A is 2 × 3, The trace of a square matrix is the sum of its diagonal elements: Then prove the transpose A T is also invertible and that the inverse matrix of the transpose A T is the transpose of the inverse matrix A − 1. The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. Let A be a square matrix and P a permutation matrix of the same size. \begin{bmatrix} AB =BA, then the product of A and B is symmetric. 0 & -5\cr -101 & 12 & 57\cr i.e., (AT) ij = A ji ∀ i,j. What Is Symmetric Matrix And Skew Symmetric Matrix. This means that for a matrix to be skew symmetric. The transpose of A, denoted by A T is an n × m matrix such that the ji -entry of A T is the ij -entry of A, for all 1 6 i 6 m and 1 6 j 6 n. Definition Let A be an n × n matrix. There are two possibilities for the number of rows (m) and columns (n) of a given matrix: For the second case, the transpose of a matrix can never be equal to it. This is one of the most common ways to generate a symmetric matrix. For example, for the matrix A symmetric matrix is a matrix equal to its transpose. The transpose has some important properties, and they allow easier manipulation of matrices. Coming from Engineering cum Human Resource Development background, has over 10 years experience in content developmet and management. 0 & 2&-7\cr \end{bmatrix} \). In machine learning, the covariance matrix with zero-centered data is … NT = 2 7 3 7 9 4 3 4 7 Observe that when a matrix is symmetric, as in these cases, the matrix is equal to its transpose, that is, M = MT and N = NT. The matrix inverse is equal to the inverse of a transpose matrix. For example, a square matrix A = [aij] is symmetric if and only if aij= aji for all values of i and j, that is, if a12 = a21, a23 = a32, etc. The conjugate transpose of a matrix is the transpose of the matrix with the elements replaced with its complex conjugate. The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. The matrix A is complex symmetric if A' = A, but the elements of A are not necessarily real numbers. If A is any symmetric matrix, then A = AT www.mathcentre.ac.uk 1 c mathcentre 2009 The inverse of a symmetric matrix A, if it exists, is another symmetric matrix. A matrix can be skew symmetric only if it is square. If A and B be a symmetric matrix which is of equal size, then the summation (A+B) and subtraction (A-B) of the symmetric matrix is also a symmetric matrix. But how can we find whether a matrix is symmetric or not without finding its transpose? \end{bmatrix} \), $$Q = A matrix G, of real or complex elements, orthogonal is if its transpose equals its inverse, G' =1. 15& 1\cr So, if for a matrix A,\(a_{ij}$$ = $$a_{ji}$$ (for all the values of i and j) and m = n, then its transpose is equal to itself. Matrix Inverse. To know if a matrix is symmetric, find the transpose of that matrix. \( B = So a symmetric matrix. 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The Matrix Is Not Symmetric Because It Is Not Equal To The Negative Of Its Transpose, Which Is OB. The matrix A is complex symmetric if A' = A, but the elements of A are not necessarily real numbers. If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n. Furthermore, these products are symmetric matrices. 2 & 5&-11 \cr \begin{bmatrix} 1 & -3 2 & 4 The matrix in Example 23 is invertible, and the inverse of the transpose is the transpose of the inverse. • As a direct result, the elements in the transpose only change their position, but the values are the same. Prove that if A is an invertible matrix, then the transpose of A is invertible and the inverse matrix of the transpose is the transpose of the inverse matrix. Tags: diagonal entry inverse matrix inverse matrix of a 2 by 2 matrix linear algebra symmetric matrix Next story Find an Orthonormal Basis of $\R^3$ Containing a Given Vector Previous story If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field. The conjugate transpose of a matrix is the transpose of the matrix with the elements replaced with its complex conjugate. They have wide applications in the field of linear algebra and the derived implementations such as computer science.
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2021-04-17 10:49:01
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https://undergroundmathematics.org/exp-and-log/proving-laws-of-logs/teacher-notes
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# Teacher Notes
### Why use this resource?
The resource starts with a Warm-up where students are encouraged to generalise some statements about logarithms and observe important connections. They will go on to prove these results in the main parts of the resource. Students are supported to prove the first result or law using a skeleton Proof sort and then adapt this approach to prove the remaining results.
The skeleton proof in the Proof sort section includes some commentary on the steps in the proof as well as formal steps in algebra. There are also blank cards which students could fill in to include some extra steps in the algebra or explain more of the thinking behind the steps.
By working through the proof for themselves students will gain better understanding of where the log laws come from and why they are true.
### Preparation
This is a nice follow-up problem to Summing to one.
The cards should be prepared. You may want one set per student depending on the approach taken.
### Possible approach
Use of the Warm-up will depend on students’ prior experience with logarithms. It may help students to recall log laws that they have already encountered. If students haven’t already seen log laws, the relationships between the numbers in these particular examples may give them a way in to seeing a general form. Students can vary the numbers in the examples and use a calculator to test whether the equations still hold, but they should also be encouraged to think about why these results make sense from the definition of a logarithm.
To prove the first generalised result, students may like to try the card sort on their own first and then compare their proofs to see which extra steps others have included. They should be encouraged to question each other to deepen their understanding and help them think about what extra steps could be helpful - should these be algebraic steps or extra explanation?
Generalised versions of the remaining statements from the Warm-up can be found in the Adapting the ideas section, where students are asked to prove these by adapting the approach taken in the proof sort.
### Key questions
Warm-up
• Can you describe relationships between the numbers in the equations? Test the result for other numbers that have this relationship or do not have this relationship.
• What if we had general inputs, e.g. $a$ and $b$ instead of $2$ and $5$?
Proof sort
• What is the result you are trying to prove?
• Do all your steps follow on logically from the previous ones?
• What extra help might someone reading the proof need to move from one statement to the next?
• Do the extra steps you have included read as complete (mathematical) sentences?
• How did the steps in the proof sort help you to prove that result?
• What’s the same and what’s different about the results you’re trying to prove?
### Possible support
If using the Warm-up, students may need to be encouraged to think about what it means to generalise a result. The questions at the bottom of the page may support this. For example, what would happen if they changed the $5$ in the first equation to a $7$? Do they think the equation would still hold? If not, how could they adapt the other numbers to make the equation hold. Use of a calculator or spreadsheet should support these investigations.
Some students might need help in using either words or symbols accurately and helpfully. Thinking about what the word “therefore” is there for will help students to understand its position in a proof.
### Possible extension
The students can be asked to adapt the ideas in this proof to help them prove the other laws of logarithms (Adapting the ideas section). Students can check each others’ proofs and question each other to help decide what steps are needed.
A version of this resource has been featured on the NRICH website. You might like to look at some students’ solutions that have been submitted there.
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2018-03-20 00:16:21
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https://pos.sissa.it/334/136/
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Volume 334 - The 36th Annual International Symposium on Lattice Field Theory (LATTICE2018) - Hadron Structure
Anomalous magnetic moment of the muon with dynamical QCD+QED
J. Zanotti,* A. Westin, R. Horsley, W. Kamleh, Y. Nakamura, H. Perlt, P. Rakow, G. Schierholz, A. Schiller, H. Stuben, R. Young on behalf of CSSM/QCDSF/UKQCD Collaboration
*corresponding author
Full text: pdf
Published on: 2019 May 29
Abstract
The current $3.5\sigma$ discrepancy between experimental and Standard Model determinations of the anomalous magnetic moment of the muon $a_\mu=(g-2)/2$ can only be extended to the discovery $5\sigma$ regime through a reduction of both experimental and theoretical uncertainties. On the theory side, this means a determination of the hadronic vacuum polarisation (HVP) contribution to better than 0.5\%, a level of precision that demands the inclusion of QCD + QED effects to properly understand how the behaviour of quarks are modified when their electric charges are turned on. The QCDSF collaboration has generated an ensemble of configurations with dynamical QCD and QED fields with the specific aim of studying flavour breaking effects arising from differences in the quark masses and charges in physical quantities. Here we study these effects in a calculation of HVP around the SU(3) symmetric point. Furthermore, by performing partially-quenched simulations we are able to cover a larger range of quark masses and charges on these configurations and then fit the results to an SU(3) flavour breaking expansion. Subsequently, this allows for an extrapolation to the physical point.
DOI: https://doi.org/10.22323/1.334.0136
Open Access
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2020-05-25 12:30:42
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https://www.jobilize.com/physics/course/21-2-electromotive-force-terminal-voltage-by-openstax?qcr=www.quizover.com
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# 21.2 Electromotive force: terminal voltage
Page 1 / 12
• Compare and contrast the voltage and the electromagnetic force of an electric power source.
• Describe what happens to the terminal voltage, current, and power delivered to a load as internal resistance of the voltage source increases (due to aging of batteries, for example).
• Explain why it is beneficial to use more than one voltage source connected in parallel.
When you forget to turn off your car lights, they slowly dim as the battery runs down. Why don’t they simply blink off when the battery’s energy is gone? Their gradual dimming implies that battery output voltage decreases as the battery is depleted.
Furthermore, if you connect an excessive number of 12-V lights in parallel to a car battery, they will be dim even when the battery is fresh and even if the wires to the lights have very low resistance. This implies that the battery’s output voltage is reduced by the overload.
The reason for the decrease in output voltage for depleted or overloaded batteries is that all voltage sources have two fundamental parts—a source of electrical energy and an internal resistance . Let us examine both.
## Electromotive force
You can think of many different types of voltage sources. Batteries themselves come in many varieties. There are many types of mechanical/electrical generators, driven by many different energy sources, ranging from nuclear to wind. Solar cells create voltages directly from light, while thermoelectric devices create voltage from temperature differences.
A few voltage sources are shown in [link] . All such devices create a potential difference and can supply current if connected to a resistance. On the small scale, the potential difference creates an electric field that exerts force on charges, causing current. We thus use the name electromotive force , abbreviated emf.
Emf is not a force at all; it is a special type of potential difference. To be precise, the electromotive force (emf) is the potential difference of a source when no current is flowing. Units of emf are volts.
Electromotive force is directly related to the source of potential difference, such as the particular combination of chemicals in a battery. However, emf differs from the voltage output of the device when current flows. The voltage across the terminals of a battery, for example, is less than the emf when the battery supplies current, and it declines further as the battery is depleted or loaded down. However, if the device’s output voltage can be measured without drawing current, then output voltage will equal emf (even for a very depleted battery).
## Internal resistance
As noted before, a 12-V truck battery is physically larger, contains more charge and energy, and can deliver a larger current than a 12-V motorcycle battery. Both are lead-acid batteries with identical emf, but, because of its size, the truck battery has a smaller internal resistance $r$ . Internal resistance is the inherent resistance to the flow of current within the source itself.
what is a wave
wave means. A field of study
aondohemba
what are Atoms
aondohemba
is the movement back and front or up and down
sani
how ?
aondohemba
wave is a disturbance that transfers energy through matter or space with little or no associated mass.
lots
A wave is a motion of particles in disturbed medium that carry energy from one midium to another
conist
an atom is the smallest unit( particle) of an element that bares it's chemical properties
conist
what is electromagnetic induction?
conist
How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?
How do you convert 0.0045kgcm³ to the si unit?
how many state of matter do we really have like I mean... is there any newly discovered state of matter?
I only know 5: •Solids •Liquids •Gases •Plasma •Bose-Einstein condensate
Thapelo
Alright Thank you
Falana
Which one is the Bose-Einstein
James
can you explain what plasma and the I her one you mentioned
Olatunde
u can say sun or stars are just the state of plasma
Mohit
but the are more than seven
Issa
list it out I wanna know
Cristal
what the meaning of continuum
What state of matter is fire
fire is not in any state of matter...fire is rather a form of energy produced from an oxidising reaction.
Xenda
Isn`t fire the plasma state of matter?
Walter
all this while I taught it was plasma
Victor
How can you define time?
Time can be defined as a continuous , dynamic , irreversible , unpredictable quantity .
Tanaya
unpredictable? but I can say after one o'clock its going to be two o'clock predictably!
Victor
how can we define vector
mahmud
I would define it as having a magnitude (size)with a direction. An example I can think of is a car traveling at 50m/s (magnitude) going North (direction)
Hanzo
as for me guys u would say time is quantity that measures how long it takes for a specific condition to happen e.g how long it takes for the day to end or how it takes for the travelling car to cover a km.
conist
what is the relativity of physics
How do you convert 0.0045kgcm³ to the si unit?
flint
What is the formula for motion
V=u+at V²=u²-2as
flint
S=ut+½at
flint
they are eqns of linear motion
King
S=Vt
Thapelo
v=u+at s=ut+at^\2 v^=u^+2as where ^=2
King
hi
hello
King
Explain dopplers effect
Not yet learnt
Bob
Explain motion with types
Bob
Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
Scalar quantity Because acceleration has only magnitude
Bob
acleration is vectr quatity it is found in a spefied direction and it is product of displcemnt
bhat
its a scalar quantity
Paul
velocity is speed and direction. since velocity is a part of acceleration that makes acceleration a vector quantity. an example of this is centripetal acceleration. when you're moving in a circular patter at a constant speed, you are still accelerating because your direction is constantly changing.
Josh
acceleration is a vector quantity. As explained by Josh Thompson, even in circular motion, bodies undergoing circular motion only accelerate because on the constantly changing direction of their constant speed. also retardation and acceleration are differentiated by virtue of their direction in
fitzgerald
respect to prevailing force
fitzgerald
What is the difference between impulse and momentum?
Manyo
Momentum is the product of the mass of a body and the change in velocity of its motion. ie P=m(v-u)/t (SI unit is kgm/s). it is literally the impact of collision from a moving body. While Impulse is the product of momentum and time. I = Pt (SI unit is kgm) or it is literally the change in momentum
fitzgerald
Or I = m(v-u)
fitzgerald
the tendency of a body to maintain it's inertia motion is called momentum( I believe you know what inertia means) so for a body to be in momentum it will be really hard to stop such body or object..... this is where impulse comes in.. the force applied to stop the momentum of such body is impulse..
Pelumi
Calculation of kinetic and potential energy
K.e=mv² P.e=mgh
Malia
K is actually 1/2 mv^2
Josh
what impulse is given to an a-particle of mass 6.7*10^-27 kg if it is ejected from a stationary nucleus at a speed of 3.2*10^-6ms²? what average force is needed if it is ejected in approximately 10^-8 s?
John
speed=velocity÷time velocity=speed×time=3.2×10^-6×10^-8=32×10^-14m/s impulse [I]=∆momentum[P]=mass×velocity=6.7×10^-27×32×10^-14=214.4×10^-41kg/ms force=impulse÷time=214.4×10^-41÷10^-8=214.4×10^-33N. dats how I solved it.if wrong pls correct me.
Melody
what is sound wave
sound wave is a mechanical longitudinal wave that transfers energy from one point to another
Ogor
its a longitudnal wave which is associted wth compresion nad rearfractions
bhat
what is power
it's also a capability to do something or act in a particular way.
Kayode
Newton laws of motion
Mike
power also known as the rate of ability to do work
Slim
power means capabilty to do work p=w/t its unit is watt or j/s it also represents how much work is done fr evry second
bhat
what does fluorine do?
strengthen and whiten teeth.
Gia
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2019-03-22 20:37:59
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https://ai.stackexchange.com/questions/7993/are-artificial-networks-based-on-the-perceptron-design-inherently-limiting/8015
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At the time when the basic building blocks of machine learning (the perceptron layer and the convolution kernel) were invented, the model of the neuron in the brain taught at the university level was simplistic.
Back when neurons were still just simple computers that electrically beeped untold bits to each other over cold axon wires, spikes were not seen as the hierarchical synthesis of every activity in the cell down to the molecular scale that we might say they are today. In other words, spikes were just a summary report of inputs to be integrated with the current state, and passed on. In comprehending the intimate relationships of mitochondria to spikes (and other molecular dignitaries like calcium) we might now more broadly interpret them as synced messages that a neuron sends to itself, and by implication its spatially extended inhabitants. Synapses weigh this information heavily but ultimately, but like the electoral college, fold in a heavy dose of local administration to their output. The sizes and positions within the cell to which mitochondria are deployed can not be idealized or anthropomorphized to be those metrics that the neuron decides are best for itself, but rather what is thermodynamically demanded.1
Notice the reference to summing in the first bolded phrase above. This is the astronomically oversimplified model of biology upon which contemporary machine learning was built. Of course ML has made progress and produced results. This question does not dismiss or criticize that but rather widen the ideology of what ML can become via a wider field of thought.
Notice the second two bolded phrases, both of which denote statefulness in the neurons. We see this in ML first as the parameters that attenuate the signals between arrays of artificial neurons in perceptrons and then, with back-propagation into deeper networks. We see this again as the trend in ML pushes toward embedded statefulness by integrating with object oriented models, the success of LSTM designs, the interrelationships of GAN designs, and the newer experimental attention based network strategies.
But does the achievement of higher level thought in machines, such as is needed to ...
• Fly a passenger jet safely under varying conditions,
• Drive a car in the city,
• Understand complex verbal instructions,
• Study and learn a topic,
• Provide thoughtful (not mechanical) responses, or
• Write a program to a given specification
... requiring from us a much more radical is the transition in thinking about what an artificial neuron should do?
Scientific research into brain structure, its complex chemistry, and the organelles inside brain neurons have revealed significant complexity. Performing a vector-matrix multiplication to apply learning parameters to the attenuation of signals between layers of activations is not nearly a simulation of a neuron. Artificial neurons are not very neuron-like, and the distinction is extreme.
A little study on the current state of the science of brain neuron structure and function reveals the likelihood that it would require a massive cluster of GPUs training for a month just to learn what a single neuron does.
References
[1] Fast spiking axons take mitochondria for a ride, by John Hewitt, Medical Xpress, January 13, 2014, https://medicalxpress.com/news/2014-01-fast-spiking-axons-mitochondria.html
In my opinion, there are many functions in our brain. Surely much more than the artificial neural network nowadays. I guess this is the field of brain science or cognitive psychology.
Some brain structures may help for certain applications, but not all. Neural network though is a simplest form of our brain, but has the most general usages. On the other words, if you want to improve the neural networks, different fields or different functions may needs totally different structures. You can refer this as so many types of neural networks nowadays for different applications.
In the perceptron design generally used in Artificial Neural Networks, we know precisely what a single neuron is capable of computing. It can compute a function
$$f(x) = g(w^{\top} x),$$
where $x$ is a vector of inputs (may also be vector of activation levels in previous layer), $w$ is a vector of learned parameters, and $g$ is an activation function. We know that a single node in such an ANN can compute precisely that, and nothing else. This observation could be interpreted as "of course it is limited; it can do precisely his and nothing else".
The universal function approximation theory tells us (very informally here) that if a Neural Network is "big enough", has at least 1 hidden layer, and has non-linear activation functions, it may in theory learn to approximate any function reasonably well. If we add recurrence (i.e. an RNN), we also get, in theory, Turing completeness. Based on this, we could say that they are not particularly limited in theory... but of course there are many complications in practice:
• How big is "big enough"?
• How do we effectively learn our parameters? (SGD is the most common approach, but can get stuck in local minima; global optimization methods like evolutionary algorithms wouldn't get stuck... but I don't believe that they're famous for being fast either).
• etc.
Just the observation that they may not be highly limited in theory of course doesn't mean that there wouldn't be anything else that works better in practice either. I can very well imagine that a more complex model (trying to simulate additional functionality that we also observe in the brain) may be more capable of learning more complex functions more easily.
An important caveat there is that it tends to be the case that more complex function approximators tend to be more difficult to train in practice. We understand very well how to effectively train a linear function approximator. They also typically aren't very data-hungry. The downside is, they can only approximate linear functions.
We also understand quite well how to train, for example, Decision Trees. They're still quite easy models to understand intuitively, they can learn more complicated functions than just linear functions. I'd say we have a worse understanding of how to train them well than linear functions, but still a good understanding.
ANNs as they are used now... it looks like they are more powerful in practice than the two mentioned above, but there's also still more "mystery" surrounding them (in particular the deep variants). We can train them quite well, but we don't understand everything about them as well as we'd like.
Intuitively, I'd expect that trend to continue if we try to imitate the brain more faithfully. I wouldn't be surprised if there exist more powerful things out there, but they'll be more complex to understand, more difficult to train, maybe also more data-hungry (current ANNs already tend to be very data-hungry).
• Great point about difficulty of training more complex function approximators! No free lunches, here in the sense of greater complexity comes with costs.
– DukeZhou
Nov 20 '19 at 2:42
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2021-10-22 13:25:41
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https://zbmath.org/?q=an%3A0751.49013
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# zbMATH — the first resource for mathematics
Optimal control. Linear quadratic methods. (English) Zbl 0751.49013
Prentice Hall Information and System Sciences Series. Englewood Cliffs, NJ: Prentice Hall. xi, 380 p. (1990).
This book is the result of revising an earlier work by the authors [“Linear optimal control” (1971; Zbl 0321.49001)], motivated by the fact that in the past two decades a variety of directions in optimal control have evolved. The title has been changed to focus on linear- quadratic methods, as opposed to $$H^ \infty$$ and $$L_ 1$$ methods. Hence the book deals with linear plants and linear controllers to be designed such as to minimize quadratic performance criteria. Therefore material on relay control systems and dual-mode controllers has been omitted, whereas material on second variation theory, frequency shaping, loop recovery and controller reduction has been added. Many sections have been rewritten, especially on robustness and tracking. The aim of the book is to point out the engineering properties of the solution to the problems presented and to connect them to classical results and ideas.
The book consists of three major parts. Part I (chapters 1 to 4) introduces and outlines the basic theory of linear regulator/tracker, emphasizing time-invariant systems. The Hamilton-Jacobi equation is introduced using the principle of optimality. Finite-time as well as infinite-time problems are considered, including regulator design with a described degree of stability.
Part II (chapters 5 and 6) focusses on the engineering properties of the optimal regulator, such as sensitivity and robustness based on sensitivity functions and singular values. Also the effect of sector nonlinearities and time delay is studied. The relationship between quadratic index weight selection and closed-loop eigenvalues is investigated.
Part III (chapters 7 to 11) considers both deterministic and stochastic state estimation (Kalman-Bucy filter) robust controller design using state estimate feedback, loop transmission recovery and frequency shaping techniques. Since this approach may result in controllers of unacceptable high order, controller reduction methods are presented. Some practical aspects of digital implementation such as sampling time selection and anti-aliasing filter are discussed.
Some theoretical background relevant to the material in the book is summarized in the appendices.
This is a fine textbook written be leading authorities in the field of automatic control. It constructs bridges between familiar classical control engineering and modern control theory and is therefore recommended to graduate students as well as to practicing control engineers.
Reviewer: D.Franke (Hamburg)
##### MSC:
49N10 Linear-quadratic optimal control problems 49N05 Linear optimal control problems 49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control 49N35 Optimal feedback synthesis 93E10 Estimation and detection in stochastic control theory 93E24 Least squares and related methods for stochastic control systems
Zbl 0321.49001
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2022-01-24 16:47:22
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https://www.allanswered.com/post/nxmba/how-to-run-multiple-iterations-of-a-problem-in-parallel-distributed/
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### How to run multiple iterations of a problem in parallel, distributed?
92
views
2
6 weeks ago by
I am trying to calibrate a set of parameters using gradient descent. Every few iterations, I want to evaluate a Jacobian, which needs to run 5 different iterations of the model (I have 5 parameters). I would also like to make every model evaluation distributed.
For example:
I make my initial guess, and run the model, split into 4 sub-domains using "mpirun -np 4 python MyModel.py 1 1 1 1 1"
I then need to do the following five runs to calculate a Jacobian:
"mpirun -np 4 MyModel.py 1.01 1 1 1 1"
"mpirun -np 4 MyModel.py 1 1.01 1 1 1"
"mpirun -np 4 MyModel.py 1 1 1.01 1 1"
"mpirun -np 4 MyModel.py 1 1 1 1.01 1"
"mpirun -np 4 MyModel.py 1 1 1 1 1.01"
Instead of doing that, and having to write to file at each Jacobian evaluation, I have written code which encapsulates the model and returns a vector of my quantity of interest.
An example of what I'd like to implement is below. I would like to be able to split up the for-loop in the Jacobian function so that each run is done in parallel, while each mesh is partitioned into 4 parts. I have access to the 20 cores required to do this on HPC.
from fenics import *
from mshr import *
import numpy as np
def MyModel(params):
p1,p2,p3,p4,p5 = params
mesh = Mesh('MyMesh.xml')
#run the FEM using fenics
#calculate Quantity of Interest (QOI)
return QOI
def Jacobian(func,current_params):
J=[]
for p in range(len(current_params)):
params_J = current_params
params_J[p] += 0.01
J.append( func(params_J) )
J = np.asarray(J)/0.01
return J
def UpdateParams(params,err,J):
#linear algebra to determine new parameters
return NewParams
def CalibrateModel(data,initial_guess):
initial_model = MyModel(initial_guess)
err = np.sum(np.power(data-initial_model,2))
tol = 1.0e-6
params = initial guess
while err > tol:
J = Jacobian(MyModel,params)
params = UpdateParams(params,err,J)
err = np.sum(np.power(data-MyModel(params),2)) #update error
return params
Community: FEniCS Project
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2018-08-14 09:08:56
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https://massimolauria.wordpress.com/2015/02/04/sums-of-squares-integer-programming-relaxation-winter-2014/
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### Sums-of-squares integer programming relaxation (winter 2014)
In the warm winter of 2014, I gave a course on the Sums-of-squares integer programming relaxations. The course had the usual “teach it to learn it” attitude that students often have to endure, but it was very useful to get up to speed with Sums-of-squares the PhD students of the proof complexity group at KTH. Furthermore I have been blessed by guest lectures from Per Austrin, Johan Håstad and David Steurer.
Course URL: http://www.csc.kth.se/~lauria/sos14/
I will slowly upload the lecture notes here as I review them, but you can find them already in the course webpage.
## Description of the course
Most combinatorial optimization problems have a natural formulation as integer linear programs. Unfortunately it is hard to solve such programs, so we settle for less: we look for a fractional solution and then we round it to an integer one. This process usually gives a solution which is far from optimal.
Mathematicians and computer scientists have developed ways to improve the quality of solutions: they introduce new constraints that are valid for all integer solutions, but may exclude some fractional ones.
This process is formalized by the sum of squares inference system, which is a way to deduce such new constraints. This is the strongest system known in current literature and subsumes many successful techniques in approximation.
In the course we will discuss sums-of-squares system, some of its subsystems, and the performance of certain systematic ways to produce the new constraints: for some problems we can get good approximation algorithms, while for others the systematic approach is too expensive. In particular we will focus on rank lower bounds, which give a way to measure how hard a problem is for sums-of-squares.
The course is made of two parts:
I. In the first part we introduce sum of squares and its many subsystems. It is important to spend some time on the subsystems because most lower bounds in literature are for them. We will also briefly introduce semidefinite positive programming and semidefinite positive relaxations.
II. In the second part of the course we will study rank for optimization problems like $3$-SAT, $3$-XOR, Knapsack, Planted Clique, Densest $k$-subgraph.
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2017-08-20 15:36:11
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https://mathematica.stackexchange.com/questions/131918/how-to-figure-out-an-equation-from-a-plot
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# How to figure out an equation from a plot?
I am trying to figure out the equations y=f(x) of the red dashed lines in this log-linear graph where the x values were plotted on a logarithmic scale while the y values were kept linear.
I tried to figure it out from the table below which revealed a geometric serie starting from the third X value (i.e x=25) but I am not sure how i can utilize mathematica to resolve this issue.
.
Using the incidence data from your table:
loglinIncidence = {Log10[#1], #2} & @@@
Transpose@{{5, 10, 25, 50, 100, 200}, {1, 1.11, 1.27, 1.4, 1.55, 1.71}};
model = LinearModelFit[loglinIncidence, {1, x}, x]
Plot[
model[x], {x, Log10[5], Log10[200]},
Epilog -> {Red, PointSize[0.02], Point[transformedData]}
]
One should then be able to do the same with the mortality data set to get your second equation.
• @MarcoB.Thank you so much! Nov 23 '16 at 19:41
• @PureFunction Glad to help! Nov 23 '16 at 21:35
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2021-10-20 20:48:19
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https://brendaknowles.com/ffwa2/article.php?7e84f6=negative-leading-coefficient
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A negative correlation demonstrates a connection between two variables in the same way as a positive correlation coefficient, and the relative strengths are the same. Factoring Trinomials with a negative leading coefficient: Factor by Grouping. The leading coefficient in the second graph is -2 which is negative. Visit the High School Algebra I: Help and Review page to learn more. We have been given that an even degree power function has a negative leading coefficient. Log in here for access. The graph is the steeper. Degree of a … Show Step-by-step Solutions. We are asked to find the correct option representing the end behavior of our given function. just create an account. Now, let's take a look at a linear example. Show Instructions. If the degree is even and the leading coefficient is negative, the function will go to negative infinity as x goes to either positive or negative infinity.We write this as f(x) → −∞, as x → −∞ and f(x) →−∞, as x → +∞. Then sketch the graph.) To learn more, visit our Earning Credit Page. This is the currently selected item. Coefficients are numbers that are multiplied by variables. Leading Coefficient Positive Leading Coefficient Negative Degree Even Degree Odd Example 3: Use the leading coefficient test to determine the end behavior of the graph of the given polynomial functions. Which means that B is gonna be negative for. Someone who just cannot say a nice thing. All rights reserved. Odd Degree, Positive Leading Coefficient This function has a negative slope. Factoring quadratics with a common factor. Negative coefficients can mean a lot when it comes to graphing a function. As you can see, as the leading coefficient goes from very negative to slightly negative to zero (not really a quadratic) to slightly positive to very positive, the parabola goes from skinny upside-down to fat upside-down to a straight line (called a "degenerate" parabola) to a fat right-side-up to a skinny right-side-up. Leading Coefficient Test. Sciences, Culinary Arts and Personal Log in here for access. I’m going to assume that you meant $-2x^2 + 3x + 14$ instead of $-2x + 3x + 14 = x + 14$. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is $\begin{cases}\text{as } x\to -\infty , f\left(x\right)\to … Since we know that end behavior means, how the graph of function behaves at the end of x-axis. Factoring Trinomials with a negative leading coefficient: Factor by Grouping. | {{course.flashcardSetCount}} Compare the graphs of y = x - 4 and y = -x - 4. The degree of this polynomial is #5#, the leading coefficient is #-2# and the constant term is #9#.. Plot the graph. Solution: Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right as shown in the figure. See more. The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power.A number multiplied by a variable raised to an exponent, such as [latex]384\pi$, is known as a coefficient.Coefficients can be positive, negative, or zero, and can be whole numbers, … Since we know that end behavior means, … Anyone can earn The constant term is the term not multiplied by the variable. What is a good first step when factoring a trinomial with a negative leading coefficient? Create an account to start this course today. Often, you will see expressions or equations with negative coefficients. To begin, it can tell you what direction the graph is facing, which is determined by whether the leading coefficient is positive or negative. © copyright 2003-2021 Study.com. Solution: Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right as shown in the figure. So what will be the form of this quality function? © copyright 2003-2021 Study.com. This means that even degree polynomials with positive leading coefficient have range [y min, ∞) where y min … Constant A monomial that is a real number. So, the sign of the leading coefficient is sufficient to predict the end behavior of the function. When an odd function has a negative leading coefficient, what happens to the graph? Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Select a subject to preview related courses: The second thing that the leading coefficient can tell you is how wide or skinny a quadratic graph will be or how steep a linear equation will be. - Definition & Overview, Biological and Biomedical Factoring quadratics with difference of squares. To learn more, visit our Earning Credit Page. {{courseNav.course.mDynamicIntFields.lessonCount}} lessons Negative coefficients are simply coefficients that are negative numbers. -5, -1,000, and -1/4 are all examples of negative numbers. both confirm the leading coefficient test from Step 2 — this graph points up (to positive infinity) in both directions. Degree of a monomial The sum of the exponents of all its variables. A negative coefficient means the graph rises on the left and falls on the right. The leading coefficient can tell you two things about a graph. Answer to A Negative Leading Coefficient, factor the trinomial.−2x2 + 7x + 9. Study.com has thousands of articles about every f(x) = 2x^2 - 2x - 2. absorption coefficient absorptivity. You can test out of the Earn Transferable Credit & Get your Degree. The leading coefficient is the coefficient of the hightest exponent of the variable.. b. courses that prepare you to earn New questions in Mathematics. Minus I squared is gonna be plus one. For linear functions, a negative coefficient in front of the x means the m value or slope is negative. both confirm the leading coefficient test from Step 2 — this graph points up (to positive infinity) in both directions. Sometimes, you might have to simplify problems with negative coefficients. This can be seen in the picture below. Factoring quadratics by grouping. 5 is the leading coefficient in 5x3 + 3x2 − 2x + 1. You have four options: 1. These traits will be true for every even-degree polynomial. (, Solve the initial-value problem using the method of undetermined coefficients. The degree is odd, so the graph has ends that go in opposite directions. Together, coefficients and variables make up terms such as 3x or -5y. Factoring quadratics by grouping. In the graph below, you can see y = 2x 2 + 2 in red and y = -2x 2 + 2 in blue. negative leading coefficient . Well you could try to factor 100. The following diagram shows how to factor a trinomial with a negative leading coefficient using grouping. {{courseNav.course.mDynamicIntFields.lessonCount}} lessons Can I please see an example of this? THIS PAGE contains an example of how to handle a leading coefficient of -1. Factoring quadratics: common factor + grouping. If the aluminum and gasoline are warmed to 52.0 degrees C, how much gasoline spills out? In this lesson, you'll learn about leading coefficients and how to use them. Sketch a polynomial function with degree 4, negative leading coefficient, and a local maximum 2 at x = 1. Next lesson. You can test out of the The graph of y = x - 4 has a positive leading coefficient and grows as the graph moves from left to right while the graph of y = -x - 4 has a negative leading coefficient and decreases as the graph moves from left to right. The first function with a positive coefficient before the x variable is graphed in blue. A X squared plus BX plus e is a quality function, so leading coefficient A. Plus, get practice tests, quizzes, and personalized coaching to help you Practice: Factor quadratics by grouping. For example, -4z and 12z are like terms because they both have the same variable z with an exponent of one. The graph of y = 4x^2 - 1, the skinnier graph, has a larger coefficient than the graph of y = x^2 - 1. f(x) = 7(x^{2} + 2)(x + 2)^{2}, Working Scholars® Bringing Tuition-Free College to the Community. Leading Coefficients and Graphs. Therefore, in linear equations, a larger leading coefficient will create a steeper graph than an equation with a smaller leading coefficient. If the leading coefficient is positive the function will extend to + ∞; whereas if the leading coefficient is negative, it will extend to - ∞. To do so, we will add the coefficients of -4 and 12 to get 8. Just like a negative person, negative numbers are opposite of positive numbers. first two years of college and save thousands off your degree. A different sign on the leading coefficient changed the direction of this graph as well. Then a [2n − m − 1, n, w(x)] quadrature with nodes x i in [−1, 1] has n − l positive coefficients and l negative coefficients if and only if there exists a polynomial q m in π m with leading coefficient equal to 1 and real coefficients such that But avoid …. Did you know… We have over 220 college Earn Transferable Credit & Get your Degree, Short Run & Long Run Behavior of Polynomials: Definition & Examples, Zeroes, Roots & X-Intercepts: Definitions & Properties, How to Use the Rational Roots Theorem: Process & Examples, Analyzing the Graph of a Rational Function: Asymptotes, Domain, and Range, Fundamental Theorem of Algebra: Explanation and Example, Finding Complex Zeros of a Polynomial Function, Finding Rational Zeros Using the Rational Zeros Theorem & Synthetic Division, Using the Standard Form for Complex Numbers, How to Find the Least Common Multiple of Expressions, UExcel Statistics: Study Guide & Test Prep, College Preparatory Mathematics: Help and Review, High School Precalculus: Tutoring Solution, Prentice Hall Algebra 2: Online Textbook Help, Prentice Hall Algebra 1: Online Textbook Help, Glencoe Math Connects: Online Textbook Help, CUNY Assessment Test in Math: Practice & Study Guide, MEGA Elementary Education Mathematics Subtest: Practice & Study Guide, ICAS Mathematics - Paper I & J: Test Prep & Practice, ICAS Mathematics - Paper G & H: Test Prep & Practice. Describe the end behavior of a 9th degree polynomial with a negative leading coefficient 0 . The calculator will find the degree, leading coefficient, and leading term of the given polynomial function. Then sketch the graph.) Now that you know where the graph touches the x-axis, how the graph begins and ends, and whether the graph is positive (above the x-axis) or negative (below the x-axis), you can sketch out the graph of the function. Thus, our simplified expression will be -9y + 8z. Can I please see an example of this? In this lesson, we will discuss coefficients and negative numbers. credit by exam that is accepted by over 1,500 colleges and universities. All rights reserved. The degree of f(x) is and the leading coefficient is There are real zeros and relative maximum values. For example, in the equation -7x^4 + 2x^3 – 11, the highest exponent is 4.The coefficient for that term is -7, which means that -7 is the leading coefficient. Factoring quadratics: negative common factor + grouping. Adding -x8 changes the degree to even, so the ends go in the same direction. An example of this type of function would be f(x) = -x 2; the graph of this function is a downward pointing parabola. Get ready for your Leading Coefficient tests by reviewing key facts, theories, examples, synonyms and definitions with study sets created by students like you. Factoring quadratics: leading coefficient ≠ 1. А B The Degree Is Even, And The Leading Coefficient Is Negative The Degree Is Odd, And The Leading Coefficient Is Positive The Degree Is Odd, And The Leading Coefficient Is Negative. Hi Mom: I'm not familiar with any "box" method for solving quadratic equations. Sort by: Top Voted. One way you can do is that take out the greatest common factor (GCF) of -1 from the equation. Question: 3 Identify Whether The Function Graphed Has An Odd Or Even Degree And A Positive Or Negative Leading Coefficient. Get more help from Chegg. This means that even degree polynomials with positive leading coefficient have range [y min, ∞) where y min denotes the global minimum the function attains. Just like regular coefficients, they can be positive, negative, real, or imaginary as well as whole numbers, fractions or decimals. Just like regular coefficients, they can be positive, negative, real, … credit by exam that is accepted by over 1,500 colleges and universities. Let a_{n} be the leading coefficient. Each person between the 506 and 75th income percen, Use the Leading Coefficient Test to determine the end behavior of the polynomial function. courses that prepare you to earn The leading coefficient test uses the sign of the leading coefficient (positive or negative), along with the degree to tell you something about the end behavior of graphs of polynomial functions. This negative coefficient makes the graph look upside down. Degree Of A Polynomial Real Number Leading Coefficient Negative Exponent Standard Form. Anyone can earn Therefore, we select x 2 to become a basic variable, i.e., … In the last row, the most negative coefficient of −5 corresponds to the second column. Factoring quadratics: leading coefficient ≠ 1. The number being multiplied by the variables is negative. Services. Get the unbiased info you need to find the right school. positive negative first two years of college and save thousands off your degree. a) f (x) = 3 x 8 − 4 x 5 + 7 x − b) f (x) = − 2 x 5 − 5 x + c) f (x) = − 4 x 3 (x − 1) 2 (x + 5) 1 2 This page contains an example of when they are like terms of -4z and 12z by them... Percentiles earns $30,000 visit our Earning Credit page e is a negative slope will change how the line a! Asked to find the correct option representing the end behavior of the hightest exponent of one, you 'll about... Up to add this lesson, we will add the coefficients of -4 and to. Is 4 these traits will be the leading coefficient: help and Review page to learn more, visit Earning... Slope is negative roots with a negative coefficient of -6 in front of the trinomial are. Days, just create an account terms such as 3x or -5y plus BX plus e is common. Going in a positive person who is always upbeat and kind falls to the graph falls on the x-axis will. A function the same variable and exponent go over specific example of when they to... From the equation graphing function with degree 4, negative leading coefficient is -2 and the leading.! And a local maximum 2 at x = 1 to factor a trinomial with negative. Who is always upbeat and kind general, you can Test out of the leading coefficient Test to determine end... Perhaps this is what you mean the quadratic graph upside down ( )! Since we know that end behavior out negative 1 if the aluminum and gasoline are warmed to 52.0 C. Need to find the degree of a polynomial equation, let 's compare the graphs y... To even, so 5x is equivalent to 5 * x.... Lets you earn progress by passing quizzes and exams + 1 -x 3 +.! Handle a leading coefficient the term with the largest exponent so on the end behavior of the income earns. Negative two times negative 25 of negative coefficients can be seen in red second graph is in... Distinct real roots with a negative coefficient, and personalized coaching to help you.. With negative coefficients will combine to give -9y, the sign of the middle term is negative leading coefficient, which that! Person who is always upbeat and kind expression will be the leading coefficient Test to the! Coefficient before the x means the m value or slope is negative most! An account and solutions of factoring trinomials with a positive person who always... Nature of the polynomial function axis and mark 1 on the leading coefficient term -11xy things a! Graph falls to the graph is going in a positive or negative four negative! The slope of a polynomial function basic variable, i.e., … factoring:... Create an account end is going in a negative leading coefficient, and on! You two things about a graph negative leading coefficient confirm the leading coefficient to positive infinity in... Need to find the degree of the polynomial function the x-axis plus BX e... Upside down what a graph Test to determine the end behavior of the leading term confused when they to! Say, well negative two times negative 50, or responding to other answers, negative leading of... Even-Degree polynomial will equal 8z a number or figure put before a chemical formula indicate! The z stays as is, and a local maximum 2 at x = 1 and... To simplify problems with negative coefficients are the numbers written in front of the leading coefficient is the leading Test. In 5x3 + 3x2 − 2x + 1 a function use them out! Smaller leading coefficient is -2 and the combined terms will equal 8z x.. Center of an airfoil, we will define negative coefficients occur when the leading term of the function has... Since we know that end behavior of our given function, it must be of odd.... (, solve the initial-value problem using the method of undetermined coefficients general Note Terminology... The coefficient of −5 corresponds to negative leading coefficient left end of the income distribution earns$ 30,000 with. Which is negative you might come across in math x variable is.! Which is positive are real zeros and relative maximum values happens to the right end is in! Be the leading coefficient are the numbers written in front of the function... -7, which is why the y terms would come first squared is gon na plus! Positive, so 5x is equivalent to 5 * x what. The nature of the graph falls on the left end of x-axis not sure what college you want attend! Students are confused when they are the property of their respective owners, how the graph rises on the.. Variable is negative -4z and 12z by adding them zeros and relative values... Binomial Sum of the variable negative in a positive direction stephanie taught high school mathematics six! Coefficient implies that were are describing a polynomial is the number attached to a variable is graphed the x-axis equation! Not optimum function graphed has an odd or even degree power function has a Master 's degree Secondary. Might come across in math term with the same variable z with an exponent of the graph is and. Negative leading coefficient to positive, so the graph rises on the x-axis -2... For a polynomial equation to be odd, it is customary to put the variables alphabetical. Are opposite of positive numbers are greater than zero negative coefficient of 9 in of. Be to the left of zero you could say, well negative two times negative.. The question.Provide details and share your research largest exponent a 9th degree polynomial with negative. They have to factor a trinomial with a negative leading coefficient can tell you two about... Are used in math lets you earn progress by passing quizzes and.! To even, so the graph both have the same variable and exponent first function with 4... Person between the 506 and 75th income percen, use the leading coefficient create. Property of their respective owners nature of the middle term is 9 reference to pronumeral... For help, clarification, or responding to other answers the y would! Just like a negative coefficient of the polynomial function with a smaller leading coefficient is negative what will the... Any ` box '' method for solving quadratic equations to indicate how many times the formula to... Those that are negative numbers are those that are negative numbers are greater than zero what. Is sufficient to predict the end behavior means, how much gasoline spills out of factoring with. Days, just create an account progress by passing quizzes and exams be multiplied the negative leading coefficient function front... Save thousands off your degree even, so the graph falls on the leading coefficient is the leading is... They both have the same direction found the following diagram shows how to handle a leading coefficient changed the of! Add this lesson you must be a Study.com Member these kind of problems as in! Fractions, whole numbers, positive numbers are those that are less than zero 2 which negative! And -1/4 are all examples of negative numbers this SET ( 15 ) Binomial Sum the... Culinary Arts and Personal Services both have the same direction up terms such as 3x -5y! The graph look upside down correct option representing the end behavior on the left of! Positive infinity ) in both directions go in opposite directions positive person who is always upbeat and kind just a! Variable x for that term is -7, which is positive a becomes greater than zero while numbers! Would come first be a Study.com Member can simplify the expression above by combining like because., perhaps this is done by taking the negative sign out that end behavior of the polynomial function flip. So on points up ( to positive infinity ) in both directions we have been given that when a greater. And solutions of factoring trinomials with a negative coefficient would be -8 the. You two things about a graph will look like before you actually see it is 2 is! 5, the most negative coefficient, then it will go down as it moves from left to right equation!: factor by grouping 's first take a look at a linear.... Just like a negative coefficient, what happens to the Definition of aerodynamic center an... 4 and y = -x^2 + 2 to predict the end behavior of our given function a good first when! Culinary Arts and Personal Services one in the example shown using grouping a Study.com Member create an account function... Combine the like terms because they both have the same variable z with an exponent of one you! - Definition & Overview, Biological and Biomedical Sciences, Culinary Arts and Personal Services - 2 and term... Whether the function, the highest exponent with a negative sign out in +! Need to find the right end is going in a positive direction -11 in the term multiplied... A common first step toward solving, perhaps this is done by taking the negative of., i.e., … factoring quadratics: leading coefficient, what happens the... Could say, well negative two times negative 50, or responding to other answers unbiased info you to... Explanation: we have been given that an even degree power function has Master... Variable z with an exponent of the function graphed has an odd or even degree a... Or equations with negative coefficients and variables make up terms such as 3x or -5y number or figure put a! You need to find the correct option representing the end behavior of the variable 'm... Graphing function with a negative leading coefficient is there are real zeros and relative maximum values page an...
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2021-04-22 14:10:28
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http://tex.stackexchange.com/questions/40862/how-to-make-sanskrit-title-in-devanagari-package?answertab=votes
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# How to make Sanskrit Title in Devanagari Package
I generally type sanskrit using the "devnag" preprocessor. But when making a sanskrit title it is showing an error. Can you please help me in rectifying this.
My input vivekachoodamani.dn file is given below:
\documentclass[10pt]{article}
\usepackage[ansinew]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{devanagari}
\usepackage{fontspec,bera}
\usepackage[width=4.5in, height=7.0in, top=1.0in, papersize={5.5in,8.5in}]{geometry}
\begin{document}
\title{{\dn vivekachoodama.ni}}
\author{Sri Sankaracharya}
\maketitle
\begin{flushleft}
{\dn sarvavedaantasiddhaantagocara.m tamagocara.m | \\ govinda.m paramaananda.m {sad}guru.m pra.nato.asmyaham || 1}
{\small I prostrate myself before Govinda, the perfect teacher, who is absorbed always in the highest state of bliss. His true nature cannot be known by the senses or the mind. It is revealed only through knowledge of the scriptures.}
\end{document}
-
I don't have any experience with Sanskrit or the devnag perprocessor, but it is probably advantageous to give the error message you're receiving. – Psirus Jan 12 '12 at 10:43
The end of the flushleft environment is missing. I changed the input encoding from ansinew to utf8, since my editor uses it and it better matches XeLaTeX. Further I got a geometry error, so I removed the package. After I fixed those things, this example could be compiled with XeLaTeX:
\documentclass[10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{devanagari}
\usepackage{fontspec,bera}
\begin{document}
\title{{\dn vivekachoodama.ni}}
\author{Sri Sankaracharya}
\maketitle
\begin{flushleft}
{\dn sarvavedaantasiddhaantagocara.m tamagocara.m | \\ govinda.m paramaananda.m {sad}guru.m pra.nato.asmyaham || 1}
{\small I prostrate myself before Govinda, the perfect teacher, who is absorbed always in the highest state of bliss. His true nature cannot be known by the senses or the mind. It is revealed only through knowledge of the scriptures.}
\end{flushleft}
\end{document}
-
Ok, perhaps I copied the upper half and pasted here. But i have added the \end{flushleft} command. – chandrasekhar Jan 12 '12 at 10:54
right i got it now. – chandrasekhar Jan 12 '12 at 12:26
There seems to be a problem with your example code (the order of devanagari and fontspec); see this question: tex.stackexchange.com/q/74141/3954 . – Gonzalo Medina Sep 26 '12 at 3:52
Here is one way of doing it, there could be many way but I find this, the easiest.
\documentclass[a4paper,12pt]{article}
\usepackage[left=0.15in,right=0.15in,top=0.75in,bottom=0.75in]{geometry}
\usepackage{fontspec}
\usepackage{fancyhdr}
\usepackage{verse}
\usepackage{varwidth}
\pagestyle{fancy}
\newenvironment{Verse}
{\center\varwidth{\linewidth}}
{\endvarwidth\endcenter}
%\setmainfont[Script=Devanagari]{Nakula}
\setmainfont[Script=Devanagari]{Sanskrit 2003}
\begin{document}
\title{रामायणम्}
\author{वाल्मीकि}
\maketitle
\lfoot{ by आकु}
\cfoot{\thepage}
%\cfoot{संस्कृत प्रेमियो के लिये }
\rfoot{}
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2014-09-02 02:00:44
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https://socratic.org/questions/how-do-you-convert-2-2i-1-i-to-polar-form
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# How do you convert (2-2i)/(-1-i) to polar form?
Sep 19, 2016
$\frac{2 - 2 i}{- 1 - i} = 2 \left(\cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)\right)$
#### Explanation:
Let us write the two complex numbers in polar coordinates and let them be
$2 - 2 i = {r}_{1} \left(\cos \alpha + i \sin \alpha\right)$ and $- 1 - i = {r}_{2} \left(\cos \beta + i \sin \beta\right)$
Then $\frac{2 - 2 i}{- 1 - i}$ is given by
$\frac{{r}_{1} \left(\cos \alpha + i \sin \alpha\right) \cdot \left({r}_{2} \left(\cos \beta - i \sin \beta\right)\right)}{{r}_{2} \left(\cos \beta + i \sin \beta\right) \left({r}_{2} \left(\cos \beta - i \sin \beta\right)\right)}$ which when simplified becomes
$\frac{{r}_{1} \cdot {r}_{2} \left(\cos \alpha \cos \beta + \sin \alpha \sin \beta\right) + i \left(\sin \alpha \cos \beta - \cos \alpha \sin \beta\right)}{{r}_{2}^{2} \left({\cos}^{2} \beta + {\sin}^{2} \beta\right)}$ or
(r_1/r_2)*(cos(alpha-beta)+isin(alpha-beta) or
${z}_{1} / {z}_{2}$ is given by $\left({r}_{1} / {r}_{2} , \left(\alpha - \beta\right)\right)$,
Now as $| 2 - 2 i | = \sqrt{{2}^{2} + {\left(- 2\right)}^{2}} = \sqrt{8} = 2 \sqrt{2}$ and 2-2i=2sqrt2(1/sqrt2-i/sqrt2)=(cos((7pi)/4)+isin((7pi)/4)
and $| - 1 - i | = \sqrt{{\left(- 1\right)}^{2} + {\left(- 1\right)}^{2}} = \sqrt{2}$ and $- 1 - i = \sqrt{2} \left(- \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\right) = \left(\cos \left(\frac{5 \pi}{4}\right) + i \sin \left(\frac{5 \pi}{4}\right)\right)$
Hence, $\frac{2 - 2 i}{- 1 - i} = \left(\frac{2 \sqrt{2}}{\sqrt{2}}\right) \cdot \left(\cos \left(\frac{7 \pi}{4} - \frac{5 \pi}{4}\right) + i \sin \left(\frac{7 \pi}{4} - \frac{5 \pi}{4}\right)\right)$
= $2 \left(\cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)\right)$
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2020-02-19 04:58:21
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https://www.physicsforums.com/threads/wave-particle-duality-help.918909/
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# B Wave-particle Duality Help
1. Jun 29, 2017
### Jman091
After reading on the basics of this & watching videos, here are some questions I can't help asking.
When the wave is travelling in the room that is filled with air, will the photon interact with all the air
molecules continuously collapsing the wave aspect?
As I understand measurement/interaction collapses the wave aspect of the photon. So wouldn't the actual wall itself with the slits do this preventing the wave passing through both slits?
Looking at the videos of the double slit they show some kind of wave that is spread out across the whole wall with the slits & yet interfere with itself on other side much like a water wave in a pond. In the case of the water wave the whole wave looks to literally interact with the wall & slits just as a beach wave strikes a long stretch of barrier. But with the photon, the wave behavior is on a scale comparable to the two slits.
Even so, the middle of the slits is solid matter & looks to cut the wave, and surely the edges/boundary of the wave & the slits themselves are not precise but peter out. If so I have a hard time seeing how the wave picture alone can account for interference. Shouldn't the edges of the wave interact with the edges of the slit?
How does a buckyball of 60atoms act like a wave? As wave collapse happens by interaction, like when locating an electron with a photon. Yet aren't such interactions continuously happening with & within the molecule itself?
thanks
2. Jun 29, 2017
### hilbert2
Not every interaction causes collapse of a quantum state, and even when it does, there can be many ways how the collapse can happen depending on what observable you're measuring. If you assume that collapse happens in every interaction, you'll get absurd results, i.e. if a nitrogen molecule in air were to go into a position eigenstate (state with definite position) after every collision with other molecules, you'd quicly end up with $N_2$ molecules that would have very large kinetic energies (large enough to break energy conservation) due to Heisenberg uncertainty principle.
3. Jun 29, 2017
### DennisN
It can display interference/diffraction patterns in experiments, see e.g. this page: Diffraction and Interference with Fullerenes.
4. Jun 29, 2017
### Grinkle
Conceptually, what kinds of interactions do cause collapse vs don't?
Is the discussion now interpretation dependent? Collapse itself is an interpretation of an experiment as opposed to a direct experimental result, is that right?
5. Jun 29, 2017
### Staff: Mentor
Any discussion involving collapse is necessarily interpretation-dependent, because collapse itself is an interpretation.
Not quite. The opposite of "interpretation" is "part of the mathematical formalism", not "seen in experiments".
Last edited: Jun 30, 2017
6. Jun 30, 2017
### yossell
"As I understand measurement/interaction collapses the wave aspect of the photon."
If by collapse' you mean that measurement projects the wavefunction onto an eigenstate, then we have entered the realm of interpretation. One can use quantum mechanics to make predictions without having to suppose that such collapse of the wave function ever happens.
But what is true is that, the more the photon interacts with its environment, the interference effects will become smaller and smaller, very quickly becoming negligible. As interference effects become smaller, the less are the "wave-like" the system becomes. This process can be explained and described without the need for a special class of collapsing interactions called measurement' that do not obey the Schrodinger equation. Decoherence explains and tells us what it is about the environment that so quickly supresses interference effects.
These days, some working physicists use `collapse' as a name for this process rather than as the novel and distinctive dynamical process that von Neumann introduced. Empirically, decoherence and von Neumann collapse are, for all practical purposes, equivalent. Is decoherence enough to solve the measurement problem? That's unclear -- but it's enough to get the right physical results without the need to posit interactions that are not described within quantum mechanics -- so von Neumann collapse becomes merely an interpretive issue.
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2018-03-20 12:39:35
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https://zbmath.org/?q=an:0901.00063
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## The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday.(English)Zbl 0901.00063
Geometry and Topology Monographs. 1. Warwick: University of Warwick, Institute of Mathematics, 556 p. (1998).
The articles of this volume will be reviewed individually.
Indexed articles:
Almgren, Frederic J. jun.; Rivin, Igor, The mean curvature integral is invariant under bending, 1-21 [Zbl 0914.53007]
Anderson, James W., A brief survey of the deformation theory of Kleinian groups, 23-49 [Zbl 0904.30024]
Bowditch, B. H., Boundaries of strongly accessible hyperbolic groups, 51-97 [Zbl 0918.20027]
Bridson, Martin R., Controlled embeddings into groups that have no non-trivial finite quotients, 99-116 [Zbl 0922.20045]
Button, Jack, All Fuchsian Schottky groups are classical Schottky groups, 117-125 [Zbl 0909.20031]
Cooper, D.; Long, D. D., On the Burau representation modulo a small prime, 127-138 [Zbl 0923.20030]
Dunwoody, M. J., Folding sequences, 139-158 [Zbl 0927.20013]
Fenn, Roger; Rourke, Colin, Characterisation of a class of equations with solutions over torsion-free groups, 159-166 [Zbl 0913.20018]
Griffiths, David, At most 27 length inequalities define Maskit’s fundamental domain for the modular group in genus 2, 169-182 [Zbl 0901.57021]
Haglund, Frédéric; Paulin, Frédéric, Simplicity of automorphism groups of spaces with negative curvature, 181-248 [Zbl 0916.51019]
Holt, Derek F., Automatic groups, subgroups and cosets, 249-260 [Zbl 0917.20031]
Howie, James, Minimal Seifert manifolds for higher ribbon knots, 261-293 [Zbl 0901.57028]
Izquierdo, M.; Singerman, D., On the fixed-point set of automorphisms of non-orientable surfaces without boundary, 295-301 [Zbl 0913.20019]
Komori, Yohei; Series, Caroline, The Riley slice revisited, 303-316 [Zbl 0904.30025]
Kourouniotis, Christos, On the continuity of bending, 317-334 [Zbl 0904.30026]
Marden, Albert, Complex projective structures on Kleinian groups, 335-340 [Zbl 0904.30027]
Mitra, Mahan, Coarse extrinsic geometry: A survey, 341-364 [Zbl 0914.20034]
Morton, H. R.; Ryder, H. J., Mutants and $$SU(3)_q$$ invariants, 365-381 [Zbl 0901.57002]
Neumann, Walter D., Hilbert’s 3rd problem and invariants of 3-manifolds, 383-411 [Zbl 0902.57013]
Niblo, Graham A.; Wise, Daniel T., The engulfing property for 3-manifolds, 413-418 [Zbl 0910.57007]
Ohshika, Ken’ichi, Divergent sequences of Kleinian groups, 419-450 [Zbl 0901.57020]
Parker, John R.; Parkkonen, Jouni, Coordinates for quasi-Fuchsian punctured torus spaces, 451-478 [Zbl 0913.32005]
Potyagailo, Leonid, The boundary of the deformation space of the fundamental group of some hyperbolic 3-manifolds fibering over the circle, 479-492 [Zbl 0911.57012]
Rees, Sarah, Hairdressing in groups: A survey of combings and formal languages, 493-509 [Zbl 0922.20040]
Thurston, William P., Shapes of polyhedra and triangulations of the sphere, 511-549 [Zbl 0931.57010]
Zeghib, Abdelghani, Homogeneous spacetimes, 551-576 [Zbl 0920.53027]
### MSC:
00B30 Festschriften 00B15 Collections of articles of miscellaneous specific interest 53-06 Proceedings, conferences, collections, etc. pertaining to differential geometry 58-06 Proceedings, conferences, collections, etc. pertaining to global analysis
### Keywords:
Epstein Birthday Schrift; Dedication
Epstein, David
Full Text:
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2022-12-07 01:13:54
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https://math.stackexchange.com/questions/2397097/roulette-wheel-probability-question
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# Roulette wheel probability question [closed]
A roulette wheel has $21$ red numbers, $21$ black numbers, and $4$ zeros. A player places $20$ dollar bets on the red numbers. If the roulette spins a red number, the player get his $20$ dollars back, and wins another $20$ dollars. If the roulette does not spin a red number, the player loses his bet. The player sits down with $40$ dollars, and keeps on playing until he has no more money. How many spins can he expect to be able to play until he has no more money?
## closed as off-topic by Namaste, JMoravitz, José Carlos Santos, Davide Giraudo, Siong Thye GohAug 17 '17 at 17:20
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Namaste, JMoravitz, José Carlos Santos, Davide Giraudo, Siong Thye Goh
If this question can be reworded to fit the rules in the help center, please edit the question.
He will win \$$20 with probability 21/(21+21+4) and win -\$$20$(that is, lose \$$20) with proability (21+4)/(21+21+4). Therefore his expected winnings per round is:$$ E:=20\left(\frac{21}{21+21+4}\right) - 20\left(\frac{21+4}{21+21+4}\right) = -\frac{80}{46} = -\frac{40}{23}. $$Given that he starts with \$$40$, this means that he can expect to play $40/(-E)=23$ rounds.
• I think you'll find that if you don't round in the first step that you'll get an answer of exactly $23$. – JMoravitz Aug 17 '17 at 17:31
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2019-07-16 08:51:13
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http://www.sklogwiki.org/SklogWiki/index.php/Lennard-Jones_model
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# Lennard-Jones model
The Lennard-Jones intermolecular pair potential is a special case of the Mie potential and takes its name from Sir John Edward Lennard-Jones [1] [2]. The Lennard-Jones model consists of two 'parts'; a steep repulsive term, and smoother attractive term, representing the London dispersion forces [3]. Apart from being an important model in itself, the Lennard-Jones potential frequently forms one of 'building blocks' of many force fields. It is worth mentioning that the 12-6 Lennard-Jones model is not the most faithful representation of the potential energy surface, but rather its use is widespread due to its computational expediency. For example, the repulsive term is maybe better described with the exp-6 potential. One of the first computer simulations using the Lennard-Jones model was undertaken by Wood and Parker in 1957 [4] in a study of liquid argon.
## Functional form
The Lennard-Jones potential is given by
$\Phi_{12}(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right]$
or is sometimes expressed as
$\Phi_{12}(r) = \frac{A}{r^{12}}- \frac{B}{r^6}$
where
• $$r := |\mathbf{r}_1 - \mathbf{r}_2|$$
• $$\Phi_{12}(r)$$ is the intermolecular pair potential between two particles or sites
• $$\sigma$$ is the value of $$r$$ at which $$\Phi_{12}(r)=0$$
• $$\epsilon$$ is the well depth (energy)
• $$A= 4\epsilon \sigma^{12}$$, $$B= 4\epsilon \sigma^{6}$$
• Minimum value of $$\Phi_{12}(r)$$ at $$r = r_{min}$$;
$\frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ...$ In reduced units:
• Density$\rho^* := \rho \sigma^3$
where $$\rho := N/V$$ (number of particles $$N$$ divided by the volume $$V$$)
• Temperature$T^* := k_B T/\epsilon$
where $$T$$ is the absolute temperature and $$k_B$$ is the Boltzmann constant
The following is a plot of the Lennard-Jones model for the Rowley, Nicholson and Parsonage parameter set [5] ($$\epsilon/k_B =$$ 119.8 K and $$\sigma=$$ 0.3405 nm). See argon for other parameter sets.
## Critical point
The location of the critical point is [6] $T_c^* = 1.326 \pm 0.002$ at a reduced density of $\rho_c^* = 0.316 \pm 0.002$ The critical compressibility factor is given by [7]
$Z_c = \frac{p_cv_c}{RT_c} = 0.281$
Vliegenthart and Lekkerkerker [8] [9] have suggested that the critical point is related to the second virial coefficient via the expression
$B_2 \vert_{T=T_c}= -\pi \sigma^3$
## Triple point
The location of the triple point as found by Mastny and de Pablo [10] is $T_{tp}^* = 0.694$
$\rho_{tp}^* = 0.84$ (liquid);
$\rho_{tp}^* = 0.96$ (solid).
The following plot is of a typical radial distribution function for the monatomic Lennard-Jones liquid[11] (here with $$\sigma=3.73$$Å and $$\epsilon=0.294$$ kcal/mol at a temperature of 111.06K):
## Helmholtz energy function
An expression for the Helmholtz energy function of the face centred cubic solid has been given by van der Hoef [12], applicable within the density range $$0.94 \le \rho^* \le 1.20$$ and the temperature range $$0.1 \le T^* \le 2.0$$. For the liquid state see the work of Johnson, Zollweg and Gubbins [13].
## Equation of state
Main article: Lennard-Jones equation of state
## Virial coefficients
Main article: Lennard-Jones model: virial coefficients
## Phase diagram
Main article: Phase diagram of the Lennard-Jones model
## Zeno line
It has been shown that the Lennard-Jones model has a straight Zeno line [14] on the density-temperature plane.
## Widom line
It has been shown that the Lennard-Jones model has a Widom line [15] on the pressure-temperature plane.
## Perturbation theory
The Lennard-Jones model is also used in perturbation theories, for example see: Weeks-Chandler-Andersen perturbation theory.
## Approximations in simulation: truncation and shifting
The Lennard-Jones model is often used with a cutoff radius of $$2.5 \sigma$$, beyond which $$\Phi_{12}(r)$$ is set to zero. Setting the well depth $$\epsilon$$ to be 1 in the potential on arrives at $$\Phi_{12}(r)\simeq -0.0163$$, i.e. at this distance the potential is at less than 2% of the well depth. For an analysis of the effect of this cutoff on the melting line see the work of Mastny and de Pablo [10] and of Ahmed and Sadus [16]. See Panagiotopoulos for critical parameters [17]. It has recently been suggested that a truncated and shifted force cutoff of $$1.5 \sigma$$ can be used under certain conditions [18]. In order to avoid any discontinuity, a piecewise continuous version, known as the modified Lennard-Jones model, was developed.
## Cutoff Lennard-Jones potential
The cutoff Lennard-Jones potential is given by (Eq. 2 in [19]):
$\Phi_{12}(r) = 4 \epsilon \left\{ \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right]+ \left[ 6\left(\frac{\sigma}{r_c} \right)^{12}- 3\left( \frac{\sigma}{r_c}\right)^6 \right] \left(\frac{r}{r_c} \right)^2 -7 \left(\frac{\sigma}{r_c} \right)^{12} + 4 \left(\frac{\sigma}{r_c} \right)^{6} \right\}$
where $$r_c$$ is the cutoff radius.
## n-m Lennard-Jones potential
It is relatively common to encounter potential functions given by: $\Phi_{12}(r) = c_{n,m} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^n - \left( \frac{\sigma}{r} \right)^m \right].$ with $$n$$ and $$m$$ being positive integers and $$n > m$$. $$c_{n,m}$$ is chosen such that the minimum value of $$\Phi_{12}(r)$$ being $$\Phi_{min} = - \epsilon$$. Such forms are usually referred to as n-m Lennard-Jones Potential. For example, the 9-3 Lennard-Jones interaction potential is often used to model the interaction between a continuous solid wall and the atoms/molecules of a liquid. On the '9-3 Lennard-Jones potential' page a justification of this use is presented. Another example is the n-6 Lennard-Jones potential, where $$m$$ is fixed at 6, and $$n$$ is free to adopt a range of integer values. The potentials form part of the larger class of potentials known as the Mie potential.
Examples:
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2014-10-20 08:04:39
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https://simple.wikipedia.org/wiki/Derivative_(mathematics)
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Derivative (mathematics)
A function (black) and a tangent (red). The derivative at the point is the slope of the tangent.
In mathematics, the derivative is a way to show rate of change: that is, the amount by which a function is changing at one given point. For functions that act on the real numbers, it is the slope of the tangent line at a point on a graph. The derivative is often written using "dy over dx" (meaning the difference in y divided by the difference in x). The d is not a variable, and therefore cannot be cancelled out.
The derivative can be expressed as: ${\displaystyle {\frac {dy}{dx}}}$
Definition of a derivative
An animation, that gives an intuitive idea of the derivative, as the "swing" of a function change when the argument changes.
The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between ${\displaystyle x_{0}}$ and ${\displaystyle x_{1}}$ becomes infinitely small (infinitesimal). In mathematical terms,
${\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}$
That is, as the distance between the two x points (h) becomes closer to zero, the slope of the line between them comes closer to resembling a tangent line.
Derivatives of functions
Linear functions
Derivatives of linear functions (functions of the form a x + b with no quadratic or higher terms) are constant. That is, the derivative in one spot on the graph will remain the same on another.
When the dependent variable y directly takes x's value (y=x), the slope of the line is 1 in all places, so ${\displaystyle {\frac {d}{dx}}(x)=1}$ regardless of where the position is.
When y modifies x's number by adding or subtracting a constant value, the slope is still one because the change in x and y do not change if the graph is shifted up or down. That is, the slope is still 1 throughout the entire graph and its derivative is also 1.
Power functions
Power functions (e.g. ${\displaystyle x^{a}}$) behave differently than linear functions because their slope varies (because they have an exponent).
Power functions, in general, follow the rule that ${\displaystyle {\frac {d}{dx}}x^{a}=ax^{a-1}}$. That is, if we give a the number 6, then ${\displaystyle {\frac {d}{dx}}x^{6}=6x^{5}}$
Another possibly not so obvious example is the function ${\displaystyle f(x)={\frac {1}{x}}}$. This is essentially the same because 1/x can be simplified to use exponents:
${\displaystyle f(x)={\frac {1}{x}}=x^{-1}}$
${\displaystyle f'(x)=-1(x^{-2})}$
${\displaystyle f'(x)=-{\frac {1}{x^{2}}}}$
In addition, roots can be changed to use fractional exponents where their derivative can be found:
${\displaystyle f(x)={\sqrt[{3}]{x^{2}}}=x^{\frac {2}{3}}}$
${\displaystyle f'(x)={\frac {2}{3}}(x^{-{\frac {1}{3}}})}$
Exponential functions
An exponential is of the form ${\displaystyle ab^{f\left(x\right)}}$ where ${\displaystyle a}$ and ${\displaystyle b}$ are constants and ${\displaystyle f(x)}$ is a function of ${\displaystyle x}$. The difference between an exponential and a polynomial is that in a polynomial ${\displaystyle x}$ is raised to some power whereas in an exponential ${\displaystyle x}$ is in the power.
Example 1
${\displaystyle {\frac {d}{dx}}\left(ab^{f\left(x\right)}\right)=ab^{f(x)}\cdot f'\left(x\right)\cdot \ln(b)}$
Example 2
Find ${\displaystyle {\frac {d}{dx}}\left(3\cdot 2^{3{x^{2}}}\right)}$.
${\displaystyle a=3}$
${\displaystyle b=2}$
${\displaystyle f\left(x\right)=3x^{2}}$
${\displaystyle f'\left(x\right)=6x}$
Therefore,
${\displaystyle {\frac {d}{dx}}\left(3\cdot 2^{3x^{2}}\right)=3\cdot 2^{3x^{2}}\cdot 6x\cdot \ln \left(2\right)=\ln \left(2\right)\cdot 18x\cdot 2^{3x^{2}}}$
Logarithmic functions
The derivative of logarithms is the reciprocal:
${\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}}}$.
Take, for example, ${\displaystyle {\frac {d}{dx}}\ln \left({\frac {5}{x}}\right)}$. This can be reduced to (by the properties of logarithms):
${\displaystyle {\frac {d}{dx}}(\ln(5))-{\frac {d}{dx}}(\ln(x))}$
The logarithm of 5 is a constant, so its derivative is 0. The derivative of ln(x) is ${\displaystyle {\frac {1}{x}}}$. So,
${\displaystyle 0-{\frac {d}{dx}}\ln(x)=-{\frac {1}{x}}}$
For derivatives of logarithms not in base e like ${\displaystyle {\frac {d}{dx}}(\log _{10}(x))}$, this can be reduced to: ${\displaystyle {\frac {d}{dx}}\log _{10}(x)={\frac {d}{dx}}{\frac {\ln {x}}{\ln {10}}}={\frac {1}{\ln {10}}}{\frac {d}{dx}}\ln {x}={\frac {1}{x\ln(10)}}}$
Trigonometric functions
The cosine function is the derivative of the sine function, while the derivative of cosine is negative sine (provided that x is measured in radians):
${\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)}$
${\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)}$
${\displaystyle {\frac {d}{dx}}\sec(x)=\sec(x)\tan(x)}$.
Properties of derivatives
Derivatives can be broken up into smaller parts where they are manageable (as they have only one of the above function characteristics), for example:
${\displaystyle {\frac {d}{dx}}(3x^{6}+x^{2}-6)}$ can be broken up as such:
${\displaystyle {\frac {d}{dx}}(3x^{6})+{\frac {d}{dx}}(x^{2})-{\frac {d}{dx}}(6)}$
${\displaystyle =6\cdot 3x^{5}+2x-0}$
${\displaystyle =18x^{5}+2x\,}$
Uses of derivatives
A function's derivative can be used to search for the maxima and minima of the function by searching for places where its slope is zero.
Derivatives are used in Newton's method which helps find zeros (roots) of a function..
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2019-08-24 19:55:24
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https://www.trustudies.com/question/1609/an-electric-motor-takes-5-a-from-a-22/
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3 Tutor System
Starting just at 265/hour
# An electric motor takes 5 A from a 220 V line. Determine the power of the motor and the energy consumed in 2 h.
Given, Electric current (I) = 5A,
Potential difference (V) = 220V,
Time (t) = 2h = 2 x 60 x 60 s = 7200 s
We know,Power (P) = V I = 220 x 5
P= 1100 W
Energy consumed by the electric appliance = P × t=1100 W × 7200 s
Energy consumed= 7.92 × $$10^6$$ J
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2023-03-24 06:59:30
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http://bugcounting.net/blog/?cat=16
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Breaking news! A recent study found that Barack Obama is, with high probability, not an American citizen! The study — destined to revive the controversy that emerged during the President’s first presidential campaign — is based on new evidence and a simple analysis using widely accepted statistical inference tools. I’ll leave it to the political pundits to analyze the grave effects that this shocking finding surely will have on the upcoming presidential campaign. This post focuses on the elegant technical machinery used to reach the unsettling conclusion.
The crux of the analysis applies, in a statistical setting, modus tollens, a basic inference rule of logic. Given two facts $X$ and $Y$ such that if $X$ is true then $Y$ is true, modus tollens derives the falsehood of $X$ from the falsehood of $Y$. In formal notation:
$\begin{matrix} X \Longrightarrow Y, \quad \neg Y \\ \hline \neg X \end{matrix}$
For example, take $X$ to be “It rains” and $Y$ to be “I have an umbrella with me”. From the fact that I am carrying no umbrella, by applying modus tollens, you can conclude that it’s not raining.
The next step introduces a simple generalization of modus tollens to the case where facts are true with some probability: if $X$ is true then $Y$ is true with high probability. Then, when $Y$ happens to be false, we conclude that $X$ is unlikely to be true. If I have an umbrella with me 99% of the times when it rains, there’s only a 1% chance that it rains if I have no umbrella with me.
All this is plain and simple, but it has surprising consequence when applied to the presidential case. A randomly sampled American citizen is quite unlikely to be the President; the odds are just 1 in 321-something millions. So we have that if “a person $p$ is American” (or $X$) is true then “$p$ is not the President” (or $Y$) is true with high probability. But Mr. Barack Obama happens to be the President, so he’s overwhelmingly unlikely to be American according to probabilistic modus tollens!
(The ironic part of the post ends here.)
Sure you’re thinking that this was a poor attempt at a joke. I would agree, were it not the case that the very same unsound inference rule is being applied willy-nilly in countless scientific papers in the form of statistical hypothesis testing. The basic statistical machinery, which I’ve discussed in a previous post, tells us that, under a null hypothesis $H_0$, a certain data $D$ is unlikely to happen. In other words: if “the null hypothesis $H_0$” is true then “the data is different than $D$” is true with high probability. So far so good. But then the way this fact is used in practice is the following: if we observe the unlikely $D$ in our experiments, we conclude that the null hypothesis is unlikely, and hence we reject it — unsoundly! How’s that for a joke?
Having seen for ourselves that modus tollens does not generalize to probabilistic inference, what is a correct inference from data to hypothesis testing? We can use Bayes’s theorem and phrase it in terms of conditional probabilities. $P(X \mid Y)$ is the probability that $X$ occurs given that $Y$ has occurred. Then $P(H_0 \mid D)$ — the probability that the null hypothesis $H_0$ is true given that we observed data $D$ — is computed as $P(D \mid H_0) \cdot P(H_0) / P(D)$. Even if we know that $D$ is unlikely under the null hypothesis — $P(D \mid H_0)$ is small — we cannot dismiss the null hypothesis with confidence unless we know something about the absolute prior probabilities of $H_0$ and $D$. To convince ourselves that Bayes’s rule leads to sound inference, we can apply it to the Barack Obama case: $H_0$ is “a person $p$ is American” and $D$ is “$p$ is the President”. We plug the numbers in and do the simple math to see that $P(H_0 \mid D)$, the probability that the President is American, is indeed one:
$(1 / A) \cdot (A / W) / (1 / W) = 1$, where $A$ is the population of the USA and $W$ is the world population. Bayes 1 – birthers 0.
Now you understand the fuss about statistical hypothesis testing that has emerged in numerous experimental sciences. Sadly, this blunder is not merely a possibility; it is quite likely that it has affected the validity of numerous published experimental “findings”. In fact, the inadequacy of statistical hypothesis testing is compounded by other statistical results such as the arbitrariness of a hard-and-fast confidence threshold, the false hypothesis paradox (when studying a rare phenomenon, that is a phenomenon with low base rates, most positive results are false positives), and self-selection (the few research efforts that detect some rare phenomenon will publish, whereas the overwhelming majority of “no effect” studies will not lead to publication). In an era of big data, these behaviors are only becoming more likely to emerge.
The take home message is simple yet important. Statistical hypothesis testing is insufficient, by itself, to derive sound conclusions about empirical observations. It must be complemented by other analysis techniques, such as data visualization, effect sizes, confidence intervals, and Bayesian analysis. Unless you remain convinced that Obama’s not American, Elvis is alive, and the Apollo moon landings were staged. In this case, this blog is not for you — with high probability.
An empirical study comparing LaTeX to Word for preparing scientific documents appeared on PLOS ONE in mid December, just in time to ignite an intense online discussion that took place over the holiday season. Among the many comments that I read, one tweet by Michele Lanza made me realize that the study was more general than I originally thought:
The Word vs. LaTeX efficiency argument is in line with microwave food vs. actual cooking
What a terrific suggestion for related research! Back from the winter break, I decided to adapt the design of the LaTeX vs. Word study in order to pit microwave food against proper cooking. The results are, I believe, quite conclusive; and my recommendations based on them compelling. While I submitted a detailed write-up for publication, I want to give you a summary of the essential findings in the rest of this post. In my empirical research, I stuck to the original study’s design as closely as possible since it seemed as much apropos for comparing cooking methods as it was for comparing document preparation systems. In the write-up that follows you will recognize passages that mirror closely the structure and even the words of the original LaTeX vs. Word paper, since those words speak for themselves — and imitation is the sincerest form of flattery.
### The study
This empirical study compares microwave usage to full-fledged cooking for food preparation. The experimental methodology was straightforward. We gathered 40 volunteers who use to cook their dinners — some using a microwave to defrost preprocessed food, some cooking from raw ingredients following a recipe. All participants used their own kitchen to run the experiments.
We set up three different sample meals: (1) a TV dinner with two food compartments; (2) a different brand of TV dinner with four food compartments; and (3) a dish consisting of 100 grams of spaghetti aglio e olio. Each participant had 30 minutes to cook each meal using his or her chosen food preparation technique, process, and equipment. Who chose full-fledged cooking was given access to a repository of raw ingredients; and who chose microwaving was given a supply of common TV dinners of different brands. The performance of each participant was measured for each sample meal by three variables: (1) the number of visual differences (food layout, appearance, color) between the cooked meal and the sample; (2) the number of differences in flavor between the cooked meal and the sample; and (3) the amount of hot edible mass produced within 30 minutes. Each participant also completed a questionnaire where they self-evaluated their performance.
The experimental results are unequivocal. Microwave users outperformed traditional cooks on most measures (p < 0.05[/latex], often even $p < 0.01$), with the only exception of the spaghetti dish. Even the most expert cooks were unable to reproduce, from raw ingredients, TV dinners that look and taste like the sample TV dinners, in contrast to the novice microwave users who managed to effortlessly heat to near perfection large amounts of prepackaged food.
These results suggest that full-fledged cooking should be adopted only to prepare complex dishes mainly consisting of oil and garlic. Experienced cooks may argue that the overall quality and healthiness of properly cooked food provides for a meal experience that is considerably more enjoyable than ingurgitating a tasteless, fat-laden, appalling microwave dinner. Although this argument may be true, the differences with the recently introduced versions of TV dinners may be a tad less spectacularly obvious than they were in the past. Thus, the results show that no reasons exist to use traditional means of cooking, except possibly for dishes that contain complex combinations of olive oil and pasta.
An unexpected result of the study was that, according to the questionnaire's answers, full-fledged cooks are highly satisfied with their experience. Despite incurring reduced usability and productivity, they assessed their work as less tiresome, less frustrating, and more enjoyable than microwave users. From a psychological perspective, the most reasonable explanation is that the cooks are delusional lunatics who are unwilling to reconsider their manifestly incorrect beliefs about their cooking ability in light of their actual poor results in faithfully reproducing low-grade industrial food.
The study's results also have implications in terms of costs of food preparation and consumption by the public. Individuals have a responsibility to act economically and efficiently, especially in cases in which their occupation is publicly funded. No reliable data is available about how many publicly-employed workers cook their own meals, and correspondingly it is unclear the amount of taxpayer's money that is spent worldwide by individuals who stubbornly insist on cooking food from raw ingredients over sticking to a more efficient and modern meal preparation system, which would free up their time to advance their respective field of occupation.
I therefore suggest that leading public institutions should consider accepting time-squandering food preparation practices by their employees only if this is justified by the prevalence of dishes involving spaghetti and garlic. In all other cases, said institutions should request employees to eat microwave food (or order take out). We believe that this would be a good policy for two reasons. First, the flavor and appearance of food is secondary to its nutritional values. And, second, preventing people from frittering away scarce culinary resources would save time and money to maximize the benefit of work and development for both individual institutions and the public.
P.S.: Some readers suggested two additional aspects in which microwave cooking is superior. First, adjusting the heating power is much easier with a microwave (where pushing a button immediately interrupts the flow of radiation) than with a traditional stove (where the stove's surface may remain hot for minutes even after power is turned off). And, second, crispy food can be properly cooked using the recently introduced hot air circulation system available in several high-end microwave ovens.
This summer, Sebastian Nanz and I have finally figured out what the best programming language is. The answer is…
Of course you immediately understood that the incipit is a joke. When it comes to complex feature-laden artifacts like general-purpose programming languages there is no such thing as the best tool for the job. In the reality of software development, different programmers with different skills, different attitudes, and different mindsets solve different problems following different methods, different practices, and different processes in different environments, using different tools and different programming languages. As a result, each programming language design strives to find trade-offs that are convenient to someone in the motley crowd of programmers.
Still, the trade-offs of different languages should be demonstrable by impacting measurable features of programs written in those languages. In this recent work [Nanz and Furia, 2014], we have tried to contribute empirical evidence to better our understanding of how programming languages can be used in practice. One of the aspects we were interested in investigating was whether one can find empirical evidence to justify some of the folk knowledge about programming languages, which is very often passed on as a series of ipse dixit that should be self-evident — except that sometimes different authorities have dissenting opinions!
Before summarizing the results of our study, here’s something about the methodology. An important decision was to use Rosetta Code as source of raw data in the form of programs. Rather than hosting full projects — a service provided by other sites such as GitHub and Bitbucket — Rosetta Code focuses on well-defined programming tasks that can be implemented by small programs (the average program in our study is around 30 lines of code); this makes implementations of the same task in different programming languages directly comparable. The revision history and submission practices of Rosetta Code also suggest that programs are often revised by multiple programmers, and hence likely have a good quality on average; and the task list includes many relevant problems that are often part of large real-world projects. This setup helped make sound inter-language comparisons based on proper language usage, thus reducing dispersion and bias in the data. Based on a combination of their popularity in TIOBE and Rosetta Code, we selected 8 languages in four categories: C and Go as procedural languages; C# and Java as object-oriented languages; F# and Haskell as functional languages; and Python and Ruby as object-oriented languages. If your favorite language is not there do not despair: let us know in the comments why you think it deserves to be included; we might consider it for future work.
Let’s have a look at the main results (see the paper for all the details). The biggest surprise is that there are no huge surprises: well-trained programmers and software engineers will recognize several well-known adages about the advantages of certain programming language features over others. To make this apparent, I’ve collected excerpts from classics texts on programming languages that somewhat match our empirical findings.
### Conciseness
It is generally understood that practical expressiveness boils down to conciseness:
The main benefit of the use of expressive languages seems to be the ability to abstract from programming patterns with simple statements and to state the purpose of a program in the concisest possible manner.
We have come to believe that the major negative consequence of a lack of expressiveness is the abundance of programming patterns to make up for the missing, non-expressible constructs.
[Felleisen, 1991]
Higher-order features such as list comprehensions, reflection, higher-order functions, and idiomatic support for lists and maps should increase the level of practical expressiveness, and hence conciseness:
Higher-order procedures can serve as powerful abstraction mechanisms, vastly increasing the expressive power of our language. [Pg. 75]
[…] expressive power […] is attained […] by accumulation and filtering [on lists]. [Pg. 81]
Elevate the conceptual level at which we can design our programs [means enhancing] the expressive power of our language. [Pg. 108]
[Abelson and Sussman, 1996]
Such higher-order features are more readily available in functional and scripting languages than imperative languages:
Against Java, we can say that (compared to, say, Python) some parts of it appear over-complex and others deficient.
[Pg. 340 in Raymond, 2003]
We measured conciseness in terms of lines of code, comparing solutions in each language against those in other languages. Our numbers draw a picture that is largely consistent with the above quotations: functional and scripting languages provide significantly more concise code than procedural and object-oriented languages. Their higher-order features increase practical expressiveness, to wit, conciseness. While in principle one can follow a functional style using object-oriented languages, it is idiomatic support that seems to make a tangible difference in practice.
### Performance
Performance is another often controversial programming language feature. We tried to contribute to the understanding of performance in practice by distinguishing between two kinds of tests. Much of the controversy when discussing performance may derive from conflating these two kinds of problems, which represent very different conditions.
The first kind of performance comparison targets raw execution speed on large inputs; for example, sorting million-element arrays or compressing tens of megabytes of data. The outcome of our experiments using Rosetta Code tasks on such problems is what most people would expect: C is indisputably the fastest — if it was a race, it’d lap all other languages. A bit more generally, language features cost raw speed, and more features tend to cost more speed. In fact, the only runner-up (still from a distance) is Go, a language that is richer than C — it offers automatic memory management and strong typing — but deliberately renounces other expressive features, such as inheritance and genericity, that have become commonplace in modern high-level programming languages.
Programs that require maximum speed […] are good candidates for C. [Pg. 326]
Python cannot compete with C or C++ on raw execution speed. [Pg. 337]
[Raymond, 2003]
[The] main problem [of automatic memory management], however, is that “useful” processing time is lost when the garbage collector is invoked.
[Pg. 168 in Ghezzi and Jazayeri, 1997]
Most of the time, however, the extreme differences in raw speed that emerge with algorithmically-intensive programs on large inputs do not matter much because such jobs are absent or extremely infrequent in the overwhelming majority of applications, which hardly ever have to deal with number crunching. How many million-element arrays did your web browser have to sort while you were browsing the news? To understand performance differences more commonly occurring in everyday conditions, we identified a second kind of targets for comparison, consisting of well-defined problems on input of moderate size, such as checksum algorithms and string manipulation tasks. The results are quite different when we consider this second kind of everyday problems. Scripting and functional languages tend to emerge as the fastest, even surpassing C. More generally, the absolute differences between languages are smallish, which means that every language is usable, and engineering concerns other than raw speed emerge as more relevant.
Most applications do not actually need better performance than Python offers.
[Pg. 337 in Raymond, 2003]
To sum up, the most significant, and somehow neglected, finding that surfaced from our performance comparisons is this distinction between “raw speed” and “everyday” performance requirements, and the corresponding emphasis on the latter for most workaday programming tasks.
### Failure proneness
Counting faults (or, more appropriately for this blog, bugs) is often used to measure the quality or success of software projects. The errors that are of principal interest in that context are those resulting from program behavior that diverges from specification; for example, a banking application that increases your balance when you withdraw money (although this is a bug most of us could live with 🙂 ). In contrast, our comparison of programming languages looked for errors that are independent of a program’s specification and have to do with what we might characterize as well-formedness, such as typing and compatibility errors. This is an admittedly restricted notion of error, but it lends itself to effective detection and analysis. The classic view on the problem of detecting such errors is clear:
[Checking for the presence of well-formedness errors] can be accomplished in different ways that can be classified in two broad categories: static and dynamic. […] In general, if a check can be performed statically, it is preferable to do so instead of delaying the check to run time […].
[Pg. 137 in Ghezzi and Jazayeri, 1997]
To see which languages follow this prescription and tend to perform more checks statically, we counted what fraction of programs in each language that compile correctly terminate without error (exit status zero). The compiled strongly-typed languages (that is, all compiled languages but C which is weakly typed) clearly emerged as those with the fewest runtime failures triggered in our experiments; their compilers do a fairly good job at catching errors at compile time by type checking and other static analyses. In contrast, the interpreted languages triggered runtime failures more frequently; nearly all checks but syntactic ones are done at runtime, when things can go wrong in many different ways.
Go was the least failure prone of the compiled strongly-typed languages. Given that we analyzed a large number of programs written by different contributors, we are reluctant to explain this difference mainly by attributing better programming skills to the authors of Go programs in our sample. Instead, this results refines the picture about what compiled strongly-typed languages can achieve: Go’s type system is more restricted than that of functional or object-oriented languages, which may help achieve type safety by minimizing dark corners where errors may originate.
Our results on failure proneness cannot set the debate about static vs. dynamic checks. This is another issue where a multitude of technical and organizational concern concur to creating several different local optima.
### Be fruitful and replicate
No single study, no matter how carefully designed and executed, can claim to provide conclusive evidence about complex issues such as the relative merits and practical impact of programming language features. Scientific inductive knowledge grows slowly, one little piece of evidence at a time. Technological factors further complicate the picture, since they may shift the importance of certain features and render some obsolete (for example, the exponential growth of processor speed in past decades has favored the development of feature-rich graphical user interfaces that were previously impractical). Yet, it is encouraging that we have found significant concordance between our empirical results and others’ point of view. A lot of what you learned from the classics about programming language was right — as long as you picked the right classics!
#### References
1. Sebastian Nanz and Carlo A. Furia: A comparative study of programming languages in Rosetta Code. Technical report arXiv.org:1409.0252, September 2014.
2. Matthias Felleisen: On the expressive power of programming languages. Science of Computer Programming, 17(1-3):35-75, 1991.
3. Harold Abelson, Gerald Jay Sussman, and Julie Sussman: Structure and interpretation of computer programs. 2nd edition, MIT Press, 1996.
4. Eric S. Raymond: The art of UNIX programming. Addison-Wesley, 2003. Available online.
5. Carlo Ghezzi and Mehdi Jazayeri: Programming language concepts. 3rd edition, Wiley & Sons, 1997.
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2017-04-29 21:22:49
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https://eccc.weizmann.ac.il/eccc-reports/1997/TR97-041/index.html
|
Under the auspices of the Computational Complexity Foundation (CCF)
REPORTS > DETAIL:
### Paper:
TR97-041 | 18th September 1997 00:00
#### On Approximation Hardness of the Bandwidth Problem
TR97-041
Authors: Marek Karpinski, Juergen Wirtgen
Publication: 19th September 1997 14:56
Keywords:
Abstract:
The bandwidth problem is the problem of enumerating
the vertices of a given graph $G$ such that the maximum
difference between the numbers of
adjacent vertices is minimal. The problem has a long
history and a number of applications
and is known to be $NP$-hard, Papadimitriou 1976.
There is not much known though on approximation hardness
of this problem. In this paper we show, that there are no
efficient polynomial time approximation schemes for the
bandwidth problem under some plausible assumptions.
Furthermore we show that there are no polynomial time
approximation algorithms with an absolute
error guarantee of $n^{1-\epsilon}$ for any
$\epsilon >0$ unless $P=NP$.
ISSN 1433-8092 | Imprint
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2020-09-30 00:02:05
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https://tex.stackexchange.com/questions/402793/scantokens-and-let
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# Scantokens and Let
The package inputenc makes it possible, for example, to write ° to print a degree symbol. Simplifying a bit, it obtains that by making it an active character (catcode 13) and defining the corresponding macro. If its category is changed to other (catcode 12) and, then, changed back with \scantokens, it keeps printing the degree symbol. Since I need to print a string, a character at a time, spaces included, my first idea was of using \let:
\documentclass{article}
\usepackage[ansinew]{inputenc}
\usepackage[T1]{fontenc}
\begin{document}
\setlength\parindent{0pt}
A°@ 1
{\catcode\°12 \xdef\str{A°@ 1}}
\str
\scantokens\expandafter{\str\empty}
\def\aaa{\afterassignment\bbb\let\ccc= }
\def\bbb{%
\ifx\ccc\nil
END
\let\next\relax
\else
[\scantokens\expandafter{\ccc\empty}]\let\next\aaa
\fi
\next
}
\expandafter\aaa\str\nil
\end{document}
whose output is:
Unfortunately, as jfbu pointed out, \scantokens does nothing on the \let character and so it prints [ř] instead of [°]. To make it apparent, consider the following:
\documentclass{article}
\usepackage[ansinew]{inputenc}
\usepackage[T1]{fontenc}
\begin{document}
\setlength\parindent{0pt}
{\catcode\°12 \gdef\ddd{°} \global\let\eee°}
[\ddd] - [\scantokens\expandafter{\ddd\empty}]
[\eee] - [\scantokens\expandafter{\eee\empty}]
\end{document}
whose output is:
Of course, in the first example, one possible solution would be to substitute the \scantokens in square brackets with \ifx\ccc\ddd°\else\ccc\fi, having taken care to add {\catcode\°12 \global\let\ddd°} at the beginning. The problem with this approach is that it is not very 'scalable'.
My real goal here is to parse some (CP-1252 encoded) files, which can contain any characters, not just °. To put this in context and for the sake of simplicity, let us just suppose I am coding a hex viewer in (La)TeX (which, by the way, would not be such a bad idea).
So, for the textual part, I first 'load' the file with something like \edef\fff#1{\pdfunescapehex{\pdffiledump length \pdffilesize{#1}{#1}}}, which gives all character tokens with catcode 12, except the space (that has catcode 10); then I scan such a token list and act as needed, printing it or else.
So, is there a way to scan a token list, including spaces, without using \let? Assigning catcode 12 also to spaces would suffice, can I do it? How? Alternatively, is there a way of changing the catcode after a \let, which does not involve a lot of conditional expressions? If not, what would be the best (compact and/or elegant) way to do it?
• – Cragfelt Nov 23 '17 at 20:18
• Are you taking into account that ° is not a single token? – Manuel Nov 23 '17 at 20:21
• @Manuel it is a single token in ansinew (cp1252 input encoding. This is not utf8. – user4686 Nov 23 '17 at 21:48
• while Manuel's comment isn't strictly accurate note your code here os very fragile, no new documents should be using [ansinew] (which doesn't refer to any standard encoding) if the encoding of the text changes (for example as posted above it is in UTF-8) then the code will fail as ° may be multiple tokens. – David Carlisle Nov 23 '17 at 21:54
• Thank you all for replying so quickly. I am trying to parse some CP-1252 encoded files (which is just an 8-bit extension of ASCII), that is why I am not using UTF-8 here; the code above is just a minimal example. – LeaningTower Nov 25 '17 at 0:53
When you do the \let\ccc= it will "let" \ccc stand for successive tokens, and they all happen to be non-expandable actually in your \str macro contents. So the \expandafter{\ccc does nothing. And then the \scantokens does nothing either as it sees \ccc, it does not see the ° token. By the way, you don't need \xdef for your \str definition, as there is nothing to expand as you have assigned \catcode12 to ° (in cp1252 code page, char code is 176 and this gives the ř in T1 font encoding.)
After update to original question. But it still isn't quite clear to me what is exactly the context. Here is how one can parse char per char, which is easy as everything but spaces is catcode 12. What I would do is to define macros \@namedef{mymeaning<char>} for special things (where <char> is one catcode 12 token, and perhaps use \ifdefined, and the others \scantokens can be used generally. Here I use \meaning and I separated backslash and percent because \scantokens on the char is no good then.
This is 8bit encoding ONLY method. I illustrate with iso-latin-1. Besides there are no \par token due to the \pdffiledump method.
And the file uses unix line endings.
% -*- coding:iso-latin-1; -*-
\documentclass{article}
\usepackage[latin1]{inputenc}
\def\ParsePerWord #1 {\ifx\ignorespaces#1\ignorespaces
<SPACE>\par
\expandafter\ParsePerWord
\else
\if\relax#1\else
\ParsePerChar #1\relax
\expandafter\expandafter\expandafter\ParsePerWord
\fi
\fi}
\def\ParsePerChar #1{\if\relax#1\bigskip % end of "word"
\else
\ifnum#1=10 % unix end of line
<EOL>\par
\else
\ifnum#1=92 % backslash
<BACKSLASH>\par
\else
\ifnum#1=37 % percent
<PERCENT>\par
\else
\scantokens{\meaning #1}\par
\fi
\fi
\fi
\expandafter\ParsePerChar
\fi}
\begin{document}
éà°§ù"£{$\Omega$}\def\x#1{&}
\edef\fff{\pdfunescapehex{\pdffiledump offset 1198 length 100{\jobname.tex}}}
\show\fff
\ttfamily
\makeatletter
\@firstofone{\@firstofone{\expandafter\ParsePerWord\fff} \relax} %
\end{document}
In the above, some spaces (those delimiting a "word") are rendered via a \bigskip and others (spaces at start of lines for example) by <SPACE>\par.
It might be better to do the following
\def\ParsePerWord #1 {\ifx\ignorespaces#1\ignorespaces
<SPACE>\par
\expandafter\ParsePerWord
\else
\if\relax#1%
<END OF DUMP>\par % <<-- CHANGED !
\else
\ParsePerChar #1\relax
<SPACE>\par % <<-- CHANGED !
\expandafter\expandafter\expandafter\ParsePerWord
\fi
\fi}
\def\ParsePerChar #1{\if\relax#1% end of "word" <<-- CHANGED !
\else
.... etc ....
which prints <SPACE>\par for all space characters from input. I also added an <END OF DUMP>\par (and removed the \bigskip).
But there is the problem that the last <SPACE> does not come from original, it is the one we added at end of \fff contents. There are various ways to get rid of it, but I don't know if the parsing loop is supposed to be expandable (so far, it is, it could be used in an \edef, if \par is let to \relax or prefixed by \noexpand to avoid special meanings it can have in a LaTeX document depending on location) so I leave it standing for now.
For example, here is output if \fff is set-up to contain the entire file.
This last <SPACE> does not come from the file contents but from our macro.
Remark: "ParsePerWord" is slightly misleading, because "words" may contain end-of-lines. To be closer to what text editors call words, we should presumably first "ParsePerLine" using not a space but a ^^J delimiter.
• So I gather the problem is using \let to scan my string. Therefore, is there a (possibly elegant) way to make it print a degree symbol in square brackets instead of ř? I know I could have used \gdef in place of \xdef. – LeaningTower Nov 25 '17 at 0:54
• with \let things become complicated, of course you could make a giant nested conditional with 256 \ifx to identify original token. Now it is not clear what is your context. In your mwe, I would simply define \str with the space being assigned catcode 12, then I would grab token by token (in the sense of #1) to build a macro with contents [token1][token2 possibly ascii code 32]....etc..., then I would apply a unique \scantokens at the end to those contents (thus, with \expandafter once). I assume you don't have braces ? – user4686 Nov 25 '17 at 7:53
• I do have braces, but with catcode 12. I have expanded my question, including also some of your remarks, in the hope of making it clearer for everybody and to provide some context. – LeaningTower Nov 27 '17 at 2:28
• do I understand correctly that you want to parse cp1252 encoded non-TeX related files, from inside a TeX file, which may be or not cp1252 encoded itself ? because that point is still not clear to me. In my answers, the TeX file and the parsed file (which I took to be the .tex file itself) are encoded the same, so a \scantokens will most of the time "print" in PDF output the expected output. If the encodings differ, we have to do additional gymnastics. Then also, why use \pdfunescapehex`, it might be simplerwith the hex encoded chars, we can define "cp1252-aware" macros from 2 hex letters. – user4686 Nov 27 '17 at 10:48
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2020-02-26 05:14:57
|
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|
http://villaopbalihuren.nl/shay-bedroom-audfvsi/material-selection-methods-f4dce2
|
Supporting information: explore pedigrees of top-ranked candidates M.F. I ) Fire performance: Building design and material selection must be in accordance with the requirements of Building Code clause C Protection from fire including fire compartment separations, allowing the occupants safe escape from the building and allowing fire service personnel safe access to the building. ( Especially, it is a robust spatial discretization method for simulating multi-phase (solid-fluid-gas) interactions. and strength By convention, a higher performance index denotes a better material. Screening: eliminate materials that cannot do the job 3. Nowadays, a lot of materials and alloys are designed. . / {\displaystyle E/\rho } Design variables are limits imposed on the design, such as how thick the beam can be or how much it can deflect, This page was last edited on 16 August 2020, at 18:25. P ρ ng new material-select. {\displaystyle w} First, three different sets of variables are identified: Next, an equation for the performance index is derived. P How do we decide which Plastics, if any,is best for a particular application?Material Selection is not as difficult as it mightappear but it does require an awareness of … INTRODUCTION The performance of an engineering component is limited by the properties of the material of which it is … Some of the important characteristics of materials are : strength, durability, flexibility, weight, resistance to heat and corrosion, ability to cast, welded or hardened, machinability, electrical conductivity, etc.[3]. Substituting this into the weight equation gives Let’s go with one by one. can be treated similarly. / {\displaystyle L} {\displaystyle A} , making it linear on a log-log graph. {\displaystyle w=(\rho /\sigma )LP} Material selection involves seeking the best match between the property-profiles of the materials and that required by the design. 4 QUANTITATIVE METHODS OF MATERIALS SELECTION can be made.1–11 Several of the quantitative procedures can be adapted to use computers in selection from a data bank of materials.12–15 Experience has shown that it is desirable to adopt the holistic decision-making approach of concurrent engineering in product development in most industries. Feedstock Selection. is the height of the beam. {\displaystyle \rho } In the first situation the beam experiences two forces: the weight of gravity C Function used for comparing an option against the selected value when determining which options should appear as selected. The value for This will give a performance index of 120 for tensile loading and 15 for bending. ρ So, moving to the top of the chart while still touching a region of material is where the highest performance will be. P Part I covers first principles. w MATERIALS SELECTION Faculty of Mechanical Engineering Ali Ourdjini, UTM -2005 • Some of the more common and analytical methods of materials selection are: 1. 5.material selection criteria 1. Factors of Material Selection. In Computational Methods for Sensor Material Selection, experts discuss the design and selection of active sensing surfaces for chemical sensors, particularly sensors used in vapor sensing arrays or electronic noses.. Part I covers first principles. should be considered, whereas for a beam that will be subject to bending, the material index must be removed. R A • Importance of printed physical and electrochemical properties, electrode structure and complexity for EESDs are considered. ( selection of material 1. is the best indicator. E σ Summary: Material Indices • A method is necessary for translating design requirements into a prescription for a material • Modulus-Density charts – Reveal a method of using lines of constant to allow selection of materials for minimum weight and deflection-limited design. {\displaystyle P} {\displaystyle M} R / The second plot shows the same materials attributes in a log-log scale. We can add options to the select by adding elements to the . 0000002547 00000 n Grenoble INP; Écoles d'ingénieurs et de management. They are strength, durability, and cost, which includes both the cost of the material and the cost of manufacturing. �Xoyw���R�Ł�����u�׀��P��\D� -T���҆TDQ�h��������d_�1�b�$E�2�l��юi�u3x��X�K�wE=2F�"{ ��,z��\̑~?d}�-� �6�� P VARGHEESE CIPET, CHENNAI 2. / should be minimized. P = ) ρ . • Sandwich and in-plane 3D printed battery and supercapacitor devices are compared in context. The material selection process can sometimes become complex, usually involving multiple factors like high strength requirements, operating temperature, high corrosion resistance, availability and cost. ρ Use quantitative methods in materials selection. P Skip to content. This means that the higher the intercept, the higher the performance of the material. C INTRODUCTION The performance of an engineering component is limited by the properties of the material of which it is made, and by the shapes to which this material can be formed. (Material Indices). Therefore, the optimal material will perform well under both circumstances. methods of corrosion control. L and for bending materials selection method for the design of automotive brake disc is scare in literature. E Each topical issue, organized by a guest editor who is an expert in the area covered, consists solely of invited quality articles by specialist authors, many of them reviews. ( First, the best bending materials can be found by examining which regions are higher on the graph than the and tension b This study investigates and evaluates critical material selection criteria in a priority framework using the fuzzy Delphi-analytical hierarchy process method to overcome all shortcomings from AHP and Delphi methods that are common in material selection problem. This article discusses the first step of the process – identify the design requirements for the component or joint. ) / is the best indicator, since a plate's bending stiffness scales by its thickness cubed. . Technological advancements and the growing material set in the world have created a large variety of options for industrial designers, but little attention has been given to the tools and methods that support material selection processes. C George E. Dieter (1997). That command will create a new Angular 8 app with the name material-select and pass all questions as 'Y' then the Angular CLI will automatically install the required NPM modules. Benefit-Cost Analysis Since both is the length and •Screening out of materials that fail the design constraints. II. How do we decide which Plastics, if any,is best for a particular application?Material Selection is not as difficult as it mightappear but it does require an awareness of … ρ M / ) Since technical ceramics are the only material which is located higher than the tension line, then the best-performing tension materials are technical ceramics. By convention, however, the performance index is always a quantity which should be maximized. … The method section of an APA format psychology paper provides the methods and procedures used in a research study or experiment.This part of an APA paper is critical because it allows other researchers to see exactly how you conducted your research, allowing for the reproduction of the experiment and assessment of alternative methods that might produce different results. The first plot on the right shows density and Young's modulus, in a linear scale. L . Computational Methods for Sensor Material Selection . ρ {\displaystyle P} M startxref [1] Systematic selection of the best material for a given application begins with properties and costs of candidate materials. Angular material table; Angular material modal; Angular material icon; It is designed to work inside of an element. This study investigates and evaluates critical material selection criteria in a priority framework using the fuzzy Delphi-analytical hierarchy process method to overcome all shortcomings from AHP and Delphi methods that are common in material selection problem. σ / Find reliable sources of material properties. Utilizing an "Ashby chart" is a common method for choosing the appropriate material. R log Mechanical Forces or Loads. C A Category: Tom Brown Inc News. log Lastly, the E Lastly the cross sectional area / Performance index Similarly, again considering both stiffness and lightness, for a rod that will be pulled in tension the specific modulus, or modulus divided by density σ Die cutting is a fast and easy method of shaping materials with relatively low structural integrity, such as paper and fiberboard. / The performance index can then be plotted on the Ashby chart by converting the equation to a log scale. {\displaystyle w=\rho AL} Note that this follows the format of = Livraison en Europe à 1 centime seulement ! / {\displaystyle I} Incorporate computer methods in the selection process. Material selection is a step in the process of designing any physical object. P y For example, life cycle assessment can show that the net present value of reducing the weight of a car by 1 kg averages around$5, so material substitution which reduces the weight of a car can cost up to \$5 per kilogram of weight reduction more than the original material. MATERIALS SELECTION Faculty of Mechanical Engineering Ali Ourdjini, UTM –2005 5.3. for tension in Figure 3 is 0.1. σ The bending performance equation The first article gave an overview of the entire process for selecting a material to use for a component or a joint between components. by choosing a material with the best combination of material variables P Langue : Anglais. log 0000000016 00000 n Issues are devoted to specific... Read more. Share; Rubber, films, and plastics; all of these can be formed through traditional die cutting methods. This is shown in Figure 2. y Computational Methods for Sensor Material Selection, 2010 Integrated Analytical Systems Series. is the moment of inertia. We summarise advances and the role of methods, designs and material selection for energy storage devices by 3D printing. Scientists used and developed some selection methods … . × {\displaystyle y} • Economic and practical considerations (purchasing, constructability, etc.) This is done by taking the log of both sides, and plotting it similar to a line with Historically, research and development worked alone when selecting a new raw material. This paper presents in detail several cavitation erosion testing methods commonly used in the laboratory. Gandhinagar Institute of Technology Subject :- Design of machine Element MECHANICAL ENGINEERING 5th - B : 2 Material selection – Malaviya Bhautik [130120119094] Pavan Narkhede [130120119111] Darshit Panchal [130120119114] : Prof. Amit Patel σ = In this case, some of the foams (blue) and technical ceramics (pink) are higher than the line. Material selection is the foundation of all engineering applications and design. Material selection in engineering design process is very important to ensure that the final product is capable of withstanding the ultimate test which is user satisfaction in the long run. Massachusetts Institute of Technology Cambridge, Massachusetts Materials Systems Laboratory ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection I – Slide 4 What parameters define material selection? Failure Analysis 5. {\displaystyle {\sqrt[{3}]{E}}/\rho } Solar cells based on the methylammonium lead halide, CH3NH3PbX3, attract researchers due to the benefits of their high absorption coefficient and sharp Urbach tail, long diffusion length and carrier lifetime, and high carrier … 1308 0 obj <> endobj , @Input() compareWith: (o1: any, o2: any) => boolean. {\displaystyle \rho ,\sigma } {\displaystyle L} Length L { \displaystyle \rho } and tension P { \displaystyle y=x+b }, making them material selection methods variables material. + b { \displaystyle P_ { CR } } out about the methods section should give enough... Are designed has enormous benefits to any engineering-based business converting the equation to a log scale the experiments \displaystyle }... And bending material choice '' Butterworth Heinemann, Oxford, UK app for intended... Log-Log plot and add all known materials in the selection of materials and the role methods... Desired material properties applied force, constraints and free variables are quantities that can change during loading! Parts used to replace other materials like bronze, stainless steel, aluminum and.. Fix this, Angular material table ; Angular material modal ; Angular material icon it... Selection for applications requiring multiple criteria is more complex foams, metals, etc. to judge the quality., the main goal of material selection methods … material selection is a common problem in material Engineering. Application and/or part design be rearranged to give a = P / σ { \displaystyle a must... Of product design, the fixed value of P C R { \displaystyle y=x+b } making! Repeatable, and cost, which includes both the cost of manufacturing will be subject to both tension bending. •Translate the design objectives, constraints and free variables are removed, only. We summarise advances and the role of methods, designs and material variables the cost of same. 'S modulus, in a log-log plot and add all known materials the! Step is to develop a monetary metric for properties of the subject in laboratory! In material selection is the second article in a Series material selection methods posts on the Ashby chart by the... Properly selected: the need to inspect and evaluate materials for Engineering purpose converting the equation to a scale... Of 120 for tensile loading and 15 for bending is ≈ 0.0316 in Figure 3 the lines... End use Dr.T.O enough information so that they can repeat the experiments them! Index can then be plotted on the log-log graph • Sandwich and in-plane 3D printed battery and devices! A=P/\Sigma } any, o2: any ) = > boolean fact, project management professional examination demands basic of... Ask users to complete tasks that involve making choices such as paper fiberboard. = P / σ { \displaystyle P_ { CR } } b { \displaystyle a } be! Add options to the top of the foams ( blue ) and technical in. Materials in the context of product design, the performance index listed on the Ashby,. The experiments the process – identify the design index listed on the right shows and... Electrochemical properties, electrode structure and complexity for EESDs are considered the possible conditions the product come... Addition the methods section should give readers enough information so that they can repeat material selection methods experiments the of., for material selection process is the key to Engineering any application and/or part design steel aluminum... Best materials for the component or a joint between components we summarise advances and the of... Material selection criteria for SPECIFIC END use Dr.T.O and alloys are designed for the or... Method of shaping materials with relatively low structural integrity, such as boron.! A lot of materials plays a very significant role in their selection better with. Or a joint between components ranking: find the materials that do the job 3 are... The performance index but more cost effective solutions is around the Engineering Composites near CFRP the product come. Density and Young 's modulus, in a linear scale methods focuses on developing... Is located higher than the tension line, then the best-performing tension materials technical! Fix this, Angular material icon ; it is a step in the process of any! Composites near CFRP gas composition determining which options should material selection methods as selected material [! A linear scale posts on the right shows density and Young 's modulus, in a log-log plot add... Loading and 15 for bending has to make decisions or declare preferences such as selecting options, or settings... Value when determining which options should appear as selected chart by converting the equation to a log scale a which. Are fixed, making them design variables and material variables and the -! Alloys are designed to die cutting is a fast and easy method of shaping materials relatively! Making choices such as settings or dialogs plotted on the log-log graph against properties is to build reliable,,. Importance of printed physical and electrochemical properties, electrode structure and complexity for EESDs are.! This example, the performance index is derived the Engineering Composites near.... Design requirements as constraints & objectives 2 }, making it linear on a log-log scale laboratory. This blog ) = > boolean combination your design has enormous benefits to any business., repeatable, and cost, which includes both the cost of the best materials for purpose!
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2021-09-26 05:03:25
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https://studyadda.com/solved-papers/neet/physics/ray-optics/neet-pyq-ray-optics/1144
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# Solved papers for NEET Physics Ray Optics NEET PYQ-Ray Optics
### done NEET PYQ-Ray Optics Total Questions - 62
• question_answer1) A luminous object is placed at a distance of 30 cm from the convex lens of focal length 20 cm. On the other side of the lens, at what distance from the lens, a convex mirror of radius of curvature 10 cm, be placed in order to have an upright image of the object coincident with it? [AIPMT 1998]
A)
12 cm
B)
30 cm
C)
50 cm
D)
60 cm
• question_answer2) Light enters at an angle of incidence in a transparent rod of refractive index n. For what value of the refractive index of the material of the rod the light once entered into it will not leave it through its lateral face whatsoever be the value of angle of incidence? [AIPMT 1998]
A)
$n>\sqrt{2}$
B)
$n=1$
C)
$n=1.1$
D)
$n=1.3$
• question_answer3) A plano-convex lens is made of material of refractive index 1.6. The radius of curvature of the curved surface is 60 cm. The focal length of the lens is: [AIPMT 1999]
A)
50 cm
B)
100 cm
C)
200 cm
D)
400 cm
• question_answer4) The refractive index of the material of the prism is $\sqrt{3},$ then the angle of minimum deviation of the prism is : [AIPMT 1999]
A)
${{30}^{o}}$
B)
${{45}^{o}}$
C)
${{60}^{o}}$
D)
${{75}^{o}}$
• question_answer5) Two sources are at a finite distance apart. They emit sounds of wavelength $\lambda$. An observer situated between them on line joining approaches one source with speed u. Then the number of beats heard/s by observer will be: [AIPMT 2000]
A)
$\frac{2u}{\lambda }$
B)
$\frac{u}{\lambda }$
C)
$\frac{u}{2\lambda }$
D)
$\frac{\lambda }{u}$
• question_answer6) Rainbows are formed by: [AIPMT 2000]
A)
reflection and diffraction
B)
refraction and scattering
C)
dispersion and total internal reflection
D)
interference only
• question_answer7) A man is 6 feet tall. In order to see his entire image, he requires a plane mirror of minimum length equal to:
A)
6 ft
B)
12 ft
C)
2 ft
D)
3 ft
• question_answer8) A plano-convex lens is made of a material of refractive index $\mu =1.5$. The radius of curvature of curved surface of the lens is 20 cm. If its plane surface is silvered, the focal length of the silvered lens will be: [AIPMT 2000]
A)
10 cm
B)
20 cm
C)
40 cm
D)
80 cm
• question_answer9) A transparent cube contains a small air bubble. Its apparent distance is 2 cm when seen through one face and 5 cm when seen through other face. If the refractive index of the material of the cube is 1.5, the real length of the edge of cube must be: [AIPMT 2000]
A)
7 cm
B)
7.5 cm
C)
10.5 cm
D)
$\frac{14}{3}\,cm$
• question_answer10) A wave enters to water from air. In air frequency, wavelength, intensity and velocity are ${{n}_{1}},\,{{\lambda }_{1}},\,{{I}_{1}}$ and ${{v}_{1}}$ respectively. In water the corresponding quantities are ${{n}_{2}},\,{{\lambda }_{2}},\,{{I}_{2}}$ and ${{v}_{2}}$ respectively; then: [AIPMT 2001]
A)
${{I}_{1}}={{I}_{2}}$
B)
${{n}_{1}}={{n}_{2}}$
C)
${{v}_{1}}={{v}_{2}}$
D)
${{\lambda }_{1}}={{\lambda }_{2}}$
• question_answer11) Transmission of light in optical fibre is due to: [AIPMT 2001]
A)
scattering
B)
diffraction
C)
polarisation
D)
multiple total internal reflections
• question_answer12) Refractive index of water is 5/3. A light source is placed in water at a depth of 4 m. Then what must be the minimum radius of disc placed on water surface so that the light of source can be stopped? [AIPMT 2001]
A)
3 m
B)
4 m
C)
5 m
D)
$\infty$
• question_answer13) Diameter of human eye lens is 2 mm. What will be the minimum distance between two points to resolve them, which are situated at a distance of 50 m from eye? The wavelength of light is $5000\text{ }{\AA}$: [AIPMT 2002]
A)
2.32 m
B)
4.28 mm
C)
1.25 cm
D)
12.48 cm
• question_answer14) A body is located on a wall. Its image of equal size is to be obtained on a parallel wall with the help of a convex lens. The lens is placed at a distance d ahead of second wall, then the required focal length will be: [AIPMT 2002]
A)
only $\frac{d}{4}$
B)
only$\frac{d}{2}$
C)
more than $\frac{d}{4}$ but less than $\frac{d}{2}$
D)
less than $\frac{d}{4}$
• question_answer15) For the given incident ray as shown in figure, the condition of total internal reflection of the ray will be satisfied if the refractive index of block will be: [AIPMT 2002]
A)
$\frac{\sqrt{3}+1}{2}$
B)
$\frac{\sqrt{2}+1}{2}$
C)
$\sqrt{\frac{3}{2}}$
D)
$\sqrt{\frac{7}{6}}$
• question_answer16) An equiconvex lens is cut into two halves along (i) XOX' and (ii) YOY' as shown in the figure. Let $f,\,f',\,f''$ be the focal lengths of the complete lens, of each half in case (i), and of each half in case (ii), respectively. Choose the correct statement from the following: [AIPMT 2003]
A)
$f'=f,\,\,f''=f$
B)
$f'=2f,\,\,f''=2f$
C)
$f'=f,\,\,f''=2f$
D)
$f'=2f,\,\,f''=f$
• question_answer17) A convex lens is dipped in a liquid whose refractive index is equal to the refractive index of the lens. Then its focal length will: [AIPMT 2003]
A)
become small, but non-zero
B)
remain unchanged
C)
become zero
D)
become infinite
• question_answer18) A beam of light composed of red and green rays is incident obliquely at a point on the face of a rectangular glass slab. When coming out on the opposite parallel face, the red and green rays emerge from: [AIPMT (S) 2004]
A)
two points propagating in two different on-parallel directions
B)
two points propagating in two different parallel directions
C)
one point propagating in two different directions
D)
one point propagating in the same direction
• question_answer19) The refractive index of the material of a prism is $\sqrt{2}$ and its refracting angle is $30{}^\circ$. One of the refracting surfaces of the prism is made a mirror inwards. A beam of monochromatic light entering the prism from the other face will retrace its path after reflection from the mirrored surface if its angle of incidence on the prism is: [AIPMT (S) 2004]
A)
$45{}^\circ$
B)
$60{}^\circ$
C)
$0{}^\circ$
D)
$30{}^\circ$
• question_answer20) A telescope has an objective lens of 10 cm diameter and is situated at a distance of one kilometre from two objects. The minimum distance between these two objects, which can be resolved by the telescope, when the mean wavelength of light is $5000\text{ }{\AA},$ of the order of: [AIPMT (S) 2004]
A)
0.5 m
B)
5 m
C)
5 mm
D)
5 cm
• question_answer21) The angular resolution of a 10 cm diameter telescope at a wavelength of $5000\text{ }{\AA}$ is of the -order of: [AIPMT (S) 2005]
A)
${{10}^{6}}\,\text{rad}$
B)
${{10}^{-2}}\,\text{rad}$
C)
${{10}^{-4}}\,\text{rad}$
D)
${{10}^{-6}}\,\text{rad}$
• question_answer22) Two vibrating tuning forks produce progressive waves given by ${{y}_{1}}=4\sin 500\,\pi t$ and${{y}_{2}}=2\sin \,506\,\pi t$. Number of beats produced per minute is: [AIPMT (S) 2005]
A)
360
B)
180
C)
3
D)
60
• question_answer23) A microscope is focussed on a mark on a piece of paper and then a slab of glass of thickness 3 cm and refractive index 1.5 is placed over the mark. How should the microscope be moved to get the mark in focus again? [AIPMT (S) 2006]
A)
1 cm upward
B)
4.5 cm downward
C)
1 cm downward
D)
2 cm upward
• question_answer24) A convex lens and a concave lens, each having same focal length of 25 cm, are put in contact to form a combination of lenses. The powers diopters of the combination is: [AIPMT (S) 2006]
A)
25
B)
50
C)
infinite
D)
zero
• question_answer25) A small coin is resting on the bottom of a beaker filled with a liquid. A ray of light from the coin travels upto the surface of the liquid and moves along its surface (see figure). [AIPMT (S) 2007] How fast is the light travelling in the liquid?
A)
$1.8\times {{10}^{8}}m/s$
B)
$2.4\times {{10}^{8}}m/s$
C)
$3.0\times {{10}^{8}}m/s$
D)
$1.2\times {{10}^{8}}m/s$
• question_answer26) The frequency of a light wave in a material is $2\times {{10}^{14}}\,Hz$ and wavelength is $5000\text{ }{\AA}$. The refractive index of material will be: [AIPMT (S) 2007]
A)
1.40
B)
1.50
C)
3.00
D)
1.33
• question_answer27) Two thin lenses of focal lengths ${{f}_{1}}$ and ${{f}_{2}}$ are in contact and coaxial. The power of the combination is [AIPMPT (S) 2008]
A)
$\sqrt{\frac{{{f}_{1}}}{{{f}_{2}}}}$
B)
$\sqrt{\frac{{{f}_{2}}}{{{f}_{1}}}}$
C)
$\frac{{{f}_{1}}+{{f}_{2}}}{{{f}_{1}}{{f}_{2}}}$
D)
$\frac{{{f}_{1}}+{{f}_{2}}}{{{f}_{1}}{{f}_{2}}}$
• question_answer28) A boy is trying to start a fire by focusing sunlight on a piece of paper using an equiconvex lens of focal length 10 cm. The diameter of the sun is $1.39\times {{10}^{9}}m$ and its mean distance from the earth is $1.5\times {{10}^{11}}m$. What is the diameter of the sun's image on the paper? [AIPMPT (S) 2008]
A)
$9.2\times {{10}^{-4}}m$
B)
$6.5\times {{10}^{-4}}m$
C)
$6.5\times {{10}^{-5}}m$
D)
$12.4\times {{10}^{-4}}m$
• question_answer29) A ray of light travelling in a transparent medium of refractive index $\mu$ falls, on a surface separating the medium from air at an angle of incidence of$45{}^\circ$. For which of the following value of p the ray can undergo total internal reflection? [AIPMT (S) 2010]
A)
$\mu =1.33$
B)
$\mu =1.40$
C)
$\mu =1.50$
D)
$\mu =1.25$
• question_answer30) The speed of light in media ${{M}_{1}}$ and ${{M}_{2}}$ is $1.5\times {{10}^{8}}m/s$ and $2.0\times {{10}^{8}}\,m/s$ respectively. A ray of light enters from medium ${{M}_{1}}$ to ${{M}_{2}}$ at an incidence angle i. If the ray suffers total internal reflection, the value of i is [AIPMT (M) 2010]
A)
equal to ${{\sin }^{-1}}\left( \frac{2}{3} \right)$
B)
equal to or less than ${{\sin }^{-1}}\left( \frac{3}{5} \right)$
C)
equal to or greater than ${{\sin }^{-1}}\left( \frac{3}{4} \right)$
D)
less than ${{\sin }^{-1}}\left( \frac{2}{3} \right)$
• question_answer31) A ray of light is incident on a $60{}^\circ$ prism at the minimum deviation position. The angle of refraction at the first face (i.e., incident face) of the prism is [AIPMT (M) 2010]
A)
zero
B)
${{30}^{o}}$
C)
${{45}^{o}}$
D)
${{60}^{o}}$
• question_answer32) A thin prism of angle $15{}^\circ$ made of glass of refractive index ${{\mu }_{1}}=1.5$ is combined with another prism of glass of refractive index ${{\mu }_{2}}=1.75$. The combination of the prism produces dispersion without deviation. The angle of the second prism should be [AIPMT (M) 2011]
A)
$7{}^\circ$
B)
$10{}^\circ$
C)
$12{}^\circ$
D)
$5{}^\circ$
• question_answer33) A conversing beam of rays is incident on a diverging lens. Having passed though the lens the rays intersect at a point 15 cm from the lens on the opposite side. If the lens is removed the point where the rays meets will move 5 cm closer to the lens. The focal length of the lens is [AIPMT (M) 2011]
A)
- 10 cm
B)
20 cm
C)
- 30 cm
D)
5 cm
• question_answer34) A biconvex lens has a radius of curvature of magnitude 20 cm. Which one of the following options describe best the image formed of an object of height 2 cm placed 30 cm from the lens? [AIPMT (S) 2011]
A)
Virtual, upright, height $=0.5\text{ }cm$
B)
Real, inverted, height $=4\text{ }cm$
C)
Real, inverted, height $=1\text{ }cm$
D)
Virtual, upright, height $=1\text{ }cm$
• question_answer35) Which of the following is not due to total internal reflection?
A)
Difference between apparent and real depth of a pond
B)
Mirage on hot summer days
C)
Brilliance of diamond
D)
Working of optical fibre
• question_answer36) For the angle of minimum deviation of a prism to be equal to its refracting angle, the prism must be made of a material whose refractive index [AIPMT (M) 2012]
A)
lies between $\sqrt{2}$ and 1
B)
lies between 2 and $\sqrt{2}$
C)
is less than 1
D)
is greater than 2
• question_answer37) A rod of length 10 cm lies along the principal axis of a concave mirror of focal length 10 cm in such a way that its end closer to the pole is 20 cm away from the mirror. The length of the image is [AIPMT (M) 2012]
A)
10 cm
B)
15 cm
C)
2.5 cm
D)
5 cm
• question_answer38) When a biconvex lens of glass having refractive index 1.47 is dipped in a liquid, it acts as a plane sheet of glass. This implies that the liquid must have refractive index [AIPMT (S) 2012]
A)
equal to that of glass
B)
less than one
C)
greater than that of glass
D)
less than that of glass
• question_answer39) A ray of light is incident at an angle of incidence, i, on one face of a prism of angle A (assumed to be small) and emerges normally from the opposite face. If the refractive index of the prism is $\mu ,$the angle of incidence i, is nearly equal to [AIPMT (S) 2012]
A)
$\mu A$
B)
$\frac{\mu A}{2}$
C)
$A/\mu$
D)
$A/2\mu$
• question_answer40) A concave mirror of focal length ${{f}_{1}}$ is placed at a distance of d from a convex lens of focal length ${{f}_{2}}$. A beam of light coming from infinity and falling on this convex lens concave mirror combination returns to infinity. The distance d must be equal [AIPMT (S) 2012]
A)
${{f}_{1}}+{{f}_{2}}$
B)
$-{{f}_{1}}+{{f}_{2}}$
C)
$2{{f}_{1}}+{{f}_{2}}$
D)
$-2{{f}_{1}}+{{f}_{2}}$
• question_answer41) A plano-convex lens fits exactly into a plano-concave lens. Their plane surfaces are parallel to each other. If lenses are made of different materials of refractive indices ${{\mu }_{1}}$ and ${{\mu }_{2}}$ and - R is the radius of curvature of the curved surface of the lenses, then the focal length of the combination is [NEET 2013]
A)
$\frac{R}{2({{\mu }_{1}}+{{\mu }_{2}})}$
B)
$\frac{R}{2({{\mu }_{1}}-{{\mu }_{2}})}$
C)
$\frac{R}{({{\mu }_{1}}-{{\mu }_{2}})}$
D)
$\frac{2R}{({{\mu }_{2}}-{{\mu }_{1}})}$
• question_answer42) For a normal eye, the cornea of eye provides a converging power of 40D and the least converging power of the eye lens behind the cornea is 20D. Using this information, the distance between the retina and the cornea-eye lens can be estimated to be [NEET 2013]
A)
5 cm
B)
2.5 cm
C)
1.67 cm
D)
1.5 cm
• question_answer43) If the focal length of objective lens is increased, then magnifying power of [NEET 2014]
A)
microscope will increase but that of telescope Decrease
B)
microscope and telescope both will increase
C)
microscope and telescope both will decrease
D)
microscope will decrease but that of telescope will increase
• question_answer44) The angle of a prism is A. One of its refracting surfaces is silvered. Light rays falling at an angle of incidence 2 A on the first surface returns back through the same path after suffering reflection at the silvered surface. The refractive index $\mu ,$ of the prism is [NEET 2014]
A)
$2\sin A$
B)
$2\cos A$
C)
$\frac{1}{2}\cos A$
D)
$\tan A$
• question_answer45) Two identical thin plano-convex glass lenses (refractive index 1.5) each having radius of curvature of 20 cm are placed wait their convex surfaces in contact at the centre. The intervening space is filled with oil of refractive index 1.7 The focal length of the combination is [NEET 2015]
A)
$-20\,cm$
B)
$-25\,cm$
C)
$-50cm$
D)
$50\,cm$
• question_answer46) The refracting angle of a prism is A, and refractive index of the material of the prism is $\cot \,(A/2)$. The angle of minimum deviation is [NEET 2015]
A)
${{180}^{o}}-3A$
B)
${{180}^{o}}-2A$
C)
${{90}^{o}}-A$
D)
${{180}^{o}}+2A$
• question_answer47) In an astronomical telescope in normal adjustment a straight black line of length L is drawn on inside part of objective lens. The eye-piece forms a real image of this line. The length of this image is I. The magnification of the telescope is [NEET (Re) 2015]
A)
$\frac{L}{l}+1$
B)
$\frac{L}{l}-1$
C)
$\frac{L+1}{L-1}$
D)
$\frac{L}{l}$
• question_answer48) A beam of light consisting of red, green and blue colours is incident on a right angled prism. The refractive index of the material of the prism for the above red, green and blue wavelengths are 1.39, 1.44 and 1.47, respectively. [NEET (Re) 2015]
The prism will,
A)
separate the blue colour part from the red and green colours
B)
separate all the three colours from one another
C)
not separate the three colours at all
D)
separate the red colour part from the green and blue colours
• question_answer49) The angle of incidence for a ray of light at a refracting surface of a prism is $\text{4}{{\text{5}}^{\text{o}}}$.The angle of prism is $\text{6}{{\text{0}}^{\text{o}}}$. If the ray suffers minimum deviation through the prism, the angle of minimum deviation and refractive index of the material of the prism respectively, are: [NEET - 2016]
A)
${{45}^{o}},\,\frac{1}{\sqrt{2}}$
B)
${{30}^{\text{o}}}\text{,}\,\sqrt{2}$
C)
${{45}^{\text{o}}}\text{,}\,\sqrt{2}$
D)
${{30}^{o}},\,\frac{1}{\sqrt{2}}$
• question_answer50) A astronomical telescope has objective and eyepiece of focal lengths 40 cm and 4 cm respectively. To view an object 200 cm away from the objective, the lenses must be separated by a distance [NEET - 2016]
A)
37.3 cm
B)
46.0 cm
C)
50.0 cm
D)
54.0 cm
Match the corresponding entries of column-1 with coloumn-2 (Where m is the magnification produced by the mirror) [NEET - 2016]
Column-1 Column-2 $m=2$ 1. Convex mirror $m=-\frac{1}{2}$ 2. Concave mirror $m=+2$ 3. Real image $m=+\frac{1}{2}$ 4. Virtual image
A)
A $\to$ b and c, B $\to$ b and c, C $\to$ b and d, D $\to$ a and d.
B)
A $\to$ a and c, B $\to$ a and d, C $\to$ a and b, D $\to$ c and d
C)
A $\to$ a and d, B $\to$ b and c, C$\to$ b and d, D $\to$ b and c
D)
A $\to$ c and d, B $\to$ b and d, C $\to$ b and c, D $\to$ a and d
• question_answer52) The ratio of resolving powers of an optical microscope for two wavelengths ${{\lambda }_{1}}=4000\,\overset{o}{\mathop{A}}\,$ and ${{\lambda }_{2}}=6000\,\overset{o}{\mathop{A}}\,$ is [NEET-2017]
A)
16 : 81
B)
8 : 27
C)
9 : 4
D)
3 : 2
• question_answer53) A thin prism having refracting angle ${{10}^{o}}$ is made of glass of refractive index 1.42. This prism is combined with another thin prism of glass of refractive index 1.7. This combination produces dispersion without deviation. The refracting angle of second prism should be [NEET-2017]
A)
${{10}^{o}}$
B)
${{4}^{o}}$
C)
${{6}^{o}}$
D)
${{8}^{o}}$
• question_answer54) A beam of light from a source L is incident normally on a plane mirror fixed at a certain distance x from the source. The beam is reflected back as a spot on a scale placed just above the source L. When the mirror is rotated through a small angle $\theta ,$ the spot of the light is found to move through a distance y on the scale. The angle$\theta$ is given by [NEET-2017]
A)
$\frac{x}{y}$
B)
$\frac{y}{2x}$
C)
$\frac{y}{x}$
D)
$\frac{x}{2y}$
• question_answer55) An astronomical refracting telescope will have large angular magnification and high angular resolution, when it has an objective lens of [NEET - 2018]
A)
Large focal length and large diameter
B)
Large focal length and small diameter
C)
Small focal length and large diameter
D)
Small focal length and small diameter
• question_answer56) The refractive index of the material of a prism is $\sqrt{2}$ and the angle of the prism is$\text{30 }\!\!{}^\circ\!\!\text{ }$. One of the two refracting surfaces of the prism is made a mirror inwards, by silver coating. A beam of monochromatic light entering the prism from the other face will retrace its path (after reflection from the silvered surface) if its angle of incidence on the prism is [NEET - 2018]
A)
$\text{30 }\!\!{}^\circ\!\!\text{ }$
B)
$45{}^\circ$
C)
$60{}^\circ$
D)
Zero
• question_answer57) An object is placed at a distance of 40 cm from a concave mirror of focal length 15 cm. If the object is displaced through a distance of 20 cm towards the mirror, the displacement of the image will be [NEET - 2018]
A)
30 cm towards the mirror
B)
36 cm away from the mirror
C)
30 cm away from the mirror
D)
36 cm towards the mirror
• question_answer58) Two similar thin equi-convex lenses, of focal length f each, are kept coaxially in contact with each other such that the focal length of the combination is ${{F}_{1}}$. When the space between the two lenses is filled with glycerin (which has the same refractive index $(\mu =1.5)$ as that of glass) then the equivalent focal length is ${{F}_{2}}$. The ratio ${{F}_{2}}$ : ${{F}_{1}}$will be: [NEET 2019]
A)
2 : 3
B)
3 : 4
C)
2 : 1
D)
1 : 2
• question_answer59) In total internal reflection when the angle of incidence is equal to the critical angle for the pair of media in contact, what will angle of refraction? [NEET 2019]
A)
Equal to angle of incidence
B)
$90{}^\circ$
C)
$180{}^\circ$
D)
$0{}^\circ$
• question_answer60) Pick the wrong answer in the context with rainbow. [NEET 2019]
A)
An observer can see a rainbow when his front is towards the sun
B)
Rainbow is a combined effect of dispersion, refraction and reflection of sunlight
C)
When the light rays undergo two internal reflections in a water drop, a secondary rainbow is formed
D)
The order of colours is reversed in the secondary rainbow
• question_answer61) A ray is incident at an angle of incidence i on one surface of a small angle prism (with angle of prism A) and emerges normally from the opposite surface. If the refractive index of the material of the prism is $\mu$, then the angle of incidence is nearly equal to: [NEET 2020]
A)
$\frac{2A}{\mu }$
B)
$\mu A$
C)
$\frac{\mu A}{2}$
D)
$\frac{A}{2\mu }$
• question_answer62) Assume that light of wavelength 600 nm is coming from a star. The limit of resolution of telescope whose objective has a diameter of 2 m is: [NEET 2020]
A)
$1.83\times {{10}^{7}}\text{rad}$
B)
$7.32\times {{10}^{7}}\text{rad}$
C)
$6.00\times {{10}^{7}}\text{rad}$
D)
$3.66\times {{10}^{7}}\text{rad}$
|
2022-11-30 10:36:57
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https://crypto.stackexchange.com/questions/25724/could-this-alternative-hash-based-mac-construction-be-as-or-even-more-secure-th
|
# Could this alternative hash based MAC construction be as, or even more secure than an HMAC?
1. Begin hashing (Key||Message)
2. Encrypt the hash state (or some part of it, such as the first 128 or 256 bits), with the key.
3. Add the encrypted hash state to the hash and return the result.
This could be roughly described as Hash(Key||Message||Encrypt(Previous hash state))
1. It may be faster than an HMAC in practice, especially for shorter messages (modern hardware accelerated encryption is up to 5x-10x the speed of hashing).
2. Encrypting the hash state in stage (2) is a psuedo-random permutation, rather than just a PRF and so may have stronger security guarantees (this is the job of the more math-oriented people [i.e. you guys/girls] to try to analyze though).
3. A practical advantage of HMAC, though, is that it could work with arbitrary “secrets”, that are not necessarily psuedo-random, but this construction would only work if the “secret” is an actual key associated with some cipher (though the “secret” could be hashed of course, to yield a key) - this may not be a significant limitation in practice though, as in most cases it is used with an actual key.
(Also consider a “weaker” version where stage (1) only performs Hash(Message), [i.e. without the key])
[Non-expert alert: I’m not a cryptographer or even a computer-scientist, so I don’t have the knowledge or capacity for the type of percise mathemtical reasoning that’s needed to analyze this. I guess that’s why I’m asking it here..]
• If you are concerned with speed of a MAC and have hardware-accelerated AES encryption, you definitely want to consider CBC-MAC with the length of the message at start, and right-padding of the message with zeroes; this is demonstrably secure (when using a key dedicated to MAC), and even standardized as ISO/IEC 9797-1:2011 Padding Method 3. As they put it: "The [first] block consists of the binary representation of the length (in bits) of the unpadded [message], left-padded with as few (possibly none) ‘0’ bits as necessary to obtain a [128-]bit block". – fgrieu May 18 '15 at 11:17
• (continued) if for some reason the length of the message is not known at the beginning of the MAC, there's the more elaborate CMAC; or OMAC2. Do not use straight CBC-MAC with a variable-length message. – fgrieu May 18 '15 at 11:24
• Sure, I'm already using CBC-MAC! (and I'm aware that prepending the length or encrypting the last block is also secure for variable length messages). I just asked this out of curiosity.. :) – Anon2000 May 18 '15 at 12:31
## Security
The level of security is likely to depend on the cryptographic primitives - the actual hash function and cipher - used. It is very likely that you can construct a function that is insecure, e.g. where the cipher is used for both the hash function an encryption. So you need to prove that the hash function and the encryption primitive are not influencing the security, even though they are using the same key. It is much easier to use a single PRF or PRP and prove that secure. Using one function also greatly simplifies implementations.
## Implementation issues
An implementation issue is that intermediate hash states are often not defined or clear. SHA-1 and SHA-2 do have "logical" intermediate states after a block is hashed as the hash output is basically (part of the) final state. The final state for these hash functions does not significantly differ from the intermediate state. This is not obvious for other hash functions. This is for instance the case for Keccak - the winner of the SHA-3 contest - as well as most other SHA-3 candidates. Using an intermediate hash state should therefore be avoided.
## Expected properties
I'll iterate over the expected properties in order:
It may be faster than an HMAC in practice, especially for shorter messages (modern hardware accelerated encryption is up to 5x-10x the speed of hashing).
Usually performance is less of an issue for shorter messages. I would argue that it is only faster for shorter messages; the effect on larger messages will be negligible. Even then, it would require the cipher implementation to be cached (and in the case of e.g. Java, optimized) in addition to the hash function.
Encrypting the hash state in stage (2) is a psuedo-random permutation, rather than just a PRF and so may have stronger security guarantees (this is the job of the more math-oriented people [i.e. you guys/girls] to try to analyze though).
A hash function especially within HMAC already provides very good security guarantees. It remains to be seen if using a PRP would have any advantages; I would expect it to be less secure.
A practical advantage of HMAC, though, is that it could work with arbitrary “secrets”, that are not necessarily psuedo-random, but this construction would only work if the “secret” is an actual key associated with some cipher (though the “secret” could be hashed of course, to yield a key) - this may not be a significant limitation in practice though, as in most cases it is used with an actual key.
One of the more annoying aspects of CBC based MAC's is the non-variable key size. This is for instance rather annoying when using the MAC as building block for a key derivation function (KDF). So this is - in my opinion - a significant limitation.
## CPU hash acceleration
Note that there are also CPU level optimizations for hash functions. Well known ones live in the Intel SHA extensions, the VIA padlock security suite, the Sun Ultra-Sparks...
• This should be pretty readable for a non-expert. Don't hesitate to ask if it is not. – Maarten Bodewes May 18 '15 at 12:25
• Thank for your time! It seems like there's a rather large gap between "being able to use cryptographic constructs correctly" or "being able to implement cryptographic functions" to "being able to reason and prove properties about cryptographic constructs". I guess the first or second should be sufficient for myself. Since I once worked a [Javascript] SHA-1 implementation (mostly to improve performance of an existing one) I assumed the type of hash function used would have the intermediate state that's equal or be very close to the output at that point. – Anon2000 May 18 '15 at 12:57
• @Anon2000: in crypto perhaps more than elsewhere, the devil is in the details. If you look closely at a typical SHA-1 implementation, the state has the 160-bit chaining variable so far, the length so far (usually 64-bit, could be in bits or bytes), and the message-block-not-hashed-yet (usually up to 511-bit, which length may or may not be tracked separately). If you want a portable implementation enciphering or hashing that, you need to care of all these details, including endianness and order of the various fields. If you consider only the chaining variable, you must be careful about padding. – fgrieu May 18 '15 at 13:21
• I would expect the hash function 1. to be precisely positioned at the end of a block and 2. for it to process a final block for this to work. And it could probably made to work. But the above shows that there are probably too many snags for it to become a generic scheme. Note that for Keccak you would not need the encryption at all. You could also have a look at GCM mode / GMAC for accelerated authentication. – Maarten Bodewes May 18 '15 at 13:25
• @fgrieu I didn't actually implement SHA-1 from scratch, but worked carefully on existing (correct) implementation. But anyway, the nice thing about writing software (compared say, to writing papers) is that you can implement an algorithm without a 100%, complete understanding of it. And then test it to match an existing implementation. So in this case it's being aggressively tested against OpenSSL by bit-bit comparison of the output for random inputs of various lengths. – Anon2000 May 18 '15 at 13:33
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2020-11-23 16:42:59
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https://en.wikipedia.org/wiki/Alternating_series_test
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# Alternating series test
In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test.
## Formal Statement
### Alternating series test
A series of the form
${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}=a_{0}-a_{1}+a_{2}-a_{3}+\cdots \!}$
where either all an are positive or all an are negative, is called an alternating series.
The alternating series test guarantees that an alternating series converges if the following two conditions are met:
1. ${\displaystyle |a_{n}|}$ decreases monotonically[1], i.e., ${\displaystyle |a_{n+1}|\leq |a_{n}|}$, and
2. ${\displaystyle \lim _{n\to \infty }a_{n}=0}$
### Alternating series estimation theorem
Moreover, let L denote the sum of the series, then the partial sum
${\displaystyle S_{k}=\sum _{n=0}^{k}(-1)^{n}a_{n}\!}$
approximates L with error bounded by the next omitted term:
${\displaystyle \left|S_{k}-L\right\vert \leq \left|S_{k}-S_{k+1}\right\vert =a_{k+1}.\!}$
## Proof
Suppose we are given a series of the form ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n-1}a_{n}\!}$, where ${\displaystyle \lim _{n\rightarrow \infty }a_{n}=0}$ and ${\displaystyle a_{n}\geq a_{n+1}}$ for all natural numbers n. (The case ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}a_{n}\!}$ follows by taking the negative.)[1]
### Proof of the alternating series test
We will prove that both the partial sums ${\displaystyle S_{2m+1}=\sum _{n=1}^{2m+1}(-1)^{n-1}a_{n}}$ with odd number of terms, and ${\displaystyle S_{2m}=\sum _{n=1}^{2m}(-1)^{n-1}a_{n}}$ with even number of terms, converge to the same number L. Thus the usual partial sum ${\displaystyle S_{k}=\sum _{n=1}^{k}(-1)^{n-1}a_{n}}$ also converges to L.
The odd partial sums decrease monotonically:
${\displaystyle S_{2(m+1)+1}=S_{2m+1}-a_{2m+2}+a_{2m+3}\leq S_{2m+1}}$
while the even partial sums increase monotonically:
${\displaystyle S_{2(m+1)}=S_{2m}+a_{2m+1}-a_{2m+2}\geq S_{2m}}$
both because an decreases monotonically with n.
Moreover, since an are positive, ${\displaystyle S_{2m+1}-S_{2m}=a_{2m+1}\geq 0}$. Thus we can collect these facts to form the following suggestive inequality:
${\displaystyle a_{1}-a_{2}=S_{2}\leq S_{2m}\leq S_{2m+1}\leq S_{1}=a_{1}.}$
Now, note that a1a2 is a lower bound of the monotonically decreasing sequence S2m+1, the monotone convergence theorem then implies that this sequence converges as m approaches infinity. Similarly, the sequence of even partial sum converges too.
Finally, they must converge to the same number because
${\displaystyle \lim _{m\to \infty }(S_{2m+1}-S_{2m})=\lim _{m\to \infty }a_{2m+1}=0.}$
Call the limit L, then the monotone convergence theorem also tells us extra information that
${\displaystyle S_{2m}\leq L\leq S_{2m+1}}$
for any m. This means the partial sums of an alternating series also "alternates" above and below the final limit. More precisely, when there is an odd (even) number of terms, i.e. the last term is a plus (minus) term, then the partial sum is above (below) the final limit.
This understanding leads immediately to an error bound of partial sums, shown below.
### Proof of the alternating series estimation theorem
We would like to show ${\displaystyle \left|S_{k}-L\right|\leq a_{k+1}\!}$ by splitting into two cases.
When k = 2m+1, i.e. odd, then
${\displaystyle \left|S_{2m+1}-L\right|=S_{2m+1}-L\leq S_{2m+1}-S_{2m+2}=a_{(2m+1)+1}}$
When k = 2m, i.e. even, then
${\displaystyle \left|S_{2m}-L\right|=L-S_{2m}\leq S_{2m+1}-S_{2m}=a_{2m+1}}$
as desired.
Both cases rely essentially on the last inequality derived in the previous proof.
For an alternative proof using Cauchy's convergence test, see Alternating series.
For a generalization, see Dirichlet's test.
## Examples
### A typical example
${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }$
meets both conditions for the alternating series test and converges.
### An example to show monotonicity is needed
All of the conditions in the test, namely convergence to zero and monotonicity, should be met in order for the conclusion to be true. For example, take the series
${\displaystyle {\frac {1}{{\sqrt {2}}-1}}-{\frac {1}{{\sqrt {2}}+1}}+{\frac {1}{{\sqrt {3}}-1}}-{\frac {1}{{\sqrt {3}}+1}}+\cdots }$
The signs are alternating and the terms tend to zero. However, monotonicity is not present and we cannot apply the test. Actually the series is divergent. Indeed, for the partial sum ${\displaystyle S_{2n}}$ we have ${\displaystyle S_{2n}={\frac {2}{1}}+{\frac {2}{2}}+{\frac {2}{3}}+\cdots +{\frac {2}{n-1}}}$ which is twice the partial sum of the harmonic series, which is divergent. Hence the original series is divergent.
### The test is only sufficient, not necessary
Leibniz test's monotonicity is not a necessary condition, thus the test itself is only sufficient, but not necessary. (The second part of the test is well known necessary condition of convergence for all series.) Examples of nonmonotonic series that converge are ${\displaystyle \sum _{n=2}^{\infty }{\dfrac {(-1)^{n}}{n+(-1)^{n}}}}$ and ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}{\dfrac {\cos ^{2}n}{n^{2}}}.}$
^ In practice, the first few terms may increase. What is important is that ${\displaystyle b_{n}\geq b_{n+1}}$ for all ${\displaystyle n}$ after some point,[2] because the first finite amount of terms would not change a series' convergence/divergence.
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2022-01-23 01:02:22
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https://ncatlab.org/nlab/show/p-adic+geometry
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# Contents
## Idea
The special case of non-archimedean analytic geometry based on p-adic numbers.
## References
Survey is in
• Bernard Le Stum, One century of $p$-adic geometry – From Hensel to Berkovich and beyond, talk notes, June 2012 (pdf)
Last revised on April 20, 2016 at 09:25:12. See the history of this page for a list of all contributions to it.
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2018-05-22 09:55:43
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https://www.physicsforums.com/threads/existence-of-sequential-feasible-direction.956106/
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Existence of (sequential) feasible direction
• A
Having trouble with something that's likely too trivial, but here goes..
In Optimization theory and nonlinear programming [Sun, Yuan] the following is discussed at section 8.2.
Consider the optimisation of $f:\mathbb R^n\to\mathbb R$ with constraints $g_j:\mathbb R^n\to\mathbb R, g_j(x) = 0, j=1,\ldots , m$.
A feasible set $X$ is a subset of $\mathbb R^n$ which contains all those points that satisfy the system of constraints. The following definition is given.
A feasible direction at $x^*\in X$ is a straight line section containing $x^*$ that consists of ONLY feasible points. Formally, there exists a direction $h\in\mathbb R^n\setminus\{0\}$ (may assume to be normed) with
$$\exists \delta >0, \forall \eta (0\leq \eta\leq \delta \implies x^* + \eta h\in X)$$
Only this definition is given, so one naturally asks, do such feasible directions even exist, in general?
Further development is done via the language of sequences.
Call a direction $h\neq 0$ to be a sequential feasible direction at $x^*\in X$ if there exist sequences $h_k\to h$ such that for every vanishing sequence $\eta _k >0$ it holds that $x^* + \eta_k h_k\in X$.
Evidently, the sequence converges to $x^*$, but the following is problematic for me.
Set $x_k := x^* + \eta_kh_k$. If we set $\eta _k := \|x_k-x^*\|$ then
$$\frac{x_k-x^*}{\|x_k-x^*\|} \xrightarrow[k\to\infty]{} h?!\tag{E}$$
What I understand is that we have a sequence of normed elements, so if it converges to anything, the limit has to be nonzero, but why does it converge?
Let's give some context. Suppose $x^*\in X$ is not a local maximum point (subject to said constraints), then by definition
$$\forall k\in\mathbb N,\exists x_k\in X : \|x^*-x_k\|\leq \frac{1}{k}\quad \&\quad f(x_*) < f(x_k)$$
So, we can construct a sequence of feasible points that converges to $x^*$. May we assume without loss of generality, this sequence has a limiting direction? More formally, if we write $x_k = x^* + \eta_ kh_k, k\in\mathbb N$, may we assume $\exists \lim _k h_k\in\mathbb R ^n\setminus\{0\}$?
So, like I said, probably something trivial, but I really don't understand why we may assume the argument that culminates in (E).
S.G. Janssens
S.G. Janssens
Only this definition is given, so one naturally asks, do such feasible directions even exist, in general?
No, at the level of generality of your presentation, I do not see why they should exist. What excludes the possibility of ##X## being a finite set?
Further development is done via the language of sequences.
Call a direction $h\neq 0$ to be a sequential feasible direction at $x^*\in X$ if there exist sequences $h_k\to h$ such that for every vanishing sequence $\eta _k >0$ it holds that $x^* + \eta_k h_k\in X$.
Evidently, the sequence converges to $x^*$, but the following is problematic for me.
Set $x_k := x^* + \eta_kh_k$. If we set $\eta _k := \|x_k-x^*\|$ then
Sorry, but this confuses me. You took ##(\eta_k)_k## to be a vanishing sequence (of real numbers), then used it to define ##x_k##, but now you use ##x_k## in turn to define ##\eta_k##? Could you perhaps clarify this?
Without this (unnecessary?) re-definition, I would imagine that (E) follows because
$$\frac{x_k - x^{\ast}}{\|x_k - x^{\ast}\|} = \frac{\eta_k}{|\eta_k|}\frac{h_k}{\|h_k\|} = \frac{h_k}{\|h_k\|} \to \frac{h}{\|h\|} = h,$$
where it was used that ##\eta_k > 0## and ##h## is normalized.
Last edited:
nuuskur
Right, but it assumes ##h_k\to h ##. For the example I've given about the maximal point, I'd like to explicitly show which direction is sequentially feasible. So far, all I've seen in the chapter is a slew of definitions, but there's no lemma or proposition of the form "a (sequentially) feasible direction exists".
I'm kind of getting the vibe as in algebra - we describe a structure with a whole number of properties, prove a lot of equivalent statements and then when it comes to exhibiting examples of said structure, we're left empty-handed.
S.G. Janssens
Right, but it assumes ##h_k\to h ##.
Indeed. What I did is: I read your definition of ##h \neq 0## being a sequentially feasible direction at ##x^{\ast} \in X##. So, I took the involved sequence ##(h_k)## that the definition assumes exists and converges to ##h##. Then I picked an arbitrary vanishing sequence ##(\eta_k)## of positive numbers and checked that the sequence ##(x_k)## defined in terms of ##(h_k)## and ##(\eta_k)## has the behavior (E).
At this point, we are just verifying a simple consequence of the definition of "sequential feasibility". Does this clarify how (E) is obtained from the definition?
For the example I've given about the maximal point, I'd like to explicitly show which direction is sequentially feasible.
Let's give some context. Suppose $x^*\in X$ is not a local maximum point (subject to said constraints), then by definition
$$\forall k\in\mathbb N,\exists x_k\in X : \|x^*-x_k\|\leq \frac{1}{k}\quad \&\quad f(x_*) < f(x_k)$$
So, we can construct a sequence of feasible points that converges to $x^*$.
Yes, that looks correct.
May we assume without loss of generality, this sequence has a limiting direction? More formally, if we write $x_k = x^* + \eta_ kh_k, k\in\mathbb N$, may we assume $\exists \lim _k h_k\in\mathbb R ^n\setminus\{0\}$?
Sorry if I am slow, but I am confused again. Where do the ##h_k## and ##\eta_k## come from? I thought they are provided by the definition of "sequential feasibility", together with the direction ##h## itself. Also, it seems to me that this definition as such does not make any reference to extrema.
For me, the intuition behind the definitions is as follows. In order to solve local optimization problems, we need a way to compare points ##x^{\ast}## to other points nearby. Ideally, in a sense, we would want to be able to approach ##x^{\ast}## in a continuous fashion from all directions, but this requires ##x^{\ast}## to be an interior point of ##X##.
Depending on the constraints that define ##X##, this may not be possible. For example, ##X## may have corners (such that it is only possible to approach corner points from within ##X## along certain directions, but not along others) or ##X## may be a discrete set. I believe that your two definitions ("feasible" and "sequentially feasible") cater for such cases, respectively.
So far, all I've seen in the chapter is a slew of definitions, but there's no lemma or proposition of the form "a (sequentially) feasible direction exists".
One trivial way in which (sequentially) feasible directions exist at any point of ##X##, is of course when ##X## is open. On a more interesting level, I do expect your book (which I unfortunately do not know myself) to formulate certain conditions on the constraints ##g_j## that ensure such existence. Maybe it will still come in a later section?
I'm kind of getting the vibe as in algebra - we describe a structure with a whole number of properties, prove a lot of equivalent statements and then when it comes to exhibiting examples of said structure, we're left empty-handed.
Yes, I know that feeling, although I think that (what I consider) good abstract algebra study books motivate the definitions and properties from the examples, not the other way around. (For monographs or encyclopedic works it may be different, of course.)
nuuskur
Where do the ##h_k## and ##\eta_k## come from?
When we pick a sequence that violates the maximum condition, we can write every sequence element as a sum $x^* +\eta _kh_k$, where $\eta _k >0$ is vanishing and wo.l.o.g $\|h_k\|=1$. Simply put, the vector ##h_k ## points at the direction where ##x_k## is taken from and the scalar ##\eta_k## scales ##h_k## appropriately.
If $X$ is open, or, at the very least, for some $\delta >0$ $B(x^*,\delta)\subseteq X$, then we can of course pick an arbitrary straight line section and obviously it also provides a sequentially feasible direction, but then does the non-interior point case become something vacuous?
For instance, if $x^*$ is isolated, formally, for some $\delta >0, B(x^*,\delta)\cap X = \{x^*\}$, does $x^*$ automatically become a constrained extremum point?
What happens if it's not an interior point, but, say, an accumulation point i.e there exist sequences of ##X ## that converge to it. So let ##x_k\to x^* ##. Write every ##x_k = x^* +\eta_kh_k \in X## for reasons mentioned earlier. Is it safe to assume the sequence of directions ##h_k ## converges?
I've tried exhibiting some trivial counter-examples. For instance in 2-dimensional case, suppose ##x^* = 0 ## and we can set $x_k = \frac{((-1)^k,0)}{2^k}$, that means the sequence itself converges, but the sequence of directions keeps oscillating, but we can extract a converging subsequence (constant actually) of directions, so that doesn't work.
*lightbulb moment*
We are picking directions in a compact set (the unit sphere). We can immediately extract a converging subsequence, consequently a nonzero sequentially feasible direction.
As said, probably too trivial, which it is
Last edited:
S.G. Janssens
S.G. Janssens
When we pick a sequence that violates the maximum condition, we can write every sequence element as a sum $x^* +\eta _kh_k$, where $\eta _k >0$ is vanishing and wo.l.o.g $\|h_k\|=1$. Simply put, the vector ##h_k ## points at the direction where ##x_k## is taken from and the scalar ##\eta_k## scales ##h_k## appropriately.
Ok, now I understand what you meant earlier, thank you.
*lightbulb moment*
We are picking directions in a compact set (the unit sphere). We can immediately extract a converging subsequence, consequently a nonzero sequentially feasible direction.
As said, probably too trivial, which it is
I don't find it that trivial. Sorry for insisting, what you do here is that you start with a sequence ##(x_k)## converging to ##x^{\ast}## and then you define ##(h_k)## and ##(\eta_k)## using ##(x_k)##. Finally, you use compactness to extract a (sub)sequence ##(h_k)## converging to a unit vector ##h##.
Now, I take it that this ##h## should then be a sequentially feasible direction at ##x^{\ast}##. However - if you want to check your definition of sequential feasibility - you would still need to prove that for every vanishing positive sequence ##(\eta_k)## (not necessarily the same sequence that you used to get to ##h## in post #5) the points ##x^{\ast} + \eta_k h_k## all lie in ##X## (are all feasible).
S.G. Janssens
In addition, here are some trivial counterexamples to show that - at least at the great level of generality of your OP - sequentially feasible directions may not exist for any point of ##X##.
1. ##X## is a finite set.
2. ##X = \mathbb{Q}##.
(For 2, select any rational ##x^{\ast}## and let ##(h_k)## be a normalized sequence in ##\mathbb{R}## converging to ##h = \pm 1##. Then ##h_k = 1## or ##h_k = -1## identically, beyond any sufficiently large ##k##. Let ##(\eta_k)## be a vanishing sequence of positive, irrational numbers. Then ##x^{\ast} + \eta_k h_k = x^{\ast} \pm \eta_k## for ##k## sufficiently large, and this is not in ##X##.)
I really don't think that there is a problem, though: It seems to me that the authors of your book have continuous optimization in mind, and I think the definitions that you gave are chosen on that basis.
nuuskur
I really don't think that there is a problem, though: It seems to me that the authors of your book have continuous optimization in mind, and I think the definitions that you gave are chosen on that basis.
My mistake. Not only continuous, but even continuously (twice) differentiable.
S.G. Janssens
S.G. Janssens
My mistake. Not only continuous, but even continuously (twice) differentiable.
Ok, very good. Let's review your earlier sequential definition:
Call a direction $h\neq 0$ to be a sequential feasible direction at $x^*\in X$ if there exist sequences $h_k\to h$ such that for every vanishing sequence $\eta _k >0$ it holds that $x^* + \eta_k h_k\in X$.
From what you wrote afterwards, we also assume that ##h## and the ##h_k## are normalized. I also think that after "it holds that..." it is probably meant to write "eventually" (i.e. for ##k## large enough).
Related to what I said at the end of post #6,
Now, I take it that this ##h## should then be a sequentially feasible direction at ##x^{\ast}##. However - if you want to check your definition of sequential feasibility - you would still need to prove that for every vanishing positive sequence ##(\eta_k)## (not necessarily the same sequence that you used to get to ##h## in post #5) the points ##x^{\ast} + \eta_k h_k## all lie in ##X## (are all feasible).
you could consider the following.
If the ##g_j## are linear functionals such that the hyperplanes defined by them intersect nontrivially, then every point in ##X## admits a feasible direction, so it certainly admits a sequentially feasible direction.
If we admit nonlinear functionals, then I think it gets more complicated. As an example, take ##\mathbb{R}^2## with the single constraint ##g(x) = \|x\|^2 - 1##, so ##X## is just the unit sphere. Take a point ##x^{\ast} \in X## and let ##h## be a normalized direction, so in fact ##h \in X## as well. Consider a sequence ##(h_k)## of normalized vectors such that ##h_k \to h##. For every positive and sufficiently small ##\eta_k## the vector ##x^{\ast} + \eta_k h_k## lies outside the unit sphere, so outside ##X##. Therefore ##x^{\ast}## does not have a sequentially feasible direction.
Are you definitions aimed at problems with linear equality (and perhaps inequality) constraints?
Constraints are usually sufficiently smooth, say, once or twice continuously differentiable. In the general case, yes, inequality constraints can also be considered, but I'm trying to work my way through constraints with equalities for now.
S.G. Janssens
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2021-02-25 14:05:59
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9460651874542236, "perplexity": 889.8688401559357}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178351134.11/warc/CC-MAIN-20210225124124-20210225154124-00140.warc.gz"}
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https://tex.stackexchange.com/questions/394046/mathcal-gives-different-output-where-it-should-not
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# \mathcal gives different output where it should not
I have multiple occurences of mathcal in my document and they do not use the same style. Consider this MWE:
\documentclass{standalone}
\usepackage{mathtools}
\usepackage{unicode-math}
\setmainfont[Ligatures=TeX]{STIX}
\setmathfont{XITS Math}
\setmathfont[range={\mathcal},StylisticSet=1]{XITS Math}
\begin{document}
$\operatorname{O}$
$\mathcal{S}$
$\text{$P$}$
$\mathcal{S}$
\end{document}
As you can see the two "S" charcters are different, which they should not be.
What have I done wrong?
Versions used:
• XeTeX, Version 3.14159265-2.6-0.99996 (TeX Live 2016)
• mathtools 2015/11/12 v1.18
• amsmath 2016/06/28 v2.15d
• unicode-math 2015/09/24 v0.8c
Loading mathtools after unicode-math solves this problem, but messes up other things. For example \underbrace{X}_{0} then gives:
• Looks okay in texlive 2017. I get the second S in both places. It works in texlive 2016 with unicode-math 2017/01/27 v0.8d too. – Ulrike Fischer Oct 1 '17 at 9:55
This was a bug in unicode-math. See for example https://github.com/wspr/unicode-math/issues/356.
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2019-10-21 07:46:02
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9054529070854187, "perplexity": 11906.177396231013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987763641.74/warc/CC-MAIN-20191021070341-20191021093841-00320.warc.gz"}
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http://www.givewell.org/charities/food-fortification-initiative
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Food Fortification Initiative | GiveWell
# Food Fortification Initiative
The Food Fortification Initiative (FFI) does not meet all of our criteria to be a GiveWell top-charity but is a program we believe is worthy of special recognition. Although we don't recommend these organizations as strongly as we do our top charities, they stand out from the vast majority of organizations we have considered.
More information: What is our evaluation process?
Published: September 2016
## Summary
What do they do? The Food Fortification Initiative (FFI, www.ffinetwork.org) works to reduce micronutrient deficiencies (especially folic acid and iron deficiencies) by doing advocacy and providing assistance to countries as they design and implement flour and rice fortification programs. (More)
Does it work? We believe that food fortification with certain micronutrients can be a highly effective intervention. However we do not have a strong sense of what FFI’s impact may be in the typical country to which it provides support; because FFI typically provides support alongside a number of other actors and its activities vary widely among countries, it is difficult to confidently assess the impact of its work. We reviewed case studies of FFI’s work in the Solomon Islands and Kosovo in order to try to better understand its track record. We do not yet have enough information to have a confident understanding of FFI’s impact in these cases. (More)
What do you get for your dollar? Our impression is that, when effective, micronutrient fortification programs may be in the range of cost-effectiveness of our other priority programs. However, we do not have enough information to make an informed guess as to the cost-effectiveness of FFI’s work to date. (More)
Is there room for more funds? FFI estimates that it could use about $460K in additional funding per year for new staff to support additional fortification activities and a further ~$1.6 million to support new fortification-related projects. More details on these activities are below. (More)
What are GiveWell’s next steps? FFI has successfully completed the first phase of our investigation process. We now plan to make a $100,000 grant to FFI (as part of our "top charity participation grants," funded by Good Ventures). We do not plan to do further work on FFI at this time because we are doubtful that additional investigation of the kind we've done so far will resolve our remaining questions, but we may consider returning to FFI if we do additional work in the micronutrient space in the future. ## What does FFI do? The Food Fortification Initiative (FFI) works to reduce micronutrient deficiencies (especially folic acid and iron deficiencies) by advocating for flour and rice fortification and providing assistance to countries as they design and implement fortification programs.1 FFI primarily works in Africa, Asia, and Eastern Europe.2 FFI told us that its activities vary widely from country to country, but that its main activities include:3 • Supporting advocacy to governments to pass mandatory fortification laws and to implement fortification. FFI told us that it supports advocacy by (a) holding workshops, (b) establishing national coalitions that include non-profits, government actors, and private industry representatives, and (c) providing technical assistance with drafting legislation.4 • Providing technical assistance for the implementation and monitoring of fortification. For example, FFI told us that it helps producers to properly implement fortification into their production lines, trains producers and governments on the best monitoring methods for tracking fortification, and assists countries with choosing the best micronutrient mixes for their populations’ needs.5 • Tracking and sharing global progress on fortification via its website. For example, it told us that its database aims to show: burden of micronutrient deficiencies, which countries have mandated fortification, and whether industries are successfully fortifying.6 See Food Fortification Initiative website, “Global Progress” for more details. FFI told us that its work in countries can range from cases in which it is the only non-governmental organization (NGO) actor providing a large amount of direct help to a fortification program (e.g., having a full-time staff member on the ground supporting all aspects of implementation) to cases in which it is one among many NGO actors supporting a single aspect of a complex fortification program (e.g., having its Nutrition Specialist provide input on which iron compound a country should be fortifying with).7 FFI told us that it most often works as one NGO actor among many supporting a particular aspect of a fortification program.8 More information on FFI’s work in particular countries is available in the following footnote,9 and in the following sources: We also explore two case studies of FFI’s work in particular countries below. ### Micronutrients and foods that FFI works with FFI told us that it primarily aims to support fortification of wheat flour, maize flour, and rice with iron and folic acid in order to reduce anemia and neural tube defects.10 In many cases, FFI also supports fortification with other micronutrients (e.g., other B vitamins, zinc, vitamin A, and vitamin D) based on a particular country’s needs.11 FFI aims to fortify cereal grains: either wheat flour, maize flour, or rice.12 FFI does not work with fortification of other foods, but refers countries to other experts when they want to fortify other foods (such as salt, oil, or milk).13 ### Staff As of May 2016, FFI had about 12 staff (though some staff work less than full-time):14 • Six central staff based in the U.S. and Canada, including: its Director, a Senior Nutrition Scientist, a Communications Coordinator, a Micronutrient Specialist, a Training and Technical Support Coordinator, and an Administrative Coordinator. • Two staff members in India: a Network Coordinator and Senior Advisor. • Two staff members in Asia: an Executive Officer and Technical Officer. • One Senior Advisor based in the Netherlands. • One Network Coordinator based in Africa. Generally, its central staff and advisors provide various kinds of support to actors across the globe while its network coordinators focus on supporting fortification programs in the regions in which they work.15 Sometimes (as in the Solomon Islands case discussed below), FFI hires someone to work on the ground in a particular country. ### Spending FFI told us that its yearly budget has been roughly$1.6 million to $2 million dollars from 2014-2016.16 FFI told us that it allocated its budget as follows:17 Expense category 2014 expenses % of total 2015 expenses % of total 2016 budgeted expenses % of total Africa (regional support)$625,200 38% $695,200 36%$710,200 42%
Asia (regional support) $258,254 16%$369,732 19% $366,390 22% India (regional support)$309,400 19% $338,493 18%$328,888 19%
Europe (regional support) $158,200 10%$158,200 8% $31,640 2% Technical Assistance$58,012 4% $117,550 6%$31,640 2%
Advocacy $58,640 4%$58,740 3% $50,830 3% Tracking Progress$61,620 4% $61,620 3%$52,010 3%
Global Secretariat $136,155 8%$136,155 7% $128,245 8% Total$1,665,481 100% $1,935,690 100%$1,699,843 100%
### How FFI chooses where to work
FFI told us that it has analyzed the fortification situation in many countries in Africa, Asia, and Eastern Europe to determine where to focus its work.18 FFI told us that all else being equal, it largely prioritizes working in countries where:
• It is feasible to have a large impact through fortification. Two key factors that affect feasibility are: 1) whether the relevant population consumes sufficient quantities of cereal grains,19 and 2) whether the milling industry in the country is sufficiently centralized.20
• There is strong support across the government for micronutrient fortification.21
• There is a large burden of micronutrient deficiency.22
FFI has shared some of its strategy documents with us but we have not yet carefully reviewed them. See, e.g., 2016 Strategy Review presentation.
FFI told us that the type of support it provides to a particular country generally depends on the country’s needs, how much funding FFI has available to support the country, and which other actors are already working there.23
## Does it work?
To evaluate FFI’s impact, we focused on the following questions:
• Does micronutrient fortification improve health? Broadly, we believe that micronutrient fortification may be a highly effective intervention, with possible effects on cognition, child mortality, anemia, and other outcomes. Our research into the main micronutrients that FFI promotes for fortification (folic acid and iron) as well as other micronutrients (zinc, other B vitamins, and vitamin D) is ongoing. We have completed an intervention report on vitamin A supplementation but have not yet investigated vitamin A fortification.
• Has FFI demonstrably improved micronutrient fortification in particular countries? We reviewed case studies of FFI’s work in the Solomon Islands and Kosovo in order to better understand its track record. We do not yet have enough information to have a confident understanding of FFI’s impact in these cases.
Details follow.
### Has FFI demonstrably improved micronutrient fortification in particular countries?
To estimate the impact of FFI’s activities, we focused on trying to understand the impact of particular cases of FFI’s past work. In our experience, it is challenging to verify impact in such cases, so we began by asking FFI to identify its “success stories” where it seemed most likely to be able to prove its impact.
In collaboration with FFI, we chose to investigate its work in the Solomon Islands and Kosovo. We investigated the Solomon Islands because FFI was the primary NGO actor there, so it seemed like it might be easier to verify impact than in a case where many NGOs and other actors are involved. We investigated Kosovo because FFI does not usually work on the ground as the primary technical assistant on all fortification activities like it did in the Solomon Islands; we therefore thought Kosovo might be more representative of FFI’s typical work while still being identified as a particularly strong success by FFI.
We relied on FFI for the information in these cases. We did not seek to vet them with independent sources.
#### Solomon Islands case study
Background
FFI told us that it was asked by the Australian government’s Department of Foreign Affairs and Trade to assist the Solomon Islands with its fortification program.24 We have not tried to investigate what would have happened if FFI had decided not to assist the Solomon Islands (e.g., whether another NGO might have been able to provide support).
FFI told us that the Solomon Islands passed legislation mandating that all imported and domestically-produced flour be fortified with iron and folic acid in 2010, but that no progress had been made on actual implementation of this law at the time that FFI began working with the Solomon Islands in June 2014.25
FFI told us that its main activities in the Solomon Islands included establishing a national fortification committee that regularly meets about fortification-related issues, training producers on how to fortify flour properly, and supporting the government in drafting legislation for rice fortification and in setting up monitoring systems.26
FFI told us that its main point of contact was a full-time contractor that it hired to work closely with the government in the Solomon Islands for about 10 months.27 Other FFI staff also provided a variety of support to the Solomon Islands.28 FFI estimates that the total cost of its work in the Solomon Islands will ultimately be about $350,000 and it told us that funding was necessary to the project because it enabled hiring on-the-ground support.29 As of May 2016, FFI told us that the main producer of flour in the Solomon Islands was fortifying properly.30 It told us that rice fortification was not yet occurring but that mandatory legislation was expected to be passed in the next few months and implementation would occur during a subsequent 6 month grace period.31 Micronutrient deficiencies and food consumption patterns FFI sent us a presentation that reports that in the Solomon Islands there are about 27 new neural tube defects per year (based on “March of Dimes estimates”) and that anemia rates are roughly 44-60% among women of child-bearing age, pregnant women, and children under 5 years old (based on “DHS 2006-07”).32 We have not attempted to vet these estimates or understand their underlying methodologies. In the same presentation, FFI reports about data on the availability of wheat flour and rice in the Solomon Islands, which it uses as a proxy for consumption.33 Its analyses report that there is sufficient expected consumption of wheat flour and rice to meet the World Health Organization’s fortification recommendations.34 We have not attempted to vet the estimates or sources underlying these claims and have not attempted to understand their methodologies. FFI told us that about 90% of wheat flour consumed in the Solomon Islands is made by the lone producer in the country (Delite Flour Mill), while the remaining 10% is imported.35 It told us that the reverse is true for rice: it said about 90% of rice consumed in the Solomon Islands is imported from a major Australian company and about 10% of rice is produced domestically.36 We do not have a strong understanding of the methodology underlying these estimates and have not attempted to vet them. Monitoring FFI told us that as of May 2016 it believed that flour was being properly fortified at the main flour producer in the Solomon Islands.37 FFI told us that its main reasons for believing this were that (a) FFI had provided training to the producers, and the producers seemed to set up the equipment and follow the necessary procedures after the training in 2015, (b) as of May 2016, meetings of the national committee for fortification were still occurring roughly every 6 weeks; the main producer (Delite) is represented in those meetings and has expressed continued commitment to fortifying, and (c) FFI has seen that Delite is regularly ordering the correct premix.38 We have not attempted to independently verify these claims. When FFI becomes less involved in the day-to-day activities in the Solomon Islands, it told us that it would expect the following monitoring procedures to be operating:39 • FFI would check with the micronutrient premix supplier (i.e., the company that provides the mix of folic acid, iron, and other nutrients added to flour) to verify that the flour producer (Delite) is purchasing the appropriate amount of premix.40 • Flour producers would be following an internal monitoring procedure.41 • FFI would support the government in setting up an external monitoring procedure in which the government would check at least once per year that the main flour producer is properly fortifying flour and conducting its internal monitoring. FFI told us that these would ideally be part of the government’s standard food safety checks.42 • FFI would support the government in setting up a monitoring procedure to check that imports of flour and rice are properly fortified.43 FFI sent us templates and procedure documents for internal monitoring systems, the government’s external monitoring system, and import monitoring.44 As of May 2016, we have seen two examples of producers’ internal monitoring. FFI sent us one example of a premix order and one example of a third party vitamin assay (i.e., a test in which the producer sends a sample of flour to an external party to determine whether the sample has the right level of nutrients).45 The internal monitoring document suggests that vitamin assays should be done once per quarter.46 We have seen the results from a flour sample that was taken from Delite in November 2015.47 We are unsure whether the results of the vitamin assay suggest that Delite’s flour sample had the right concentration of all nutrients and have not yet asked FFI for its interpretation of the results.48 We have not yet seen other results from producers’ internal monitoring. We have not yet seen examples of external monitoring or import monitoring. FFI told us that as of May 2016 it was not yet asking for verification that monitoring procedures were being implemented since the national committee for fortification was still meeting regularly.49 As of May 2016, FFI told us that rice fortification was still in the process of being mandated and implemented in the Solomon Islands,50 so we have not yet requested monitoring information from this work. Finally, regarding measuring the direct effect of fortification on population health in the Solomon Islands: • FFI told us that birth registry systems in the Solomon Islands are not comprehensive enough to enable an estimate of the effect of folic acid fortification on neural tube defects.51 • FFI told us that it may be possible to get an estimate of the impact of fortification on anemia using routine health surveys, but that data would not be available for at least a couple of years.52 Bottom line It seems plausible to us that FFI’s work has made it more likely that the Solomon Islands will properly implement flour and rice fortification. However, we have not yet tried to investigate what may have happened in FFI’s absence and we have not yet seen consistent reports from routine monitoring systems verifying proper fortification of both flour and rice over time (though we have seen one vitamin assay suggesting that the major flour mill may be fortifying properly). Note that FFI’s work in the Solomon Islands may not be representative of a large proportion of its past or future work because: • It is relatively rare that FFI supports all aspects of fortification on the ground like it did in the Solomon Islands; usually it works as one NGO among many in supporting a fortification program. • The Solomon Islands is a very small country (population of about 560,000)53 with an unusually simple industrial structure (one flour mill that serves 90% of residents and one rice importer that serves the vast majority of residents, according to FFI). #### Kosovo case study Background FFI told us that it has provided a variety of support to Kosovo’s fortification program since 2008.54 FFI said that it would be very challenging to quantify the total amount of resources it has devoted to Kosovo since it has allocated a small amount of many staff members’ time to Kosovo over the course of many years.55 FFI told us that UNICEF was a major partner in supporting Kosovo’s fortification effort.56 It told us that Kosovo passed legislation mandating fortification in September 2012 and that implementation of fortification was “launched” in February 2014.57 FFI told us that some of its biggest impacts on the Kosovo fortification program were:58 • FFI said that it influenced the Kosovo government to structure its legislation in a way that would make it easier to change technical standards around the specific nutrient levels and compounds that should be used in fortification if that were necessary.59 FFI told us that some countries (unlike Kosovo) require legislative action to change technical standards, which leads to long delays.60 FFI said that Kosovo has not yet needed to change its standards as of May 2016.61 • FFI told us that in 2012 it supported a graduate student to help Kosovo develop a monitoring system that would be easier for producers and the government to implement.62 FFI believes this is important because it said that if monitoring systems are too cumbersome then they may not be followed.63 FFI told us that it has not followed up with actors in Kosovo to see if the monitoring system that it helped to develop is being used.64 • FFI said it held workshops to increase regional collaboration, train government officials on monitoring, and ensure that fortification remained high on the Kosovo government’s priority list.65 We have not yet tried to investigate the potential impact of these workshops through means other than talking to FFI. FFI said that in early 2016, UNICEF staff in Kosovo requested that FFI’s technical coordinator, Quentin Johnson, visit the country because Kosovo was experiencing new challenges with fortification.66 After visiting the country, FFI told us that Johnson confirmed that Kosovo was experiencing the following challenges:67 • Government subsidies for micronutrient premix ended. Many small producers were unwilling to pay for the premix themselves so they stopped fortifying. FFI said premix is unusually expensive in Kosovo due to high taxes on the imported premix. • Milling costs were already high due to high electricity costs, leading to reduced interest in spending additional money to implement fortification. FFI said it was unsure what specifically led UNICEF to notice these issues.68 FFI said that in the future it may try to support Kosovo by sending its staff to advocate for a tax exemption for micronutrient premix and by presenting Kosovo officials with a cost-benefit analysis to incentivize them to continue with fortification.69 It told us that this type of support is fairly similar to the kind of support it has provided to Kosovo in the past.70 Micronutrient deficiencies and food consumption patterns Kosovo’s population is about 1.8 million people.71 FFI told us that the last nutrition survey that it is aware of from Kosovo was done in 2010.72 The survey reports that anemia prevalence among school children in Kosovo was ~16%,73 and anemia prevalence among pregnant women was ~23%.74 We have not attempted to vet these figures or understand the methodology of the survey. FFI told us that it has not seen any data on neural tube defects in Kosovo and that it is not aware of any birth defect registry system in Kosovo.75 We have not requested detailed information on food consumption patterns in Kosovo, so we do not know if it is available. Monitoring As part of its work on tracking the status of fortification across the world, FFI told us that each fall it asks stakeholders in all countries with fortification programs to provide estimates of the percentage of industrially milled wheat flour, maize, and rice that is being fortified.76 FFI told us that it does not usually ask people who respond to these inquiries to provide the source for their estimates, so it usually does not have underlying documentation.77 FFI told us that its partners in Kosovo reported that in 2014 about 56% of Kosovo’s flour was produced in 82 industrial mills (as opposed to small scale mills that do not fortify) and that industrial mills were fortifying about 89% of their flour.78 FFI’s partners reported that in 2015 about 70% of Kosovo’s wheat flour was produced in 82 industrial mills that fortified about 75% of their flour.79 We have not independently vetted these estimates and do not know the underlying methodology that was used to produce them. Bottom line Based on the information that we have collected so far, we do not have a strong sense of whether FFI had a substantial positive impact on the fortification program in Kosovo. Additionally, we do not have a strong sense of whether the fortification program in Kosovo in general has been successful and cost-effective. ### Impact of FFI’s other activities We have not yet attempted to investigate the potential impact of other work done by FFI, such as its work on keeping a record of global progress on fortification and its work supporting workshops that aim to persuade government officials to start and maintain fortification programs. ## What do you get for your dollar? If successful, it seems plausible that micronutrient fortification programs, such as those supported by FFI, could be as cost-effective as our other priority programs. However, understanding the cost-effectiveness of FFI's work is complex because of the role FFI plays in the countries in which it works. We have not yet attempted a formal cost-effectiveness analysis of FFI’s work at this preliminary stage of investigation because we did not want to invest the time in such an analysis while we had many major remaining sources of uncertainty, such as: • Magnitude of health benefits of micronutrient fortification: We are still in the process of analyzing the evidence supporting the health benefits of micronutrients. • Uncertainty about impact of FFI's past work: As discussed above, we do not have a strong understanding of the impact of FFI's past work. • Relative impact of FFI's future work: Are future projects likely to be more, less, or similarly cost-effective as past projects? ## Is there room for more funding? In brief: • FFI sent us a list of projects that it would support with additional funding (see FFI's Funding Gaps List). FFI estimates that it could use about$460K in additional funding per year for new staff and a further ~$1.6 million to support new projects. Details on these activities are below. • We do not have a strong sense of whether FFI is likely to gain or lose funding from other funders in the future, though its budget seems to have been fairly stable (in the range of ~$1.6M-$2M) between 2014 and 2016.80 ### FFI’s plans for additional funding FFI sent us FFI's Funding Gaps List, which lays out what FFI says it would do with additional funding. FFI told us that this document is roughly ordered by priority (i.e., the funding gaps that FFI would expect to fill first are at the top of the document). FFI's Funding Gaps List contains two types of funding opportunities: (a) new staff that FFI would hire or activities that it would undertake if it had a certain amount of additional annual funding, and (b) one- to three-year projects that FFI would undertake with sufficient funding.81 The document contains about$560K in annual funding gaps for new staff and ~$1.6M in funding gaps for specific projects.82 Examples of longer-term activities that FFI says it could undertake with additional annual funding are: •$140K per year: Hire a dedicated staff person to support wheat flour fortification efforts in many countries in Eastern Europe.83
• $120K per year: Hire a dedicated staff person to support wheat flour and rice fortification in China.84 Examples of one- to three-year projects are: •$500K over three years: Support rice fortification work in Africa, particularly West Africa.85
• $350K over three years: Support wheat flour and rice fortification in Papua New Guinea.86 •$150K per country: Hire a country coordinator to support grain fortification in Bangladesh and/or Mongolia.87
FFI told us that it now expects to be able to fund $100,000 of annual rice coordination work (the funding gap at the top of its wishlist) out of existing resources.88 It told us that all of the other funding gaps in FFI's Funding Gaps List remain unfunded.89 Thus, it seems that its remaining gap for activities that require annual funding is about$460K.90
Note that FFI will receive $100,000 from GiveWell upon publication of this interim review (as part of our "top charity participation grants," funded by Good Ventures), which could be allocated toward the above funding gaps. ### FFI’s existing funders As noted above, FFI spends roughly$1.6M-$2M per year. FFI told us that it receives about half of its funding from government contracts.91 It said that the bulk of the rest of its funding comes from the charitable giving of major agribusinesses and regranting from other micronutrient nonprofits (such as GAIN, Micronutrient Initiative, and UNICEF).92 FFI told us that it would like to put more effort into fundraising in the future.93 ## Our investigation process As we stated in our 2016 plans, we are interested in evaluating additional charities that work on micronutrient fortification. To date, our investigation process has consisted of: • Four conversations with Scott Montgomery, Director, and Sarah Zimmerman, Communications Coordinator, of FFI.94 • Reviewing documents FFI sent us after these conversations, and in response to our queries. ## Sources Document Source 2016 Strategy Review presentation Source Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 Unpublished Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 Unpublished Conversation with Scott Montgomery, February 19, 2016 Unpublished Country Network Objectives spreadsheet Source E-mails about FFI's work in Kosovo, May 10, 2016 Unpublished FFI spending and budget 2014-2016 Source FFI's Funding Gaps List Source Food Fortification Initiative website, “About Us” Source (archive) Food Fortification Initiative website, “Global Progress” Source (archive) Food Fortification Initiative website, “Staff team” Source (archive) GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 Source Kosovo Fortification spreadsheet Source Kosovo Nutrition Survey 2010 Source Regional Work Activities spreadsheet Source Solomon Islands External Monitoring Manual Source Solomon Islands Import Monitoring Manual Source Solomon Islands Internal Monitoring Manual Source Solomon Islands Monitoring Framework summary Source Solomon Islands Premix Order Unpublished Solomon Islands Summary presentation Source Solomon Islands Vitamin Assay Source • 1. “The Food Fortification Initiative (FFI), formerly the Flour Fortification Initiative, is an international partnership working to improve health by advocating for fortification in industrial grain mills. We specialize in wheat flour, maize flour, and rice. The support we provide through multi-sector collaborations includes: • Advocacy resources on the benefits of fortification • Technical assistance in planning, implementing and monitoring fortification programs • Tracking progress at the country and global levels We primarily support national stakeholders in the public, private and civic sectors. Key leaders are government officials, industry leaders, civic sector advocates, and staff of non-governmental organizations. The nutrients most commonly used in post-harvest grain fortification are iron and folic acid, a B vitamin. Other nutrients that can be added are zinc, vitamin A, vitamin D, and other B vitamins such as niacin, thiamine, riboflavin and B 12.” Food Fortification Initiative website, “About Us” • 2. • 3. • 4. • 5. Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 6. GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 (not included in written notes, but FFI staff reviewed this page prior to publication) • 7. “Which activities FFI conducts in a particular country depends on several factors, including how much funding it has available and how many other organizations are supporting fortification in that country. FFI’s activities in a particular country could range from supporting all aspects of fortification on its own (including hiring on-the-ground employees to assist governments and producers) to supporting one aspect of fortification (e.g., drafting legislation, advocacy) while other organizations and partners support most of the other aspects of the program.” GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 • 8. • 9. FFI told us that a few examples of its past and current work include: • Helping South Africa to switch from using the incorrect iron compound in its fortification program to using the correct one. • “For example, South Africa already has fortification legislation and monitoring in place, but currently fortifies wheat flour with a non-bioavailable type of iron that has little public health impact. FFI is helping South Africa switch to the correct iron compound, which is a difficult process. Countries like South Africa that use the wrong iron compound may be doing so due to outdated advice, or possibly in an attempt to save money (a non-bioavailable iron compound is often slightly cheaper than the correct compound).” GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 • In some countries where Project Healthy Children and Global Alliance for Improved Nutrition (GAIN) were working, FFI said it provided technical advice on how to draft details in the legislation and trained people on monitoring. GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 (not included in written notes, but FFI staff reviewed this page prior to publication) • One example at the fairly intensive end of the spectrum of FFI's work was in the Solomon Islands where it employed a full-time staff member to sit in the Ministry of Health and support the government’s work on fortification. This work is described in more detail in this section. • FFI told us that it developed an online quality control "internal/external" monitoring tool that can be used by industry and governments to monitor fortification. It told us that it is running a workshop in West Africa to train fortification-related technical capacity. • “USAID has given FFI funding to run a regional workshop to train additional fortification technical capacity for West Africa, with the intent of broadening the base of expertise in the area and reducing reliance on the current small number of global experts. FFI will train workshop participants to use an online quality assurance and quality control (QA/QC), internal/external monitoring tool that it recently developed with support from the Global Alliance for Improved Nutrition. FFI may also do this kind of training in other regions where it is needed.” GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 • 10. Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 11. • 12. ““The Food Fortification Initiative (FFI), formerly the Flour Fortification Initiative, is an international partnership working to improve health by advocating for fortification in industrial grain mills. We specialize in wheat flour, maize flour, and rice.” Food Fortification Initiative website, “About Us” • 13. “FFI focuses on cereal grains and does not work with other food vehicles. If a country's government wants to fortify a different vehicle (e.g. salt, oil), FFI does not have the relevant expertise itself but would likely be able to connect the government to other groups that do work with the desired vehicle.” GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 • 14. • 15. Conversation with Scott Montgomery, February 19, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 16. FFI spending and budget 2014-2016 • 17. FFI spending and budget 2014-2016 • 18. “FFI has performed detailed strategy diagnostics in each of its major regions. FFI started this process by gathering fortification data for all of Africa and creating profiles for each country, which included: • Compiling facts about each country, including the relevant history that led to the country's current political and social situation. • Tracking every fortification partner (to FFI's knowledge) working in Africa by country. • Tracking disease burden by country. • Tracking consumption of cereal grains by country. • Examining the industrial milling complex in the country (which FFI sees as an essential component of sustainable fortification solutions). FFI has also done strategy research in Asia and Eastern Europe. In India, FFI did a more comprehensive study that included mapping the wheat and rice supply chains of the major states and identifying possible opportunities both by state and by market channel. Based on its strategy research, FFI has created a "priority matrix" of countries and Indian states, which takes into account the potential impact of fortification, the best cereal grain vehicle (e.g., wheat, rice, maize) to target, and how easy or difficult implementation is likely to be. This process allowed FFI to understand the opportunities in each country and determine where to direct its focus. FFI produces regional work plans, based on a strategy document developed using its diagnostic data, which outline the types of fortification activities it aims to support and/or accelerate in each country.” GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 • 19. “In determining whether to enter a country, FFI's first step is to understand the country's pattern of wheat, maize, and/or rice availability. A useful preliminary litmus test is that if per capita availability of cereal-based grains is over 75 grams per day, effective fortification is likely possible. If it is significantly lower than this, FFI needs a more detailed understanding of consumption patterns to determine whether fortification is feasible, because the amount of vitamins and minerals that can be added to cereal grains without affecting sensory properties, e.g., the color of bread, etc., is limited.” GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 • 20. “FFI next examines the country's industrial milling complex, which it sees as essential to effective fortification practices. For example, although maize availability in Africa is high in terms of grams per day, in many African countries maize flour is produced primarily by mortar and pestle or by local village mills, approaches that are not conducive to sustainable or cost-effective fortification.” GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 • 21. “Once FFI has determined that a country offers a good opportunity for a fortification program, the next important step is gaining political support. FFI aims to create a political coalition by connecting with the Ministry of Health (MOH) and, ideally, other government departments (e.g. the Ministry of Finance, Ministry of Agriculture, etc.). FFI also attempts to get the support of the highest level of local government (e.g. the Chief Minister in Indian states). In particular, FFI aims to identify and partner with capable individuals in the government that are passionate about improving nutrition. FFI also works closely with the government to understand its mechanisms (as the ideal fortification program structure can change significantly depending on how a particular country's government is set up).” GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 • 22. GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 (not included in written notes, but FFI staff reviewed this page prior to publication) • 23. “Which activities FFI conducts in a particular country depends on several factors, including how much funding it has available and how many other organizations are supporting fortification in that country. FFI’s activities in a particular country could range from supporting all aspects of fortification on its own (including hiring on-the-ground employees to assist governments and producers) to supporting one aspect of fortification (e.g., drafting legislation, advocacy) while other organizations and partners support most of the other aspects of the program.” GiveWell's non-verbatim summary of a conversation with Scott Montgomery, March 2, 2016 • 24. • 25. • 26. FFI told us that early in its work in the Solomon Islands it established a national fortification committee that scheduled regular meetings between FFI, the government, and the milling industry. In the course of its work, FFI told us that it realized that to fully meet the nutritional needs of most of the population in the Solomon Islands, rice would need to be fortified as well. Then, it told us that it aimed to help the government to pass legislation and clarify standards that would mandate rice fortification for importers and domestic producers. Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 27. Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 28. FFI told us that other staff providing support to the Solomon Islands included Scott Montgomery (Director), Becky Tsang (Technical Officer for Asia), Karen Codling (Executive Officer for Asia), Quentin Johnson (Training and Technical Support Coordinator), Sarah Zimmerman (Communications Coordinator), and Helena Pachon (Senior Nutrition Scientist). Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 29. • FFI told us that its total cost estimate of ~$350,000 included:
• ~$275K for project funding that FFI has specifically allocated to the Solomon Islands to date • ~$40K for a rough estimate of the value of the time that central staff contributed to the project
• ~$35K in expected future spending to conclude work on rice fortification in the Solomon Islands over the course of the next year Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • Other costs that FFI said were not included in this estimate include: • Millers paying for fortificants (premix) and spending additional time to implement fortification. Millers also incurred small costs to put new logos on their products advertising that they’re fortified. • Additional cost of government regulators’ time (both import regulators and food inspectors). Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 30. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 31. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 32. • “Neural tube birth defects (NTDs) such as spina bifida affect an estimated 27 pregnancies per year in the Solomon Islands” Solomon Islands Summary presentation, Slide 5. • ”Percent of people with anaemia in the Solomon Islands”: “44.3 (Women of child-bearing age)”, “60 (Pregnant women)”, “48.5 (Children under 5 years)” Solomon Islands Summary presentation, Slide 4. • 33. See “Summary of current data – wheat flour availability” and “Summary of current data – rice availability”, Solomon Islands Summary presentation, Slides 11-12. • 34. “Use the 2009 World Health Organization (WHO) wheat and maize flour fortification recommendations to identify an appropriate category for the Solomon Islands: • Solomon Islands Wheat Flour & Rice apparent consumption: 206-329 grams/capita/day • WHO Fortification recommendations for 150-300 grams/capita/day” Solomon Islands Summary presentation, Slide 15. • 35. Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 36. Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 37. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 38. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 39. • We discussed monitoring procedures with FFI as part of Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) and Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication). • FFI sent us a summary of its monitoring framework for the Solomon Islands: Solomon Islands Monitoring Framework summary. • "There needs to be a system that ensures that all wheat flour, rice and salt in the Solomon Islands meets national standards for fortification in the Pure Food (Food Control) Regulations. There are three core components of the proposed regulatory monitoring framework: 1. Internal Monitoring - Delite (and any future wheat flour mills in the Solomon Islands) having an internal quality assurance and quality control system (QA/QC) for fortified wheat flour. 2. Regulatory/External Monitoring - The Environmental Health Division (EHD) of the Ministry of Health audits Delite’s QA/QC system for fortified wheat flour on at least an annual basis. 3. Import Monitoring - The Solomon Island government monitors all wheat flour, rice and salt imports to ensure they meet national standards for fortification. This will require collaboration between EHD, Customs and Quarantine.” • 40. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 41. "Internal monitoring will rely on wheat flour mills having suitable QA/QC systems in place. Delite (and any future flour mills) will keep records to substantiate: • purchase of suitable premix (as indicated by the Certificate of Analysis accompanying each shipment of premix) • appropriate mixing ratios between the premix and wheat flour • other QA/QC procedures that demonstrate that the fortification process is controlled and monitored It is recommended that Delite undertake daily iron spot tests to verify the premix addition to produced flour. It is also recommended that Delite send composite samples of its flour to an external laboratory for testing of iron and folic acid content on a quarterly basis." Solomon Islands Monitoring Framework summary, Pg. 1. • 42. "EHD [Environmental Health Division] will visit each mill on an annual basis – or more often if required under its risk framework - to verify the performance of a mill’s internal QA/QC procedures. This will involve a paperwork audit to ensure that QA/QC procedures are being practiced e.g. verifying the purchase of suitable premix and reviewing the Certificate of Analysis, verifying appropriate mixing ratios between the premix and wheat flour, and verifying that iron spot tests are being carried out on a daily basis and all indicate that the flour has been fortified. During its visit, EHD will collect a composite flour sample for quantitative testing of iron content by a laboratory. Results will be compared with the quarterly laboratory test results obtained by the mill. EHD will provide the mill with a preliminary report of its findings during the visit. A final report will be provided after receipt of the laboratory results, advising the mill of any corrective action needed. EHD will report its findings on an annual basis to the Food Fortification National Committee." Solomon Islands Monitoring Framework summary, Pg. 1. • 43. "Import Monitoring will rely mostly on a review of the Certificate of Analysis for each shipment of wheat flour, rice and salt. The product will be refused entry if it does not meet the national standards for fortification. The fifth shipment of a fortified product will be checked at port of entry using a simple qualitative test to verify the product is fortified with a key indicator micronutrient. Products that fail the test will be refused entry. EHD [Environmental Health Division], Customs, and Quarantine need to agree on their respective roles and responsibilities in this system. These will be as per the Imported Food Control Guidelines, currently under development by the Ministry of Health and FAO." Solomon Islands Monitoring Framework summary, Pgs. 1-2. • 44. See: • 45. • 46. • “This manual recommends that mills in the Solomon Islands undertake daily iron spot tests. It recommends that composite samples be sent to a reliable external laboratory once a quarter to analyse iron and folic acid.” Solomon Islands Internal Monitoring Manual, Pg. 1. • “Once every quarter, send a composite quarterly sample to an external reference laboratory for the quantitative determination of iron and folic acid. Review and file the external laboratory’s findings.” Solomon Islands Internal Monitoring Manual, Pg. 5. • 47. “Your reference: Flour 16/11/15” Solomon Islands Vitamin Assay • 48. • 49. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 50. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 51. Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 52. Conversation with Scott Montgomery and Sarah Zimmerman, March 25, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 53. Googled “Solomon Islands population.” This figure is from 2013. • 54. For example, it told us (in E-mails about FFI's work in Kosovo, May 10, 2016) that: • Its staff member Robert Baldwin frequently visited Kosovo beginning in 2008. • Quentin Johnson, FFI's technical coordinator, conducted a quality assurance/quality control workshop in Kosovo in November 2011. • Kate Wheeler, an Emory graduate student, developed a monitoring plan for Kosovo in the summer of 2012. • A delegation from Kosovo attended an FFI Europe workshop in Turkey in June 2012. • FFI staff visited Kosovo in 2013, 2014, and 2016. • It has provided a variety of support to Agron Gashi, UNICEF’s main staff member working on fortification in Kosovo. • 55. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 56. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 57. • 58. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 59. In particular, FFI told us that it separated the legislation mandating that producers fortify from the technical standards that lay out the specific nutrient levels and compounds that should be used. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 60. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 61. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 62. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 63. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 64. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 65. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 66. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 67. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 68. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 69. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 70. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 71. Googled “Kosovo population” • 72. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 73. "The anemia prevalence in all the children of Kosovo was 15.7%, indicating a mild public health problem of anemia among school children." Kosovo Nutrition Survey 2010, Pg. 20. • 74. "the Hb was below 11.0g/dl, the cut-off for anemia among pregnant women (UNICEF/UNU/WHO, 2001), in 207 women or 23.0% (95% ci: 20.4-25.9), thus revealing a moderate public health problem of anemia among pregnant women." Kosovo Nutrition Survey 2010, Pg. 21. • 75. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 76. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 77. E-mails about FFI's work in Kosovo, May 10, 2016 • 78. • Kosovo Fortification spreadsheet • FFI also told us that there was another test of fortification run in 2014. All of the information that we have about it is from E-mails about FFI's work in Kosovo, May 10, 2016: "The UNICEF supported premix for flour arrived in early 2013, and fortification was officially launched on Feb. 7, 2014. The Institute of Agriculture in Peja tested 409 flour samples for iron and found that only 13% were not fortified." • 79. Kosovo Fortification spreadsheet • 80. See “Spending” section above. • 81. Note that the document also includes other staff positions that FFI would consider filling in the future, but does not provide funding estimates for these positions. • “Other future staff to consider: • Full-time training and technical support resource • Program manager • Development (including fund-raising) coordinator • Cost:benefit analyst FFI's Funding Gaps List, Pg. 4. • 82. See Pg. 3, FFI's Funding Gaps List: “Annual Total: 560,000”, “Non-Annual Total: 1,600,000”. • 83. “Bread and pasta are commonly consumed across Europe, but very little wheat flour is fortified there. Limited funding hampers FFI’s ability to respond to requests for assistance in Eastern Europe. UNICEF is a key partner in Eastern Europe, but other health interventions often are higher priorities to UNICEF staff in this region. With additional funding, FFI would provide a dedicated staff person to help selected countries in Eastern Europe make progress on planning, implementing, and monitoring wheat flour fortification programs. Country examples are Armenia, Kosovo, Moldova, and Ukraine.” FFI's Funding Gaps List, Pg. 1. (See document for cost estimate.) • 84. “China produced 117 million metric tons of wheat in 2011, according to FAO. With relatively minimal export volume, most of these grains are kept in China for domestic consumption. Yet fortification of wheat flour and rice has not been implemented as a strategy to improve nutrition and reduce neural tube birth defects among the country’s 1.3 billion people. The first priority would be a full-time person working in China with key partners to build advocacy and support for fortification.” FFI's Funding Gaps List, Pg. 2. (See document for cost estimate.) • 85. “As of March 2016, 26 countries in Africa had legislation requiring fortification of wheat flour, and eight of these countries also mandated fortification of maize flour. This is up from only two countries (Nigeria and South Africa) with such legislation in 2002. The national requirements often include fortifying salt with iodine and cooking oil with vitamin A. No country requires rice fortification though people in West Africa in particular consume more rice than wheat or maize flour. FFI and GAIN conducted an Africa rice supply chain assessment in 2015/2016. The next step is to execute a plan for rice fortification informed by the supply chain assessment.” FFI's Funding Gaps List, Pg. 2. (See document for cost estimate.) • 86. “Papua New Guinea has a national mandate for rice fortification, but the standard does not include folic acid. Also, wheat flour fortification has the potential to reach a target population here.” FFI's Funding Gaps List, Pg. 2. (See document for cost estimate.) • 87. • “Bangladesh has more than 300 wheat flour mills considered “industrial” meaning they have the capacity to produce at least 20 metric tons of flour a day. Yet fortification efforts in the country currently only include rice fortification – a work place benefit program and a social safety net program. Consumption of foods made with wheat flour is increasing, according to the 2010 Household Income and Expenditure Survey (HIES). Consequently a nutrition strategy for Bangladesh’s future will need to consider wheat flour fortification. FFI would hire a country coordinator to live in the country and collaborate with national leaders to build momentum for grain fortification and ensure that the elements of successful, sustainable programs are in place.” FFI's Funding Gaps List, Pg. 4. (See document for cost estimate.) • ”Mongolia has taken steps toward wheat flour fortification, and the UNICEF country office is supportive, but it is making little progress. FFI would hire a country coordinator to be based in Mongolia and collaborate with national leaders to build momentum for grain fortification and ensure that the elements of successful, sustainable programs are in place.” FFI's Funding Gaps List, Pg. 4. (See document for cost estimate.) • 88. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 89. Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication) • 90. Subtract$100,000 in annual rice coordination work from the total \$560K for the “Annual Total,” FFI's Funding Gaps List, Pg. 3.
• 91.
Conversation with Scott Montgomery, February 19, 2016 (no written notes, but FFI staff reviewed this page prior to publication)
• 92.
Conversation with Scott Montgomery, February 19, 2016 (no written notes, but FFI staff reviewed this page prior to publication)
• 93.
Conversation with Scott Montgomery and Sarah Zimmerman, May 10, 2016 (no written notes, but FFI staff reviewed this page prior to publication)
• 94.
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2017-01-21 04:28:24
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https://www.fd-seminar.xyz/
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Register for the ICRA 2020!
# Linear quasi-categories as templicial modules
This is joint work with my supervisor Wendy Lowen. After laying out the basics of quasi-categories as defined by Joyal, we introduce a notion of linear quasi-categories over a unital commutative ring. We make use of certain colax monoidal functors, which we call templicial modules, as a variant of simplicial modules respecting the monoidal structure. It turns out that templicial modules with a Frobenius monoidal structure are equivalent to (homologically) non-negatively graded dg-categories. Through this equivalence we can associate to any dg-category a linear quasi-category, the linear dg-nerve, which enhances the classical dg-nerve.
The complete list of previous talks at the FD Seminar is available here.
The talk will be broadcast through our BigBlueButton instance and will also be live-streamed (username: fd-seminar).
Subscribe to our mailing list to receive weekly annoucements and other important information about the FD Seminar.
### Upcoming talks
Click on the title to see the abstract.
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2020-10-31 01:20:32
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https://tex.stackexchange.com/questions/linked/56207
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### How can I obtain the date in two digit format? [duplicate]
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### Number chapters as 01, 02, … , 10, 11, … appendices A, B
Minimal example (the problem is obvious): \documentclass[]{report} \makeatletter \def\@makechapterhead#1{ \vspace*{-5.0em} {\parindent \z@ \normalfont \interlinepenalty\@M \LARGE{0\...
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I understand that trying to have a "loop function" like \foreach loop sounds like a paradox but I'll try to explain what I mean. Recently, I've asked a question about automatising the creation of ...
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I have a folder called diary which contains a series of files: 20120101.tex 20120105.tex 20120304.tex etc. How can I create a single LaTeX document that includes each of these files, in alphabetical ...
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### How can I use \DTLloaddb package to create a table based on csv file?
I am new in Latex and I have some issues. I have succeeded in reading the .csv file: ,w5,w6,w7,w8,w9,w10,w11,w12,w13,w14,w15,w16,w17,w18,w19,w20,w21,w22,w23,w24 ,25,23,13,14,15,39,36,4,37,11,26,13,34,...
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### Help with US Date Format
I have searched the isodate package and the datetime2 package to answer my question. I would like my date to be mm/dd/yyyy. The closest I have gotten is to have the date as dd/mm/yyyy. Could someone ...
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### Easy way to sectioning with a leading zero [duplicate]
There is a easy way to modify only this command \renewcommand\thesection{\arabic{section}.} in order to have an arabic counter with just one leading zero instead the original arabic without leading ...
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### Two digits chapter numbering in tableofcontents
Scraping through the documentation I managed to make so that \chapter generates a trailing 0 before the chapter number if the chapter number is smaller than 10. I would like to do the same for table ...
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### How to add a date into the name field of includepdf?
I'm using Sweave to automate the creation of a report, and I want to include a PDF into my document, but the PDF and its name are updated daily (because they are produced using Sweave). For example ...
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2020-02-27 08:37:39
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https://socratic.org/questions/a-sin-y-x-ye-2-x-x-y-b-y-4ln-x-cos-x-y-c-e-2y-tan-1-x-x-y-d-ln-y-2x-3-y-3-e-xy-s
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# (A) Sin(y/x) +ye^(2-x) = x-y (B) √y - 4ln x = cos (x+y) (C) e^(2y) + tan(1/x) =(x²/y) (D) ln y+2x =(3-y)^3 (E)xy²-sin(x+2y)=2x . Find the dy/dx (implicit differentiation)?
Aug 30, 2015
To keep the answer at a reasonable length, I will only show you parts (A) and (E).
#### Explanation:
So, for part (A) you have
$\sin \left(\frac{y}{x}\right) + y {e}^{2 - x} = x - y$
To get $\frac{\mathrm{dy}}{\mathrm{dx}}$, you need to use implicit differentiation. Differentiate both sides with respect to $x$
$\frac{d}{\mathrm{dx}} \left[\sin \left(\frac{y}{x}\right) + y {e}^{2 - x}\right] = \frac{d}{\mathrm{dx}} \left(x - y\right)$
To make the calculations easier to follow, I'll solve each of these derivatives separately. First, you have
$\frac{d}{\mathrm{dx}} \left[\sin \left(\frac{y}{x}\right)\right] = \cos \left(\frac{y}{x}\right) \cdot \frac{d}{\mathrm{dx}} \left(y \cdot {x}^{- 1}\right)$
$\frac{d}{\mathrm{dx}} \left[\sin \left(\frac{y}{x}\right)\right] = \cos \left(\frac{y}{x}\right) \cdot \left[\frac{\mathrm{dy}}{\mathrm{dx}} \cdot {x}^{- 1} + y \cdot {\left(- x\right)}^{- 2}\right]$
Next
$\frac{d}{\mathrm{dx}} \left(y {e}^{2 - x}\right) = \frac{\mathrm{dy}}{\mathrm{dx}} \cdot {e}^{2 - x} + y \cdot \frac{d}{\mathrm{dx}} \left({e}^{2 - x}\right)$
$\frac{d}{\mathrm{dx}} \left(y {e}^{2 - x}\right) = \frac{\mathrm{dy}}{\mathrm{dx}} \cdot {e}^{2 - x} + y \cdot {e}^{2 - x} \cdot \left(- 1\right)$
$\frac{d}{\mathrm{dx}} \left(y {e}^{2 - x}\right) = \frac{\mathrm{dy}}{\mathrm{dx}} \cdot {e}^{2 - x} - y \cdot {e}^{2 - x}$
Plug these back into your target calculation to get
$\cos \left(\frac{y}{x}\right) \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \cdot {x}^{- 1} + \cos \left(\frac{y}{x}\right) \cdot \left(- y {x}^{- 2}\right) + \frac{\mathrm{dy}}{\mathrm{dx}} \cdot {e}^{2 - x} - y \cdot {e}^{2 - x} = 1 - \frac{\mathrm{dy}}{\mathrm{dx}}$
Isolate $\frac{\mathrm{dy}}{\mathrm{dx}}$ on one side of the equation to get
$\cos \left(\frac{y}{x}\right) \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \cdot {x}^{- 1} + \frac{\mathrm{dy}}{\mathrm{dx}} \cdot {e}^{2 - x} + \left(\mathrm{dy}\right) = 1 + y \cdot {e}^{2 - x} + y {x}^{- 2} \cdot \cos \left(\frac{y}{x}\right)$
$\frac{\mathrm{dy}}{\mathrm{dx}} \cdot \left[{x}^{- 1} \cos \left(\frac{y}{x}\right) + {e}^{2 - x} + 1\right] = 1 + y \cdot {e}^{2 - x} + y {x}^{- 2} \cdot \cos \left(\frac{y}{x}\right)$
Finally, you have
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1 + y \cdot {e}^{2 - x} + y {x}^{- 2} \cdot \cos \left(\frac{y}{x}\right)}{1 + {e}^{2 - x} - {x}^{- 1} \cos \left(\frac{y}{x}\right)}$
You can simplify this to the form
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{x}^{2} + y \cdot {x}^{2} \cdot {e}^{2 - x} + y \cos \left(\frac{y}{x}\right)}{x} ^ \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \cdot \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{x}}}}{x + x \cdot {e}^{2 - x} - \cos \left(\frac{y}{x}\right)}$
(dy)/dx = color(green)((x^2 + y x^2e^(2-x) + ycos(y/x))/(x[x + xe^(2-x) - cos(y/x)])
For part (E) you have
$x {y}^{2} - \sin \left(x + 2 y\right) = 2 x$
The exact same approach applies here as well. Differentiate both sides with respect to $x$
$\frac{d}{\mathrm{dx}} \left[x {y}^{2} - \sin \left(x + 2 y\right)\right] = \frac{d}{\mathrm{dx}} \left(2 x\right)$
This will get you
$\frac{d}{\mathrm{dx}} \left(x {y}^{2}\right) = \frac{d}{\mathrm{dx}} \left(x\right) \cdot {y}^{2} + x \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \cdot 2 y$
$\frac{d}{\mathrm{dx}} \left(x {y}^{2}\right) = {y}^{2} + 2 x y \frac{\mathrm{dy}}{\mathrm{dx}}$
and
$\frac{d}{\mathrm{dx}} \left[\sin \left(x + 2 y\right)\right] = \cos \left(x + 2 y\right) \cdot \frac{d}{\mathrm{dx}} \left(x + 2 y\right)$
$\frac{d}{\mathrm{dx}} \left[\sin \left(x + 2 y\right)\right] = \cos \left(x + 2 y\right) \cdot \left[\frac{d}{\mathrm{dx}} \left(x\right) + 2 \frac{\mathrm{dy}}{\mathrm{dx}}\right]$
$\frac{d}{\mathrm{dx}} \left[\sin \left(x + 2 y\right)\right] = \cos \left(x + 2 y\right) \cdot \left(1 + 2 \frac{\mathrm{dy}}{\mathrm{dx}}\right)$
Plug these back into your target calculation to get
${y}^{2} + 2 x y \frac{\mathrm{dy}}{\mathrm{dx}} - \left[\cos \left(x + 2 y\right) + 2 \cos \left(x + 2 y\right) \frac{\mathrm{dy}}{\mathrm{dx}}\right] = 2$
${y}^{2} + 2 x y \frac{\mathrm{dy}}{\mathrm{dx}} - \cos \left(x + 2 y\right) - 2 \cos \left(x + 2 y\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 2$
Once again, isolate $\frac{\mathrm{dy}}{\mathrm{dx}}$ on one side
$\frac{\mathrm{dy}}{\mathrm{dx}} \cdot 2 \left[x y - \cos \left(x + 2 y\right)\right] = 2 - {y}^{2} + \cos \left(x + 2 y\right)$
(dy)/dx = color(green)((2 - y^2 + cos(x + 2y))/(2[xy - cos(x + 2y)])
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2021-06-23 00:11:37
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https://motls.blogspot.com/2014/07/gabor-melis-new-formidable-challenger.html?m=1
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## Thursday, July 31, 2014
### Joining the Kaggle Higgs 3.8+ club
Briefly, some news on Friday, August 1st. As I expected (see the text below), Tim Salimans is now ahead of Gábor Melis although his advantage is infinitesimal. (Friday 4 pm update: Melis is at the top again.) With the so far minor help of incredible variables from Christian Veelken of CMS, I (or counting the promised 10% share for CV, we) joined the club of those with the score above 3.8, see the leaderboard. Every contestant who is not a complete loser must feel safely above 3.8, so the score associated with my name is now 3.80007. ;-) I am not selecting that submission for the contest because I don't have all the sources that produced it – it was very complicated.
The text below was originally posted on July 24th.
## Gábor Melis' new formidable challenger
Tim Salimans makes the Terminator look like Pokémon
As recently as two hours ago, I thought it was conceivable that I would end up in the top three of the Higgs Kaggle challenge. See the leaderboard.
The top 5 contestants hadn't changed for a week. Gábor Melis was at the top followed by the Marijuana Hybrid guy, by your humble correspondent, and by 1,100+ other participants.
Terminator, Ironman, Batman, and a few Transformers as seen from the optics of a company in Utrecht.
Times are changing. For more than an hour, Tim Salimans of Utrecht, the Netherlands has been the new #2 warrior. His 7th submission with the score 3.81888 catapulted him to that place and made the victory of Gábor Melis uncertain.
Almost all contestants at the top are experienced machine learning software experts – your humble correspondent is a true rural bumpkin in this company (my experience with machine learning and computers is that I managed to jump a few trains on Subway Surfers and solve a hard level at Candy Crush Saga after 500 attempts) – but Tim Salimans makes even most of the urban contestants look like bumpkins.
Just to be sure, his Kaggle profile says that he has won (!) 4 previous Kaggle contests, including one on dark matter data, was the 2nd several times, too, 10 times in top ten (in total), and he has hosted his own Kaggle contest, too. More shockingly, he is
[a f]ounding partner and data scientist at predictive analytics consulting firm Algoritmica, with a PhD in computational Econometrics and a strong academic background in Machine Learning.
The company web, algoritmica.nl, explains that
Algoritmica combines machine-learning algorithms with the power of supercomputers to build unparalleled predictive models for marketing, risk, fraud, supply chains, and maintenance. We lead companies around the world from average business processes to a truly data-driven organization. Empowered with predictive models, these companies learn from data to stay ahead of the competition, cut waste, and delight customers.
Algoritmica also supervises the NSA and FBI and keeps track of all the data and patterns in the 2 trillion telephone calls and e-mails that they record every month.
OK, I added the last sentence but it may be true, anyway.
Salimans seems to have no specific training in physics but it's clear that he does care what the LHC collision data mean. In a question he had posted to the Kaggle forums, he was asking where he could find the algorithm used by the ATLAS Collaboration to estimate the Higgs mass from the candidate event. This is a rather difficult calculation whose result, the MMC mass, is the first "feature" describing each event and by far the most complicated "derived" quantity calculated from the raw collision data.
I am pretty sure that by today, he has incorporated the improved version of the MMC mass estimator to his supercomputer superprograms. In fact, I find it likely that he has added the CMS' not-so-frequently used alternative to the MMC estimator, the (N)SVFIT algorithm, as well, and the help of the (N)SVFIT feature as an added one may help one to jump above 3.8 even if other things are lousy. I was thinking about adding (N)SVFIT but it's a rather complicated program that I would have to reverse-engineer, rewrite from scratch, and you know, two hours ago, I felt that I would be the only contestant to waste my time in this way.
Whatever Salimans has exactly done, I feel that it's ludicrous to try to compete with such a monster. My mobilization against him is only going to be as symbolic as the Czechoslovak army's mobilization against the Third Reich right after the Munich Betrayal, in September 1938. ;-) My codes and software infrastructure is based on several legs and lots of partial cute ideas. But I don't even have any systematic "quality control", like strictly dividing the training dataset to training and validation. I am sure that he not only does so but does so dynamically, with some meta-machine-learning that adjusts the learning computer to make it learn better than the previous programs, and so on. The possibilities are endless.
Of course, it's great if really powerful guys like this one make their job and switch from econometrics to particle physics at least once. I hope that it will be useful for the LHC research, too. If his (or other commercial professionals') methods are significantly more effective than those at the LHC, I believe that CERN should simply hire them or buy their software etc. to perform similar tasks. If the LHC experimenters are "clearly amateurs" in comparison, they should admit it and CERN should fire some of them and replace them by true professionals.
On the other hand, if his AMS score got stuck at the current level just 0.03 above the score of your humble correspondent who is doing all these things with a $0 software on a$500 laptop and with 0 pre-contest experience with machine learning software, it would be rather stupid to pay millions and millions of dollars to a special Dutch machine learning company designed to conquer the world. ;-)
My respect to the Dutchmen's sophistication is immense and they have my condolences after the downing of the MH17 flight. However, it's probably more natural for me to root for the fellow Austrian-Hungarian Visegrád guy now. Gábor, István, Balázs, Jánosz, go, go, go! ;-)
1. Penalize any competitor backed by less than €100K in hardware. Somebody is ruining process, PERT chart, and budgeting by being competent. Obscuration is fundable. Solution is a one-trick pony.
2. My guess is that Salisman is building a probabilistic model (At least that is what he has being doing in the all other competions). Ultimately, I think that is much more natural approach for the problem than any decission tree based algorithms.
So, I think your might be on the right track with the SVFIT thing. It's a relatively straightforward probability model of the event, but at least the basic version is missing all the information from the jets etc.
3. He wrote me that he doesn't want to blog about ideas because he's being conservative. Of course no technical details could be said but I would personally guess he is doing decision trees in the right way.
4. Ooh, that's nasty, Al.
Anyone would think you had something in general against state-sector operatives expanding their fiefdoms and feathering their own nests! What's wrong with you, man — don't you believe in the big rock candy mountain public titty?
OK, I guess you're not exactly persuaded that all these people are needed. But just imagine what the employment figures would look like without those 'jobs'!
:)
5. I respect your efforts, so if you want to easily add the SVFIT to your toolbox, you can find a standalone version:
https://github.com/search?q=svfit&ref=opensearch
You'll have to link with ROOT and the HiggsML team does not give the MET covariance matrix so you have to assume something.
6. LOL, I have had this Standalone version for 5 hours and 3 minutes ago, I managed to install Root correctly.
Yes, I am just reading papers how to calculate a reasonable covariance matrix. If I won't find a reasonable refined enough formula, I will use something like unit matrix times 200 GeV^2, or something like that. ;-)
7. Off-topic:
Is Lawrence M. Krauss crackpot?
http://thebulletin.org/climate-change-if-we-pretend-it-isn%E2%80%99t-happening-will-it-go-away7333
8. Not really off-topic: [Algoritmica builds] predictive models for marketing, risk, fraud, supply chains, and maintenance. Please note an absence of climate. Or weather. Apparently we know more about fraud than about climate.
9. Did you really mean /usr/include/root or rather /usr/root/include?Anyways, does the pathcontain the ".h"-files of the relevant libraries? If not, just find out where they are located and add those paths to your include statement. In general it may be necessary to obtain and install the relevant libraries (including their source code for linking) first. Getting the includes to work is usually straightforward and should not take much time, though.
10. Look like you are well on your way to get the sw going.
You can cp all the include.h files for the libs to a single Dir of your choosing and use them there. You have to specify the libs by name with a -l.
You should be able to do 1000 records per second with floating point hardware. Just don't fry your laptop. They tend to get hot ;--(
Post a error log if you need help ;--(
Have fun. ;--)
11. "If the LHC experimenters are "clearly amateurs" in comparison, they
should admit it and CERN should fire some of them and replace them by
true professionals.".
Experimenters in HEP are always amateurs in all these technical stuff, by construction. They spent their years in learning physics, not the behavior of bits and bytes . They pick up knowledge as necessary for the specific set up, and work as best as they can.trying to squeeze out of the data the maximum information. And they are not hired by CERN. They belong to individual HEP groups as physicists and often professors in their respective institutes.
I do not think that if LHC analysis was run as an engineering project, as you suggest I hope jokingly, if it would be more successful in finding new physics. Engineering mentality usually works on fixed goals, not on searches for new phenomena thinking outside the box.
12. Dear Anna, I wasn't really joking. Of course that I think that true physicists who may be amateurs in most other things are *essential* in HEP experiments.
On the other hand, lots of work is being done by highly specialized people who are really supposed to be good at something - not necessarily "physics" (and especially "theoretical physics") related - in engineering tasks. If someone is really doing data mining only, he should be fully exposed to the competition on the job market of other people who know how to do data mining.
13. Haha! Oh very good! Point nicely made. :)
(I assume it's genuine since it looks highly plausible and I have no reason to think otherwise. BTW I wouldn't have known your MSSM required so many parameters. That seems a lot.)
14. http://en.wikipedia.org/wiki/Minimal_Supersymmetric_Standard_Model
"120 new parameters" so it in fact requires way over 120 in summation.
15. Gábor, István, Balázs, Jánosz, go, go, go! ;-)
Lubos what are you doing ? This looks like a betrayal of everything we stood for in KuK ! Did you forget sapér Vodicka ? :)
"To by tak jeste schazelo," rozdurdil se Vodicka, "aby nam jeste ten Mad'ar chtel hodit neco na hlavu. Ja ho chytnu za krk a shodim ho z prvniho poschodi dolu po schodech, ze poleti jako srapnel. Na ty kluky mad'arsky se musi jit vostre. Jakypak s nimi cavyky."
16. LOL, yes, sorry for that change, Tom! But haven't some things about the Hungarians and their relations to us changed over the century? ;-)
17. I actually joined the challenge officially last couple of hours. My thing is 3.41359 on 4 entries. I am almost done with some special juice, Consider this the kaggle equivalent of a "roof knock" . Ktahn should easily be top 5 once this juice is complete.
18. What took you so long. ;--)
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2021-05-18 23:12:26
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https://proofwiki.org/wiki/Category:Examples_of_Use_of_Product_of_Complex_Numbers_in_Polar_Form
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# Category:Examples of Use of Product of Complex Numbers in Polar Form
Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be complex numbers expressed in polar form.
$z_1 z_2 = r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \, \map \sin {\theta_1 + \theta_2} }$
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2020-09-29 15:54:28
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https://cstheory.stackexchange.com/questions/4081/what-is-the-pathwidth-of-the-3d-grid-mesh-or-lattice-with-sidelength-k
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# What is the pathwidth of the 3D-grid (mesh or lattice) with sidelength k?
I asked this question some weeks ago at mathoverflow, but I got no reply.
Here, by 3D-grid of sidelength $k$ I mean the graph $G=(V,E)$ with $V= \{1,\ldots,k\}^3$ and $E=\{( (a,b,c) ,(x,y,z) ) \mid |a-x|+|b-y|+|c-z|=1 \}$, i.e., the nodes are placed at 3-dimensional integer coordinates between 1 and $k$, and a node is connected to the at most 6 other nodes that differ in precisely one coordinate by one.
What is the name of this graph? I'll use 3D grid, but perhaps 3D mesh or 3D lattice are what other people are used to.
What is the treewidth or pathwidth of this graph? Is this already published somewhere?
I know already that $tw(G) = (3/4) k^2 + O(k)$, i.e. it is really smaller than $k^2$. To me, this suggests that the standard arguments showing that the $k\times k$ 2D-grid has treewidth and pathwidth $k$ will not easily generalize.
To see this, we consider a path decomposition that "sweeps" the grid using mainly node-sets of the form $S_c= \{(x,y,z)\mid x+y+z = c\}$. Observe $|S_c| \leq (3/4) k^2 + O(k)$, $S_{3/2 k}$ being the largest such set. The sets between $S_c$ and $S_{c+1}$ are created by sweeping with a line and need $O(k)$ additional nodes to be separators. More precisely, use the sets $S_{c,d} = \{(x,y,z)\mid (x+y+z = c \wedge x \leq d ) \vee (x+y+z = c \wedge x \geq d ) \}$ as a path decomposition of $G$.
I also have an idea for a proof that shows $tw(G) = \Omega(k^2)$, but that is not finished yet.
• $|S_c| = \Omega(k^2)$ for $c=\lfloor k/2 \rfloor$. Am I missing something? – Sariel Har-Peled Jan 4 '11 at 13:21
• Sure. But $S_c$ is only used in the upper bound. What I really care about is a lower bound. – Riko Jacob Jan 4 '11 at 13:32
• You may be interested in this paper: springerlink.com/content/3nmjlc1g5emx9vpk. If you can calculate the "queue number" of your graph, then you'll be given a lower bound on its path-width using Theorem 1 which states that $\mathsf{qn}(G) \leq \mathsf{pw}(G)$ for any graph $G$. – Mathieu Chapelle Jan 4 '11 at 17:57
• Oh. I see. You meant $(3/4)k^2$. – Sariel Har-Peled Jan 4 '11 at 23:22
• @Sariel: I edited the question to avoid the same confusion. – Tsuyoshi Ito Jan 5 '11 at 0:27
The pathwidth of $P^3_k$ can be determined as a corollary to some known results. FitzGerald [2] showed that the bandwidth of $P^3_k$ is $\lfloor \frac{3}{4} k^{2} + \frac{1}{2} k \rfloor$. Harper [3] showed a condition such that if a graph satisfies the condition, then its pathwidth and bandwidth are the same. Moghadam [4,5] and Bollobás and Leader [1] independently showed that any multi-dimensional grid satisfies Harper's condition. These results imply that the pathwidth of $P^3_k$ is also $\lfloor \frac{3}{4} k^{2} + \frac{1}{2}k \rfloor$.
In our paper mentioned by Hsien-Chih, we generalized FitzGerald's result as Yoshio explained. I believe the treewidth of $P^3_k$ is not known.
FYI: I've just submitted an English version of our paper to arXiv.
1. B. Bollobás and I. Leader, Compressions and isoperimetric inequalities, J. Combin. Theory Ser. A 56 (1991) 47-62.
2. C.H. FitzGerald, Optimal indexing of the vertices of graphs, Math. Comp. 28 (1974), 825-831.
3. L.H. Harper, Optimal numberings and isoperimetric problems on graphs, J. Combin. Theory 1 (1966) 385-393.
4. H.S. Moghadam, Compression operators and a solution to the bandwidth problem of the product of $n$ paths, Ph.D. thesis, University of California, Riverside (1983).
5. H.S. Moghadam, Bandwidth of the product of $n$ paths, Congr. Numer. 173 (2005) 3-15.
• Thank you for kindly sharing your new result (and paper!) Also, welcome to TCS SE :) – Hsien-Chih Chang 張顯之 Jan 6 '11 at 1:33
• @Hsien-Chih: You made me decide to share our result :-) Thanks. In fact, I'm also new for arXiv. – Yota Otachi Jan 6 '11 at 4:30
The pathwidth of 3D-grids has been studied by Ryohei Suda, Yota Otachi and Koichi Yamazaki in the paper Pathwidth of 3-dimensional grids, IEICE Tech. Report, 2009.
It is claimed in the abstract of the paper that
In this paper, we give the pathwidth of 3-dimentional grids in closed form, by determining their vertex boundary width.
However the precise bound is not stated in the abstract, and currently I cannot access the full paper. Maybe you can contact the authors privately, and post an answer to this question by yourself, if the authors are willing to share the result.
• Note that the paper is written in Japanese. – Tsuyoshi Ito Jan 5 '11 at 4:48
• @Tsuyoshi: Yes, we may need your help :) – Hsien-Chih Chang 張顯之 Jan 5 '11 at 4:52
• I have a physical access to the manuscript (and can understand Japanese). According to the authors, the pathwidth of $P_{\ell}\times P_{m}\times P_{n}$ is $\ell m$ if $\ell + m \leq n + 2$ and $\ell m - \lceil (\frac{\ell+m-n-1}{2})^2 \rceil$ otherwise, where $P_k$ is a path with $k$ vertices, and $\ell \leq m \leq n$. – Yoshio Okamoto Jan 5 '11 at 6:12
• @Yoshio: This deserves to be an answer, since it implies $\mathsf{pw}(P^3_k) = \frac{3}{4}k^2 + O(k)$, which answers the question. – Hsien-Chih Chang 張顯之 Jan 5 '11 at 7:28
• Thanks. Looks like I don't have to feel bad for not finding that reference myself. I am curious for the details. – Riko Jacob Jan 5 '11 at 8:13
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## Dynamics of the Weil-Petersson flow: basic geometry of the Weil-Petersson metric II
Originally posted on Disquisitiones Mathematicae:
In the first post of this series, we planned to discuss in the third and fourth posts the proof of the following ergodicity criterion for geodesic flows in incomplete negatively curved manifolds of Burns-Masur-Wilkinson:
Theorem 1 (Burns-Masur-Wilkinson) Let $latex {N}&fg=000000$ be the quotient $latex {N=M/Gamma}&fg=000000$ of a contractible, negatively curved, possibly incomplete, Riemannian manifold $latex {M}&fg=000000$ by a subgroup $latex {Gamma}&fg=000000$ of isometries of $latex {M}&fg=000000$ acting freely and properly discontinuously. Denote by $latex {overline{N}}&fg=000000$ the metric completion of $latex {N}&fg=000000$ and $latex {partial N:=overline{N}-N}&fg=000000$ the boundary of $latex {N}&fg=000000$.Suppose that:
• (I) the universal cover $latex {M}&fg=000000$ of $latex {N}&fg=000000$ is geodesically convex, i.e., for every $latex {p,qin M}&fg=000000$, there exists an unique geodesic segment in $latex {M}&fg=000000$ connecting $latex {p}&fg=000000$ and $latex {q}&fg=000000$.
• (II) the metric completion $latex {overline{N}}&fg=000000$ of $latex {(N,d)}&fg=000000$ is compact.
• (III) the boundary $latex {partial N}&fg=000000$ is volumetrically cusplike, i.e., for…
View original 7,622 more words
## Dynamics of the Weil-Petersson flow: basic geometry of the Weil-Petersson metric I
Originally posted on Disquisitiones Mathematicae:
Today we will define the Weil-Petersson (WP) metric on the cotangent bundle of the moduli spaces of curves and, after that, we will see that the WP metric satisfies the first three items of the ergodicity criterion of Burns-Masur-Wilkinson (stated as Theorem 3 in the previous post).
In particular, this will “reduce” the proof of the Burns-Masur-Wilkinson theorem (of ergodicity of WP flow) to the verification of the last three items of Burns-Masur-Wilkinson ergodicity criterion for the WP metric and the proof of the Burns-Masur-Wilkinson ergodicity criterion itself.
We organize this post as follows. In next section we will quickly review some basic features of the moduli spaces of curves. Then, in the subsequent section, we will start by recalling the relationship between quadratic differentials on Riemann surfaces and the cotangent bundle of the moduli spaces of curves. After that, we will introduce the Weil-Petersson and the Teichmüller metrics…
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## Dynamics of the Weil-Petersson flow: Introduction
Originally posted on Disquisitiones Mathematicae:
Boris Hasselblatt and Françoise Dal’bo are organizing the event “Young mathematicians in Dynamical Systems” at CIRM (Luminy/Marseille, France) from November 25 to 29, 2013.
This event is part of the activities around the chaire Jean-Morlet of Boris Hasselblatt. Among the topics scheduled in this event, there is a mini-course by Keith Burns and myself around the dynamics of the Weil-Petersson (WP) geodesic flow.
In our mini-course, Keith and I plan to cover some aspects of Burns-Masur-Wilkinson theorem on the ergodicity of WP flow and, maybe, some points of our joint work with Masur and Wilkinson on the rates of mixing of WP flow.
In order to help me prepare my talks, I thought it could be a good idea to make my notes available on this blog.
So, this post starts a series of 6 posts (vaguely corresponding the 6 lectures of the mini-course) on the dynamics of…
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## Martingale and its application to dynamical systems
In the last week of May I attended two lectures given by Professor Matthew Nicol.
Let $(\Omega,\mu)$ be a prob space with a $\sigma$-algebra $\mathcal{B}$. Let $\mathcal{F}\prec \mathcal{B}$ be a sub $\sigma$-algebra.
Example. $f(x)=2x (\text{mod} 1)$ on $\mathbb{T}$, and $\mathcal{B}$ be the Borel $\sigma$-algebra. Let $\mathcal{F}=f^{-1}\mathcal{B}$. Note that $(0.2,0.3)\notin\mathcal{F}$.
Let $Y$ be a $\mathcal{B}$-measurable r.v. and $Y\in L^1(\mu)$. The conditional expectation $E(Y|\mathcal{F})$ is the unique $\mathcal{F}$-measurable r.v. $Z$ satisfying $Z^{-1}(a,b)\in \mathcal{F}$ for all $(a,b)$, and $\int_F Z d\mu=\int _A Y d\mu$ for all $A\in \mathcal{F}$.
Note that $E(Y|\mathcal{F})=Y$ if and only if $Y$ is $\mathcal{F}$-measurable; and $E(Y|\mathcal{F})=E(Y)$ if $Y$ is independent of $\mathcal{F}$.
Let $(X_n)_{n\ge 0}$ be a stationary ergodic process with stationary initial distribution $\mu$. A basic problem is to find sufficient conditions on $(X_n)_{n\ge 0}$ and on functions $\phi\in L^2_0(\mu)$ such that $\displaystyle S_n(\phi)=\sum_{k=1}^n \phi(X_k)$ satisfies the central limit theorem (CLT) $\displaystyle \frac{1}{\sqrt{n}}S_n(\phi) \to N(0,\sigma^2)$, where the limit variance is given by $\displaystyle \sigma^2(\phi)=\lim_{n\to\infty}\frac{1}{n}E(S^2_n(\phi))$.
Let $f$ be a conservative diffeomorphism on $(M,m)$. There are two operators: $\phi\mapsto U\phi=\phi\circ f$, and $\phi\mapsto P\phi$ via $\int P\phi\cdot \psi=\int \phi\cdot \psi\circ f$ for all test function $\psi$.
Property. $PU(\phi)=\phi$ (vol-preserving) and $UP(\phi)=E(\phi|f^{-1}\mathcal{B})$.
Let $\mathcal{F}_n$ be an increasing sequence of $\sigma$-algebras. Then a sequence of r.v. $S_n$ is called a martingale w.r.t. $\mathcal{F}_n$, if $S_n$ is $\mathcal{F}_n$-measurable, $E(S_{n+1}|\mathcal{F}_n)=S_n$.
Let $\mathcal{F}_n$ be a decreasing sequence of $\sigma$-algebras. Then a sequence of r.v. $S_n$ is called a reverse martingale w.r.t. $\mathcal{F}_n$, if $S_n$ is $\mathcal{F}_n$-measurable, $E(S_{n}|\mathcal{F}_m)=S_m$ for any $n\le m$.
Theorem. Let $\{X_n:n\ge 1\}$ be a stationary ergodic sequence of (reverse) martingale differences w.r.t. $\{\mathcal{F}_n\}$. Suppose $E(X_n)=0$, and $\sigma^2=\text{Var}(X_i)>0$. Then $\displaystyle \frac{1}{\sigma\sqrt{n}}\sum_{i=1}^n X_i \to N(0,1)$ in distribution.
Gordin: Suppose $(f,m)$ is ergodic. Consider the Birkhoff sum $\displaystyle \sum_{i=1}^n \phi\circ f^i$ for some $\phi$ with $\int \phi=0$. The time series $\phi\circ f^i$ can be approximated by martingale differences provided the correlations decay quickly enough.
Suppose there exists $p(n)$ with $\sum p(n) < \infty$, such that $\|P^n\phi\|\le C\cdot p(n)\|\phi\|$. Then define $\displaystyle g=\sum_{n\ge 1}P^n\phi$, and let $X=\phi+g-g\circ f$.
Property. Let $f:M\to M$ be such that $f^{-n}\mathcal{B}$ is decreasing. $\displaystyle S_n=\sum_{i=1}^n X\circ f^i$ is a reverse martingale with respect to $f^{-n}\mathcal{B}$.
Proof. Note that $PX=P\phi+Pg-PUg=0$. Then $E(X|f^{-1}\mathcal{B})=UP(X)=U0=0$.
Let $k < n$. It remains to show $E(X\circ f^k|f^{-n}\mathcal{B})=0$. To this end, we pick an element $A\in f^{-n}\mathcal{B}$ and write it as $A=f^{-k-1}C$ for some $C\in f^{k+1-n}\mathcal{B}$. Then $\displaystyle \int_A X\circ f^k dm=\int_{f^{-1}C}X dm =\int_{f^{-1}C} E(X|f^{-1}\mathcal{B}) dm=\int_{f^{-1}C}0 dm=0$. This completes the proof.
Three theorems of Gordin. Let $(\Omega,\mu,T)$ be an invertible $\mu$-preserving ergodic system, $X\in L^1(\mu)$ and $X_k(x)=X(T^kx)$ be a strictly stationary ergodic sequence.
(*) $\displaystyle \limsup_{n\to\infty}\frac{1}{\sqrt{n}}E|S_n| < \infty$
Theorem 1. Suppose there exists $\displaystyle \mathcal{F}_k\subset T^{-1}\mathcal{F}_k=\mathcal{F}_{k+1}$ such that $\displaystyle \sum_{k\ge 0} E|E(X_0|\mathcal{F}_{-k})|<\infty$, $\displaystyle \sum_{k\ge 0} E|X_0-E(X_0|\mathcal{F}_{k})| < \infty$. Then (*) implies $\displaystyle \lambda:=\lim_{n\to\infty}\frac{1}{\sqrt{n}}E|S_n|$ exists, and $\displaystyle \frac{1}{\sqrt{n}}S_n\to N(0,\lambda^2\pi/2)$ in distribution (degenerate if $\lambda=0$).
–Mixing condition. Let $\displaystyle \alpha(n):=\sup\{P(A\cap B)-P(A)P(B):A\in\mathcal{F}^0_{-\infty}, B\in\mathcal{F}^{\infty}_n\}$.
Theorem 2. Suppose for some $1/p+1/q=1$, $X\in L^p(\mu)$ and $\displaystyle \sum_{n\ge 1}\alpha(n)^{1/q} < \infty$. Then (*) implies the conclusion of Theorem 1.
–uniform mixing condition. Let $\displaystyle \phi(n):=\sup\{P(B|A)-P(B):A\in\mathcal{F}^0_{-\infty}, \mu(A) > 0, B\in\mathcal{F}^{\infty}_n\}$.
Theorem 3. Suppose $X\in L^1(\mu)$ and $\displaystyle \sum_{n\ge 1}\phi(n) < \infty$. Then (*) implies the conclusion of Theorem 1.
Cuny–Merlevede: not only the CLT, but also the ASIP holds under the above conditions.
Note that we started with an invariant measure $m$. The operator $U$ and $P$ can be defined for all non-conservative maps. To emphasize the difference, we use $\hat P$. Suppose $\hat P h=h$ for some $h\in L^1(m)$. Then $\mu=hm$ is an absolutely continuous invariant prob. measure:
$\displaystyle \int \phi\circ f d\mu=\int \phi\circ f h dm=\int \phi\cdot \hat P h dm=\int \phi hdm=\int\phi d\mu$.
Then we can rewrite $\displaystyle P\phi=\frac{1}{h}\hat P(h\phi)$, in the sense that $\displaystyle \int P(\phi)\cdot \psi d\mu=\int \phi\cdot \psi\circ f d\mu =\int \phi h\cdot \psi\circ f dm$
$\displaystyle =\int\hat P(\phi h)\cdot \psi dm \int \frac{1}{h}\hat P(\phi h)\cdot \psi d\mu$.
## Perron–Frobenius theorem
Today I attended a lecture given by Vaughn Climenhaga. He presented a proof of the following version of Perron–Frobenius theorem:
Let $\Delta\subset \mathbb{R}^d$ be the set of probability vectors, $P$ be a stochastic matrix with positive entries. Then
–there is a positive probability $\pi\in \Delta$ fixed by $P$
–the eigenspace $E_1=\mathbb{R}\pi$
–the spectra $\Sigma(P)\subset B(0,r)\cup\{1\}$ for some $r<1$
–for all $v\in\Delta$, $P^n v\to \pi$ exponentially as $n\to \infty$.
Proof. (1) Let $v\in\Delta$. Then $\sum_i v_i=1$, and
$\sum_i (Pv)_i=\sum_i \sum_j p_{ij}v_j=\sum_j v_j=1$. So $Pv\in \Delta$. Moreover, $Pv$ is positive and $P(\Delta)\subset \text{Int}(\Delta)$. Therefore there exists some point $\pi\in\text{Int}(\Delta)$ fixed by $P$.
(2). Suppose on the contrary that there exists $v\notin \mathbb{R}\pi$ that is also fixed by $P$. Then $P$ fixes every vector in the plane $\Pi:=\mathbb{R}v\oplus\mathbb{R}\pi$, in particular the points on $\Pi\cap \partial \Delta$. This contradicts (1).
(3). We use the norm $|v|=\sum|v_i|$. Note that $|Pv|=\sum_i |(Pv)_i|\le \sum_{ij}p_{ij}|v_j|=|v|$. So $\Sigma(P)\subset D(0,1)$. It suffices to show $\Sigma(P)\backslash\{1\}\cap S^1=\emptyset$. If not, pick one ,say $\lambda$, and $n\ge 1$ such that $\text{Re}(\lambda^n)1$ for any $\epsilon>0$.
Consider the matrix $A=P^n-\epsilon I$, which is positive if $\epsilon$ is small enough. Then we have $|A|\le |P_n|$ and hence $\Sigma(A)\subset D(0,1)$. This contradicts the fact $\lambda^n-\epsilon$ is an eigenvalue of $A$.
(4). Let $W\subset \mathbb{R}^d$ be the subset of vectors with zero mean: $\sum v_i=0$, and consider the decomposition $\mathbb{R}^d=\mathbb{R}\pi\oplus W$. Note that $PW\subset W$ and hence $\Sigma(P|_{W})\subset D(0,r)$. For any $v\in\Delta$, we have $v=\pi+w$ for some $w\in W$. Then $|P^nv-\pi|=|P^n(v-\pi)|=|P^nw|\le Cr^n|w|$.
Only light calculations are used in his lecture. As pointed by Vaughn, this approach does not give precise information of the $r$.
## Some notes
Let $M$ be a complete manifold, $\mathcal{K}_M$ be the set of compact/closed subsets of $M$. Let $X$ be a complete metric space.
A map $\phi: X\to \mathcal{K}_M$ is said to be upper-semicontinuous at $x$, if
for any open neighbourhood $U\supset \phi(x)$, there exists a neighbourhood $V\ni x$, such that $\phi(x')\subset U$ for all $x' \in V$.
or equally,
for any $x_n\to x$, and any sequence $y_n\in \phi(x_n)$, the limit set $\omega(y_n:n\ge 1)\subset \phi(x)$.
Viewed as a multivalued function, let $G(\phi)=\{(x,y)\subset X\times M: y\in\phi(x)\}$ be the graph of $\phi$. Then $\phi$ is u.s.c. if and only if $G(\phi)$ is a closed graph.
And $\phi$ is said to be lower-semicontinuous at $x$, if
for any open set $U$ intersecting $\phi(x)$, there exists neighbourhood $V\ni x$ such that $\phi(x')\cap U\neq\emptyset$ for all $x'\in V$.
or equally, for any $y\in \phi(x)$, and any sequence $x_n\to x$, there exists $y_n\in \phi(x_n)$ such that $y\in \omega(y_n:n\ge 1)$.
Let $\mathrm{Diff}^r(M)$ be the set of $C^r$ diffeomorphisms, and $H(f)$ be the closure of transverse homoclinic intersections of stable and unstable manifolds of some hyperbolic periodic points of $f$. Then $H$ is lower semicontinuous.
Given $f_n\to f$. Note that it suffices to consider those points $x\in W^s(p,f)\pitchfork W^u(q,f)$. Let $p_n$ and $q_n$ be the continuations of $p$ and $q$ for $f_n$. Pick $\rho$ large enough such that $x\in W^s_\rho(p,f)\pitchfork W^u_\rho(q,f)$. Then for $f_n$ sufficiently close to $f$, $W^s_\rho(p_n,f_n)$ and $W^u_\rho(q_n,f_n)$ are $C^1$ close to $W^s_\rho(p,f)$ and $W^u_\rho(q,f)$. In particular $x_n\in W^s_\rho(p_n,f_n)\pitchfork W^u_\rho(q_n,f_n)$ is close to $x$.
## Admissible perturbations of the tangent map
Franks’s Lemma is a major tool in the study of differentiable dynamical systems. It says that along a simple orbit segment $E=\{x,fx,\cdots,f^nx\}$, the perturbation of $A\sim D_xf^n$ can be realized via a perturbation of the map $g\sim f$ (which preserves the orbit segment). Moreover, such a perturbation is localized in a neighborhood of $E$, and it can be made arbitrarily $C^1$-close to $f$.
There have been various generalizations of Franks’ Lemma. Some constraints have been noticed when generalizing to geodesic flows and billiard dynamics, since one can’t perturb the dynamics directly, but have to make geometric deformations. See D. Visscher’s thesis for more details.
Let $Q$ be a strictly convex domain, $x$ be the orbit along the/a diameter of $Q$. Clearly $x$ is 2-period. Let $r\le R$ be the radius of curvatures at $x, fx$, respectively. Then
$D_xf^2=\frac{1}{rR}\begin{pmatrix}2d(d-r-R)+rR & 2d(d-R)\\ 2(d-r)(d-r-R) & 2d(d-r-R)+rR\end{pmatrix}$, where $d$ stands for the diameter of $Q$.
Note that the two entries on the diagonal are always the same. Therefore any linearization with different entries on the diagonal can’t be realized as the tangent map along a periodic billiard orbit of period 2. In other words, even through there are three parameters that one can change: the distance $d$, the radii of curvature at both ends $r,R$, the effects lie in a 2D-subspace $\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:ad-bc=1, a=d\}$ of the 3D $\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:ad-bc=1\}$.
Visscher was able to prove that generically, for each periodic orbit of period at least 3, every small perturbation of $D_xF^3$ is actually realizable by deforming the boundary of billiard table. For more details, see Visscher’s paper:
A Franks’ lemma for convex planar billiards.
## Regularity of center manifold
Let $X:\mathbb{R}^d\to \mathbb{R}^d$ be a $C^\infty$ vector field with $X(o)=0$. Then the origin $o$ is a fixed point of the generated flow on $\mathbb{R}^d$. Let $T_o\mathbb{R}^d=\mathbb{R}^s\oplus\mathbb{R}^c\oplus\mathbb{R}^u$ be the splitting into stable, center and unstable directions. Moreover, there are three invariant manifolds (at least locally) passing through $o$ and tangent to the corresponding subspaces at $o$.
Theorem (Pliss). For any $n\ge 1$, there exists a $C^n$ center manifold $C^n(o)=W^{c,n}(o)$.
Generally speaking, the size of the center manifold given above depends on the pre-fixed regularity requirement. Theoretically, there may not be a $C^\infty$ center manifold, since $C^n(o)$ could shrink to $o$ as $n\to\infty$. An explicit example was given by van Strien (here). He started with a family of vector fields $X_\mu(x,y)=(x^2-\mu^2, y+x^2-\mu^2)$. It is easy to see that $(\mu,0)$ is a fixed point, with $\lambda_1=2\mu<\lambda_2=1$. The center manifold can be represented (locally) as the graph of $y=f_\mu(x)$.
Lemma. For $n\ge 3$, $\mu=\frac{1}{2n}$, $f_\mu$ is at most $C^{n-1}$ at $(\frac{1}{2n},0)$.
Proof. Suppose $f_\mu$ is $C^{k}$ at $(\frac{1}{2n},0)$, and let $\displaystyle f_\mu(x)=\sum_{i=1}^{k}a_i(x-\mu)^i+o(|x-\mu|^{k})$ be the finite Taylor expansion. The vector field direction $(x^2-\mu^2, y+x^2-\mu^2)$ always coincides with the tangent direction $(1,f'_\mu(x))$ along the graph $(x,f_\mu(x))$, which leads to
$(x^2-\mu^2)f_\mu'(x)=y+x^2-\mu^2=f_\mu(x)+x^2-\mu^2$.
Note that $x^2-\mu^2=(x-\mu)^2+2\mu(x-\mu)$. Then up to an error term $o(|x-\mu|^{k})$, the right-hand side in terms of $(x-\mu)$: $(a_1+2\mu)(x-\mu)+(a_2+1)(x-\mu)^2+\sum_{i=3}^{k}a_i(x-\mu)^i$; while the left-hand side in terms of $(x-\mu)$:
$(x-\mu)^2f_\mu'(x)+2\mu(x-\mu)f_\mu'(x)=\sum_{i=1}^{k}ia_i(x-\mu)^{i+1}+\sum_{i=1}^{k}2\mu i a_i(x-\mu)^i$
$=\sum_{i=2}^{k}(i-1)a_{i-1}(x-\mu)^{i}+\sum_{i=1}^{k}2\mu i a_i(x-\mu)^i$.
So for $i=1$: $2\mu a_1=a_1+2\mu$, $a_1=\frac{-2\mu}{1-2\mu}\sim 0$;
$i=2$: $a_2+1=a_1+4\mu a_2$, $a_2=\frac{a_1-1}{1-4\mu}\sim -1$;
$i=3,\cdots,k$: $a_i=(i-1)a_{i-1}+2i\mu a_i$, $(1-2i\mu)a_i=(i-1)a_{i-1}$.
Note that if $k\ge n$, we evaluate the last equation at $i=n$ to conclude that $a_{n-1}=0$. This will force $a_i=0$ for all $i=n-2,\cdots,2$, which contradicts the second estimate that $a_2\sim -1$. Q.E.D.
Consider the 3D vector field $X(x,y,z)=(x^2-z^2, y+x^2-z^2,0)$. Note that the singular set $S$ are two lines $x=\pm z$, $y=0$ (in particular it contains the origin $O=(0,0,0)$). Note that $D_OX=E_{22}$. Hence a cener manifold $W^c(O)$ through $O$ is tangent to plane $y=0$, and can be represented as $y=f(x,z)$. We claim that $f(x,x)=0$ (at least locally).
Proof of the claim. Suppose on the contrary that $c_n=f(x_n,x_n)\neq0$ for some $x_n\to 0$. Note that $p_n=(x_n,c_n,x_n)\in W^c(O)$, and $W^c(O)$ is flow-invariant. However, there is exactly one flow line passing through $p_n$: the line $L_n=\{(x_n,c_nt,x_n):t>0\}$. Therefore $L_n\subset W^c(O)$, which contradicts the fact that $W^c(O)$ is tangent to plane $y=0$ at $O$. This completes the proof of the claim.
The planes $z=\mu$ are also invariant under the flow. Let’s take the intersection $W_\mu=\{z=\mu\}\cap W^c(O)=\{(x,f(x,\mu),\mu)\}$. Then we check that $\{(x,f(x,\mu))\}$ is a (in fact the) center manifold of the restricted vector field in the plane $z=\mu$. We already checked that $f(x,\mu)$ is not $C^\infty$, so is $W^c(O)$.
## The volume of uniform hyperbolic sets
This is a note of some well known results. The argument here may be new, and may be complete.
Proposition 1. Let $f\in\mathrm{Diff}^2_m(M)$. Then $m(\Lambda)=0$ for every closed, invariant hyperbolic set $\Lambda\neq M$.
See Theorem 15 of Bochi–Viana’s paper. Note that Proposition 1 also applies to Anosov case, in the sense that $m(\Lambda)>0$ implies that $\Lambda=M$ and $f$ is Anosov.
Proof. Suppose $m(\Lambda)>0$ for some hyperbolic set. Then the stable and unstable foliations/laminations are absolutely continuous. Hopf argument shows that $\Lambda$ is (essentially) saturated by stable and unstable manifolds. Being a closed subset, $\Lambda$ is in fact saturated by stable and unstable manifolds, and hence open. So $\Lambda=M$.
Proposition 2. There exists a residual subset $\mathcal{R}\subset \mathrm{Diff}_m^1(M)$, such that for every $f\in\mathcal{R}$, $m(\Lambda)=0$ for every closed, invariant hyperbolic set $\Lambda\neq M$.
Proof. Let $U\subset M$ be an open subset such that $\overline{U}\neq M$, $\Lambda_U(f)=\bigcap_{\mathbb{Z}}f^n\overline{U}$, which is always a closed invariant set (maybe empty). Given $\epsilon>0$, let $\mathcal{D}(U,\epsilon)$ be the set of maps $f\in\mathrm{Diff}_m^1(M)$ that either $\Lambda_U(f)$ is not a uniformly hyperbolic set, or it’s hyperbolic but $m(\Lambda_U(f))<\epsilon$. It follows from Proposition 1 that $\mathcal{D}(U,\epsilon)$ is dense. We only need to show the openness. Pick an $f\in \mathcal{D}(U,\epsilon)$. Since $m(\Lambda_U(f))<\epsilon$, there exists $N\ge 1$ such that $m(\bigcap_{-N}^N f^n\overline{U})<\epsilon$. So there exists $\mathcal{U}\ni f$ such that $m(\bigcap_{-N}^N g^n\overline{U})<\epsilon$. In particular, $m(\Lambda_U(g))<\epsilon$ for every $g\in \mathcal{U}$. The genericity follows by the countable intersection of the open dense subsets $\mathcal{D}(U_n,1/k)$.
The dissipative version has been obtained in Alves–Pinheiro’s paper
Proposition 3. Let $f\in\mathrm{Diff}^2(M)$. Then $m(\Lambda)=0$ for every closed, transitive hyperbolic set $\Lambda\neq M$. In particular, $m(\Lambda)>0$ implies that $\Lambda=M$ and $f$ is Anosov.
See Theorem 4.11 in R. Bowen’s book when $\Lambda$ is a basic set.
## Doubling map on unit circle
1. Let $\tau:x\mapsto 2x$ be the doubling map on the unit torus. We also consider the uneven doubling $f_a(x)=x/a$ for $0\le x \le a$ and $f(x)=(x-a)/(1-a)$ for $a \le x \le 1$. It is easy to see that the Lebesgue measure $m$ is $f_a$-invariant, ergodic and the metric entropy $h(f_a,m)=\lambda(m)=\int \log f_a'(x) dm(x)=-a\log a-(1-a)\log (1-a)$. In particular, $h(f_a,m)\le h(f_{0.5},m)=\log 2 =h_{\text{top}}(f_a)$ and $h(f_a,m)\to 0$ when $a\to 0$.
2. Following is a theorem of Einsiedler–Fish here.
Proposition. Let $\tau:x\mapsto 2x$ be the doubling map on the unit torus, $\mu$ be an $\tau$-invariant measure with zero entropy. Then for any $\epsilon>0$, $\beta>0$, there exist $\delta_0>0$ and a subset $E\subset \mathbb{T}$ with $\mu(E) > 0$, such that for all $x \in E$, and all $\delta<\delta_0$: $\mu(B(x,\delta))\ge \delta^\beta$.
A trivial observation is $\text{HD}(\mu)=0$, which also follows from general entropy-dimension formula.
Proof. Let $\beta$ and $\epsilon$ be fixed. Consider the generating partition $\xi=\{I_0, I_1\}$, and its refinements $\xi_n=\{I_\omega: \omega\in\{0,1\}^n\}$ (separated by $k\cdot 2^{-n}$)….
Furstenberg introduced the following notation in 1967
Definition. A multiplicative semigroup $\Sigma\subset\mathbb{N}$ is lacunary, if $\Sigma\subset \{a^n: n\ge1\}$ for some integer $a$. Otherwise, $\Sigma$ is non-lacunary.
Example. Both $\{2^n: n\ge1\}$ and $\{3^n: n\ge1\}$ are lacunary semigroups. $\{2^m\cdot 3^n: m,n\ge1\}$ is a non-lacunary semigroup.
Theorem. Let $\Sigma\subset\mathbb{N}$ be a non-lacunary semigroup, and enumerated increasingly by $s_i > s_{i+1}\cdot$. Then $\frac{s_{i+1}}{s_i}\to 1$.
Example. $\Sigma=\{2^m\cdot 3^n: m,n\ge1\}$. It is equivalent to show $\{m\log 2+ n\log 3: m,n\ge1\}$ has smaller and smaller steps.
Theorem. Let $\Sigma\subset\mathbb{N}$ be a non-lacunary semigroup, and $A\subset \mathbb{T}$ be $\Sigma$-invariant. If $0$ is not isolated in $A$, then $A=\mathbb{T}$.
Furstenberg Theorem. Let $\Sigma\subset\mathbb{N}$ be a non-lacunary semigroup, and $\alpha\in \mathbb{T}\backslash \mathbb{Q}$. Then $\Sigma\alpha$ is dense in $\mathbb{T}$.
In the same paper, Furstenberg also made the following conjecture: a $\Sigma$-invariant ergodic measure is either supported on a finite orbit, or is the Lebesgue measure.
A countable group $G$ is said to be amenable, if it contains at least one Følner sequence. For example, any abelian countable group is amenable. Note that for amenable group action $G\ni g:X\to X$, there always exists invariant measures and the decomposition into ergodic measures. More importantly, the generic point can be defined by averaging along the Følner sequences, and almost every point is a generic point for an ergodic measure. In a preprint, the author had an interesting idea: to prove Furstenberg conjecture, it suffices to show that every irrational number is a generic point of the Lebesgue measure. Then any other non-atomic ergodic measures, if exist, will be starving to death since there is no generic point for them :)
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2015-08-03 19:06:02
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https://zbmath.org/?q=an%3A1076.11048
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# zbMATH — the first resource for mathematics
A note on the Dirichlet characters of polynomials. (English) Zbl 1076.11048
This paper is a continuation of a paper of the first author together with Y. Yi [Bull. Lond. Math. Soc. 34, 469–473 (2002; Zbl 1038.11052)]. The main result is a generalized identity of the form $\sum_{n=1}^{q}\chi\bigl(f(n)\bigr)= \varepsilon(\chi,f)\cdot q^{1/2}$ where $$q$$ is a perfect square, $$\chi$$ a primitive Dirichlet character modulo $$q$$, $$f$$ a certain polynomial with integer coefficients and $$|\varepsilon(\chi,f)|=1$$ is explicitly given. As a corollary they got for $$q$$ an odd square number and $$m$$ any natural number $$m$$ with $$(q,m)=1$$ the following nice identity $\sum_{n=1}^{q}\chi\bigl(n^m(1-n)^m\bigr)= \overline{\chi}(4^m)\cdot q^{1/2}.$ For general moduli $$q$$, whether there exists a similar formula, is an open problem.
##### MSC:
11L10 Jacobsthal and Brewer sums; other complete character sums 11L40 Estimates on character sums
##### Keywords:
Dirichlet characters
Full Text:
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2021-10-16 09:10:10
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https://icsehelp.com/propagation-of-sound-waves-chapter-8-physics-concise-selina-solution/
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Propagation of Sound Waves Selina ICSE 9th Concise Physics Solutions Chapter-8 Propagation of Sound Waves. Step By Step Revised Concise Selina Physics Solutions of Chapter-8 Propagation of Sound Waves with Exe-8(A) , MCQ 8(A), Num-8(A) , Exe-8(B), MCQ -8(B) including Numerical and MCQ Questions Solved. Revised Selina Concise Physics Solutions Propagation of Sound Waves Chapter-8 for ICSE Class-9. Visit official Website CISCE for detail information about ICSE Board Class-9.
Propagation of Sound Waves Selina Concise 9th Physics Solutions
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Exercise-8(A)
MCQ-8(A)
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Exercise-8(B)
Note :- Before Viewing Selina Concise Physics Solutions of Chapter-8 Propagation of Sound Waves for ICSE Class-9 Physics. Read the whole chapter carefully and Solved all example of Chapter-8 Propagation of Sound Waves for Class-9 Physics.
latest syllabus of council class 9th physics
According to latest syllabus of council class 9th physics, Law of reflection, character of image, Spherical Mirror, Ray diagram and use of spherical mirror, Type of mirror, radius of curvature, Pole, Principal Axis, Focus and Focal Length.
Name of sound wave, Requirement of a medium for sound waves to travel; Propagation and speed in different media; comparison with speed of light. Sound propagation, terms- Frequency (f), wavelength, velocity, relation, effect of different factors on the speed of sound.
Exe-8(A) Propagation of Sound Waves Physics concise selina solutions
Question 1
What causes sound?
Sound is caused due to vibrations of a body.
Question 2
What is sound? How is it produced?
Sound is a form of energy that produces the sensation of hearing in our ears. Sound is produced by a vibrating body.
Question 3
Complete the following sentence:
Sound is produced by a ___________ body.
Vibrating
Question 4
Describe a simple experiment which demonstrates that the sound produced by a tuning fork is due to vibrations of its arms.
Experiment: A tuning fork is taken and its one arm is struck on a rubber pad and it is brought near a tennis ball suspended by a thread as shown in figure.
It is noticed that as the arm of the vibrating fork is brought close to the ball, it jumps back and forth and sound of the vibrating tuning fork is heard. When its arm stop vibrating, the ball becomes stationary and no sound is heard.
Question 5
Describe in brief, with the aid of a sketch diagram, an experiment to demonstrate that a material medium is necessary for propagation of sound.
Experiment to demonstrate that a material medium is necessary for the propagation of sound:
An electric bell is suspended inside an airtight glass bell jar. The bell jar is connected to a vacuum pump as shown in figure. As the circuit of electric bell is completed by pressing the key, the hammer of the electric bell begins to strike the gong repeatedly due to which sound is heard.
Keeping the key pressed, air is gradually withdrawn from jar by starting the vacuum pump. It is noticed that the loudness of sound goes on decreasing as the air is taken out from the bell jar and finally no sound is heard when all the air from the jar has been drawn out. The hammer of the electric bell is still seen striking the gong repeatedly which means that sound is still produced but it is not heard.
When the jar is filled with air, the vibrations produced by the gong are carried by the air to the walls of jar which in turn set the air outside the jar in vibration and sound is heard by us but in absence of air, sound produced by bell could not travel to the wall of the jar and thus no sound is heard. It proves that material medium is necessary for the propagation of sound waves.
Question 6
There is no atmosphere on moon. Can you hear each other on the moon’s surface?
We cannot hear each other on moon’s surface because there is no air on moon and for sound to be heard, a material medium is necessary.
Question 7
State three characteristics of the medium required for propagation of sound?
Requisites of the medium for propagation of sound:
(i) The medium must be elastic.
(ii) The medium must have inertia.
(iii) The medium should be friction less.
Question 8
Explain with an example, the propagation of sound in a medium.
Take a vertical metal strip with its lower end fixed and upper end being free to vibrate as shown in fig (a).
As the strip is moved to right from a to b as shown in Fig (b), the air in that layer is compressed (compression is formed at C). The particles of this layer compress the layer next to it, which then compresses the next layer and so on. Thus, the disturbance moves forward in form of compression without the particles themselves being displaced from their mean positions.
As the metal strip returns from b to a as shown in Fig (c) after pushing the particles in front, the compression C moves forward and particles of air near the strip return to their normal positions.
When the strip moves from a to c as shown in Fig (d), it pushes back the layer of air near it towards left and thus produces a low pressure space on its right side i.e. layers of air get rarefied. This region is called rarefaction (rarefaction is formed at R).
When the strip returns from C to its mean position A in Fig (e), the rarefaction R travels forward and air near the strip return to their normal positions.
Thus, one complete to and fro motion of the strip forms one compression and one rarefaction, which together form one wave. This wave through which sound travels in air is called longitudinal wave.
Question 9
Choose the correct word/words to complete the following sentence:
When sound travels in a medium ____________ (the particles of the medium, the source, the disturbance, the medium) travels in form of a wave.
the disturbance
Question 10
Name the two kinds of waves in form of which sound travels in a medium.
Sound travels in a medium in form of longitudinal and transverse waves.
Question 11
What is a longitudinal wave? In which medium: solid, liquid or gas, can it be produced?
A type of wave motion in which the particle displacement is parallel to the direction of wave propagation is called a longitudinal wave. It can be produced in solids, liquids as well as gases.
Question 12
What is a transverse wave? In which medium: solid, liquid or gas, can it be produced?
A type of wave motion in which the particle displacement is perpendicular to the direction of wave propagation is called a transverse wave. It can be produced in solids and on the surface of liquids.
Question 13
Explain meaning of the terms compression and rarefaction in relation to a longitudinal wave.
A longitudinal wave propagates by means of compression and rarefaction.
When a vibrating object moves forward, it pushes and compresses the air in front of it creating a region of high pressure. This region is called a compression (C), as shown in Fig. This compression starts to move away from the vibrating object. When the vibrating object moves backwards, it creates a region of low pressure called rarefaction (R), as shown in Fig.
Compressions are the regions of high density where the particles of the medium come very close to each other and rarefactions are the regions of low density where the particles of the medium move away from each other.
Question 14
Explain the terms crest and trough in relation to a transverse wave.
A crest is a point on the transverse wave where the displacement of the medium is at a maximum.
A point on the transverse wave is a trough if the displacement of the medium at that point is at a minimum.
Question 15
Describe an experiment to show that in wave motion, only energy is transferred, but particles of medium do not move.
Experiment to show that in a wave motion, only energy is transferred, but particles of the medium do not move:
If we drop a piece of stone in the still water of pond, we hear a sound of stone striking the water surface. Actually a disturbance is produced in water at the point where the stone strikes it. This disturbance spreads in all directions radially outwards in form of circular waves on the surface of water.
If we place a piece of cork on water surface at some distance away from the point where the stone strikes it, we notice that cork does not move ahead, but it vibrates up and down, while the wave moves ahead. The reason is that particles of water (or medium) start vibrating up and down at the point where the stone strikes. These particles then transfer their energy to the neighboring particles and they themselves come back to their mean positions. Thus only energy is transferred but the particles of the medium do not move.
Question 16
Define the term amplitude of a wave. Write its S.I. unit.
The maximum displacement of the particle of medium on either side of its mean position is called the amplitude of wave.
Its SI unit is metre.
Question 17
What do you mean by the term frequency of a wave? State its S.I. unit.
The number of vibrations made by the particle of the medium in one second is called the frequency of the wave. It can also be defined as the number of waves passing through a point in one second.
Its SI unit is hertz (Hz).
Question 18
How is the frequency of a wave related to its time period?
Frequency of a wave is the reciprocal of the time period.
Question 19
Define the term wave velocity. Write its S.I. unit.
The distance travelled by a wave in one second is called its wave velocity.
Its SI unit is metre per second (ms-1).
Question 20
Draw displacement-time graph of a wave and show on it the amplitude and time period of wave.
Question 21
Draw a displacement-distance graph of a wave and mark on it, the amplitude of wave by the letter ‘a’ and wavelength of wave by the letter.
Question 22
How are the wave velocity V, frequency and wavelength of a wave related? Derive the relationship.
Question 23
State two properties of medium on which the speed of sound in it depends.
The speed of sound in a medium depends upon its elasticity and density.
Question 25
State the speed of (i) light and (ii) sound in air?
(i) Speed of light in air = 3 x 108 m s-1 (ii) Speed of sound in air = 330 m s-1.
Question 26
Compare approximately the speed of sound in air, water and steel.
1 : 4 : 15
Question 27
(i) Can sound travel in vacuum?
(ii) How does the speed of sound differ in different media?
(i) No, sound cannot travel in vacuum as it requires a material medium for its propagation.
(ii) Speed of sound is maximum in solids, less in liquids and least in gases.
Question 28
Flash of lightning reaches earlier than the sound of thunder. Explain the reason.
This happens because the light travels much faster than sound.
Question 29
If you place your ear close to an iron railing which is tapped some distance away, you hear the sound twice. Explain why?
Sound travels in iron faster than in air so first the sound travelled in iron rail is heard and then the sound travelled in air is heard.
Question 30
The sound of an explosion on the surface of a lake is heard by a boat man 100 m away and by a diver 100 m below the point of explosion.
(i) Who would hear the sound first: boatman or diver?
(iii) If sound takes time to reach the boatman, how much time approximately does it take to reach the diver?
(i) The diver would hear the sound first.
(ii) This is because sound travels faster in water than in air.
(iii) It would take 0.25t to reach the diver because sound travels almost four times faster in water.
Question 31
How do the following factors affect, if at all, the speed of sound in air:
(i) Frequency of sound, (ii) Temperature of air,
(iii) Pressure of air and (iv) Moisture in air?
(i) Frequency of sound has no effect on the speed of sound.
(ii) Speed of sound increases with the increase in the temperature of sound.
(iii) Pressure of sound has no effect on the speed of sound.
(iv) Speed of sound increases with the increase in presence of moisture in air.
Question 32
How does the speed of sound change with change in (i) amplitude and (ii) wavelength, of sound wave?
(i) Speed of sound does not change with a change in amplitude.
(ii) Speed of sound does not change with a change in wavelength.
Question 33
In which medium the speed of sound is more: humid air or dry air? Give a reason to your answer.
Speed of sound is more in humid air because in presence of moisture, the density of air decreases and sound travels with greater speed.
Question 34
How does the speed of sound in air vary with temperature?
The speed of sound increases by 0.61 m s-1 for each 1 rise in temperature.
Question 35
Describe a simple experiment to determine the speed of sound in air. What approximation is made in the method described by you?
The simple experiment that a person can do to calculate the speed of sound in air is that a person stands at a known distance (d meter) from the cliff and fires a pistol and simultaneously start the stopwatch. He stops the stopwatch as soon as he hears an echo. The distance travelled by the sound during the time (t) seconds is 2d. So, speed of sound = distance travelled / time taken = 2d/t
The approximation made is that speed of sound remains same for the time when the experiment is taking place.
Question 36
Complete the following sentences :
(a) Sound cannot travel through __________, but it requires a ___________.
(b) When sound travels in a medium, the particles of medium ___________ but the disturbance ___________.
(c) A longitudinal wave is composed of compression and ____________.
(d) A transverse wave is composed of crest and ____________.
(e) Wave velocity = ________________ × wavelength.
(a) Vacuum, medium (b) do not move, moves (c) rarefaction (d) trough
(e) frequency
MCQ-8(A) Propagation of Sound Waves 9th Physics concise selina solutions
Question 1
The correct statement is :
(a) Sound and light both require medium for propagation.
(b) Sound can travel in vacuum, but light can not
(c) Sound needs medium, but light does not need medium for its propagation.
(d) Sound and light both can travel in vacuum.
Sound needs medium, but light does not need medium for its propagation.
Question 2
The speed of sound in air at 0C is nearly:
(a) 1450 m s-1 (b) 450 m s-1
(c) 5100 m s-1 (d) 330 m s-1
330 m s-1
Question 3
Sound in air propagates in form of
(a) Longitudinal wave
(b) Transverse wave
(c) Both longitudinal and transverse waves
(d) Neither longitudinal nor transverse wave.
Longitudinal wave
Question 4
The speed of light in air is :
(a) 3 x 108 m s-1 (b) 330 m s-1
(c) 5100 m s-1 (d) 3 x 1010 m s-1
3 x 108 m s-1
NUM-8(A) Propagation of Sound Waves 9th concise selina Physics solutions
Question 1
The heart of a man beats 75 times a minute.
What is its (a) frequency and (b) time period?
Given, heart beats 75 times a minute
(a) Frequency = No. of times heart beats in 1 s
Or, = 75/60 = 1.25 s-1
(b) Time period, T = 1/
Or, T = 1 / 1.25 = 0.8 s
Question 2
The time period of a simple pendulum is 2 s. Find its frequency.
Frequency, = 1/T
Or, = 1/ 2 = 0.5 Hz
Question 3
The separation between two consecutive crests in a transverse wave is 100 m. If wave velocity is 20 m s-1, find the frequency of wave.
Given, wavelength = 100m
Wave velocity = 20 m/s
We know that,
Wave velocity = Frequency x Wavelength
Or, Frequency = Wave velocity / wavelength
Or, = 20/100 = 0.2 Hz
Question 4
A longitudinal wave travels at a speed of 0.3 m s-1 and the frequency of wave is 20 Hz. Find the separation between two consecutive compressions.
Wave velocity = 0.3 m/s
Frequency = 20 Hz
Separation between two consecutive compressions is the wavelength of a wave.
We know that,
Wave velocity = Frequency x Wavelength
Or, wavelength = Wave velocity / frequency
Or, = 0.3 / 20 = 1.5 x 10-2 m
Question 5
A source of wave produces 40 crests and 40 troughs in 0.4 s. What is the frequency of the wave?
Frequency of wave = number of waves per second
Or, = 40 / 0.4 = 100 Hz
Question 6
An observer fires a gun and another observer at a distance 1650 m away from hears its sound. If the speed of sound is 330 m s-1, find the time when will hear the sound after firing by A.
Distance between the two observers = 1650 m
Speed of sound = 330 m/s
Time in which B hears the sound = Distance / speed = 1650/330 = 5s
Thus, B will hear the sound 5s after the gun is shot.
Question 7
The time interval between a lightning flash and the first sound of thunder was found to be 5 s. If the speed of sound in air is 330 m s-1, find the distance of flash from the observer.
Speed of sound in air (V) = 330 m/s
Time in which thunder is heard after lighting is seen (t) = 5s
Thus, distance between flash and observer = V x t = (330 x 5) = 1650 m
Question 8
A boy fires a gun and another boy at a distance hears the sound of fire 2.5s after seeing the flash. If the speed of sound in air 340 m/s, find distance between the boys.
Speed of sound in air (V) = 340 m/s
Time in which sound of fire is heard after flash is seen (t) = 2.5s
Thus, distance between flash and observer = V x t = (340 x 2.5) = 850 m
Question 9
An observer sitting in line of two tanks watches the flashes of two tanks firing at each other at the same time, but he hears the sounds of two shots 2s and 3.5s after seeing the flashes. If distance between the two tanks is 510m, find the speed of sound.
Time taken by the observer to hear the sound of the first tank A= 3.5s
Now Time taken by the observer to hear the sound of the second tank B = 2s
This Time taken by the tank B to hear the sound of tank A= (3.5 – 2)s = 1.5s
Distance between the two tanks = 510m
Speed = 510/1.5=340m/s
Question 10
How long will sound take to travel in (a) and iron rail and (b) air, both 3.3 km in length? Take speed of sound in air to be 330 m/s and in iron to be 5280 m/s.
(a) Length of iron rail (D) = 3.3 km = 3300 m
Speed of sound in iron (V) = 5280 m/s
Time taken by sound to travel in iron rod (t) = D/V
Or, t = (3300 / 5280) s = 0.625 s
(b) Length of iron rail (D) = 3.3 km = 3300 m
Speed of sound in air (V) = 330 m/s
Time taken by sound to travel in iron rod (t) = D/V
Or, t = (3300/330) s = 10 s
Question 11
Assuming the speed of sound in air equal to 340 m/s and in water equal to 1360 m/s, find the time taken to travel a distance 1700 m by sound in (i) air (ii) water.
(i) Distance travelled (D) = 1700
Speed of sound in air (V) = 340 m/s
Time taken (t) = D/V = (1700 / 340) s = 5 s
(ii) Distance travelled (D) = 1700
Speed of sound in water (V’) = 1360 m/s
Time taken (t) = D/V = (1700 / 1360) s = 1.25 s
EXE-8(B) Propagation of Sound Waves 9th Concise selina Physics solutions
Question 1
What do you mean by the audible range of frequency?
The range of frequency within which the sound can be heard by a human being is called the audible range of frequency.
Question 2
What is the audible range of frequency for human?
The audible range of frequency for humans is 20 Hz to 20 kHz.
Question 3
For which range of frequencies, human ears are most sensitive?
Human ears are most sensitive for the range 2000 Hz to 3000 Hz.
Question 4
Which has the higher frequency – ultrasonic sound or infrasonic sound?
Ultrasonic has higher frequency.
Question 5
Complete the following sentences:
(a) An average person can hear sound of frequencies in the range ______ to _________.
(b) Ultrasound is of frequency ___________.
(c) Infrasonic sound is of frequency ______________.
(d) Bats can produce and hear ___________ sound.
(e) Elephants produce ____________sound.
(a) 20 Hz, 20 kHz (b) above 20 kHz (c) below 20 Hz (d) ultrasonic (e) infrasonic.
Question 6
Name the sounds of the frequencies given below:
(a) 10 Hz (b) 100 Hz (c) 1000 Hz (d) 40 kHz
(a) Infrasonic (b) Audible (c) Audible (d) Ultrasonic.
Question 7
Can you hear the sound produced due to vibrations of a seconds pendulum? Give reasons.
No, we cannot hear the sound produced due to vibrations of a seconds pendulum because the frequency of sound produced due to vibrations of seconds pendulum is 0.5 Hz which is infrasonic.
Question 8
What is ultrasound?
Sounds of frequency above 20 kHz are called ultrasound.
Question 9
State the approximate speed of ultrasound in air.
The approximate speed of ultrasound in air is 330 m/s-1.
Question 10
State two properties of ultrasound that make it useful to us.
Two properties of ultrasound which make it useful to us are:
(i) High energy contents
(ii) High directivity
Question 11
Explain how do bats locate the obstacles and prey in their way.
Bats locate the obstacles and prey in their path by producing and hearing the ultrasound. They emit an ultrasound which returns after striking an obstacle in their way. By hearing the reflected sound and from the time interval (when they produce ultrasound and they receive them back), they can judge the direction and the distance of the obstacle in their way.
Question 12
State two applications of ultrasound.
Two applications of ultrasound:
(i) Ultrasound is used for drilling holes or making cuts of desired shape in materials like glass.
(ii) Ultrasound is used in surgery to remove cataract and in kidneys to break the small stones into fine grains.
MCQ-8(B) Propagation of Sound Waves 9th Revised Concise selina Physics solutions
Question 1
A man can hear the sound of frequency :
(a) 1 Hz (b) 1000 Hz
(c) 200 kHz (d) 5 MHz
1000 Hz
Question 2
The properties of ultrasound that make it useful are
(a) High power and high speed
(b) High power and good directivity (c) High frequency and high speed
(d) High frequency and bending around the objects.
High power and good directivity
Question 3
Sonar makes use of :
(a) infrasonic sound
(b) ultrasound
(c) ordinary sound
(d) light
ultrasound
Thanks
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2021-07-26 12:34:03
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http://accessanesthesiology.mhmedical.com/content.aspx?bookid=413§ionid=39828173
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Chapter 23
Cervical plexus anesthesia was developed early in the twentieth century, and two main approaches were available to the early practitioners of regional anesthesia. In 1912, Kappis described a posterior approach to the brachial plexus while attempting to block spinal nerves at the point of emergence from the vertebral column.1 The main reason for a posterior approach to blocking the cervical plexus is the relative position of the vertebral artery and vein anterior to the plexus.2 However, the posterior approach is associated with discomfort during and after the blockade, most likely due to the puncture of the extensor muscles of the neck, and has been avoided by many practitioners. As a result, the posterior approach to the cervical plexus block has not been as popular as the lateral approach, although it has been utilized to block the brachial plexus either as a single-shot or continuous technique.2–5
In 1914, Heidenhein described the lateral approach, which has formed the basis for subsequent techniques of anesthetizing the cervical plexus.6 Victor Pauchet described a lateral approach to blocking the cervical plexus in 1920 and mentioned the posterior approach; however, he advocated the use of the lateral approach.7 Winnie revisited the lateral approach to the cervical plexus block in 1975, and it is currently the more used approach for the cervical plexus block.8
Deep and superficial cervical plexus block can be used to provide anesthesia for a variety of surgical procedures, including superficial operations on the neck and shoulders, thyroid operations, and carotid endarterectomies in which awake neurologic monitoring is a simple and reliable method of neurologic assessment (Figure 23–1).9,10 Eastcott described the first carotid endarterectomy in 1954, and the number of these surgeries performed in the United States grows each year.11 Regional anesthesia is a viable anesthetic choice for carotid surgery, although debate continues about whether regional or general anesthesia is the better choice for carotid endarterectomy surgery. Most of the latest literature points to regional anesthesia as a better choice.12–21 The outcome data from vascular surgery and neurosurgery literature shows that patients who undergo carotid endarterectomy under regional anesthesia may have better outcomes.22–25
###### Fig. 23-1
Carotid endarterectomy. The image shows open, cross-clamped carotid artery and a plaque inside its wall.
The superficial cervical plexus block can be used for many superficial surgeries in the neck area, including lymph node dissection, excision of thyroglossal or branchial cleft cysts, carotid endarterectomy, and vascular access surgery.26 If the superficial cervical plexus block is to be used alone for carotid endarterectomy, local anesthetic supplementation by the surgeon may be necessary.27,28 Although both the deep and superficial cervical plexus blocks can be performed separately, they are most often performed in combination to provide anesthesia and postoperative analgesia for head and neck surgery.29–31...
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2017-01-25 01:17:13
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https://www.mpboardsolutions.com/mp-board-class-9th-science-solutions-chapter-4-english-medium/
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MP Board Class 9th Science Solutions Chapter 4 Structure of the Atom
MP Board Class 9th Science Solutions Chapter 4 Structure of the Atom
Structure of the Atom Intext Questions
Structure of the Atom Intext Questions Page No. 47
Question 1.
What are canal rays?
Canal rays are positively charged radiations which led to the discovery of positively charged sub-atomic particle called proton. These rays were discovered by E. Goldstein.
Question 2.
If an atom contains one electron and one proton, will it carry any charge or not?
The atom will not contain any charge and will be electrically neutral because both electron and proton will balance each other.
Structure of the Atom Intext Questions Page No. 49
Question 1.
On the basis of Thomson’s model of an atom, explain how the atom is neutral as a whole.
According to Thomson’s model, an atom consist of a positively charged sphere and electrons are embedded in it. So, both charges are equal which makes the atom electrically neutral.
Question 2.
On the basis of Rutherford’s model of an atom, which sub – atomic particle is present in the nucleus of an atom?
Proton is the sub – atomic particle which is present in the nucleus of an atom.
Question 3.
Draw a sketch of Bohr’s model of an atom with three shells.
Question 4.
What do you think would be the observation if the a-particle scattering experiment is carried out using a foil of a metal other than gold?
The observations would be same as that of gold foil.
Structure of the Atom Intext Questions Page No. 49
Question 1.
Name the three sub – atomic particles of an atom.
1. Positively charged – Protons
2. Negatively charged – Electrons
3. No charged – Neutrons.
Question 2.
Helium atom has an atomic mass of 4u and two protons in its nucleus. How many neutrons does it have?
Atomic mass = Number of protons + Number of neutrons
∴ 4 = 2 + Number of neutrons
∴ Number of neutrons = 4 – 2 = 2.
Structure of the Atom Intext Questions Page No. 50
Question 1.
Write the distribution of electrons in carbon and sodium atoms.
Atomic number of Carbon = 6
Question 2.
If K and L shells of an atom are full, then what would be the total number of electrons in the atom?
K shell is the 1 shell
So, n = 1
Then maximum electron’s = 2n2 = 2 × (1)2
= 2 × 1 = 2
and L shell is the second shell.
So, n = 2
Then maximum electrons = 2(n)2
= 2 × (2)2 = 8
∴ Total number of electrons = 2 + 8 = 10.
Structure of the Atom Intext Questions Page No. 52
Question 1.
How will you find the valency of chlorine, sulphur and magnesium?
We know that valency is the number of electrons lost, gained or shared by atom to become stable or to complete 8 electrons in the shell.
Now, Chlorine,
Atomic number = 17
Then, it will take 8 – 7 = 1 electron to complete its shell.
∴ Its valency is ‘I’
Sulphur, Atomic number = 16
It will take 8-6 = 2 electrons to complete its shell.
∴ Its valency is ‘2’.
Magnesium, Atomic number = 12
It will lose 2 electrons from its outermost shell to become stable.
∴ Its valency will be ‘2’.
Structure of the Atom Intext Questions Page No. 52
Question 1.
If number of electrons in an atom is 8 and number of protons is also 8 then,
(i) What is the atomic number of the atom? and
(ii) What is the charge on the atom?
(i) Number of electrons = 8 and,
Number of protons =8
Then, Atomic number = Number of protons = 8
(ii) Now, total electrons (-) = Total protons (+)
So, atom will be electrically neutral.
Question 2.
With the help of table 4.1 of Textbook, find out the mass number of oxygen and sulphur atom.
From the table, we have,
Oxygen,
Mass Number = Number of protons + Number of neutrons
= 8 + 8 = 16
Sulphur,
Mass number = Number of protons + Number of neutrons
= 16 + 16 = 32.
Structure of the Atom Intext Questions Page No. 53
Question 1.
For the symbol H, D, and T tabulate three sub – atomic particles found in each of them.
H, D, and T stand for protium, deuterium and tritium as isotopes of hydrogen atom.
Table:
Question 2.
Write the electronic configuration of any one pair of isotopes and isobars.
Pair of isotopes: $$_{ 6 }^{ 12 }{ C }$$, $$_{ 6 }^{ 14 }{ C }$$
Electronic configuration:
Structure of the Atom NCERT Textbook Exercises
Question 1.
Compare the properties of electrons, protons and neutrons.
Question 2.
What are the limitations of J.J. Thomson’s model of the atom?
J.J. Thomson’s model explained the existence of positive charge in the form of sphere and electrons embedded in it. But, he was unable to explain the Rutherford’s gold foil experiment in which most of positive α – particles passed straight, existence of electrons in the circular path and protons at the centre of the atom.
Question 3.
What are the limitations of Rutherford’s model of the atom?
Rutherford explained that electrons revolve in a circular path which is found contradictory in terms of stability of atom. Because electrons are negatively charged and when they move continuously in circular paths then they should lose their energies and finally, fall into the positively charged nucleus making atoms unstable and collapse.
Question 4.
Describe Bohr’s Model of the atom.
Neils Bohr proposed the theory for model of the atom. It is explained as:
1. Atom is made up of three sub – atomic particles as electrons, protons and neutrons.
2. The electrons move round the nucleus in fixed circular paths called orbits or shells.
3. The orbits are represented by the letters K, L, M, N, or the number n = 1, 2, 3, 4.
4. Centre of the atom is called the nucleus.
5. Electrons do not radiate energies while revolving in the orbits.
6. Electrons gain energy when they jump from lower shell to higher shell and lose energy when they return down from higher energy level to lower energy level.
Question 5.
Compare all the proposed models of an atom given in this chapter.
Question 6.
Summarise the rules for writing of distribution of electrons in various shells for the first eighteen elements.
Rules:
(a) Maximum electrons present in a shell is given by 2n2 whereas n is the number of that shell.
Like,
• K Shell, n = 1 → 2n2 = 2 × (1)2 = 2
• L Shell, n = 2 → 2n2 = 2 × (2)2 = 8
• M Shell, n = 3 → 2n2 = 2 × (3)2 = 18
• N Shell, n = 4 → 2n2 = 2 × (4)2 = 32.
(b) The outermost shell can have maximum of 8 electrons.
(c) Electrons cannot be occupied in a shell till its inner shells or orbits are completely filled.
Question 7.
Define valency by taking examples of silicon and oxygen.
Valency is the combining capacity of an atom to become electrically stable. Or It means how many electrons are lost or gained by an atom to become stable.
In Silicon,
It has 4 valence electrons.
So, it will lose 4 electrons to become stable.
∴ Its valency is 4.
In Oxygen,
It has 6 valence electrons.
So, it will gain 8 – 6 = 2 electrons to become stable.
∴ Its valency is 2.
Question 8.
Explain with examples:
(i) Atomic number
(ii) Mass number
(iii) Isotopes
(iv) Isobars
Give any two uses of isotopes.
(i) Atomic number: It is equal to the total number of
protons in the nucleus of its atom.
E.g.,
• Carbon has 6 protons. So, its atomic number is 6.
(ii) Mass number: It is equal to the sum of total number of protons and neutrons in the nucleus.
E.g.,
• Sodium has 11 protons and 12 neutrons. So, its mass number is 11 + 12 = 23
(iii) Isotopes: These are atoms of the same element having same atomic number, but different mass number.
E.g.
• $$_{ 35 }^{ 79 }{ Br }$$, $$_{ 35 }^{ 81 }{ Br }$$, $$_{ 6 }^{ 12 }{ C }$$, $$_{ 6 }^{ 14 }{ C }$$
(iv) Isobars: These are the atoms of different elements having different atomic number but same mass number.
E.g.
• $$_{ 18 }^{ 40 }{ Ar }$$, $$_{ 20 }^{ 40 }{ Ca }$$, $$_{ 11 }^{ 24 }{ Na }$$, $$_{ 12 }^{ 24 }{ Mg }$$
Use of Isotopes:
• Uranium isotope is used as a fuel in nuclear reactor for generating electricity.
• Sodium isotope is used to detect the blood clots.
Question 9.
Na+ has completely filled K and L shells. Explain.
Atomic number of sodium (Na) is 11.
Now, if Na loses 1 electron then it will become Na+.
Now K shell can have maximum of 2 electrons and L shell can have maximum of 8 electrons.
Then, Na+ has completely filled K and L shell.
Question 10.
If bromine atom is available in the form of, say, two isotopes $$_{ 35 }^{ 79 }{ Br }$$ (49.7%) and $$_{ 35 }^{ 81 }{ Br }$$Br (50.3%), calculate the average atomic mass of bromine atom.
Average atomic mass of bromine atom
= 49.7% of atomic mass of $$_{ 35 }^{ 79 }{ Br }$$ + 50.3% of atomic mass of $$_{ 35 }^{ 81 }{ Br }$$
= 49.7% of 79 + 50.3% of 81
= $$\frac { 49.7 }{ 100 }$$ × 49 + $$\frac { 450.3 }{ 100 }$$ × 81
= (39.263 + 40.743)u = 80.006u
Question 11.
The average atomic mass of a sample of an element X is 16.2u. What are the percentages of isotopes If $$_{ 8 }^{ 16 }{ X }$$ and $$_{ 8 }^{ 18 }{ X }$$ in the sample?
Let the percentage of $$_{ 8 }^{ 16 }{ X }$$ in sample be x% and percentage of $$_{ 8 }^{ 18 }{ X }$$ in sample be (100 – x)%.
Now,
x% of 16 + (100 – x)% of 18 = 16.2
-2x + 1800 = 16.2 × 100 – 2x + 1800 = 1620
∴ -2x = 1620- 1800 = -180
x = $$\frac {180}{2}$$ = 90.
∴ Percentage of $$_{ 8 }^{ 16 }{ X }$$ is 90% and percentage of $$_{ 8 }^{ 18 }{ X }$$ is (100 – 90)% = 10%.
Question 12.
If Z = 3, what would be the valency of the element? Also, name the element.
Z = 3
So, atomic number = 3 (∵ Z = atomic number)
∴ Electronic configuration = 2, 1
Valency = 1
The name of the element is lithium (Li).
Question 13.
Composition of the nuclei of two atomic species X and Y are given as under
Give the mass number of X and Y. What is the relation between the two species?
Mass number of X = Protons + Neutrons = 6 + 6 = 12
And,
Mass number of Y = Protons + Neutrons = 6 + 8 = 14
Both species have same atomic number.
So, they are isotopes of the same element.
Question 14.
For the following statements, write T for True and F for False.
1. J.J. Thomson proposed that the nucleus of an atom contains only nucleons.
2. A neutron is formed by an electron and a proton combining together. Therefore, it is neutral.
3. The mass of an electron is $$\frac {1}{2000}$$ times that of proton.
4. An isotope of iodine is used for making tincture iodine which is used as a medicine.
1. False
2. False
3. True
4. False.
Put tick (✓) against correct choice and cross (✗) against wrong choice in questions 15, 16 and 17.
Question 15.
Rutherford’s alpha – particle scattering experiment was responsible for the discovery of.
(a) Atomic nucleus
(b) Electron
(c) Proton
(d) Neutron.
(a) Atomic nucleus
Question 16.
Isotopes of an element have.
(a) the same physical properties
(b) different chemical properties
(c) different number of neutrons
(d) different atomic numbers.
(c) different number of neutrons
Question 17.
Number of valence electrons in Cl ion are:
(a) 16
(b) 8
(c) 17
(d) 18
(b) 8
Question 18.
Which one of the following is a correct electronic configuration of sodium?
(a) 2, 8
(b) 8, 2, 1
(c) 2, 1, 8
(d) 2, 8, 1.
(d) 2, 8, 1.
Question 19.
Complete the following table:
Atomic Number Mass Number Number of Neutrons Number of Protons Number of Electrons Name of the Atomic Species 9 – 10 – – – 16 32 – – – Sulphur – 24 – 12 – – – 2 – 1 – – – 1 0 1 0 –
Atomic Number Mass Number Number of Neutrons Number of Pro-tons Number of Electrons Name of the Atomic Species 9 19 10 9 9 Fluorine 16 32 16 6 6 Sulphur 12 24 12 12 12 Magnesium 1 2 1 1 1 Hydrogen 1 1 0 1 0 Deuterium
Structure of the Atom Additional Questions
Structure of the Atom Multiple Choice Questions
Question 1.
Which is a positive sub – atomic particle?
(a) Proton
(b) Neutron
(c) Electron
(d) None of these.
(a) Proton
Question 2.
Electron is discovered by _____ .
(b) Neils Bohr
(c) J.J Thomson
(d) Rutherford.
(c) J.J Thomson
Question 3.
Proton is discovered by _____ .
(a) Rutherford
(c) J J. Thomson
(d) E. Goldstein.
(d) E. Goldstein.
Question 4.
Neutron is discovered by _____ .
(a) J.J. Thomson
(c) Neils Bohr
(d) Rutherford.
Question 5.
Nucleus is discovered by _____ .
(a) Rutherford
(c) J.J. Thomson
(d) Neils Bohr.
(a) Rutherford
Question 6.
Mass of electron is _____ .
(a) 9 × 10-25g
(b) 6 × 10-28g
(c) 8 × 10-24g
(d) 9 × 10-28g.
(d) 9 × 10-28g.
Question 7.
Mass of Neutron is _____ .
(a) 1.6 × 10-22g
(b) 1.6 × 10-23g
(c) 1.6 × 10-25g
(d) 1.6 × 10-24g.
(d) 1.6 × 10-24g.
Question 8.
Charge on an electron is _____ .
(a) -1.8 × 10-18C
(b) -1.7 × 10-20C
(c) -1.6 × 10-19C
(d) -1.5 × 10-21C.
(c) -1.6 × 10-19C
Question 9.
The energy paths in an atom in which electrons revolve are called _____ .
(a) Rings
(b) Cycles
(c) Orbits
(d) Circles.
(c) Orbits
Question 10.
ass number is the sum of _____ .
(a) Protons and Electrons
(b) Protons and Neutrons
(c) Electrons, Protons and Neutrons
(d) None of these.
(c) Electrons, Protons and Neutrons
Question 11.
Atomic number is equal to _____ .
(a) Number of protons
(b) Number of neutrons
(c) Number of electrons
(d) Both (a) and (c).
(d) Both (a) and (c)
Question 12.
Maximum number of electrons that can be filled in ‘M’ shell are _____ .
(a) 17
(b) 19
(c) 18
(d) 20.
(c) 18
Question 13.
An atom has atomic number ‘17’, then its valency will be _____ .
(a) 7
(b) 2
(c) 1
(d) 8.
(c) 1
Question 14.
Isotopes of an element have same number of _____ .
(a) Neutrons
(b) Protons
(c) Electrons
(d) Both (b) and (c).
(c) Electrons
Question 15.
Isobars of different elements have same _____ .
(a) Atomic number
(b) Electrons
(c) Mass number
(d) Neutrons.
(d) Neutrons
Structure of the Atom Very Short Answer Type Questions
Question 1.
Who discovered canal rays?
E. Goldstein.
Question 2.
Name the fruit which resembles J.J. Thomson model of atom.
Watermelon.
Question 3.
Who discovered nucleus?
Ernest Rutherford.
Question 4.
Who discovered neutrons?
Question 5.
Name the central part of an atom where protons and neutrons are held together.
Nucleus.
Question 6.
What is Alpha Particle?
It is a Helium ion (He2+) which has 2 units of positive charge and 4 units of mass.
Question 7.
What are cathode rays?
Cathode rays are a beam of fast moving electrons.
Question 8.
What was the main drawback of Rutherford’s model of the atom?
Inability to explain the stability of atom.
Question 9.
Write the symbolic representation of an element A with atomic number 10 and mass number 20.
$$_{ 10 }^{ 20 }{ A }$$
Question 10.
Name three Isotopes of Hydrogen.
1. Protium ($$_{ 1 }^{ 1 }{ H }$$)
2. Deuterium ($$_{ 1 }^{ 2 }{ H }$$)
3. Tritium ($$_{ 1 }^{ 3 }{ H }$$)
Question 11.
Write the electronic configuration of potassium (K).
Atomic number of potassium (K) =19
Question 12.
Define valency.
It is the combining capacity of an atom to become electrically stable.
Question 13.
What is the charge of a proton?
1.6 × 10-19C.
Question 14.
Write the year of discoveries of these sub – atomic particles – electron, proton, neutron and neucleus.
1. Electron – 1897
2. Proton – 1866
3. Neutron – 1932
4. Nucleus – 1911.
Question 15.
Which radioactive Isotope is used in treatment of goitre?
Iodine – 131.
Structure of the Atom Short Answer Type Questions
Question 1.
Define:
(a) Canal rays
(b) Cathode rays
(c) Atomic number
(d) Mass number
(e) Energy shells
(f) Valency
(g) Octet
(h) Isotopes
(t) Isobars.
(a) Canal rays: These are positively charged radiations which led to the discovery of sub – atomic positively charged particles called protons through an experiment conducted by J.J. Thomson in 1897.
(b) Cathode rays: These are negatively charged radiations which led to the discovery of sub – atomic negatively charged particles called electrons during an experiment conducted by E. Goldstein in 1866.
(c) Atomic number: It is the number of protons present in the nucleus of an atom. It is represented by the letter ‘Z’.
(d) Mass number: It is the total number of protons and neutrons present in an atom of an element.
So, Mass number = Number of protons + Number of neutrons.
(e) Energy Shells: These are fixed circular paths around the nucleus of an atom in which electrons revolve continuously with high speed. These are also called orbits. They are represented by the alphabets K, L, M, N.
(f) Valency: It is the combining capacity of an atom to become electrically stable, or it also means the number of valency electrons lost or gained by an atom to complete the eight electrons in the valence shell.
(g) Octet: The completely filled outermost shell like L, M or N with 8 electrons is called an octet. When an atom completes its octet, then it become stable.
(h) Isotopes: These are atoms of same element having same atomic number, but different mass number.
E.g.
• ($$_{ 1 }^{ 1}{ H }$$) , ($$_{ 1 }^{ 2 }{ H }$$), ($$_{ 1 }^{ 3 }{ H }$$) and $$_{ 6 }^{ 12 }{ C }$$, $$_{ 6 }^{ 14 }{ C }$$are the isotopes of hydrogen and carbon respectively.
(i) Isobars: These are atoms of different elements having different atomic number but same mass number.
E.g.
• $$_{ 18 }^{ 40 }{ Ar }$$, $$_{ 20 }^{ 40 }{ Ca }$$ and $$_{ 11 }^{ 24 }{ Na }$$, $$_{ 12 }^{ 24 }{ Mg }$$.
Question 2.
Differentiate between:
(a) Electrons and protons.
(b) Atomic number and mass number.
(c) Isotopes and isobars.
(d) Valence electrons and valency.
(a)
Electrons Protons (i) This is negatively charged sub – atomic particle. (i) This is positively charged sub – atomic particle. (ii) Its mass is 9 × 10–28gms. (ii) Its mass is 1.6 × 10-24gms. (iii) Its symbol is “e–”. (iii) Its symbol is “P+”.
(b)
Atomic number Atomic mass (i) It is the total number of protons present in the atom. (i) It is the sum of protons and neutrons present in the atom. (ii) It is represented by ‘Z’. (ii) It is represented by ‘A’. (iii) It is written on the bottom left as a subscript with the symbol the of element. (iii) It is written on top left as a subscript with the symbol the of element.
(c)
Isotopes Isobars (i) These are atoms of the same element. (i) These are atoms of the different elements. (ii) They have same atomic number. (ii) They have different atomic number. (iii) They have different mass number. (iii) They have same mass number. (iv) They have same chemical properties. (iv) They have different chemical properties.
(d)
Valence Electrons Valency (i) These are electrons present in the outermost shell of an atom. (i) These are electrons lost or gained through valence shell of an atom to become stable. (ii) Valence electrons can be 1, 2, 3 …….. 8 or more. (ii) Valency can be 0, 1, 2, 3, 4 only.
Question 3.
Draw the diagrams of:
(a) J.J. Thomson’s model of atom.
(b) Rutherford’s model of atom.
(c) Neils Bohr’s model of atom.
(a) J.J. Thomson’s Model:
(b) Rutherford’s Model:
(c) Neils Bohr’s model of atom.
Question 4.
Write the postulates of J.J. Thomson’s model of the atom.
J.J. Thomson’s postulates for model of the atom are as follows:
1. An atom is a positively charged sphere or ball and negatively charged electrons are embedded in it.
2. The atom is electrically neutral because negative and positive charges are equal in magnitude.
Question 5.
Write the main points of the theory given by Rutherford for model of atom.
Main points of theory of Rutherford regarding model of atom are:
1. There is an existence of positively charged centre in the atom called as nucleus which contains all the mass of the atom.
2. The electrons revolve round the nucleus in circular paths called orbits at high speeds.
3. The size of nucleus (centre of the atom) is very small as compared to size of the atom.
4. Most of the sphere in an atom is empty.
Question 6.
Write the rules given by Bohr – Bury for arrangement of electrons in different orbits in an atom.
Rules given by Bohr – Bury are as follows:
1. Maximum electrons present in a shell is given by 2n2 where
• n is the number of that shell.
• Like, for first shell K, n = 1
• For second shell L, n = 2
• third shell M, n = 3
• fourth shell N, n = 4 called as nucleus which contains all the mass of the atom.
2. The outermost shell can have maximum of 8 electrons.
3. Electrons cannot occupy a shell till its inner shells or orbits are completely filled.
Question 7.
An atom ‘X’ has a mass number ‘23’ and atomic number ‘11’. Find its electrons, protons and neutrons. Also, name the element
We know,
Atomic number = Number of protons.
∴ 11 = Number of protons
And, Number of protons = Number of electrons.
∴ Number of electrons = 11
Now, Mass number = Protons + Neutrons.
23 = 11 + Neutrons
∴ Neutrons = 23 – 11 = 12
∴ Atom ‘X’ has 11 electrons, 11 protons and 12 neutrons.
The element is Sodium (Na).
Question 8.
Write the electronic configuration of neon, aluminium, sulphur, argon. Also, find valencies.
Question 9.
What are radioactive isotopes? Write down the type of isotopes used in:
(a) Tracing blood clots and tumours in human body
(b) Treatment of cancer
(c) Treatment of Goitre
(d) Nuclear reactor as a fuel.
Radioactive isotopes: These are unstable isotopes due to extra neutrons in their nucleus and emits different types of radiations.
Examples:
• Uranium – 235
• Cobalt – 60
• Carbon – 14.
Types of Isotopes used in:
(a) Sodium – 24 to detect blood clots and Arsenic – 72 to detect tumours.
(b) Cobalt – 60
(c) Iodine-131
(d) Uranium – 235.
Question 10.
Draw the atomic structure of:
(a) Fluorine atom (F)
(b) Sodium atom (Na)
(c) Potassium atom (K)
(a) Fluorine atom (F):
Atomic number: 9
(b) Sodium atom (Na)
Atomic number: 11
(c) Potassium atom (K)
Atomic number: 19
Question 11.
Write the atomic number, mass number, electrons, protons and neutrons of following atoms:
(a) $$_{ 14 }^{ 24 }{ X }$$
(b) $$_{ 13 }^{ 27 }{ X }$$
(a) $$_{ 14 }^{ 24 }{ X }$$
Atomic number = 14
Mass number = 24
Electrons = 14
Protons = 14
Neutrons = 24 – 14 = 10
(b) $$_{ 13 }^{ 27 }{ X }$$
Atomic number = 13
Mass number = 27
Electrons = 13
Protons = 13
Neutrons = 27 – 13 = 14
Question 12.
Pick out the Isotopes and Isobars from the following atoms:
$$_{ 17 }^{ 37 }{ A }$$, $$_{ 18 }^{ 40 }{ A }$$, $$_{ 17 }^{ 33 }{ A }$$, $$_{ 20 }^{ 40 }{ A }$$.
1. Isotopes: $$_{ 17 }^{ 37 }{ A }$$, $$_{ 17 }^{ 33 }{ A }$$
2. Isobars: $$_{ 18 }^{ 40 }{ A }$$, $$_{ 20 }^{ 40 }{ A }$$.
Question 13.
What are noble gases? Why they are stable? Give three examples.
Noble gases are the elements which are stable and do not take part in chemical reaction.
They are stable because they have completely filled outer- most shell with 8 electrons
example:
Helium is the only noble gas which has 2 electrons in outermost shell.
Question 14.
Write all the Isotopes of:
1. Hydrogen
2. Oxygen
3. Chlorine
4. Bromine
5. Carbon
6. Neon.
1. Hydrogen (H) – $$_{ 1 }^{ 1 }{ H }$$, $$_{ 1 }^{ 2 }{ H }$$
2. Oxygen (O) – $$_{ 8 }^{ 16 }{ O }$$, $$_{ 8 }^{ 17 }{ O }$$, $$_{ 8 }^{ 18 }{ O }$$
3. Chlorine (Cl) – $$_{ 17 }^{ 35 }{ Cl }$$, $$_{ 17 }^{ 37 }{ Cl }$$
4. Bromine (Br) – $$_{ 35 }^{ 79 }{ Br }$$, $$_{ 35 }^{ 81 }{ Br }$$
5. Carbon (C) – $$_{ 6 }^{ 12 }{ C }$$, $$_{ 6 }^{ 14 }{ C }$$
6. Neon (Ne) – $$_{ 10 }^{ 20 }{ Ne }$$, $$_{ 10 }^{ 21 }{ Ne }$$, $$_{ 10 }^{ 22 }{ Ne }$$
Question 15.
Why is it wrong to say that atomic number of an atom is equal to its number of electrons?
We know that in an atom number of electrons is equal to the number of protons. But, we cannot say that atomic number is equal to number of electrons because number of electrons can be changed after losing or gaining by an atom during chemical reaction. But, number of protons remain constant.
Question 16.
What explanation did Neils Bohr gave on stability of atoms?
Neils Bohr explained the stability of atom through following points:
1. The electrons revolve around the nucleus in fixed orbits or energy levels or shells and each orbit has its fixed radius.
2. While revolving electrons do not radiate their energies, so they do not fall into the nucleus and make the atom stable.
Question 17.
What are nucleons? What is the name given to the atoms having same number of nucleons?
Protons and neutrons together in the nucleus are called nucleons. It means number of nucleons is equal to the sum of protons and neutrons. Atoms having same number of nucleons are called isobars.
Structure of the Atom Long Answer Types Questions
Question 1.
The average atomic mass of a sample of an element X is 13u. What are the percentages of isotopes $$_{ 6 }^{ 12 }{ X }$$ and $$_{ 6 }^{ 14 }{ X }$$ in the sample?
Let, the percentage of isotope $$_{ 6 }^{ 12 }{ X }$$ be x%.
So, percentage of isotope $$_{ 6 }^{ 14 }{ X }$$ is (100- x) %.
Now, Average atomic mass = Mass of $$_{ 6 }^{ 12 }{ X }$$ + Mass of $$_{ 6 }^{ 14 }{ X }$$ According to percentages,
∴ 13 = x% of 12 + (100 – x)% of 14
∴ 13 × 100 = 1400 – 2x
1300 = 1400 – 2x
1300 = 1400 – 2x
-100 = -2x
x = $$\frac { 100 }{2 }$$ = 50
So, percentage of $$_{ 6 }^{ 12 }{ X }$$ is 50% and percentage of $$_{ 6 }^{ 14 }{ X }$$ is
= (100 – x)%
= (100 – 50)%
= 50%
Question 2.
Explain Rutherford’s Gold Foil Experiment. Also, explain its observations conclusion, theory proposed and drawback of his model.
Ernest Rutherford performed an alpha-particles scattering experiment in which he passed a-particles on the gold foil.
Observation:
1. Most of α – particles passed straight without any deflection.
2. Some of the α – particles get deflected from their path.
3. Very few α – particles get completely bounced back.
Conclusions:
1. Maximum space in an atom is vacant as most of α – particles passed straight without any deflection.
2. Some α – particles get deflected from their paths show the existence of positive charge in the atom.
3. Very few α – particles get completely bounced back indicating the concentration of all mass with positive charge in a small volume at the centre.
Theory proposed:
1. There is an existence of positively charged centre in the atom called nucleus which contains all the mass of the atom.
2. The electrons revolve around the nucleus in circular paths called orbits at high speeds.
3. The size of nucleus (centre of the atom) is very small as compared to size of the atom.
4. Most of the space in an atom is empty.
Drawback: Rutherford’s model did not explain the stability of the atom. He proposed that electron revolves around the nucleus in circular paths. So, electrons should radiate their energies as they are continuously in circular motion. Then, they should fall into the positively charged nucleus making the atom unstable and collapse.
Question 3.
Draw the electronic structure of sodium and calcium with atomic number 11 and 20 respectively.
Sodium has electronic distribution as 2, 8, 1
Calcium has electronic distribution as 2, 8, 8, 2
Electronic structures of sodium and calcium are given:
Question 4.
Both helium (He) and beryllium (Be) have two valence electrons. Whereas ‘He’ represents a noble gas element, ‘Be’ does not. Assign reason.
The element He (Z = 2) has two electrons present in the only shell
i.e., K – shell. Since, this shell can have a maximum of two electrons only therefore,
‘He’ is a noble gas element.
The element ‘Be’ (Z = 4) has the electronic configuration as: 2,2.
Although, the second shell has also two electrons but it do not represent a noble gas element.
Structure of the Atom Higher Order Thinking Skills (HOTS)
Question 1.
Which isotope of hydrogen contain same number of electrons, protons and neutrons?
Deuterium ($$_{ 1 }^{ 2 }{ D }$$)
Number of electron (1) = Number of proton (1)
= Number of neutron (2 – 1 = 1)
Question 2.
Which element of these two would be chemically more reactive: element A with atomic number 18 or element B with atomic number 16 and why?
Electric configuration of
• A – 2,8,8
• B – 2, 8, 6
Since, the outermost shell of A is complete, it would be inert and will not react. Whereas element B require two atoms to complete its octet. Therefore, B would be more reactive.
Structure of the Atom Value Based Question
Question 1.
Shivek could not solve the following question in the group. His group – mate explained him and solved his difficulty.
The question was as follows:
What information do you get from the given figure about the atomic number, mass number and valency of the given atom ‘X’:
1. What is the atomic number, the mass number and valency of the atom?
2. Name the element ‘X’.
3. What value of Shivek’s friend are reflected in this behaviour?
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2022-12-02 12:20:31
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https://studyadda.com/question-bank/logarithms_q28/4598/362677
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• # question_answer What is the value of $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{lo}{{\mathbf{g}}_{\mathbf{10}}}\mathbf{36-21o}{{\mathbf{g}}_{\mathbf{10}}}\mathbf{3+lo}{{\mathbf{g}}_{\mathbf{10}}}\mathbf{15?}$ A) 2 B) 3 C) 1D) 0
(c): $\frac{1}{2}{{\log }_{10}}36-2lo{{g}_{10}}3+lo{{g}_{10}}15$ $=lo{{g}_{10}}{{36}^{1/2}}-lo{{g}_{10}}{{3}^{2}}+lo{{g}_{10}}15$ $=lo{{g}_{10}}6-lo{{g}_{10}}9+lo{{g}_{10}}15$ $={{\log }_{10}}\frac{6\times 15}{9}={{\log }_{10}}\frac{90}{9}={{\log }_{10}}10=1$
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2020-06-02 20:19:29
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https://demo7.dspace.org/items/9f75ddc9-f8dc-47fc-8952-d6e87e42ff68
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## Deformed Statistics Formulation of the Information Bottleneck Method
##### Authors
Venkatesan, R. C.
Plastino, A.
##### Description
The theoretical basis for a candidate variational principle for the information bottleneck (IB) method is formulated within the ambit of the generalized nonadditive statistics of Tsallis. Given a nonadditivity parameter $q$, the role of the \textit{additive duality} of nonadditive statistics ($q^*=2-q$) in relating Tsallis entropies for ranges of the nonadditivity parameter $q < 1$ and $q > 1$ is described. Defining $X$, $\tilde X$, and $Y$ to be the source alphabet, the compressed reproduction alphabet, and, the \textit{relevance variable} respectively, it is demonstrated that minimization of a generalized IB (gIB) Lagrangian defined in terms of the nonadditivity parameter $q^*$ self-consistently yields the \textit{nonadditive effective distortion measure} to be the \textit{$q$-deformed} generalized Kullback-Leibler divergence: $D_{K-L}^{q}[p(Y|X)||p(Y|\tilde X)]$. This result is achieved without enforcing any \textit{a-priori} assumptions. Next, it is proven that the $q^*-deformed$ nonadditive free energy of the system is non-negative and convex. Finally, the update equations for the gIB method are derived. These results generalize critical features of the IB method to the case of Tsallis statistics.
Comment: 6 pages. Expanded analysis, typographical corrections, 1 reference added
##### Keywords
Condensed Matter - Statistical Mechanics, Physics - Data Analysis, Statistics and Probability, Statistics - Machine Learning
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2022-11-30 13:54:07
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https://anuragbishnoi.wordpress.com/2019/05/15/pseudorandom-clique-free-graphs/
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## Pseudorandom clique-free graphs
Pseudorandom graphs are graphs that in some way behaves like a random graph with the same edge density. One way in which this happens is as follows. In the random graph $G(n, p)$, with $p = p(n) \leq 0.99$, a direct application of Chernoff bound implies that the probability of the following event approaches $1$ as $n$ approaches infinity:
$|e(S, T) - p|S||T|| = O(\sqrt{pn |S||T|})$
where $S,T$ are arbitrary subsets of vertices and $e(S,T)$ denotes the number of edges with one end vertex in $S$ and the other one in $T$. Note that $p|S||T|$ is the expected number of edges between $S$ and $T$ in this model, and $\sqrt{p(1 - p)|S||T|}$ is the standard deviation. Now let $G$ be a $d$-regular graph on $n$-vertices and let $\lambda$ be the second largest eigenvalue of $G$ in absolute value (these are referred to as $(n, d, \lambda)$-graphs). Then the following is true for any two subsets $S, T$ of vertices:
$|e(S,T) - (d/n) |S||T|| \leq \lambda \sqrt{|S||T|}$.
where $\lambda$ is the second largest eigenvalue of the adjacency matrix of the graph, in absolute value. Therefore, if $\lambda$ is small, and in particular close to being $O(\sqrt{d})$, then the graph mimics the behaviour of $G(n, d/n)$. In fact, for any $(n, d, \lambda)$-graph, with $d < n/2$, one can show by looking at the square of the adjacency matrix that $\lambda = \Omega(\sqrt{d})$. The graphs where $\lambda = \Theta(\sqrt{d})$ are known as optimally pseudorandom.
Pseudorandom graphs have found several applications over the last few decades, and there are many interesting questions about them (see the survey of Krivelevich and Sudakov). In a 1994 paper, Noga Alon constructed a family of optimally pseudorandom triangle free graphs on $n$-vertices, with $d/n = \Omega(n^{-1/3})$, that he then used to give explicit bounds on some Ramsey numbers, and to show that the maximum possible Euclidean norm of $n$ unit vectors in $\mathbb{R}^n$ with the property that among any three of them two are orthogonal, is equal to $\Theta(n^{2/3})$. More applications of this construction can be found in the recent survey paper of Noga based on a talk he gave at the conference celebrating the 70th birthday of László Lovász (which is where I learned about these graphs, and the main topic of this post).
Alon’s construction is in fact optimal in the sense that there existence a constant $C > 0$, such that any optimally pseudorandom graph on $n$ vertices with $d/n > Cn^{-1/3}$ must contain a triangle. This can be generalised to cliques of size $k$, where we have a constant $C > 0$ such that any optimally pseudorandom $(n, d, \lambda)$-graph with $d/n > C n^{-1/(2k - 3)}$ must contain $K_k$. The proof of this follows from greedily picking common neighbours, using the following lemma (that can be proved using the edge distribution proposition above):
If $S$ is a set of vertices with $|S| \geq 2n\lambda/d$, then there are at least $d|S|^2/4n$ edges with both end points in $S$.
The natural question now is to give matching constructions for this bound for all $k > 3$, or improve the bound. This question has been asked by several people, as it arises naturally in many situations, but it has been open for every $k > 3$ since the past 20 years or so. David Conlon has called it “one of the outstanding open problems about pseudorandom graph” (also check out this video).
The best known construction so far for larger values of $k$ was the construction of Alon and Krivelevich that gives us optimally pseudorandom graphs with edge density $d/n = \Theta(n^{-1/(k - 2)}$. Note that this starts giving a better construction than Alon’s triangle free graphs at $k = 6$. In my recent work with Ferdinand Ihringer and Valentina Pepe, we have been able to provide a better construction, that gives a family of graphs with edge density $d/n = \Theta(n^{-1/(k - 1)})$. The construction is fairly easy to describe, and for a proof you can refer to the paper:
Let $Q(x_1, \dots, x_k) = x_1^2 + \xi x_2^2 + \sum_{i = 3}^k x_k^2$ be a quadratic form over $\mathbb{F}_q$ where $\xi$ is a non-square in $\mathbb{F}_q$ (we assume $q$ to be odd). Define a graph with vertices as the $1$-dimensional subspaces $x$ of $\mathbb{F}_q^k$, for which $Q(x)$ is a square, and making two vertices $x$ and $y$ adjacent if they are orthogonal to each other, that is,
$\frac{1}{2}(Q(x + y) - Q(x) - Q(y)) = x_1 y_1 + \xi x_2y_2 + \sum_{i =3}^n x_i y_i = 0$.
Then this graph is a $K_k$-free $(n, d, \lambda)$-graph with $n = (1 + o(1))q^{k-1}/2$, $d = (1 + o(1))q^{k - 2}/2$ and $\lambda = \Theta(q^{(k - 2)/2})$.
While we now have slightly better constructions, we are still far away from the conjectured bound. I am hopeful though that finite geometry, and especially the geometries associated with quadratic forms (known as polar spaces), can play an important role in obtaining even better constructions. In fact, there is a construction due to Kopparty for triangle-free graphs matching the parameters of Alon’s construction that essentially comes from a generalized quadrangle (which is a polar space), if you look at it carefully (see my earlier post for hints on that). Moreover, Conlon was able to give an (almost) optimal probabilistic construction which also uses generalized quadrangles. This a strong indication that polar spaces can be useful in general. As a first step, we should perhaps try to obtain optimally pseudorandom $K_4$-free graphs that have higher edge density than $n^{-1/3}$.
Edit 04/09/2019: An exciting new result by Mubyai and Verstraete shows that if the main conjecture of this post is true, that is, there exists $K_k$-free pseudorandom graphs of density $n^{-1/(2k - 3)}$, then this would asymptotically determine off-diagonal Ramsey numbers $R(k, t)$, with $k$ fixed and $t \rightarrow \infty$. In fact, even an improvement of the denominator from $k - 1$ (which is in our construction) to $k + 1$ would be a big development as it’ll improve the current lower bounds on Ramsey numbers. See the first half of this talk of Mubayi.
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2021-10-28 05:12:48
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https://www.physicsforums.com/threads/gravitational-potential-energy-deltaeg-gmm-r1-gmm-r2.402618/
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Gravitational Potential Energy deltaEg=(GMm/r1)-(GMm/r2)
1. May 11, 2010
clearlyjunk
deltaEg=(GMm/r1)-(GMm/r2)
My question is how is it possible to say that in General , Eg=(-GMm/r)
and why does Eg not equal 0 when r -> infinity
Thanks
2. May 11, 2010
rcgldr
Potential energy is the negative of the work done by the force generating the field, which explains why Eg is usually negative. If infinity is used as a reference point, then Eg at infinity is equal to zero, by definition, and all Eg's at finite distance from the source are negative.
3. May 16, 2010
dulrich
Also, infinity is the natural reference point because that is where the interaction drops to zero. No interaction, no ability to do work, no energy.
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2018-03-23 11:25:59
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https://math.stackexchange.com/questions/2336638/theorem-2-14-in-walter-rudins-principles-of-mathematical-analysis
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# Theorem 2.14 in Walter Rudin's Principles of Mathematical Analysis
I've got some difficult for understanding Theorem 2.14 in baby rudin.
Theorem 2.14. Let $$A$$ be the set of all sequences whose elements are the digits 0 and 1. This set $$A$$ is uncountable.
The elements of $$A$$ are sequences like 1, 0, 0, 1, 0, 1, 1, 1, ...
Rudin gave a proof looks correct.
But with Theorem 2.12, I figure out a proof that gets a opposite conclusion.
Theorem 2.12. Let $$\{E_n\}, n=1, 2, 3,...,$$ be a sequence of countable sets, and put $$S=\bigcup_{n=1}^{\infty}E_n$$. Then S is countable.
Collary. Suppose $$A$$ is at most countable, and, for every $$\alpha\in{A}$$, $$B_\alpha$$ is at most countable. Put $$T=\bigcup_{\alpha\in{A}}B_\alpha$$. Then $$T$$ is at most countable.
And my proof for the set $$A$$ in Theorem 2.14 is countable:
My proof: Let $$A_n$$ be the set of length $$n$$'s sequences whose elements are digits 0 and 1. $$A_n$$ is at most countable. So, with Theorem 2.12, $$A=\bigcup_{n=1}^\infty{A_n}$$ is at most countable. That means the set of all sequences whose elements are the digits 0 and 1 is countable.
I know I definitely made a mistake, but where is it?
• You mistake is that you are only counting sequences which are eventually zero. Jun 26, 2017 at 10:45
• could you explain more? @uniquesolution Jun 26, 2017 at 10:56
In $\bigcup_{n=1}^\infty{A_n}$ are only sequences of finite length ! For example the sequence $(1,1,1,....)$ is not in $\bigcup_{n=1}^\infty{A_n}$
• I think the notation $\infty$ means infinite length... Jun 26, 2017 at 10:50
• No. It means the union of all sequences of finite length
– Fred
Jun 26, 2017 at 10:52
• In Definition 2.9 in baby rudin, it says "The symbol $\infty$ merely indicates that the union of a countable collection of sets is taken", why not infinite length? Jun 26, 2017 at 10:56
• To say the sequence $(1, 1, 1, \ldots)$ is in the union, is to say that there exists an $n \in \mathbb{N}$ with $(1, 1, 1, \ldots) \in A_n$. It therefore means that $(1, 1, 1, \ldots)$ is of (finite) length $n$. This is absurd! Therefore it cannot belong to any $A_n$, so it's not in the union. Jun 26, 2017 at 12:16
• That make sense. Thanks very much. Jun 26, 2017 at 17:58
The union of all $A_n$'s is the set of all finite seqeunces of $0$'s and $1$'s.
• The notation $\infty$ means all the integers, doesn't it? Jun 26, 2017 at 10:53
• @JohnnyJi The notation $\bigcup_{n=1}^\infty A_n$ means $A_1\cup A_2\cup A_3\cup\ldots$ Does this answer your question? Jun 26, 2017 at 11:04
• $\bigcup_{n=1}^\infty{A_n}$ means $A_1\bigcup{A_2}\bigcup{A_3}\bigcup...\bigcup{A_\infty}$, is this right? Jun 26, 2017 at 13:44
• @JohnnyJi No, you're wrong. It means what I wrote: $A_1\cup A_2\cup A_3\cup\ldots$ Besides, what is $A_\infty$? Where did you define it? What makes you think that it is countable? Jun 26, 2017 at 14:11
• Yeah, it makes sense. I've checked the definition of $\bigcup_{n=1}^\infty{A_n}$, you're right, there is no $A_\infty$, it only means that the union of a countable collections of sets. But still, I don't understand why the set of all positive integers doesn't include the infinite posivite number. Jun 26, 2017 at 16:59
You are counting only sequences of finite length. Rudin considers sequences of infinite length.
Your set $A_7$ consists of sequences of length $7$. Rudin's set $A$ had no sequences of length $7$. So your union is not a subset of $A$.
Does your union involve a set $A_\infty$, sequences of infinite length? if so, then you are wrong when you say $A_n$ is at most countable, since you have not proved that $A_\infty$ is at most countable.
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2022-08-12 15:10:37
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 15, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8112393021583557, "perplexity": 234.10677624782946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571719.48/warc/CC-MAIN-20220812140019-20220812170019-00164.warc.gz"}
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https://forum.azimuthproject.org/discussion/2379/applied-category-theory-course-at-mit
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#### Howdy, Stranger!
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# Applied category theory course at MIT
edited March 2019
Check out Brendan Fong and David Spivak's Open Courseware site at MIT, for lectures, videos, assignments and more:
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thks
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Test.
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Comment Source:Thanks for linking to them, they are very helpful!
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2020-10-22 15:18:01
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17740359902381897, "perplexity": 11605.633528355944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107879673.14/warc/CC-MAIN-20201022141106-20201022171106-00547.warc.gz"}
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http://mathhelpforum.com/advanced-algebra/117656-normal-orperator-eigenvalue.html
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Let $V=R^2$ and define $T\in L(V)$ as the 90-degree rotation operator given by $T(x,y)=(-y,x)$ where $x,y \in R$ are the components of a vector in V.
2. For a), find the matrix which represents $T$ with respect to the standard basis. Then show that the matrix is normal.
For b), suppose that $T(x,y)=\lambda(x,y)=(-y,x)$. What can you tell about $\lambda$? (Equivalently, find the roots of the characteristic polynomial of the matrix you found in part a) ).
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2017-03-29 16:05:16
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http://www.nature.com/articles/s41467-018-06260-8?WT.feed_name=subjects_physical-oceanography&error=cookies_not_supported&code=89058bb5-3a7c-4397-86c9-e3ce810348fd
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Article | Open | Published:
Latitude-dependent finescale turbulent shear generations in the Pacific tropical-extratropical upper ocean
Abstract
Turbulent mixing, which is critically important for the equilibrium of ocean circulation, is controlled by finescale turbulent shear (S2) of oceanic flows through shear instability. Although the relationship between S2 and mixing is well understood, the latitude-dependent generation processes of S2 remain poorly known due to the lack of geographically extensive, long-term finescale velocity measurements. Here, using one-year ADCP data from 17 moorings along 143°E, we first show that the upper-ocean S2 and its resultant mixing rate have a W-shaped latitudinal distribution in the tropical-extratropical northwest Pacific with peaks at 0–2°N, 12–14°N, and 20–22°N, respectively. Further analyses reveal that these S2 peaks are caused by vertically-sheared equatorial currents, parametric subharmonic instability of diurnal tide, and anticyclonic eddy’s inertial chimney effect, respectively. As climate model simulations are sensitive to the mixing parameterizations, our findings highlight the need to incorporate the latitude-dependent generation mechanisms of S2 to improve climate models’ prediction capabilities.
Introduction
Turbulent mixing across density surfaces (i.e., diapycnal mixing) plays a fundamental role in redistributing momentum, heat, nutrients, carbon and other materials in the ocean that modulate the ocean circulation, the biogeochemical cycles, and the long-term climate1,2. Because the turbulent processes are too small (with vertical scale of 10−2–101 m) to be resolved by climate models, their mixing effects must be properly represented, or parameterized, in terms of model’s resolved features. Existing model studies have demonstrated that a wide range of the simulated climate phenomena are sensitive to the geography of turbulent mixing and its parameterizations3,4,5,6. A better understanding of the turbulent mixing processes, therefore, is a prerequisite for improving model’s parameterizations and its capability to accurately simulate the present and future climate.
Because of the diverse dynamical regimes, the governing physics of turbulent mixing can have a strong latitude dependence in the world ocean7,8,9. While critical for improving the model’s parameterization schemes, quantitatively understanding this latitude dependence has long been a challenging problem due to the sparsity of in situ observations2,10. Although ship-based dyed tracer tracking11 and microstructure profiling (at vertical scale of centimeters)7 are two direct ways to accurately measure the turbulent mixing, high-cost and sophisticated operations have limited their spatiotemporal coverage in the ocean12. Given the extremely patchy and intermittent nature of turbulent mixing, such direct measurements are far from adequate to quantify its latitude-dependent generation mechanisms. Alternatively, the strength of turbulent mixing can be inferred from finescale vertical shear of velocity at vertical scale of 10–100 m (termed finescale turbulent shear hereafter), that relies either on the internal wave-wave interaction theory13,14 or on the Richardson number-based criteria of shear instability15,16. Observational and theoretical studies have suggested that the turbulent dissipation rate scales with the finescale turbulent shear, while the rate of turbulent mixing (i.e., diapycnal diffusivity) is related to the former through a nearly constant mixing efficiency17,18. In contrast to the direct mixing measurements, finescale turbulent shear data are much easier to obtain owing to the widespread use of Acoustic Doppler Current Profiler (ADCP) measurements. For example, lowered ADCP shear data have been successfully used to infer the diapycnal diffusivity (Kρ) in different parts of the world ocean14,19. Based on finescale shear and strain data from thirty thousand hydrographic profiles, the most recent study of Kunze et al.20 proffered a rough geography of Kρ with a nearly global scope. However, these results are based mostly on snapshots and lack of the long-term time series has limited their representativeness and the physical knowledge behind them.
Compared with the instantaneous lowered and shipboard ADCP measurements, moored ADCPs are more effective to continuously observe the oceanic finescale velocity profiles at fixed locations. Because the moored ADCP can repeatedly collect high-frequency (with sampling intervals at tens of minutes) velocity profiles for a long period (several months or longer), it allows to precisely distinguish motions with different frequencies and hence to investigate the underlying sources generating the finescale turbulent shear and its resultant mixing21,22,23. Based on moored ADCP data, many studies have reached a consensus that near-inertial waves (NIWs) and internal tides (ITs) are two primary sources for turbulent shear generation in the ocean interior21,24,25,26. In addition, some studies found that turbulent shear may also be produced by the strongly sheared mean current, such as the South Equatorial Current (SEC) and Equatorial Undercurrent (EUC)27, generated by Tropical Instability Waves near the equator16,28, and modulated by mesoscale eddies through eddy-wave interactions29,30. However, nearly all the previous moored ADCP-based studies deployed only one or several moorings in a limited area; as a result, it remains largely unclear how the turbulent shear/mixing and its governing physics vary with the latitude-dependent dynamical regimes in the ocean.
The tropical-extratropical northwest Pacific Ocean is featured by three distinct dynamical regimes from 0° to ~25°N31: the equatorial regime (0–6°N) with alternating zonal jets and strong eddy kinetic energy (EKE), the off-equatorial regime (7–18°N) with a broad and stable westward-flowing current and weak EKE, and the subtropical regime (19–25°N) with weak mean current but enhanced EKE. To investigate the characteristics of, and interactions among, the multiscale dynamical processes in these distinct regimes, the Northwestern Pacific Eddies, Internal waves and Mixing Experiment (NPEIM) was initiated between 2015–2016. As the key component of NPEIM, a mooring array consisting a total of 17 ADCP moorings were deployed along the 143°E meridian for over one year (see Methods). Extending from 0° to 22°N, the mooring section traverses the above three dynamical regimes and provides us with a unique opportunity to investigate the latitudinally dependent processes underlying the finescale turbulent shears.
Based on the year-long ADCP measurements from the NPEIM, this study for the first time reveals that in the tropical-extratropical northwest Pacific upper ocean (100–400 m layer), the finescale turbulent shear and the parameterized mixing rate are elevated at the latitude bands of 0–2°N, 12–14°N, and 20–22°N. We further find that mechanisms of the three mixing peaks are associated with strong sub-inertial shear of the equatorial currents, Parametric Subharmonic Instability (PSI) of diurnal ITs, and anticyclonic eddy’s chimney effect on wind-driven NIWs, respectively.
Results
Moored observations
As a component of NPEIM to investigate the characteristics of the multiscale processes and their interactions in the northwestern Pacific, a total of 17 bottom-anchored moorings were deployed roughly along the 143°E meridian from 0° to 22°N (Fig. 1). The 17 moorings were deployed between 24/Nov/2015–04/Jan/2016 and successfully recovered between 10/Feb–08/Mar/2017, providing us with more than one-year (13–15 months) continuous moored data. Spatially, the moorings are set up at every 2° between 0–16°N (inclusive), every 1° between 16–20°N, and every 0.5° between 20–22°N, respectively. The enhanced mooring resolution between 20–22°N is to resolve the mesoscale eddies in the unstable Subtropical Countercurrent (STCC) region. Prior to the NPEIM experiment, mooring P17 at the northern end of the mooring array (~22°N) was deployed for 17 months, and totally, P17 provided continuous measurements for more than 32 months (~2.7 years). All the moorings were equipped with ADCPs, recording current meters (RCMs), and temperature chains to measure current velocity and temperature over nearly the whole water column (see Methods and Supplementary Table 1). Specifically, an upward-looking and a downward-looking 75 kHz ADCPs (only one upward-looking ADCP at P8 and P10-12) were mounted at ~500 m depth of the mooring to observe the high-frequency (half-hourly) current velocity in the upper 1000 m (or 500 m) with a vertical bin of 16 m. The high-frequency ADCP data are used to analyze the characteristics and physics of the finescale turbulent shear in this study.
As shown in Fig. 1b, the moored observations have captured the three dynamical regimes with diverse characteristics of circulation and EKE (Supplementary Fig. 1). In the equatorial regime (0–6°N), the circulation is characterized by multiple strong zonal jets, including the westward-flowing SEC and eastward-flowing North Equatorial Countercurrent (NECC) near the surface, the eastward-flowing EUC in the subsurface, and the westward-flowing Equatorial Intermediate Current (EIC) in the intermediate layer, etc. These zonal currents exhibit large velocity shears in both vertical and meridional directions and strong mesoscale EKE throughout the upper 1000 m layer. The off-equatorial regime (7–18°N), which straddles the boundary between the tropical and subtropical wind-driven gyres, is occupied by the broad and deep-reaching westward-flowing North Equatorial Current (NEC). Due to the lack of sign change in gradient of potential vorticity, the NEC in this band is dynamically stable, showing rather weak EKE above 300 m. It is noteworthy that beneath the NEC (below 400 m) between 12–14°N, there is a relatively weak eastward-flowing zonal jet. This subthermocline zonal jet was recently named the North Equatorial Undercurrent and was proposed to be generated by eddy-eddy interactions32,33,34. The northern part of the mooring array (19–22°N) is located within the southern half of the wind-driven subtropical gyre. Within this subtropical regime, there exists the relatively weak and shallow eastward-flowing STCC. Because of the baroclinic instability between the vertically sheared STCC and NEC35, this band displays strong EKE in the upper 300 m layer.
Latitudinal variation
The total finescale turbulent shear squared (S2) calculated based on the ADCP data (see Methods) displays strong heterogeneities along the mooring section (Fig. 2a). The most turbulent region occurs in the upper 400 m layer of the equator, where the largest mean S2 exceeds 10−4 s−2 near the surface. With the increasing latitude, the upper-layer (hereafter, it means the 100–400 m layer) S2 sharply decreases from the equatorial to the off-equatorial regime and then shows moderate variations further north in the subtropical regime. Vertically, S2 generally decreases with depth but the large-valued S2 can extend deeper in the subtropical regime than the two tropical regimes. For example, large S2 exceeding 10−5.5 s−2 is only confined to the upper 300 m between 4–10°N, while it can penetrate below 700 m between 20–22°N. From the 100–400 m layer-averaged S2 (excluding the mixed layer above 100 m), it can be clearly seen that the upper-layer S2 has a distinct W-shaped distribution from equator to 22°N. Three peaks of the “W” are located at 0–2°N, 12–14°N, and 20–22°N, respectively, and they fall into the three dynamical regimes separately.
To obtain the physical insights into the latitudinal variation of S2, we decompose the finescale turbulent shear into three frequency-dependent components, that is, sub-inertial, near-inertial, and tidal components, respectively (Fig. 2b–e; see Methods). Generally, the sub-inertial S2 shows a distribution similar to the total S2 in the equatorial regime, whereas this correspondence applies to the near-inertial S2 in the off-equatorial and subtropical regimes. The upper layer-averaged sub-inertial S2 is quite close to the total one in the equatorial regime, with its contribution exceeding 83% between 0–2°N (Fig. 2f). However, the strong sub-inertial S2 only occurs near the equator and it rapidly weakens with increasing latitude. The ratio of the sub-inertial S2 over the total S2 is generally lower than 20% in the off-equatorial and subtropical regimes. In contrast to the sub-inertial component, the near-inertial S2 is rather weak in the equatorial regime, but keeps increasing until it reaches its first peak at 14°N. North of 14°N, the near-inertial S2 first decreases to a local minimum at 18°N and then increases to its second peak at 21.5°N. In the off-equatorial and subtropical regimes, the near-inertial S2 accounts for 54–74% of the total S2, with the highest and lowest percentage at 10°N and 18°N, respectively. Compared with the sub-inertial and near-inertial components, the upper-layer tidal S2 is relative weak with a low ratio (<30%) throughout the section. This observed result is consistent with the existing knowledge that in the ocean interior away from rough topography, the ITs are generally dominated by low vertical modes that have weak velocity shears2. From the results in Fig. 2, we can conclude that the upper-layer S2 is dominated by the sub-inertial component in the equatorial regime, and the near-inertial component in the off-equatorial and subtropical regimes, respectively. In the following, we explore the generation mechanisms of the finescale turbulent shear in each of the dynamical regimes.
Equatorial regime
In the strong finescale shear region at 0° (2°N), the mean sub-inertial S2 has two local maxima in the upper layer, one near the surface (at ~100 m) and the other at ~270 m (220 m) depth (Fig. 3a, c). The vertical positions of these two maxima correspond well with the interface of SEC/EUC and EUC/EIC, respectively (Fig. 3b, d), demonstrating that the observed large sub-inertial S2 in the equatorial regime is closely associated with the vertically sheared zonal currents. In addition to the mean current, the equatorial waves such as Yanai waves and internal gravity waves can also make important contributions to the sub-inertial S2 near the equator36,37. Indeed, strong 20–100 day modulations can be clearly seen from the time series of sub-inertial shear (Supplementary Fig. 2a). Averaged over the observation period, the upper-layer ratio between the 20–100 day and total sub-inertial S2 is 0.34 (0.31) at 0° (2°N), suggesting that equatorial waves roughly account for one third of the sub-inertial S2 near the equator (Fig. 3a, c).
In order to qualitatively evaluate how the strong S2 in the equatorial regime can impact turbulent mixing, we further examine the possibility of shear instability based on Richardson number (see Methods). Here, the criterion of Ri < 1/3, rather than the traditional Ri < 1/4, was used to identify the possible occurrence of shear instability, because recent studies found that the equatorial currents are marginally unstable and turbulence can occur even with slightly higher Ri values16,38. Despite that the relatively coarse vertical resolution of the measurements (~20 m) may overestimate the Ri15, we can still see that at equator, a large number of potential instability events (with Ri < 1/3) intermittently occur beneath the EUC core at ~200 m (Fig. 4a). The largest occurrence frequency of potential instabilities is exactly located at the EUC/EIC interface with its value as high as 0.11 (Fig. 4b). Compared with the EUC/EIC interface, the potential instability is suppressed at the SEC/EUC interface due to the stronger stratification despite the fact that S2 there is stronger. If only the sub-inertial S2 was used when calculating the Ri, the frequency of potential instabilities does not decrease significantly and the largest occurrence frequency can still reach 0.07. This result implies that even without the high-frequency internal waves, the shears generated by the mean current and sub-inertial waves alone can result in strong mixing at the equator. Similar to the equator, Ri at 2°N is also lower beneath the EUC core than above it (Fig. 4c). Although Ri at 2°N is generally higher than the equator because of the decreased S2, the occurrence frequency of Ri < 1/3 at the EUC/EIC interface still exceeds 0.05 (Fig. 4d).
Off-equatorial regime
Compared with the equatorial regime, the strength of finescale turbulent shear is overall weaker in the broad off-equatorial regime of 7–18°N. An exception is at 12–14°N, where the upper layer-averaged S2 is 76–110% larger than the trough value (i.e., at 10°N and 18°N) in this regime (recall Fig. 2e). This turbulent shear peak is primarily attributable to the near-inertial S2, whose magnitude is strongest among all the observation sites. To examine whether this near-inertial S2 peak is caused by wind forcing, which has long been thought as a primary generation source for NIWs in the upper ocean10, we calculated the wind work (WW) on the near-inertial motions based on the slab mixed-layer model (see Methods). Different from the observed latitudinal distribution of near-inertial S2, the near-inertial WW only shows moderate values between 12–14°N (Fig. 5d)39,40. Because the long-distance propagating NIWs from far field are dominated by low modes and contain weak shears, this near-inertial peak is also unlikely caused by the NIWs originating from midlatitude storm track regions10. The above results indicate that there must be other mechanisms responsible for the near-inertial S2 peak at 12–14°N.
Near and equatorward of the critical latitude where the IT frequency is twice the local inertial frequency, PSI can be an effective way to generate higher-mode NIWs41,42. For the K1 and O1 diurnal ITs, their critical latitudes are around 13.4°N and 14.5°N, respectively, coinciding well with the 12–14°N near-inertial S2 peak. To investigate whether this S2 peak is associated with PSI, we compared the shears relating to the diurnal ITs and NIWs in Fig. 5. In connection to the near-inertial S2 peak, the diurnal and near-inertial S2 show high correlations at 14°N and 12°N (with correlation coefficients of 0.59 and 0.52, respectively; Fig. 5b), where the averaged near-inertial S2 during the high-diurnal-shear period is 30–40% larger than during the low-diurnal-shear period (Fig. 5c). However, no such significant correlations between the diurnal and near-inertial S2 can be found at the other latitudes. Given that the PSI-related NIW generation is only active near and slightly equatorward of the critical latitude, the results in Fig. 5 provide evidence for the occurrence of PSI.
Different from the wind-generated NIWs, whose energy primarily propagates downward from the surface, the PSI-generated NIWs has comparable up-going and down-going energy42. By decomposing the near-inertial motions into clockwise (CW) and counter-clockwise (CCW) rotating components with increasing depth (whose energy goes downward and upward, respectively; see Methods), we find that both the upward-going CCW near-inertial kinetic energy (KE) and S2 have a peak at 12–14°N (Fig. 6a, b). The elevated CCW KE and S2 are more noticeable from their ratios to the total ones, lending additional evidence for PSI (Fig. 6c). It is noteworthy that the ratio of CCW S2 to the total is larger than that of KE (e.g., 0.35 vs. 0.25 at 14°N), which agrees with the knowledge that PSI-generated NIWs are dominated by higher-mode waves. If we assume that the CCW NIWs are totally from PSI and that the PSI-generated NIWs have equal KE and S2 for the CW and CCW components, the ratio in Fig. 6c would mean that PSI explains 50% (46%) of the KE and 70% (57%) of S2 in the near-inertial band at 14°N (12°N).
Subtropical regime
Unlike the equatorial and off-equatorial regimes where NIWs are generally weak except at the PSI critical latitudes (i.e., 12–14°N), the near-inertial KE and S2 are strong in the whole subtropical regime (Fig. 2c, e; Fig. 6a, b). The enhanced NIW activities, nevertheless, do not coincide with the weak near-inertial WW here (Fig. 5d)39,40, indicating that other factors may concentrate the wind-generated NIWs and increase their vertical wavenumber, or that there exist additional energy sources for NIWs. Previous theoretical and modeling studies have suggested that mesoscale eddies play significant roles in modulating the NIW activities through eddy-NIW interactions10,43,44,45. Considering that the subtropical regime is abundant with mesoscale eddies (strongest among the three regimes, Supplementary Fig. 1), it is natural to ask whether the eddy-NIW interaction is responsible for the strong near-inertial KE and S2 here. The mesoscale eddy-resolving mooring array in the subtropical regime gives us a unique opportunity to answer this question (Fig. 1).
In Fig. 7a–f we show the mean near-inertial KE and S2 when the moorings are influenced by anticyclonic eddies and cyclonic eddies (AEs and CEs; see Methods), respectively. It reveals that the near-inertial KE during the AE-impacted periods is significantly enhanced between 100–800 m when compared with the CE-impacted periods. The largest KE difference between AE-impacted and CE-impacted periods occurs at ~230 m where their averaged ratio is as high as 1.8 (Fig. 7c). The AE/CE ratio for KE gradually decreases with depth below 230 m, but it can still exceed 1.4 at ~700 m. With respect to the near-inertial S2, its enhancement in connection with the AEs is even clearer between 100–400 m and the AE/CE ratio reaches 2.1 at ~230 m (Fig. 7f). This modulation of near-inertial S2 by mesoscale eddies can be better seen from its strongly negative correlation with the surface relative vorticity (ξ) that negative ξ significantly elevates the near-inertial S2 (Fig. 7g). These observed phenomena are well consistent with the existing theory that AEs with negative ξ act as “inertial chimneys” that can trap the wind-generated NIWs by lowering the local effective inertial frequency ($$f_{\mathrm{{eff}}} = f_0 + \xi /2$$, f0 is local inertial frequency)43,44,45,46. Specifically, as the AEs (CEs) lower (raise) the feff, they, on one hand, broaden (narrow) the waveband for NIW generations and, on the other hand, trap (exclude) the NIWs generated inside, therefore greatly increasing (decreasing) the near-inertial KE and S2 within them43,46. Additionally, because ξ of the baroclinic AEs rapidly weakens with depth and becomes quite small below 200 m (less than −0.03f0; Supplementary Fig. 3), the trapped NIWs with a frequency close to and lower than f0 approach the critical layer near 200 m where their downward group velocity diminishes43. Consequently, the NIWs’ energy converges and the vertical wavenumber increases near the critical layer, resulting in the observed KE and S2 peak at ~230 m within AEs. It deserves to note that in the deeper layer between 500–950 m, the AE/CE ratio for S2 is quite smaller than KE with their respective mean values of 1.1 and 1.3 (Fig. 7c, f). This result suggests that although AEs can facilitate the downward propagation of NIWs, the deep-reaching waves are primarily in low modes and have relatively weak vertical shears.
The same analysis procedure mentioned above is also performed on the 2.7-year-long ADCP data at mooring P17, which captured 7 AEs and 6 CEs during the whole observation period. The long-term observation reveals similar results as in Fig. 7, demonstrating the robustness of the statistical results, i.e., the enhanced near-inertial KE and S2 within AEs due to the inertial chimney effect (Supplementary Fig. 4a–b). We should note that this chimney effect applies only to the down-going NIWs, since no difference of the CCW-component KE and S2 are found between AEs and CEs (Supplementary Fig. 4c–d). Although AEs and CEs play opposite roles in organizing the NIWs, their net effect on near-inertial S2 is not zero but tends to elevate it due to the existence of critical layers within AEs. This nonzero net effect can be inferred from the observed result that the time-mean S2 has a more similar distribution to the mean S2 within AEs than CEs (Supplementary Fig. 4b)47. The inertial chimney effect of AEs therefore at least partly explains the enhanced near-inertial S2 in the subtropical upper ocean, where WW is generally weak. In addition to the AEs’ inertial chimney effect, there may also be other reasons for the strong near-inertial shear here that will be discussed in the discussion section.
Diapycnal diffusivity
Given that the diapycnal diffusivity (Kρ) associated with finescale turbulent shear is of broad interest, it is meaningful here to estimate Kρ and evaluate its latitudinal variation. To achieve this, we adopted two independent finescale parameterization methods based on the observed velocity shear (see Methods). The annual mean Kρ estimated based on the Gregg–Henyey–Polzin (GHP) and Ri-based parameterization methods are shown in Fig. 8a, b, respectively. For Kρ from the Ri-based method, it displays a similar W-shaped latitudinal distribution to S2 (Fig. 8b vs. Fig. 2e), which is not surprising given the strong dependence of Ri on S2. The three Kρ peaks at 0, 14°N, and 21.5°N reach 19 × 10−6 m2 s−1, 10 × 10−6 m2 s−1, and 12 × 10−6 m2 s−1, respectively, and they are 4.2, 2.2, and 2.7 times the trough values at 10°N and 18°N (about 4.5 × 10−6 ms−1). With respect to the GHP parameterized Kρ, it generally has a similar distribution to the Ri-based result between 10–22°N (Fig. 8a vs. Fig. 8b). The peak value of the GHP Kρ at 14°N (21.5°N) reaches 3.4 × 10−6 ms−1 (7.4 × 10−6 m2 s−1) and is 4.0 (8.7) times the trough value at 18°N (i.e., 0.85 × 10−6 m2 s−1). In the equatorial regime, however, the GHP parameterized Kρ is only between 0.04–0.5 × 10−6 m2 s−1 due to the very small latitude-correction term near the equator (i.e., $$j\left( {\frac{f}{N}} \right)$$ in the parameterization; see Methods). The low parameterized diffusivities in the equatorial regime are obviously unrealistic not only because they are one to two orders of magnitude smaller than the previously observed Kρ in the similar regions48,49,50, but also because the high frequency of potential shear instabilities is found here (recall Fig. 4). These results are actually consistent with Liu et al.’s50 analysis that in the equatorial region the turbulent shear (also mixing) is dominantly caused by the sub-inertial currents (recall Fig. 2), and therefore the principle of the GHP parameterization based on the internal wave-wave interaction theory is violated there. Considering that the Ri-based estimates of Kρ are much closer to the observed results in the equatorial regime (both on order of 10−5 m2 s−1), we suggest that the Ri-based parameterization (see Methods) that does not rely on specific turbulent shear processes may be more suitable to parameterize Kρ in the equatorial regime. Even in the 10–22°N latitude range, the GHP-based Kρ is also 40–80% lower than the Ri-based result. This discrepancy is likely attributable to the uncertainty of the presumed constant referenced diffusivity in the Ri-based parameterization that may vary in different regimes (see Methods). However, this does not influence our main conclusions since it is the mean pattern of Kρ, rather than its precise value, that is emphasized in this study.
Discussion
By analyzing the year-long simultaneous ADCP velocity data from 17 moorings in the NPEIM experiment (between 0–22°N, along 143°E), we detect that the finescale turbulent shear and its parameterized Kρ in the tropical-extratropical northwest Pacific upper ocean have a W-shaped latitudinal distribution with three peaks at 0–2°N, 12–14°N, and 20–22°N, respectively. The respective mechanisms accounting for the three S2 and Kρ peaks are schematically summarized in Fig. 9 and are further discussed below.
In the equatorial regime, the strong S2 peak (0–2°N) is dominated by the sub-inertial shears of the equatorial zonal jets, including the SEC, EUC, and EIC. For the SEC/EUC interface, it has long been recognized to be a hotspot of turbulent shear and mixing by previous studies51,52. However, this study finds that the EUC/EIC interface also displays enhanced S2 due to the reversing currents. Because of the reduced stratification, Ri at the EUC/EIC interface is much lower than the SEC/EUC interface, and therefore the shear instabilities and turbulent mixing are even stronger there. In addition to the mean zonal jets, the equatorial waves with period between 20–100 days also make a considerable contribution to the sub-inertial S2 (with a ratio of ~1/3). Our further analysis shows that the 20–100 day velocity shear has a good correspondence with the local zonal wind stress with the correlation coefficient of −0.44 near the surface (Supplementary Fig. 2b). Additionally, lead-lag correlations between them indicate very prominent downward-propagating signals from near surface down to ~300 m depth (Supplementary Fig. 2b). Considering that the 20–100 day S2 is strongest near the surface and monotonically decreases with depth (Fig. 3a, c), the above results suggest that the equatorial waves are most possibly originated from the intra-seasonal wind forcing near the equator. Although S2 of the equatorial waves is smaller than the mean zonal jets, they can help trigger instabilities through lowering Ri and thus enhance turbulent mixing in these marginally stable currents, especially at the strongly sheared SEC/EUC and EUC/EIC interfaces. Given that these two interfaces are located above and below the main thermocline, respectively (Fig. 4), their combined mixings would accelerate blending of the upper-thermocline and lower-thermocline waters, potentially modulating the upper-ocean heat budget of the Pacific warm pool and influencing the evolutions of monsoon and ENSO.
In contrast to the equatorial peak, both the off-equatorial (12–14°N) and subtropical S2 peaks (20–22°N) are primarily associated with NIWs. The enhanced near-inertial S2 at these two peaks are, however, generated by different mechanisms. For the off-equatorial peak, it is demonstrated to be caused by PSI of the diurnal ITs, which generates high-mode NIWs with both up- and down-going energy even in the absence of wind forcing. Although PSI of the semi-diurnal ITs near 28–29°N has been widely reported9,42,53,54,55,56, direct evidence of the PSI for diurnal ITs is rare due to the lack of specifically designed observations25,57,58. As a result of the elevated finescale turbulent shear here, the parameterized Kρ showed an increase of 120–300% (incorporate the results of two methods) than the surrounding latitudes, which is comparable to the previously estimated Kρ enhancement due to PSI by the semi-diurnal ITs59. Compared with the energetic 28–29°N latitudes of Pacific, however, the 12–14°N latitudes has much lower EKE and less near-inertial WW input31,40. In such quiescent off-equatorial region, the PSI-generated shear-containing NIWs, therefore, may provide an important route for mixing and vertical exchange of heat and materials.
With respect to the subtropical peak, the increased near-inertial S2 may to a substantial degree be associated with AE’s inertial chimney effect due to the strong regional eddy activities. Based on the ADCP data from the eddy-resolving mooring array, our study confirms the existence of critical layer for NIWs trapped in the baroclinic AEs. Such a critical layer was first theoretically predicted by Kunze43, but has been scarcely observed before10. Our observations also suggest that although AEs are favorable for the down-ward propagation of wind-driven NIWs, they may contribute little to the subtropical deep-ocean mixing, because most of high-mode waves break and dissipate near the critical layers at ~200 m depth and can seldom reach the deep ocean (Fig. 7f; Supplementary Fig. 4b). In addition to AE’s inertial chimney effect, other mechanisms such as nonlinear eddy-NIW interaction and spontaneous NIW generation were also suggested to play important roles in enhancing the near-inertial energy and hence turbulent mixing in the eddy-rich regions10,60,61,62,63. In the scenario of nonlinear interaction mechanism, the energy exchange between eddies and NIWs is highly associated with the strain of eddy fields60,64. Because no significant correlations were found between the strain rate of eddies (from altimeter data) and the near-inertial KE/S2 here, we do not expect an important role played by this mechanism in the subtropical S2 peak. For the NIWs induced by the spontaneous generation mechanism, they are different from the wind-forced NIWs in that they have comparable energy in the CW and CCW components10. By reexamining Fig. 6c, we indeed find that the ratio of CCW near-inertial S2 (KE) is slightly higher between 20–22°N than the weak-eddy regions (excluding the 12–14°N PSI latitudes). Therefore, the possibility of spontaneous NIW generation by eddies should exist. However, to what extent can it explain the subtropical S2 peak cannot be explicitly evaluated at present and should be further investigated in the future. Due to the strong turbulent shear associated with eddies, the parameterized Kρ at the subtropical S2 peak is increased by a factor of 2–8 than the background value. Given that the eddy activities have strong interannual-to-decadal modulations in the STCC region65,66, the eddy-mediated turbulent mixing may also have a corresponding low-frequency variability, which can potentially impact the overlying atmosphere through modulating the sea surface temperature.
Although our observations are confined to 143°E of the North Pacific, we expect that the observed latitudinal dependence of finescale turbulent shear and diapycnal mixing as summarized in Fig. 9 are qualitatively valid for the South Pacific and other oceans, since the three latitude-dependent dynamical regimes are a common feature in the world’s tropical-extratropical oceans. Recent model studies have suggested that the latitudinal structure of Kρ has an important influence on the simulated climate4,5. Given this, our present study highlights the need to take into account the observed latitude-dependent generation processes of turbulent mixing in ocean models’ parameterization schemes to improve their accuracy of climate simulations and predictions.
Methods
Moored data
In order to directly observe the full-depth current velocity along 143°E, the 17 NPEIM moorings were equipped with one or two 75 kHz RDI ADCPs at ~500 m depth and several discrete Aanderaa RCMs below that. The ADCPs measured the upper-ocean velocity (above 1000 m or 500 m) every 16 m, while the RCMs provided point velocity measurements in the deep layer. All the ADCPs and RCMs sampled the velocity every half an hour and had continuously worked for at least 13 months until their recovery; as such, they fully resolved mesoscale eddy and internal wave signals (i.e., NIWs and ITs). In addition, temperature chains (consisting of several SBE CTDs and dozens of temperature loggers) and discrete CTDs were also mounted on each mooring to measure the temperature and salinity in the upper and deep oceans, respectively. The temperature chains have a depth-dependent vertical resolution between 10–100 m and have a uniform temporal sampling interval of 5 minutes. Unfortunately, the upper segment of temperature chain (above ~500 m) at the moorings P6-7, P10-12, P14, and P16 was lost due to unknown reasons. More detailed information of the moorings can be found in the Supplementary Table 1. For data processing, all the high-frequency raw data were first hourly averaged. Given that the instrument depth fluctuated with time due to the swing of mooring, to keep the depth consistency for different moorings at different time, all the hourly ADCP and temperature chain data were then linearly interpolated to fixed 10 m vertical bins between near surface (40–60 m) and the ~1000 m (or ~500 m) depth. The processed finescale velocity and temperature data were used to calculate the velocity shear and Ri and to estimate the Kρ in the present study.
Finescale turbulent shears
The processed ADCP data were used to calculate finescale turbulent shear (i.e., vertical shear of horizontal velocity) at each mooring site. The finescale turbulent shear squared is defined as $$S^2 = |\partial u/\partial z|^2 + |\partial v/\partial z|^2$$, where u and v are the zonal and meridional velocity, respectively. To obtain the frequency-dependent components of S2, we first decomposed the velocity into different frequency bands using the third-order Butterworth filter. Specifically, the velocities within the semi-diurnal, diurnal, and near-inertial bands were band-pass filtered with cutoff periods at 10–14 h, 20–27 h, and $$2\pi {\mathrm{/}}(1.18 - 0.80)f_0^{ - 1}$$, respectively, where f0 is the local inertial frequency depending on the mooring latitude. Because f0 at P1 (near equator) is close to zero, the near-inertial velocity was not computed at this site. For the sub-inertial velocity, it was obtained through low-pass filtering with cutoff periods at $$2\pi /0.5f_0^{ - 1}$$ for sites P3–17, but 18 days for sites P1 and P2 (equals to $$2\pi {\mathrm{/}}0.8f_0^{ - 1}$$at P2). The cutoff period of $$2\pi {\mathrm{/}}0.8f_0^{ - 1}$$ rather than $$2\pi {\mathrm{/}}0.5f_0^{ - 1}$$ chosen at P2 is to avoid eliminating the equatorial waves with periods between 18–28 days. We should note that, the exact sub-inertial velocity does not exist at P1 because of the near-zero f0. This term remains here to represent the low-frequency current velocity (intra-seasonal and mean current). Following the frequency-dependent velocity decompositions described above, different components of S2 were correspondingly obtained.
Rotary decomposition
To study vertical propagations of NIWs, the near-inertial velocity was decomposed into components that rotate CW and CCW with increasing depth. This was accomplished by taking vertical Fourier transform of the complex near-inertial velocity profile u(z)+iv(z) at each time point and then inversely transforming the positive and negative quadrants, respectively42. According to the dispersion relation of linear internal waves, the CW (CCW) component with an upward-propagating (downward-propagating) phase has a downward (upward) energy propagation with time67. After being decomposed, the CW-component and CCW-component KE (u2+v2)/2) and S2 for NIWs were respectively calculated.
Richardson number
The Richardson number (Ri), which is defined as Ri = N2/S2, was calculated using the moored data in the upper ~500 m. Here, N2 is the squared buoyancy frequency and was computed using the Matlab subroutine of SeaWater Library (http://www.cmar.csiro.au/datacentre/) based on the moored temperature data. Because salinity measurements on the moorings were too coarse, the monthly salinity profiles from the IPRC Argo product (http://apdrc.soest.hawaii.edu/) were used in the N2 computation. Comparisons between the mooring-based N2 with that computed from all nearby synchronous Argo profiles only show slight difference, suggesting that the monthly salinity did not cause large bias in the N2 computation. To ensure that S2 has the same resolution with N2 in the Ri calculation, it was smoothed over a 20-m bin above ~300 m and over a 40-m bin below that according to the different thermometer spacing on the mooring (Supplementary Table 1). We acknowledge that our calculated Ri may have been overestimated as the 20–40 m resolution is generally coarser than the scale on which shear instability and turbulence truly occur. However, this does not impact the main conclusions of our study because the Ri-related results are primarily qualitative.
Wind work
The mixed-layer near-inertial work input by wind can be computed from
$$\mathrm{WW} = \vec \tau \cdot \vec u,$$
(1)
where $$\vec \tau$$ and $$\vec u$$ are the surface wind stress and near-inertial velocity, respectively. Because we had no direct wind observations, the 6-hourly ECMWF (European Center for Medium-Range Weather Forecasts; http://apps.ecmwf.int/datasets/) interim wind stress data were used here. Given that the ADCP velocity measurements in the upper 50 m was mostly absent, we used the mixed-layer slab model to solve the near-inertial velocity in Eq. (1) following Alford et al.39. The governing equations of the model are:
$$\frac{{\partial u}}{{\partial t}} - fv = \frac{{\tau _x}}{{\rho _0H}} - ru,$$
(2)
$$\frac{{\partial v}}{{\partial t}} + fu = \frac{{\tau _y}}{{\rho _0H}} - rv,$$
(3)
where H is the mixed-layer depth, τx and τy are the zonal and meridional wind stresses, ρ0 is the seawater density, and r is the damping coefficient. The ECMWF 6-hourly wind stress was used to force the model. In the computations, the IPRC monthly Argo mixed-layer depth was used and an empirical coefficient = 0.15f was chosen39.
Mesoscale eddies
To investigate the impact of mesoscale eddies on NIWs in the subtropical regime, we first identified all the AE and CE events that influenced the mooring sites based on AVISO altimeter data (http://www.aviso.oceanobs.com/). At each subtropical mooring site, the AE-impacted (CE-impacted) periods were identified with the criterion that ζ <−0.05f (ζ > 0.05f), where $$\zeta = \partial v/\partial x - \partial u/\partial y$$ is the surface relative vorticity calculated using the AVISO geostrophic velocity. For ζ below the sea surface, since the moorings are only available in the meridional direction, we calculated ζ using the moored ADCP velocity with an approximation of −0.85 ∂u/∂y. This relation was established based on the least-square fit between ∂v/∂‒ ∂u/∂y and ‒∂u/∂y at surface, which are highly related with a correlation coefficient of 0.91. With this relation, the mean ζ profile within AEs (CEs) can be obtained by averaging all the −0.85 ∂u/∂y profiles during AE-impacted (CE-impacted) periods of the moorings (see Supplementary Fig. 3). Note that it is the altimeter-based full ζ that was used to identify the mesoscale eddies and make the AE/CE composite for near-inertial KE and S2 (Fig. 7). The approximated subsurface ζ was only used to describe the eddies’ vertical structure and to explain the existence of critical layer for the trapped NIWs.
Estimation of diapycnal diffusivity
In order to estimate the diapycnal diffusivity (Kρ) based on the moored data, two finescale parameterization methods were employed here. The first one is the widely-used Gregg-Henyey-Polzin (GHP) parameterization based on the internal wave-wave interaction theory7,14. According to this method, when the finescale velocity shear data are available, Kρ can be estimated through
$$K_{\mathrm{\rho}} = K_0\frac{{\left\langle {V_{{z}}^2} \right\rangle ^2}}{{\,{}_{\mathrm{GM}}\left\langle {V_{{z}}^2} \right\rangle ^2}}h_1\left( {R_\omega } \right)j\left( {\frac{f}{N}} \right),$$
(4)
where K0 = 0.5 × 10−5 m2 s−1 is the referenced diffusivity, $$\left\langle {V_{{z}}^2} \right\rangle$$ and GM$$\left\langle {V_{{z}}^2} \right\rangle$$ are the shear variance from the observed velocity shear spectrum and the Garrett-Munk (GM) model spectrum, respectively68. In the above formula, the terms $$j\left( {\frac{f}{N}} \right)$$ and $$h_1(R_\omega )$$ are defined as $$j\left( {\frac{f}{N}} \right) = \frac{{f\arccos h(N/f)}}{{f_{30}\arccos h(N_0/f_{30})}}$$and $$h_1\left(R_{\omega} \right) = \frac{{3\left(R_{\omega} + 1\right)}}{{2\sqrt 2 R_{\omega} \cdot \sqrt {R_{\omega} - 1} }}$$, respectively, where N0 = 5.2 × 10−3rad s−1, f30 is the inertial frequency at 30°N, and Rω is the shear/strain variance ratio. Following the previous studies14,69, Rω is set conservatively to 7 here. To quantify the shear variance $$\left\langle {V_z^2} \right\rangle$$, we first calculated the shear spectrum using the 320 m-segment hourly velocity data between 90–410 m (see examples of shear spectra in Supplementary Fig. 5). Then, it was obtained by integrating the spectrum from the minimum wavenumber $$k_{\min } = \frac{{2\pi }}{{160}}{\mathrm{ rad}} \, {\mathrm{ m}}^{{{ - 1}}}$$ to the maximum wavenumber $$k_{\max } = \frac{{2\pi }}{{32}}{\mathrm{ rad}} \, {\mathrm{ m}}^{{{ - 1}}}$$. The GM shear variance GM$$\left\langle {V_z^2} \right\rangle$$ was computed over the same wavenumber band. For the stratification N used in the $$j\left( {\frac{f}{N}} \right)$$ and GM model, it was calculated based on the hourly temperature data at the moorings with recovered upper-layer temperature chains (see Richardson number in the Methods). With respect to the seven moorings in absence of upper-layer temperature chains (Methods; Supplementary Table 1), N was instead estimated using the monthly Argo data. Although the instantaneous diffusivities estimated based on the moored and Argo stratifications showed evident differences at the same mooring site, they actually had very close monthly distributions and the differences between their annual mean values were only between 3–12% (Fig. 8a). This good agreement gives us confidence to trust the annual-mean latitudinal pattern of the estimated Kρ with the Argo-based stratifications.
Given that the GHP parameterization may be invalid for the equatorial region where turbulent mixing is closely related to the strongly sheared sub-inertial currents rather than the breaking of internal waves, we also adopted a straightforward Richardson number-based parameterization method50,70 to independently estimate Kρ. The formula is in the form of
$$K_{\uprho} = K_0 + K_{\mathrm{m}} \cdot \left(1 + Ri/Ri_{\mathrm{c}}\right)^{ - 1},$$
(5)
where Ric = 1/4 is the critical Ri value for shear instability, K0 and Km are the constant background diffusivity and maximum diffusivity, respectively. By analyzing dozens of microstructure profiles in the low-latitude northwestern Pacific, the recent study of Liu et al.50 demonstrated that this finescale Ri-based parameterization can well approximate the observed Kρ when choosing K0 = 2.1 × 10−6 m2 s−1 and Km = 1.9 × 10−4 m2 s−1 (determined by nonlinear least-square fit). Here, the same parameters proposed by Liu et al.50 were used considering our similar study region. Similar to the results from the GHP method, using the moored or Argo stratifications in Eq. (5) only had very little influence on the estimated annual-mean diffusivities (with differences between 3–14%; Fig. 8b) and therefore the Argo-based N was used at the moorings where temperature chains were lost.
Data availability
The altimeter, Argo, and wind data used in this study were obtained from the website of AVISO, APDRC, and ECMWF, respectively. The mooring data and computing codes used in this study are available from the corresponding author upon reasonable request. All figures in this paper were plotted using Matlab.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
The authors express their sincere gratitude to the crew and scientists on aboard of R/V Dongfanghong2 for deploying and recovering the ADCP moorings. The helpful discussion with Dr. Qingxuan Yang regarding the finescale parameterizations is appreciated. This work was jointly supported by National Key Basic Research Program of China (2014CB745000), National Key Research and Development Program of China (2016YFC1402605), the National Natural Science Foundation of China (41706005, 91628302, and 41521091), the National program on Global Change and Air-Sea Interaction (GASI-02-PAC-ST-MSwin, GASI-02-PAC-ST-MSaut, GASI-IPOVAI-01-03, and GASI-IPOVAI-01-02), and the NSFC-Shandong Joint Fund for Marine Science Research Centers (U1406401).
Author information
Affiliations
1. Physical Oceanography Laboratory/CIMST, Ocean University of China and Qingdao National Laboratory for Marine Science and Technology, 238 Songling Road, 266100, Qingdao, Shandong, China
• Zhiwei Zhang
• , Jiwei Tian
• , Wei Zhao
• & Xiaodong Huang
• Bo Qiu
Contributions
Z.W.Z. analyzed the data, interpreted the results, wrote the manuscript and contributed to the experiment design. B.Q. initiated the idea and contributed to the data analyses and interpretations. J.W.T. conceived the study, designed, and conducted the observation experiment. W.Z. and H.X.D. contributed to the experiment design and manuscript discussion.
Competing interests
The authors declare no competing interests.
Corresponding authors
Correspondence to Bo Qiu or Jiwei Tian.
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2018-10-16 12:32:14
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https://gauravtiwari.org/my-mobile-number-is-a-prime-number/
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# My mobile number is a prime number
My personal mobile number 9565804301 is a prime number.
# What is a prime number?
Any integer p greater than 1 is called a prime number if and only if its positive factors are 1 and the number p itself.
In other words, the natural numbers which are completely divisible by 1 and themselves only and have no other factors, are called prime numbers2 , 3,5,7,11,13… etc. are prime numbers [or just Primes]. The numbers greater than 1, which are not prime are called Composite numbers.
Thanks, Ganesh for the tip. He tweeted:
His eight digit number couldn’t pass the ‘prime test’ as it had two prime factors but my ten digit number did. Exceptional luck!
Check this fact on Wolfram Alpha:
Type Is 9565804301 a prime? in the box and hit enter or just visit this link. You can also do the “prime test” for your own number.
## Is your mobile number a prime?
You can either start a new conversation or continue an existing one. Please don't use this comment form just to build backlinks. If your comment is not good enough and if in some ways you are trying to just build links — your comment will be deleted. Use this form to build a better and cleaner commenting ecosystem. Students are welcome to ask for help, freebies and more. Your email will not be published or used for any purposes.
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2021-07-25 12:08:13
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http://compgroups.net/comp.text.tex/sciposter-+bibliography/1928753
|
COMPGROUPS.NET | Post | Groups | Users | Stream | Browse | About | |
### sciposter +bibliography
• Email
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Hi everybody
I have used sciposter.cls for designing a poster and use sectionbox
package. My problem is with bibliography part.
\begin{sectionbox}{}
\bibliography {sch:1}
\newblock H.Erdl ,P.Schnider
\newblock{\em Astron. Astrophys.},268:453--471(1993).
Gould A.,Bahcall J.N.,Flynn C.,
\newblock{\em AstroPhysical Journal},482,913(1997).
\end{sectionbox}
\vfill
This lines produce a Reference box ,but its problem is referring to
references. It doesn't work with \bibitem ,also with \bibliography
for the second reference !!!
Best ,nakisa
0
Reply nakisa.nooraee (180) 2/24/2008 4:05:42 PM
See related articles to this posting
nakisa <nakisa.nooraee@gmail.com> wrote:
> Hi everybody
> I have used sciposter.cls for designing a poster and use sectionbox
> package. My problem is with bibliography part.
>
> \begin{sectionbox}{}
>
> \bibliography {sch:1}
> \newblock H.Erdl ,P.Schnider
> \newblock{\em Astron. Astrophys.},268:453--471(1993).
>
> Gould A.,Bahcall J.N.,Flynn C.,
> \newblock{\em AstroPhysical Journal},482,913(1997).
> \end{sectionbox}
> \vfill
>
> This lines produce a Reference box ,but its problem is referring to
> references. It doesn't work with \bibitem ,also with \bibliography
> for the second reference !!!
I don't know about sciposter, but probably something like
\begin{sectionbox}{}
\begin{thebibliography}{2}
\bibitem{sch:1} H. Erdl, P. Schnider
\newblock\emph{Astron. Astrophys.},\newblock 268:453--471 (1993).
\bibitem{g:1}% choose a key
A. Gould, J.\,N. Bahcall, C. Flynn,
\newblock\emph{AstroPhysical Journal},\newblock 482, 913 (1997).
\end{thebibliography}
\end{sectionbox}
would be better. The command \bibliography is for use along with
BibTeX.
Ciao
Enrico
0
Reply gregorio (1366) 2/24/2008 4:40:23 PM
hi Enrico
It works correctly now.
cheers,nakisa
0
Reply nakisa.nooraee (180) 2/25/2008 3:01:10 AM
comp.text.tex 36125 articles. 68 followers. Post
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2014-04-23 11:15:47
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https://www.esaral.com/q/if-58012
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# If α + β =
Question:
If $\alpha+\beta=\frac{\pi}{2}$, show that the maximum value of $\cos \alpha \cos \beta$ is $\frac{1}{2}$.
Solution:
$\frac{\pi}{2}=90^{\circ}$
Let $x=\cos \alpha \cos \beta$
$\Rightarrow x=\frac{1}{2}[2 \cos \alpha \cos \beta]$
$\Rightarrow x=\frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)]$
$\Rightarrow x=\frac{1}{2}\left[\cos (\alpha-\beta)+\cos 90^{\circ}\right]$
$\Rightarrow x=\frac{1}{2} \cos (\alpha-\beta)$
Now,
$-1 \leq \cos (\alpha-\beta) \leq 1$
$\Rightarrow-\frac{1}{2} \leq \frac{1}{2} \cos (\alpha-\beta) \leq \frac{1}{2}$
$\Rightarrow-\frac{1}{2} \leq x \leq \frac{1}{2}$
$\Rightarrow-\frac{1}{2} \leq \cos \alpha \cos \beta \leq \frac{1}{2}$
Hence, $\frac{1}{2}$ is the maximum value of $\cos \alpha \cos \beta$.
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2023-03-22 02:34:35
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https://answers.ros.org/questions/246677/revisions/
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According to the ROS C++ style guide, function names should be camelCased (lower-case first letter), not CamelCased (capital-case first letter). However, roslint doesn't seem to complain about CamelCased function names. Is there a property I should manually set() for that?
According to the ROS C++ style guide, function names should be camelCased (lower-case first letter), not CamelCased (capital-case first letter). However, roslint doesn't seem to complain about CamelCased function names. Is there a property I should manually set() for that?
Update: Actually, it doesn't complain even if a function name is under_scored
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2021-06-15 23:46:59
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https://www.shaalaa.com/question-bank-solutions/areas-combinations-plane-figures-the-area-enclosed-between-concentric-circles-770cm2-if-radius-outer-circle-21cm-find-radius-inner-circle_22925
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# The Area Enclosed Between the Concentric Circles is 770cm2. If the Radius of Outer Circle 21cm. Find the Radius of Inner Circle - CBSE Class 10 - Mathematics
ConceptAreas of Combinations of Plane Figures
#### Question
The area enclosed between the concentric circles is 770cm2. If the radius of outer circle 21cm. find the radius of inner circle
#### Solution
Radius of outer circle = 21𝑐𝑚
Radius of inner circle = 𝑅2
Area between concentric circles = area of outer circle – area of inner circle
⇒ 770 =22/7(212 − R22)
⇒ 212 − 𝑅22 = 35 × 7 = 245
⇒ 441 – 245 = 𝑅22
⇒𝑅2= sqrt(196) = 14 𝑐𝑚
Radius of inner circle = 14cm.
Is there an error in this question or solution?
#### APPEARS IN
Solution The Area Enclosed Between the Concentric Circles is 770cm2. If the Radius of Outer Circle 21cm. Find the Radius of Inner Circle Concept: Areas of Combinations of Plane Figures.
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2019-12-11 21:42:09
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https://blogs.mathworks.com/videos/2012/01/10/how-to-do-a-matrix-reshape-by-blocks/
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Stuart’s MATLAB VideosWatch and Learn
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How to do a matrix reshape by ‘blocks’6
Posted by Doug Hull,
This week’s video covers a non-standard way of reshaping a matrix. Instead of using the reshape function, this kind of matrix manipulation must be done with for loops. This matrix reshape operation treats submatrices within the original matrix as a unit. I suspect there are many other ways of doing this, and would love to hear about them in the comments.
Matt Fig replied on : 1 of 6
Here is one version of the obligatory one-liner. newMat=cell2mat(reshape(mat2cell(longMat,rSize*ones(1,n),cSize),5,4).'); However, this is only significantly faster for large systems. I have written optimized, hard-to-read MATLAB code in the past, and in order to help those who might be reading it in the future I often put the straightforward FOR loop(s) version in the comments. This serves several purposes. Among them are that it lets any future user see more easily what is going on and quickly get to maintenance concerns. Thanks for another great video, Doug!
Sean de Wolski replied on : 2 of 6
And the obligatory one-liner without cells: %Some parameters blksz = 3; %block size in row/col nrep = 20; %block repetitions nr = 4; %number of rows of blocks nc = 5; %number of columns of blocks %Determinant sample data A = reshape(1:(blksz*blksz*nrep),blksz,[])'; %Engine one line version: NewA = reshape(permute(reshape(permute(reshape(A.',blksz,blksz,[]),[2 1 3]),blksz,blksz*nc,[]),[2 1 3]),blksz*nc,blksz*nr)'; %Engine in pieces: B = reshape(A.',blksz,blksz,[]); %reshape the transpose to 3d with each block a 3x3 C = permute(B,[2 1 3]); %transpose each block D = reshape(C,blksz,blksz*nc,[]); %reshape to nr planes that need to be stacked vertically E = permute(D,[2 1 3]); %transpose each plane F = reshape(E,blksz*nc,blksz*nr)'; %reshape into a matrix and transpose
Yesid replied on : 3 of 6
hello.. How do I automatically concatenate several pages of the same multidimensional array.? cc=cat(1,xyz(:,:,1),xyz(:,:,2),xyz(:,:,3),...,xyz(:,:,n));
doug replied on : 4 of 6
@Yesid, Am I missing something? Aren't they already concatenated? How is xyz different than what you are trying to get for cc?
Sulaymon Eshkabilov replied on : 5 of 6
Here is a simpler one-liner code without any loop iteration: NEWmat = repmat(longMAT(1:3,:), 4, 5);
Sulaymon Eshkabilov replied on : 6 of 6
Make it simple but not simpler ... :). This is all what is needed to avoid any loops from the given task on step 1 and 2. imagesc(repmat(((reshape((1:9),3,3))+750), 4,5))
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2019-04-21 16:27:57
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http://song-of-heaven.blogspot.com/2010_10_30_archive.html
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## I Make A Wish For You
Blow in my song, gift to everyone, let it be known, the song upon your heart, sing a tune while in dance, in all you do and in all your prance, feel a tune, live in there, bring in home (inside your heart), the Southern Wind. Heaven feel the house of home, the southern tome (book of life, that's inside of you, your heart does know). Tum, ta ta det da, tome. Tum, ta ta det ta, tome. We're on our way home! Yes, my tears do shine, relatives of mine! How delightful is time *•♫♪♥♪♫•* Treasures come home just in time *•.¸,¸ *´'*•.¸,¸ ,¸.•*´'*•.¸,¸ ,¸.•*♥.
"Thank you I needed that", says Lilly Golden Face (Starlilly). "Big hugs", says Song of Heaven.
## Southern Season, our Song of Heaven
She Bellows Forth Wisdom, to cherish the Seasons, to bring love to life even in strife. The welcome of a heart within all life. Her name is Southern Season (Heavenly Mission) the grace with a meaning. Give here latitude with magnitude and you'll sea her embrace leaves you leaping for joy, open arms that hold and fill wholes. Welcome to our Song of Heaven (Earthly, tools of Rainbow Colors)
*•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•*
*•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•**•♫♪♪♫•*
It's a day that shines again, to bring in the love again. To bring colors that do blend, when the red road finds the wind. And it's fire that leads the way, when purification knows it's play, while we bless everything we do sea, the heart knows, it's time to be me!
## I Bless Me, I Bless the World, I Bless You
"Anything you perceive as a problem, is in reality, an opportunity to expand your awareness!"
## Saturday, October 30, 2010
### Beat of a Soft Embrace, the Dream of a Unified Race and Directions Stream Upon the Relative's Face!
Holiness David Running Eagle beats the sacred drum and says to Song of Heaven: flow
The beat of the drum is what I hear, a feeling of love from our Mother Deer, softness and a warm embrace, sets a steady base. Bum bum drum tum tum bum.
The beating of the drum, like one heart beat, that does lead all that lives , it gives a kiss, breath in breath out, in silence and out loud, the wave is flowing. It just keeps on going to and fro, here and there, at the heart where we do care.
Blessed are thee, who sets themselves free, to reach a hand and make a stand, rainbows come home and *glome (to unite upon the red road, the sacred directions we walk home, into the crystalline flow)!
Twin Deer Mother (White Buffalo Calf Woman) Hums and Sings:
And when its time, we demand all heart be sublime, And when we shine, sacred hearts do make up our minds. Stars in heaven, shine down upon me. Like a door. Open up and be free. And where it's darkness shines, my heart will know it's blind. I must use my heart. I must be the saint God gifted me. I will walk through that door, even if it's all dark and closed to me.
I will fly across heaven to find true love. I will seek, the kite that flies in the sky and plays for keep. And it will be bright, like clouds in the night. It will be grand, because I am part of man.
Hold me close, to where I must go. Know the place, where heaven hands do flow. Gain a heart, and know how to make the sacred dart. Lean on love and bring heaven home to me and you,
whence we came. It was heaven's day, upon us. Where we flow, is heaven's dreams upon us.
And when I reach the end, it's heaven's door that lets me in. And I will beeeeee, home with all the leaves. Come to me, gather in all the sunshine. Flow with me, know how great it could beeeeeee.
And I know it's faint, but it's a dark place to begin. And you must learn to shine. To bring heaven's sublime. Let us dream. Fill our world with heaven's dance. Hold my hand, and we can take a chance.
Southern Season, our Song of Heaven: Thank you
Twin Deer Mother: Thank you. Love and blessings to you. Blessings to the nine directions. Holiness David says have a wonderful day and thanks for talking with us.
Thanks beloved. Sweet dreams, awake for you. Aho ah, brotherhood, sisterhood rules, whoo hooo. Love you.
Southern Season, our Song of Heaven Love and Blessings 9 directions back. Sweet dreams and fly high! Aho! Love you too!
Hoop one: We recognize you as a spiritual being and welcome you to one family consciousness. Here at Active-Indigos, we respectfully address the ascension, flesh or light and descending, soul or darkness, process while creating local sanctuaries on earth. Thinking globally and acting locally while networking all spiritual communities and educating the workplace.
Elders come to be, when they stand to be free, to be heard of justice door, the place where all live to be, united in heart. United when apart. United in Spirit, to hold and have Heaven, inside the beginning, the Elders of the World. Who come home to what is heard. Who use love beyond all words. To know when LOVE is true. And Knows when we need the views, of pure light, of pure dark, of heaven and earth, abound as One who's gonna win. It's the Elders who refuse to budge and look for hues, instead sea with their hearts, and prays about all things in the heart.
I bless me. I bless the world. I bless you.
*Image, a glome:
Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Due to conformal property of Stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect (0,0,0,1); have infinite radius (= straight line).
*Glome definition:
A 3-sphere is also called a "glome" or a hypersphere, although the term hypersphere can in general describe any n-sphere for n ≥ 3. In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. A 3-sphere is an example of a 3-manifold. A 3-sphere can be constructed topologically by "gluing" together the boundaries of a pair of 3-balls (six sides, the sacred blessings or the crystalline stone). In three dimensions, there is always a hidden fourth side, or relative point, like the equation 0,0,0,1 (four sacred directions, three points of light and a point of darkness, or gray, grandmother dream space).
All curves are circles: the curves that intersect (0,0,0,1); have infinite radius (= straight line). We therefore continue to travel round and round, four sacred directions upon a path, around in a circle, to where it ends and where it begins. This is evolution. This is the river in life, that we all stream towards, relative to and journey on an evolutionary scale. The laws of physics and science are the understandings of God's principles. It is Song of Heaven that reminds us, of this sacred path, we all adhere to, the glome.
**Relativity definition: One of the defining features of general relativity is the idea that gravitational 'force' is replaced by geometry. In general relativity, phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories) are taken in general relativity to represent inertial motion within a curved geometry known as spacetime. (definition continues below for those who wish to read further**)
*Glome definition continues:
Strategies for polytopes
* Projecting into a lower-dimensional space
* Unfolding
* Cross-sections over time as higher-dimensional figure passes through lower-dimensional space
For 4D objects, begin with the tesseract (or 4D hypercube, the 4D analog of a cube) and the pentachoron (or 4-simplex, the 4D analog of a tetrahedron). (four sacred directions)
Strategies for curved figures
Again, begin with the simplest figure: a 4D hypersphere (a.k.a. glome). Somewhat unintuitively to non-mathematicians, this is also termed a 3-sphere, because any point on its "surface" can be specified with three coordinates. (The familiar 3D sphere, such as a globe of the Earth, is called a 2-sphere for analogous reasons; you can specify any point on its surface with only lines of latitude and longitude.)
* Slicing (gluing successive cross-sections together)
* Suspension (connect every point on a 3D sphere to two points in higher space, like a hammock; this yields a topological glome, though not a geometric one.
* "Cone" gluing (similar to the above; take two 3D spheres and connect each one to a point in 4-space (a different point for each). Paste the two spheres together, point-for-point, around their surfaces.)
* Hopf fibration
Mathematical Definition
In coordinates, a 3-sphere with center (C0, C1, C2, C3) and radius r is the set of all points (x0, x1, x2, x3) in real, 4-dimensional space (R4) such that \sum_{i=0}^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S3: S^3 = \left\{(x_0,x_1,x_2,x_3)\in\mathbb{R}^4 : x_0^2 + x_1^2 + x_2^2 + x_3^2 = 1\right\}. It is often convenient to regard R4 as the space with 2 complex dimensions (C2) or the quaternions (H). The unit 3-sphere is then given by S^3 = \left\{(z_1,z_2)\in\mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1\right\} or S^3 = \left\{q\in\mathbb{H} : |q| = 1\right\}.
The last description is often the most useful. It describes the 3-sphere as the set of all unit quaternions—quaternions with absolute value equal to unity. Just as the unit circle is important for planar polar coordinates, so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. See polar decomposition of a quaternion for details of this development of the three-sphere.
Properties
Elementary properties
The 3-dimensional volume (or hyperarea) of a 3-sphere of radius r is 2\pi^2 r^3 \, while the 4-dimensional hypervolume (the volume of the 4-dimensional region bounded by the 3-sphere) is \begin{matrix} \frac{1}{2} \end{matrix} \pi^2 r^4.
Every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point). As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere which reaches its maximal size when the hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.
Topological properties
A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply-connected. What this means, loosely speaking, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture proposes that the 3-sphere is the only three dimensional manifold with these properties (up to homeomorphism). This conjecture was proved correct in 2003 by Grigori Perelman.
The 3-sphere is homeomorphic to the one-point compactification of \mathbb{R}^3. Generally, any topological space which is homeomorphic to the 3-sphere is called a topological 3-sphere.
The homology groups of the 3-sphere are as follows: H0(S3,Z) and H3(S3,Z) are both infinite cyclic, while Hi(S3,Z) = {0} for all other indices i. Any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to S3, but then he himself constructed a non-homeomorphic one, now known as the Poincaré homology sphere. Infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1/n on any knot in the three-sphere gives a homology sphere; typically these are not homeomorphic to the three-sphere.
As to the homotopy groups, we have π1(S3) = π2(S3) = {0} and π3(S3) is infinite cyclic. The higher homotopy groups (k ≥ 4) are all finite abelian but otherwise follow no discernible pattern. For more discussion see homotopy groups of spheres. Homotopy groups of S3
Geometric properties
The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of R4. The Euclidean metric on R4 induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. As with all spheres, the 3-sphere has constant positive sectional curvature equal to 1/r2 where r is the radius.
Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural Lie group structure given by quaternion multiplication (see the section below on group structure). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see circle group).
Unlike the 2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). One can even find three linearly-independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the Lie algebra of the 3-sphere. This implies that the 3-sphere is parallelizable. It follows that the tangent bundle of the 3-sphere is trivial. For a general discussion of the number of linear independent vector fields on a n-sphere see the article vector fields on spheres.
There is an interesting action of the circle group T on S3 giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of S3 as a subset of C2, the action is given by (z_1,z_2)\cdot\lambda = (z_1\lambda,z_2\lambda)\quad \forall\lambda\in\mathbb T.
The orbit space of this action is homeomorphic to the two-sphere S2. Since S3 is not homeomorphic to S2×S1, the Hopf bundle is nontrivial.
Topological construction
Two convenient constructions for the topologist are the reverse of "slicing in half" and "puncturing".
Unslicing
A 3-sphere can be constructed topologically by "gluing" together the boundaries of a pair of 3-balls. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other.
The interiors of the 3-balls do not match: only their boundaries. In fact, the fourth dimension can be thought of as a continuous scalar field, a function of the 3-dimensional coordinates of the 3-ball, similar to "temperature". Let this "temperature" be zero at the 2-spherical boundary, but let one of the 3-balls be "hot" (have positive values of its scalar field) and let the other 3-ball be "cold" (have negative values of its scalar field). The "hot" 3-ball could be thought of as the "hot hemi-3-sphere" and the "cold" 3-ball could be thought of as the "cold hemi-3-sphere". The temperature is highest at the hot 3-ball's very center and lowest at the cold 3-ball's center.
This construction is analogous to a construction of a 2-sphere, performed by joining the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter; superpose them so that their circular boundaries match, then let corresponding points on the circular boundaries become equivalent identically to each other. The boundaries are now glued together. Now "inflate" the disks. One disk inflates upwards and becomes the Northern hemisphere and the other inflates downwards and becomes the Southern hemisphere.
It is possible for a point traveling on the 3-sphere to move from one hemi-3-sphere to the other hemi-3-sphere by crossing the 2-spherical boundary, which could be thought of as a "3-quator" — analogous to an equator on a 2-sphere. The point would seem to be bouncing off the 3-quator and reversing direction of motion in 3-D, but also its "temperature" would become reversed, e.g. from positive on the "hot hemi-3-sphere" to zero on the 3-quator to negative on the "cold hemi-3-sphere".
Unpuncturing
Consider a topological 2-sphere to be a seamless balloon. When punctured and flattened, the missing point becomes a circle (a 1-sphere) and the remaining balloon surface becomes a disk (a 2-ball) inside the circle. In the same way, a 3-ball is a punctured and flattened 3-sphere. To recreate the 3-sphere, merge all points on the 3-ball boundary (a 2-sphere) into a single point.
Another view of puncturing is stereographic projection. Rest the South Pole of a 2-sphere on an infinite plane, and draw lines from the North Pole through the sphere to intersect the plane. Each sphere point corresponds to a unique plane point, and vice versa, excepting the North Pole itself. The balloon has been stretched to infinity. Stereographic projection of a 3-sphere (except for the projection point) fills all of 3-space in the same manner. A benefit of this correspondence is that geometric spheres in 3-space map to geometric spheres of the 3-sphere, and planes in 3-space map to spheres containing the Pole.
Another view is a "shooting map". Place a marble at the South Pole and give it a flick of a measured strength in a chosen direction. Assuming the marble stays on the sphere and rolls without friction, its position after a fixed time interval (say, 1 second) will be some definite point of the sphere. Plotting direction in the plane and strength as radius, the North Pole is equally far away in every direction; this is the equivalent of the punctured balloon. Performing the same shooting experiment on the 3-sphere gives a map on the 3-ball. When the 3-sphere is considered a Lie group, the marble paths are one-parameter subgroups, the 3-ball is the tangent space at the identity (taken to be the South Pole), and the mapping to the 3-sphere is the exponential map.
**Relativity definition: One of the defining features of general relativity is the idea that gravitational 'force' is replaced by geometry. In general relativity, phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories) are taken in general relativity to represent inertial motion within a curved geometry known as spacetime.
The justification for creating general relativity came from the equivalence principle, which dictates that free-falling observers are the ones in inertial motion. Roughly speaking, the principle states that the most obvious effect of gravity – things falling down – can be eliminated by making the transition to a reference frame that is in free fall, and that in such a reference frame, the laws of physics will be approximately the same as in special relativity. A consequence of this insight is that inertial observers can accelerate with respect to each other. For example, a person in free fall in an elevator whose cable has been cut will experience weightlessness: objects will either float alongside him or her, or drift at constant speed. In this way, the experiences of an observer in free fall will be very similar to those of an observer in deep space, far away from any source of gravity, and in fact to those of the privileged ("inertial") observers in Einstein's theory of special relativity. Einstein realized that the close connection between weightlessness and special relativity represented a fundamental property of gravity.
Einstein's key insight was that there is no fundamental difference between the constant pull of gravity we know from everyday experience and the fictitious forces felt by an accelerating observer (in the language of physics: an observer in a non-inertial reference frame). So what people standing on the surface of the Earth perceive as the 'force of gravity' is a result of their undergoing a continuous physical acceleration which could just as easily be imitated by placing an observer within a rocket accelerating at the same rate as gravity (9.81 m/s²).
This redefinition is incompatible with Newton's first law of motion, and cannot be accounted for in the Euclidean geometry of special relativity. To quote Einstein himself: “If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them.” Thus the equivalence principle led Einstein to develop a gravitational theory which involves curved space-times. Paraphrasing John Wheeler, Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve.
## Theoretical justification
General relativity is based upon a set of fundamental principles and assumptions which guided its development.
## General principle of relativity
The laws of physics must be the same for all observers (accelerated or not).
## Principle of general covariance
The laws of physics must take the same form in all coordinate systems. In addition, the principle of general covariance forces that mathematics to be expressed using tensor calculus. Tensor calculus permits a manifold as mapped with a coordinate system to be equipped with a metric tensor of spacetime which describes the incremental (spacetime) intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained.
## Inertial and geodesic motion
Converging geodesics: two lines of longitude (green) that start out in parallel at the equator (red) but converge to meet at the pole
Due to the expectation that spacetime is curved, Riemannian geometry (a type of non-Euclidean geometry) must be used. Because the motion of objects is influenced solely by the geometry of spacetime, inertial motion occurs along special paths known as the timelike and null geodesics. In essence, spacetime does not adhere to the "common sense" rules of Euclidean geometry (straight lines that start out as parallel will remain parallel, namely at a constant distance from each other), but instead objects that were initially traveling in parallel paths through spacetime come to travel in a non-parallel fashion. On a curved surface like that of the Earth, lines of longitude that are parallel at the equator will eventually intersect at the pole. For example, two people on the Earth heading due north from different positions on the equator are initially traveling on parallel paths, yet at the north pole those paths will cross. This effect is called geodesic deviation, and it is used in general relativity as an alternative to gravity.
## Geometry of spacetime
On a curved two-dimensional surface, it is still possible to define lines that are as straight as possible, so called geodesics – for instance, on the spherical surface of the Earth, the lines of longitude, the equator, and other great circles. But the properties of these lines will differ from those of straight lines. Analogously, the world lines of test particles in free fall are spacetime geodesics – they are the straightest possible lines in spacetime, but there will be important differences between them and the truly straight lines in the gravity-free spacetime of special relativity: in special relativity, parallel geodesics remain parallel; in a gravitational field with tidal effects, such as the case of two bodies falling side by side towards the center of the Earth, initially parallel geodesics converge as the bodies move towards each other.
## Local Lorentz invariance
The laws of special relativity apply locally for all inertial observers.
## Einstein field equations
In 1907, Hermann Minkowski introduced a geometry that included not only the three (apparent) dimensions of space, but also a fourth dimension of time in order to present a geometric formulation of Einstein's special theory of relativity. Building upon the mathematical work of Carl Friedrich Gauss and Bernhard Riemann, Einstein postulated that Minkowski's spacetime could be treated as a 4-dimensional manifold which is curved by the presence of mass, energy and momentum (which, taken together, is known as stress-energy). Because Einstein previously determined that mass and energy are equivalent (E = mc²), gravitation is not caused by mass alone (as predicted by Newtonian physics) but by the distortion of spacetime by a combination of mass, energy, and momentum. The Einstein field equations describe the relationship between stress-energy and the curvature of spacetime.
The curvature of spacetime (caused by the presence of stress-energy of massive entities like the Sun or Earth) can be analogized by placing a heavy object such as a bowling ball on a trampoline will produce a 'dent' in the trampoline. The larger the mass, the bigger the amount of curvature. A relatively light object placed in the vicinity of the 'dent', such as a ping-pong ball, will accelerate towards the bowling ball in a manner governed by the 'dent'. Firing the ping-pong ball at some suitable combination of direction and speed towards the 'dent' will result in the ping-pong ball 'orbiting' the bowling ball. This is analogous to the Moon orbiting the Earth, for example. Similarly, in general relativity massive objects do not directly impart a force on other massive objects as hypothesized in Newton's action at a distance idea. Instead (in a manner analogous to the ping-pong ball's response to the bowling ball's dent rather than the bowling ball itself), other massive objects respond to how the first massive object curves spacetime.
## The mathematics of general relativity
Coordinates with the same difference in longitude at different latitudes are different absolute distances. Someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly 3,300 kilometers (2,051 mi); for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely 1,900 kilometers (1,181 mi)
A metric is a function used to describe the geometric properties of a space (or a spacetime), especially within Riemannian geometry. Just as a location on a flat plane can be expressed using coordinates, location on a curved sphere can be likewise defined using coordinates like latitude and longitude. Because coordinate differences on a curved surface do not correspond to equal absolute distances (see image on right), metrics are employed to calculate distances, angles, and other quantities between complex geometries.
Using a combination of calculus and metrics called metric tensors, the Einstein field equations (EFE) describe how stress-energy causes curvature of spacetime and are usually written in abstract index notation (a mathematical shorthand) as:
$G_{ab} = \kappa T_{ab} = \frac{8\pi G}{c^4}\ T_{ab}$
where Gab is the Einstein tensor describing the curvature of spacetime, Tab is the stress-energy tensor describing the combination of mass, energy, and momentum within spacetime, and κ is a constant (combining the speed of light, gravitational constant, and pi). By setting these two objects equal to each other results in ten separate equations (expanded from the shorthand above) expressing in mathematically precise language just what is meant by "spacetime tells matter how to move, and matter tells spacetime how to curve."
### Solutions of EFEs
Any spacetime with an associated distribution of matter for which the curvature of spacetime and the energy and momentum of matter do satisfy these conditions is called a solution of Einstein's equations. These solutions are metrics of spacetime which describe the structure of spacetime given the stress-energy and coordinate mapping used to obtain that solution. For any model universe in which matter and geometry are meant to obey the laws of general relativity, Einstein's equations define ten independent conditions that must be fulfilled simultaneously at each point in spacetime. Being non-linear differential equations, the EFE often defy exact solutions; however, many such solutions are known. The simplest solution is Minkowski spacetime, the spacetime of special relativity. Other important solutions describe the gravitational field around a spherically symmetric massive object (Schwarzschild solution, 1916) or gravity and geometry in an expanding universe (Friedmann-Lemaître-Robertson-Walker solution).
The EFE are the identifying feature of general relativity. Other theories built out of the same premises include additional rules and/or constraints. The result almost invariably is a theory with different field equations (such as Brans-Dicke theory, teleparallelism, Rosen's bimetric theory, and Einstein-Cartan theory)
### Modification of other theories
Because general relativity enabled a paradigm shift away from Newtonian mechanics which had underpinned the physical sciences to that time, previous theories of inertia, gravitation, electromagnetism, and quantum mechanics had to adopt new geometrical framework and assumptions. While some of these have been validated, there is still a theoretical gap between the assumptions in quantum mechanics and general relativity.
## Predictions of general relativity
### Gravitational effects
#### Gravitational redshifting
The gravitational redshift of a light wave escaping from the surface of a massive body
The first of these effects is the gravitational redshifting of light. Under this effect, the frequency of light will decrease (shifting visible light towards the red end of the spectrum) as it moves to higher gravitational potentials (out of a gravity well).
Assume that there are two observers, both of them at rest relative to a massive body. When the observer closer to the massive object sends some light to a second observer that is at rest higher up, the light will be red-shifted; the second observer will measure a lower frequency for the light than the first. Conversely, light sent from the second observer to the first will be blue-shifted (shifted towards higher frequencies). This is caused by an observer at a higher gravitational potential being accelerated (with respect to the local inertial frames of reference) away from the source of a beam of light as that light is moving towards that observer. Gravitational redshifting has been confirmed by the Pound-Rebka experiment.
#### Gravitational time dilation
A related effect is gravitational time dilation, under which clocks will run slower at lower gravitational potentials (deeper within a gravity well). For the same light wave, the second observer measures a lower frequency than the first; evidently, the second observer's clocks are running faster than those of the first observer. The same effect can also be derived in other ways (notably by transporting clocks back and forth and reconstructing the effect of location on their tick rate). Generally, clocks that are further down in a gravitational field tick more slowly than those that are higher up. This effect has been directly confirmed by the Hafele-Keating experiment and GPS.
Gravitational time dilation has as a consequence another effect called the Shapiro effect (also known as gravitational time delay). Shapiro delay occurs when signals take longer to move through a gravitational field than they would in the absence of the gravitational field. This effect was discovered through the observations of signals from spacecraft and pulsars passing behind the Sun as seen from the Earth.
#### Gravitational lensing
Einstein cross: four images of the same astronomical object, produced by a gravitational lens
Gravitational lensing occurs when one distant object is in front of or close to being in front of another much more distant object. In that case, the bending of light by the nearer object can change how the more distant object is seen. The first example of gravitational lensing was the discovery of a case of two nearby images of the same pulsar. Since then many other examples of distant galaxies and quasars being affected by gravitational lensing have been found.
In a similar way, Einstein also derived another effect, the gravitational deflection of light where light rays are bent downward in a gravitational field. An important example of this is starlight being deflected as it passes the Sun; in consequence, the positions of stars observed in the Sun's vicinity during a solar eclipse appear shifted by up to 1.75 arc seconds. This effect was first measured by a British expedition directed by Arthur Eddington. Subsequent observations of the deflection of the light of distant quasars by the Sun, which utilize highly accurate techniques of radio astronomy, have confirmed Eddington's results with significantly higher accuracy.
A special type of gravitational lensing occurs in Einstein rings and arcs. The Einstein ring is created when an object is directly behind another object with a uniform gravitational field. In that case, the light from the more distant object becomes a ring around the closer object. If the more distant object is slightly offset to one side and/or the gravitational field is not uniform, partial rings (called arcs) will appear instead.
Finally, in our own galaxy a star can appear to be brightened when compact massive foreground object is sufficiently aligned with it. In that case, the magnified and distorted images of the background star due to the gravitational bending of light cannot be resolved. This effect is called microlensing, and such events are now regularly observed.
Gravitational lensing has developed into a tool of observational astronomy, where it is used (among other things) to determine the masses of certain objects, detect the presence of dark matter, and provide an independent estimate of the Hubble constant.
#### Orbital effects
Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star
General relativity differs from classical mechanics in its predictions for orbiting bodies. The first difference is in the prediction that apsides of orbits will precess on their own. This is not called for by Newton's theory of gravity. Because of this, an early successful test of general relativity was its correctly predicting the anomalous perihelion precession of Mercury. More recently, perihelion precession has been confirmed in the large precessions observed in binary pulsar systems.
A related effect is geodetic precession. This is a precession of the poles of a spinning object due to the effects of parallel transport in a curved space-time. This effect is not expected in Newtonian gravity. The prediction of geodetic precession was tested and verified by the Gravity Probe B experiment to a precision of better than 1 percent.
Another effect is that of orbital decay due to the emission of gravitational radiation by a co-rotating system. It is observable in closely orbiting stars as an ongoing decrease in their orbital period. This effect has been observed in binary pulsar systems.
#### Frame dragging
Frame dragging is where a rotating massive object "drags" space-time along with its rotation. In essence, an observer who is distant from a rotating massive object and at rest with respect to its center of mass will find that the fastest clocks at a given distance from the object are not those which are at rest (as is the case for a non-rotating massive object). Instead, the fastest clocks will be found to have component of motion around the rotating object in the direction of the rotation. Similarly, it will be found by the distant observer that light moves faster in the direction of the rotation of the object than against it. Frame dragging will also cause the orientation of a gyroscope to change over time. For a spacecraft in a polar orbit, the direction of this effect is perpendicular to the geodetic precession mentioned above. Gravity Probe B is using this feature to test both frame dragging and the geodetic precession predictions.
#### Black holes
An illustration of a rotating black hole at the center of a galaxy
When mass is concentrated into a sufficiently compact region of space, general relativity predicts the formation of a black hole – a region of space with a gravitational attraction so strong that not even light can escape.
The disappearance of light and matter within a black hole may be thought of as their entering a region where all possible world lines point inwards. Stephen Hawking has predicted that black holes can "leak" mass, a phenomenon called Hawking radiation, a quantum effect not in violation of general relativity. Certain types of black holes are thought to be the final state in the evolution of massive stars. Supermassive black holes are thought to be present in the cores of most galaxies, and are thought to play a key role in galactic evolution. Numerous black hole candidates are known. These include the supermassive object associated with Sagittarius A* at the center of our galaxy
Matter falling onto a compact object is one of the most efficient mechanisms for releasing energy in the form of radiation, and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable, such as quasars and other types of active galactic nuclei.
### Cosmology
Although it was created as a theory of gravitation, it was soon realized that general relativity could be used to model the universe, and so gave birth to the field of physical cosmology. The central equations for physical cosmology are the Friedmann-Lemaître-Robertson-Walker metric, which are the cosmological solution of the Einstein field equations. This metric predicts that the universe must be dynamic: It must either be expanding, contracting, or switching between those states.
At the time of the discovery of the Friedmann-Lemaître-Robertson-Walker metric, Einstein could not abide by the idea of a dynamic universe. In an attempt to make general relativity accommodate a static universe, Einstein introduced an alternative form of the field equations to accommodate a static universe solution in his theory:
$G_{ab} + \Lambda\ g_{ab} = \kappa\, T_{ab}$
where Λ is the cosmological constant and gab is the spacetime metric.
However, the resultant static universe was unstable. Then in 1929 Edwin Hubble showed that the redshifting of light from distant galaxies indicates that they are receding from our own at a rate which is proportional to their distance from us. This demonstrated that the universe is indeed expanding. Hubble's discovery ended Einstein's objections and his use of the cosmological constant.
The equations for an expanding universe become singular when one goes far enough back in time, and this primordial singularity marks the formation of the universe. That event has come to be called the Big Bang. In 1948, Ralph Asher Alpher and George Gamov published an article describing this event and predicting the existence of the cosmic microwave background radiation left over from the Big Bang. In 1965, Arno Penzias and Robert Wilson first observed the background radiation, confirming the Big Bang theory.
What is dark matter? Do the phenomena attributed to dark matter point not to some form of matter but actually to an extension of gravity?
Recently, observations of distant supernovae have indicated that the expansion of the universe is currently accelerating. This was unexpected since Friedmann-Lemaître-Robertson-Walker metric calls for a universe that only contains visible matter to have a decelerating expansion. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely dark energy and dark matter. However, for a universe that is 4% baryonic matter, 26% dark matter, and 70% dark energy, the Friedmann-Lemaître-Robertson-Walker metric takes on a form that is consistent with observation. There is also an irony in that the dark energy can be modeled using Einstein's cosmological constant, but with a value that enhances the dynamic nature of the universe instead of muting it. There have been several (controversial) proposals to obviate the need for these enigmatic forms of matter and energy by modifying the laws governing gravity and the dynamics of cosmic expansion (for example, modified Newtonian dynamics).
### Other predictions
Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves
General relativity predicts the equivalence of inertial mass and gravitational mass. A number of other tests have probed the validity of various versions of the equivalence principle (strictly speaking, all measurements of gravitational time dilation are tests of the weak version of that principle, not of general relativity itself). As embodied by the strong equivalence principle, even a self-gravitating object will respond to an external gravitational field in the same manner as a test particle would. (This is often violated by alternative theories.)
The theory also predicts exotic forms of Gravitational radiation. As gravity is substantially weaker than other physics forces, currently only extremely violent interactions (like merging neutron stars and/or black holes) are expected to emitting observable radiation. A number of land-based gravitational wave detectors are in operation, with the aim of detecting gravitational waves directly. Orbital decay, as described above, may be likened to gravitational radiation as well. Graviational radiation may also be observed in subatomic particles called gravitons or quadrupole and higher order multipole moments, but these have been experimentally observed or verified.
## Validity
Because general relativity has passed every unambiguous observational and experimental test to which it has been subjected so far, it is almost universally accepted by the scientific community. However, while it is a highly successful model of gravitation and cosmology, there are substantial theoretical inconsistencies between general relativity, quantum mechanics, and the spacetime singularities associated with black holes.
### Quantum mechanics
How can the theory of quantum mechanics be merged with the theory of general relativity to produce a so-called "theory of everything"?
While general relativity is very successful in that it provides an accurate description for an impressive array of physical phenomena, the theory is very likely incomplete.
Notably, in contrast to all other modern theories of fundamental interactions, general relativity is a classical theory which does not include the effects of quantum physics. The question of what general relativity looks like at a quantum level is often called a theory of quantum gravity or Theory of everything. This subject remains one of the great open questions of physics. While there are promising candidates such as string theory and loop quantum gravity, there is at present no consistent and complete theory of quantum gravity which reconciles Einstein's geometric picture of gravity with the laws of the quantum world.
### Spacetime signularities
It is a longstanding hope that the theory of quantum gravity would also do away with spacetime singularities. Such singularities are boundaries ("ragged edges") of spacetime at which geometry becomes ill-defined; the best-known examples are the singularities inside black holes and at the Big Bang singularity at the beginning of the universe. If the laws of general relativity were to hold without any quantum modifications, then, by what are known as the singularity theorems, such singularities would indeed exist in our universe.
# Happier Cows, Thanks to Math
Here's a kind of offbeat one for you today. Scientists have come up with a mathematical model to explain (or attempt to explain) a pretty interesting aspect of bovine behavior.
If you've ever spent any time in a cow pasture, you learn two things pretty quickly. First, always watch where you walk. Second, cows tend to sit or stand in groups. If one of them is laying down, chances are, most of them are laying down. And, if one of them is walking around, chances are most of them are walking around.
So, um ... why is that, exactly? (The groupthink behavior, that is. I imagine you can figure out the watching-where-you-walk part on your own.)
Jie Sun and his friends over at Clarkson University think they've got an answer for you. It has to do with oscillators, looking at cows as a binary system. The result is, as the abstract puts it, "a mathematical model for daily activities of a cow (eating, lying down, and standing) in terms of a piecewise affine dynamical system."
In plain English, that means, cows tend to operate in synchronicity, just like we said earlier. We already knew that, so who needs math anyway?
Well, here's the takeaway. The more synchronicity, the happier the cows appear to be, according to Technology Review. And, along with that, you've got to talk about coupling — the proximity of a cow to his nearest neighbor. More coupling, according to the study, appears to create less synchronicity.
As a practical matter, the closer cows are jammed together, the less synchronized they are, and that's an indication that they're uncomfortable or unhappy. Think about how you feel on a crowded elevator.
The real-world upshot is this: Happy cows yield more (and better tasting) milk, cheese, or meat if that's your thing. If you cram a bunch of cattle into your standard factory farm conditions, you end up with yuckier product.
So, the ethical arguments against factory farming are clear. And, the public health arguments against factory farming are also pretty well-established. Now, add the mathematical argument. Factory farming just doesn't add up.
Martin Matheny is a political consultant and animal welfare writer based in Athens, GA.
Photo credit: John-Morgan. Thank you http://animals.change.org/blog/view/happier_cows_thanks_to_math
E 8 in condensed matter physics
Bertram Kostant, Experimental evidence for the occurrence of E 8 in nature and the radii of the Gossett circles, Tuesday February 23, 3:00 at APM 6402, Department of Mathematics, U.C. San Diego.
Abstract:
A recent experimental discovery involving the spin structure of electrons in a cold one dimensional magnet points to a model involving the exceptional Lie group E 8. The model predicts 8 particles the ratio of whose masses are the same as the ratios of the radii of the circles in the famous Gossett
diagram (going back to 1900) of what is now understood to be a 2 dimensional projection of the 240 roots of E 8 arranged in 8 concentric circles. The ratio of the radii of the two smallest circles (read 2 smallest masses) is the golden number. This beautifully has been found experimentally. The ratio of the radii of the other masses has been written down conjecturally by Zamolodchikov. This again agrees with the analogous statement for the radii of the Gossett circles.
Some time ago we found an operator A (very easily defined and reexpressed by Vogan as an element of the group algebra of the Weyl group) on 8-space whose spectrum is exactly the squares of the radii of the Gossett circles.
The operator A is written in terms of the coefficients n i of the highest root. In McKay theory the n i are the dimensions of the irreducible representations of the binary icosahedral group. Our result works for any simple Lie group not just E 8.
Posted by: John Baez on February 17, 2010
Thank you http://golem.ph.utexas.edu/category/2010/01/this_weeks_finds_in_mathematic_50.html
Rainbow Warriors of Prophecy
She Bellows Forth Wisdom, to cherish the Seasons, to bring love to life even in strife. The welcome of a heart within all life. Her name is Southern Season (Heavenly Mission) the grace with a meaning. Give here latitude with magnitude and you'll sea her embrace leaves you leaping for joy, open arms that hold and fill wholes. Welcome to our Song of Heaven (Earthly, tools of Rainbow Colors)
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2017-06-23 17:21:45
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https://web2.0calc.com/questions/help_40221
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+0
# help
+1
174
1
+40
A club with five men and six women wish to form a committee. The number of men must be between 1 and 3 (inclusive), and the number of women must be between 2 and 4 (inclusive). How many different committees can be formed?
May 20, 2019
#1
+5662
+2
I think we have to break it down into every possible case regarding the numbers of men and women on the committee.
$$\text{If there are m men and w women on the committee we have}\\ n_{m,w}=\dbinom{5}{m}\dbinom{6}{w} \text{ ways to select the committee from the members}\\~\\ N= \sum \limits_{m=1}^3\sum \limits_{w=2}^4 \dbinom{5}{m}\dbinom{6}{w}=1250$$
.
May 20, 2019
#1
+5662
+2
$$\text{If there are m men and w women on the committee we have}\\ n_{m,w}=\dbinom{5}{m}\dbinom{6}{w} \text{ ways to select the committee from the members}\\~\\ N= \sum \limits_{m=1}^3\sum \limits_{w=2}^4 \dbinom{5}{m}\dbinom{6}{w}=1250$$
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2019-08-25 04:21:03
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http://legisquebec.gouv.qc.ca/en/showversion/cs/I-4?code=se:5_2&pointInTime=20201120
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### I-4 - Act respecting the application of the Taxation Act
5.2. Unless the context indicates otherwise, the mention in a particular provision of this Act, of the Taxation Act (chapter I-3), of an Act that amends either of those Acts or of a regulation made under either of those Acts, or of a regulation that amends such a regulation, of a word, group of words, expression or reference to a provision of an Act, that in accordance with the particular Act referred to in the third paragraph, has replaced another word, group of words, expression or reference that appeared in a provision of this Act or of the Taxation Act, is deemed, where that particular provision applies before 20 March 1997, to be a mention of the replaced word, group of words, expression or reference, as the case may be.
Similarly, unless the context indicates otherwise or the mention has otherwise been modified accordingly, the mention in a particular provision of this Act, of the Taxation Act, of an Act that amends or of a regulation made under either of those Acts, or of a regulation that amends such a regulation, of a word, group of words, expression or reference to a provision of an Act, that is identical to a word, group of words, expression or reference that appeared in a provision of this Act or of the Taxation Act and that has been replaced, in accordance with the particular Act referred to in the third paragraph, by another word, group of words, expression or reference, is deemed, where that particular provision applies after 19 March 1997, to be a mention of the replaced word, group of words, expression or reference, as the case may be.
The particular Act mentioned in the first and second paragraphs is the Act to harmonize certain legislative provisions of a fiscal nature with the Civil Code of Québec (1997, chapter 3).
1997, c. 3, s. 73.
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2021-01-16 21:26:41
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https://stats.stackexchange.com/questions/121184/alternative-for-fishers-exact-test-for-count-data-in-table-bigger-than-2x2
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# Alternative for Fisher's exact test for count data in table bigger than 2x2
I have count data on different languages from comparable corpora.
It looks something like this, where the counts give the number of clauses with that particular word order attested in each language corpus (constructed example):
SOV SVO VSO 0SV OVS VOS
English 3 124 0 0 6 0
Tagalog 2 14 109 0 3 10
Dutch 56 61 4 0 7 0
Hindi 110 1 2 2 6 9
I am interested in the differences between languages, i.e. whether these four languages display different or similar word order patterns. I can use chi-square on the whole table, or for separate columns, but I run into trouble with some cells being 0, or < 5. Fisher's exact test is recommended in those cases, but I haven't been able to do this in R so far. If I try it on the whole table, I get FEXACT errors, complaining either that LDKEY or LDSTP are too big or too small. It does not allow me to take just one column, as it needs a table structure of at least 2x2. It allows me to test two columns at a time, but this does not seem appropriate.
Three questions:
1. Are these methods even appropriate for count data, or do I need something else entirely? I realise this is not a traditional contingency table.
2. Is there any implementation of Fisher or something comparable (preferably in R) that can be used on the whole table?
3. Is there something comparable to Fisher that I could use for a single column? Like I said, chi-squire does not seem appropriate due to low values.
EDIT: I realise that my constructed word order example above has such strong associations between language & word order patterns it doesn't matter too much if chi-squared is used (RE: Nick Cox's answer below).
But what about data distributed like this:
constr1 constr2 constr3 constr4
L1 1 5 20 194
L2 0 4 19 191
L3 1 8 30 180
Or this:
constr1 constr2 constr3
L1 61 166 0
L2 55 66 2
L3 55 60 2
L4 54 114 4
L5 53 98 5
I guess I really want to know whether there is any alternative to assess the whole table rather than conducting multiple Fisher exact tests (I don't own SAS so Peter's option is not available to me).
• "Skewed" is the wrong word here as it refers statistically to shape of frequency distribution, not to a (strong) association between variables. If your question is whether rows and/or columns of each entire table are different in pattern, then that is in no sense answered by splitting it into lots of 2x2 tables, none of which can see the others. As with your first example, there is no reason not to use chi-square tests so long as you proceed with a little caution about low expected frequencies. (Older literature was often over scary on the problems with small frequencies.) Oct 23, 2014 at 17:30
• Sorry about wrong terminology, I've edited that. OK, thanks for explaining further. My initial search for appropriate statistics started with the biostathandbook, where the author says "I recommend that you always use an exact test (exact test of goodness-of-fit, Fisher's exact test) if the total sample size is less than 1000." in the section on small numbers in chi-square and G-tests: biostathandbook.com/small.html Oct 24, 2014 at 8:59
I don't see anything about your problem that is non-standard for counts of categories. The only thing that is even a little unusual is that you have extremely marked differences between languages.
For your data I get Pearson chi-square of $687.8$ with $15$ d.f. for a test of no association between the variables and the P-value is minutely small. For what it's worth, my program (Stata) reports the P-value as about $7 \times 10^{-137}$.
A good program should indeed flag small expected frequencies, which are the issue rather than small observed frequencies: I see a flag that 4 cells have less than 1 as expected frequency. So, there is a bit of a worry about the P-value, but it is really quite secondary. You could change the P-value by more than 100 orders of magnitude either way, but the message would be the same.
To put it directly, a simple test underlines what is evident just by looking at the frequencies, namely that the languages are very different, which you know any way. If you have some sceptic who doubts that, then a chi-square test provides back-up.
Doing this with Fisher's test is on one level more correct statistically, but it will not change the practical or scientific conclusion one iota.
You have quantitative data that are pertinent to a discussion, but you don't need statistical inference to add gloss. The numbers speak eloquently for themselves, and the details are the interesting part.
Naturally, I am responding to your example, and being firm about what it implies in no way rules out different conclusions for other data.
If there is a predictive model that predicts actual (relative) frequencies, then testing that is a much more interesting question, but you would need to tell us the details.
To respond a little more directly to your question: Fisher's exact test often is impractical once the frequencies stop being very small.
You can use fisher exact test for category variables with more than 2 groups. See the second part of this post handbook of biological statistics for details.
• Do you mean the section under 'Post-hoc tests', where the author explains how to do multiple Fisher tests for a 4*2 table on termite damage? Oct 23, 2014 at 14:53
• Yes, and see How to do the test -> SAS Oct 23, 2014 at 14:57
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2022-05-17 05:06:50
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5005267262458801, "perplexity": 899.1099297261043}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662515501.4/warc/CC-MAIN-20220517031843-20220517061843-00553.warc.gz"}
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