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http://tex.stackexchange.com/questions/63710/list-of-abbreviations-acronyms-in-latex-several-issues
|
# List of Abbreviations/Acronyms in Latex. Several issues
I want a list of abbreviations in my thesis. My thesis consist of multiple parts, which are included into the final document with the "include" command. The distinct parts of the thesis introduce sometimes the same abbreviations. However, i do not want to maintain abbreviations on the level of the whole thesis, but on the level of the included parts.
Two packages are commonly used for abbreviations: nomencl and glossaries. Both have issues.
Nomencl: I cannot have duplicate abbreviations, thus i have to maintain them at the thesis' level, which is complicated and distracting. I want the distinct parts to be as independent as possible.
Glossaries: Glossaries does not work good with natbib, e.g. \gls{SOM}\citep{Kohonen2001} results in (SOM) (Kohonen, 2001), what is considered bad style. It would be nicer, if it would produce (SOM; Kohonen, 2001). So, my idea is to use the glossaries-package in a similar manner to nomencl. Only defining acronyms for the list of abbreviations, but not using \gls, glspl, etc. in the text. However, to print the acronyms in the list, they must explicitly added with /glsaddall, but this command does not work for acronyms defined in included documents. I consider this a bug.
So, the best i think would be to stick with nomencl and somehow force it to ignore redefinitions of acronyms in the included documents. Is this possible?
Alternatively, what approach would you suggest for my usecase (need abbrev list, multiple included documents)?
-
\glsaddall doesn't work in the included file for the simple reason that managing abbreviations on a per-file basis is awkward and error prone. Having the same acronym defined (perhaps in sligtly different ways) in two parts of the document can be a source for headaches. Don't do like that. – egreg Jul 18 '12 at 9:41
It's not clear if you want the citation every time you use the acronym, or just specific instances. It's also not clear what you want to happen if you try to redefine an existing acronym. Assuming specific instances of a citation and a reset when you attempt to define an existing acronym, you could achieve this as follows. First the main tex file:
\documentclass{book}
\usepackage{natbib}
\usepackage{etoolbox}
\usepackage[acronym,smallcaps]{glossaries}
\makeglossaries
\newcommand*{\provideacronym}[4][]{%
\ifglsentryexists{#2}%
{%
\glsreset{#2}%
}%
{%
\newacronym[#1]{#2}{#3}{#4}%
}%
}
\newcommand*{\provideglossaryentry}[2]{%
\ifglsentryexists{#1}%
{}%
{%
\newglossaryentry{#1}{#2}%
}%
}
\renewcommand*{\acronymfont}[1]{#1}
\defglsdisplayfirst[acronym]{%
#1 \ifstrempty{#4}{(#3)}{\citep[#3;][]{#4}}%
}
\defglsdisplay[acronym]{%
#1\ifstrempty{#4}{}{ \citep{#4}}%
}
\title{Sample Thesis}
\author{A.N. Other}
\begin{document}
\maketitle
\include{sample1}
\include{sample2}
\printglossaries
\bibliographystyle{plainnat}
\bibliography{xampl}
\end{document}
Now the first chapter (sample1.tex):
\chapter{First Sample Chapter}
\provideacronym{abc}{ABC}{Sample Acronym 1}%
\provideacronym{xyz}{XYZ}{Sample Acronym 2}%
\provideglossaryentry{sample}{name=sample,description=An example}%
An acronym: \gls{abc}. A \gls{sample}.
Sample acronym with a citation
\gls{xyz}[article-minimal].
Another instance with a citation
\gls{xyz}[article-minimal].
And the second chapter (sample2.tex):
\chapter{Second Sample Chapter}
\provideacronym{abc}{ABC}{Sample Acronym 1}%
\provideglossaryentry{sample2}{name=another sample,description=Another example}%
An acronym: \gls{abc}. \Gls{sample2}.
Sample acronym with a citation
\gls{abc}[article-minimal].
Another instance with a citation
\gls{abc}[article-minimal].
-
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2016-05-03 16:51:03
|
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https://www.sciencemadness.org/whisper/viewthread.php?tid=66023#pid446407
|
Sciencemadness Discussion Board » Fundamentals » Reagents and Apparatus Acquisition » Reagents for sale Select A Forum Fundamentals » Chemistry in General » Organic Chemistry » Reagents and Apparatus Acquisition » Beginnings » Responsible Practices » Miscellaneous » The Wiki Special topics » Technochemistry » Energetic Materials » Biochemistry » Radiochemistry » Computational Models and Techniques » Prepublication Non-chemistry » Forum Matters » Legal and Societal Issues
Pages: 1
Author: Subject: Reagents for sale
Gurt
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Posts: 42
Registered: 5-2-2016
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Reagents for sale
Hello everyone,
I am posting today to say that after a good while lurking and reading on the forums, I have come to a realization. I am currently setting the last bit of my LLC up, and have decided to do some chemical importation and sales. I arrived at this idea mostly because of the difficulty for amateur experimenters in finding certain reagents for a fair price. As well as the issue of often having to order much larger amounts than needed if one can find them. So I started looking and decided to set up an LLC and do some calling around to find out what would be required of me to set up a small chemical business. Now with everything coming together all that is really left is to decide what I should stock first.
So far I plan on ordering in:
Sodium Borohydride
Benzene
Elemental Sodium
Ethyl Acetoacetate
Acetonitrile
Acetamide
My question now is what you, the users of this forum, would like to see. What do you think would be good additions to stock? If there is enough interest in certain materials I will certainly see about either importing them, or purchasing from a manufacturer in country. Also, at this time I plan on only working within the U.S. just to see how it goes. But I would be willing to work out shipping internationally as long as it is nothing highly hazardous. Thank you, and please let me know what you would like to see!
Loptr
International Hazard
Posts: 1298
Registered: 20-5-2014
Location: USA
Member Is Offline
Mood: Grateful
Hydrides would be of interest in general, since they are more difficult to get for the amateur. Benzene seems to be similar to unobtainium unless preparing it yourself or ordering from a larger supply warehouse as the smaller ones don't seem to carry it. Sodium is quite available if you wish to order off eBay or a few select websites, but any additional supply of it is always welcomed. Ethyl acetoacetate is easily prepared at home. Acetonitrile is expensive for the most part, unless you have an account with a chemical supply warehouse. Acetamide is easily prepared at home.
Strong bases (LDA, potassium tert-butoxide, sodium amide, etc., although, I have been looking into the use of NaOH under PTC conditions)
Hydrides (hydrides, borohydrides, DIBAL, Red-Al; some of the more friendly ones would be appreicated. if it has a tendency towards becoming a fireball, an alternative might be a better choice.)
Alkylating agents
Acid chlorides
Acid anhydrides
Some of the more exotic, deprecated, and expensive reagents (TEMPO, KCN, silanes)
Amines
...
...
...
Solvents in general are a good thing to have access to, especially if they are at a reduced price. (glymes, diethyl ether, THF, 2-MeTHF, etc.)
Basically, if it isn't commonly available to the amateur, but is commonly used in a lab, it will be of interest. I am positive others here could provide a much better variety than what I have listed here from the top of my head while in a meeting.
Also, the more general the reagent in its use in chemistry, the better it would be for stocking up. There are a few different types of reagents that are more difficult to get ahold of--I would start with those.
[Edited on 21-4-2016 by Loptr]
[Edited on 21-4-2016 by Loptr]
Gurt
Harmless
Posts: 42
Registered: 5-2-2016
Location: USA
Member Is Offline
Mood: Inquisitive
Thank you very much for your input. I will make a few inquiries for supplies of solvents and some acid chlorides. My only concern with importing large quantities of solvent is gaining a skeptical eye from government agencies. Same reason why I am not interested in importing certain listed or watched chemicals, IE Benzaldehyde, Acetic Anhydride, and such. However, I think solvents may be fine, so long as I choose carefully which ones to carry. Of course good record keeping, and being careful who I sell my wares to will also help keep products from going to unethical practices.
chemrox
International Hazard
Posts: 2953
Registered: 18-1-2007
Location: UTM
Member Is Offline
Mood: LaGrangian
oxalyl chloride
phosphorous oxychloride
"When you let the dumbasses vote you end up with populism followed by autocracy and getting back is a bitch." Plato (sort of)
careysub
International Hazard
Posts: 1339
Registered: 4-8-2014
Location: Coastal Sage Scrub Biome
Member Is Offline
Mood: Lowest quantum state
Carbon disulfide
n-Butylamine
Sodium hydride
Sodium peroxide
Trifluoroacetic acid
Thionyl chloride
Gurt
Harmless
Posts: 42
Registered: 5-2-2016
Location: USA
Member Is Offline
Mood: Inquisitive
Phosphorus Oxychloride may be a bit of a challenge, as far as shipping is concerned. It is very pricey to import and ship about, from what I have seen. On the note of phosphorus compounds, is there any interest in phosphorus pentoxide? I know it can be found sometimes online here, but the prices are ridiculous, ie $30-$40 for 50gm. I could offer this chemical at less than half the price of others.
careysub
International Hazard
Posts: 1339
Registered: 4-8-2014
Location: Coastal Sage Scrub Biome
Member Is Offline
Mood: Lowest quantum state
Quote: Originally posted by Gurt Phosphorus Oxychloride may be a bit of a challenge, as far as shipping is concerned. It is very pricey to import and ship about, from what I have seen. On the note of phosphorus compounds, is there any interest in phosphorus pentoxide? I know it can be found sometimes online here, but the prices are ridiculous, ie $30-$40 for 50gm. I could offer this chemical at less than half the price of others.
Phosphorus pentoxide can be had in the U.S. from Firefox-FX for $21/lb: http://www.firefox-fx.com/ChemN-P.htm Gurt Harmless Posts: 42 Registered: 5-2-2016 Location: USA Member Is Offline Mood: Inquisitive Thank you very much for letting me know, it saves from having to sit on the stuff. I had completely forgotten that they offer it, even though I had seen mention of it in abother thread. As of right now, I will have sodium borohydride, benzene, and some Lithium aluminum hydride in stock within 2 weeks. As of now my preliminary pricing for the NaBH4 is: 1kg-$180
500g - $100 250g -$ 60
100g - $30 Do these prices seem acceptable? They will drop as I get a better understanding of what demand will be. As the cost drops sharply with larger orders from suppliers. Also if any members would be interested, I am looking to send out samples to interested parties. So they may test and verify that product is up to standard. Therefore anyone who may want to do an analysis would be sent a free sample, if they would be willing to simply post the results of their analysis. So others may have a review of sorts before buying. JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline I would say those prices for sodium borohydride are reasonable. I am interested in thionyl chloride. Texium Administrator Posts: 3808 Registered: 11-1-2014 Location: Salt Lake City Member Is Offline Mood: Triturated If the price were to drop further, I would be interested in purchasing some. As it is now, I simply don't currently have enough of a need for it to drop at least$30 on it.
Come check out the Official Sciencemadness Wiki
They're not really active right now, but here's my YouTube channel and my blog.
Gurt
Harmless
Posts: 42
Registered: 5-2-2016
Location: USA
Member Is Offline
Mood: Inquisitive
I will inquire for a quote on thionyl chloride. Zts16 how much do you need? As I haven't priced out smaller quantities yet.
gdflp
Super Moderator
Posts: 1320
Registered: 14-2-2014
Location: NY, USA
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Mood: Staring at code
Unfortunately, thionyl chloride may be near impossible to ship legally at a reasonable cost. This has already come up in a previous thread here. While you could circumvent shipping regulations, that is not going to result in a lasting business, as something will happen sooner or later.
As for the borohydride, that's quite a good price, especially if there's room for it to drop further. I'd be quite tempted to buy some. Any pricing on the other two you mentioned?
Gurt
Harmless
Posts: 42
Registered: 5-2-2016
Location: USA
Member Is Offline
Mood: Inquisitive
Hi, I will look into just how much shipping on the thionyl may be. As for the other reagents, I am still working out a final price with the supplier. I will be sure to update as soon as a arrive at a solid number. So far for benzene I am looking at potentially being able to sell for less then $80 a liter, on the high side. My target is to be able to sell it for around$50-$60 a liter, which is seeming like a fairly good possibility. Edit: After a little reading, I am doubtful that I want to carry thionyl chloride in the near future. Due to its scheduling alone. My apologies [Edited on Apr04-23-2016 by Gurt] careysub International Hazard Posts: 1339 Registered: 4-8-2014 Location: Coastal Sage Scrub Biome Member Is Offline Mood: Lowest quantum state I will for sure get some of that NaBH4, 250 g at least. [Edited on 23-4-2016 by careysub] JJay International Hazard Posts: 3440 Registered: 15-10-2015 Member Is Offline Quote: Originally posted by Gurt Hi, I will look into just how much shipping on the thionyl may be. As for the other reagents, I am still working out a final price with the supplier. I will be sure to update as soon as a arrive at a solid number. So far for benzene I am looking at potentially being able to sell for less then$80 a liter, on the high side. My target is to be able to sell it for around $50-$60 a liter, which is seeming like a fairly good possibility. Edit: After a little reading, I am doubtful that I want to carry thionyl chloride in the near future. Due to its scheduling alone. My apologies [Edited on Apr04-23-2016 by Gurt]
I don't see why the schedule would be a concern, but it does seem like it is fairly hard to obtain. I am pretty sure I could buy some from one of the local chemical supply stores, but the price would be outrageous. Oh and benzene shouldn't be more than $25/liter... I consider even that to be a high price for such a simple and ubiquituous substance. [Edited on 23-4-2016 by JJay] Gurt Harmless Posts: 42 Registered: 5-2-2016 Location: USA Member Is Offline Mood: Inquisitive I will see what I can work out for benzene, but it seems to be far more than$25/L wherever I look.
Edit: I received a quote now from a different importer, who is offering Benzene at a very attractive price. Which would allow me actually to shoot very close to the $25/L mark, however the have a rather large minimum order quantity (100L). If interested shoot me a U2U or drop a reply about how much you would be looking for, or if you are looking for continuous supply let me know what your demand would be. If it seems I can get close to their minimum order quantity as far as demand from interested parties, I will order it. [Edited on Apr04-23-2016 by Gurt] Loptr International Hazard Posts: 1298 Registered: 20-5-2014 Location: USA Member Is Offline Mood: Grateful Benzene is available on eBay for about ~$80/liter, and from supply houses I have seen it somewhere in the range of ~$180/4L. I recently purchased 4L for ~$70, but it was an old stock sale that I got it from.
And just so you know, as I am not familiar with your background, but NaBH4 is pretty hygroscopic. With deal with larger amounts exposed to the atmosphere, you will start to see it becoming wet pretty quickly. So consider this when breaking down the bulk package. You might want a way of excluding the atmosphere. It's almost sometimes easier to work with the basic aqueous solutions of NaBH4, if your substrate will tolerate the conditions.
[Edited on 25-4-2016 by Loptr]
[Edited on 25-4-2016 by Loptr]
JJay
International Hazard
Posts: 3440
Registered: 15-10-2015
Member Is Offline
I can name two major suppliers that sell benzene for under $25/liter, and I can make it at home for less, although there is quite a bit of work involved. Edit: Ugh... I just checked those vendors, and Sigma Aldrich discontinued their cheap benzene lines and the other one (which is the cheapest I am seeing) is now charging$36/liter for chromatography grade. What is the world coming to....
[Edited on 25-4-2016 by JJay]
Dr.Bob
International Hazard
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Location: USA - NC
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For the nasty liquids, you can ship up to 34 ml as a "small sample", if you package and label it correctly, and those are simpler to ship via UPS or FedEx by ground. It is not a trivial task do do it legally, but if you will be shipping any chemicals, you should be able to get the proper paperwork and find the rules on it. It is a pain, which is why I am not dealing with shipping most chemicals.
JJay
International Hazard
Posts: 3440
Registered: 15-10-2015
Member Is Offline
Quote: Originally posted by Dr.Bob For the nasty liquids, you can ship up to 34 ml as a "small sample", if you package and label it correctly, and those are simpler to ship via UPS or FedEx by ground. It is not a trivial task do do it legally, but if you will be shipping any chemicals, you should be able to get the proper paperwork and find the rules on it. It is a pain, which is why I am not dealing with shipping most chemicals.
I was wondering about that... I have often seen reputable vendors ship 30 mL vials of extremely nasty liquids (bromine, phosphorus oxychloride, etc.) without any special labeling and was wondering how they got away with it. It seems reasonable that there would be an exemption for tiny quantities.
If anyone would like to send me 30 mL of thionyl chloride, I'd like to have some for my chemistry set
Gurt
Harmless
Posts: 42
Registered: 5-2-2016
Location: USA
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Mood: Inquisitive
NaBH4 is shipped and should be in stock within 8-10 days. Benzene is on order, as are Carbon tetrachloride, DMF, and THF. As for packing, I appreciate the advice Loptr, I have set up to handle packing under inert atmosphere, and storage is in a very well climate controlled and isolated room. Packing will be in foil/mylar vacuum bags, several layers of them. When shipped all will be shipped by FedEx according to their hazard class 4.3 packing requirements.
urenthesage
Hazard to Self
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Registered: 21-2-2016
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Mood: No Mood
The only risk here, for stuff like thionyl chloride, where Im from is seizure at the border, meaning you lose money. Unless its an illegal drug the border patrol only has the ability to seize a shipment, they cant really charge you as you are only violating import restrictions. Ive spoken to law enforcement about this issue (its the OPP here) and they have no interest in busting someone for chemicals unless those chemicals are illegal drugs. They simply dont find it worth their time to prosecute someone for a hundred grams of thionyl chloride, the paperwork isnt worth the paltry fine you'd get for posession.
Loptr
International Hazard
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Registered: 20-5-2014
Location: USA
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Mood: Grateful
Quote: Originally posted by Gurt NaBH4 is shipped and should be in stock within 8-10 days. Benzene is on order, as are Carbon tetrachloride, DMF, and THF. As for packing, I appreciate the advice Loptr, I have set up to handle packing under inert atmosphere, and storage is in a very well climate controlled and isolated room. Packing will be in foil/mylar vacuum bags, several layers of them. When shipped all will be shipped by FedEx according to their hazard class 4.3 packing requirements.
Good to hear! This is starting to sound like a very interesting venture.
The first time I actually handled NaBH4 that wasn't an aqueous solution I was surprised at deliquescent it would be become. It's almost like NaOH in that regard. The powder seems to wet the easiest, which decreases as the form is provided as chunks and prills.
The wet NaBH4 hydrolyzes somewhat easily, as the BH4(-) reacts with H(+), so the alkaline NaOH solutions reduce the available H(+) capable of reacting, and there by keeping hydrolysis from proceeding.
NitreRat
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Location: Cyberspace
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Mood: No Mood
I'm not sure about the legality of shipping benzene to the EU, but if it's possible and doesn't cost an insane amount of money, I would be interested in buying around 1 to 5 liters of benzene. As an amateur chemist - despite having a degree in O-Chem and a vaguely professional looking lab in my garden shed - it's almost impossible for an amateur to obtain benzene within the EU. I'm tired of having to perform the old CaO/Benzoic acid reaction or scavenging through old stores to the find right kind of mothballs (i.e. 1,4-Dichlorobenzene -> grignard formation -> hydration) for a couple drops of benzene.
On a related note, does anyone know what sort of criteria I need to meet before I can purchase from companies like Sigma-Aldrich? Do I just need to be a registered company with a proper business address, or do I need to be thoroughly inspected by some sort of law agency to confirm that I'm a real profitable chemical company?
[Edited on 5/5/2016 by NitreRat]
Gurt
Harmless
Posts: 42
Registered: 5-2-2016
Location: USA
Member Is Offline
Mood: Inquisitive
Hi, I will look into shipping regulations for you and let you know what I find out. From what I've seen you must be a proven business with tax number, business address, company bank account, and proof or required use to order from sigma. Also I have heard but have not personally seen that you must have been established at least 2 years prior to ordering. For myself I usually just order sigma chemicals through Fisher or another 3rd party. As they are fine with shipping to my LLC and invoicing my business. Without having a long standing prior to ordering.
[Edited on May05-7-2016 by Gurt]
Pages: 1
Sciencemadness Discussion Board » Fundamentals » Reagents and Apparatus Acquisition » Reagents for sale Select A Forum Fundamentals » Chemistry in General » Organic Chemistry » Reagents and Apparatus Acquisition » Beginnings » Responsible Practices » Miscellaneous » The Wiki Special topics » Technochemistry » Energetic Materials » Biochemistry » Radiochemistry » Computational Models and Techniques » Prepublication Non-chemistry » Forum Matters » Legal and Societal Issues
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2022-07-04 11:50:23
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http://math.stackexchange.com/questions/17711/why-do-we-need-to-prove-euv-euev/17712
|
# Why do we need to prove $e^{u+v} = e^ue^v$?
In this book I'm using the author seems to feel a need to prove
$e^{u+v} = e^ue^v$
By
$\ln(e^{u+v}) = u + v = \ln(e^u) + \ln(e^v) = \ln(e^u e^v)$
Hence $e^{u+v} = e^u e^v$
But we know from basic algebra that $x^{a+b} = x^ax^b$.
Earlier in the chapter the author says that you should not assume $e^x$ "is an ordinary power of a base e with exponent x."
This is both a math and pedagogy question then, why does he do that?
So 2 questions really
1. Do we need to prove this for such a basic property?
2. If we don't need to, then why does he do it? Fun? To make it memorable? Establish more neural connections? A case of wildly uncontrolled OCD?
Also I've always taken for granted the property that $x^{a+b} = x^a x^b$. I take it as an axiom, but I actually don't know where that axiom is listed.
-
How is the function $e^x$ defined in the text? – Jon Jan 16 '11 at 16:00
@bobobobo: How is $\text{ln}(x)$ defined? – Isaac Jan 16 '11 at 16:12
@bobobobo: in other words, I think your confusion here is linguistic. You know there is an operation called exponentiation which satisfies x^{a+b} = x^a x^b, and the author is defining an operation which he calls exponentiation, so you assume it also satisfies x^{a+b} = x^a x^b. But you can't assume this. If I call something red, and you call something red, we aren't necessarily talking about the same color. We have to prove it, e.g. by agreeing on a unit of length and measuring the wavelengths of the appropriate type of light. Only then can we agree on the meaning of a statement like... – Qiaochu Yuan Jan 16 '11 at 17:19
-1 for asking a question of the form "Why did the author of my book do this?" without identifying the book in question. This makes things unnecessarily difficult and can waste a lot of people's time as they are forced to guess what the text might say. – Pete L. Clark Jan 17 '11 at 3:43
@bobobobo: there's absolutely nothing wrong with being a "math noob", and of course everyone who knows some advanced mathematics was at one point a beginner. I removed my downvote and added an upvote since you identified the book. @Jonas: "There's nothing wrong with inexpensive math books." No kidding. I'll complement your observation with: "There's something wrong with expensive math books." (Namely, they're expensive!) – Pete L. Clark Jan 19 '11 at 0:24
In fact, such a proof is often necessary, which is why many authors write the function $e^x$ as $\exp(x)$ until they establish that it's just a "normal" exponent. For instance, if the original definition is given as
$$\exp(x) = \lim_{n \to \infty} \left(1+\frac{x}{n}\right)^n,$$
then proving that $\exp(x + y) = \exp(x) \exp(y)$ is non-obvious, and certainly necessary.
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This is very true, though defining the exponential function in this way and proceeding from there is slightly masochistic! – Noldorin Jan 18 '11 at 16:29
@Noldorin: why? It is an excellent intuitive definition from the point of view of differential equations: it's exactly what you get if you use Euler's method to approximate the solution to y' = y with step size 1/n and take n to infinity. Certainly it's better motivated from first principles than the power series definition, even if the latter is easier to prove things about. – Qiaochu Yuan Jan 18 '11 at 16:39
@Quiaochu: Oh, it's a lovely definition; only, it makes the following proof less trivial than other definitions. :) – Noldorin Jan 18 '11 at 16:39
I remember in Analysis I my instructor defined $e^x$ by its Taylor series. It is easy to check with this definition that this function is increasing, $f(1)=e$ and $f(x+y)=f(x)f(y)$. It follows imediatelly from these three simple properties that this is what we would call $e^x$... I liked that approach. – N. S. Oct 12 '11 at 23:09
Mathematicians have a habit of using the same notation to denote many different concepts. To justify their overloading, they tend to point to similarities between properties of those concepts. You may think of the exponential as one concept, but it is actually a large family of related concepts:
• The exponentials $x^n$ where $x$ is an element of an arbitrary monoid and $n$ a non-negative integer,
• The exponentials $x^n$ where $x$ is an element of an arbitrary group and $n$ an integer,
• The exponentials $x^n$ where $x$ is an element of an arbitrary group and $n$ a rational (which are not guaranteed to exist in general, and are also not guaranteed to be unique in general),
• The exponential $x^n$ where $x$ is an element of a topological group where $x^n$ for $n$ rational is unique, and $n$ a real number (defined by limits as in Rudin),
• The exponential $e^x = 1 + x + \frac{x^2}{2} + ...$ where $x$ is an element of a topological ring containing $\mathbb{Q}$ and the series converges,
• etc.
It is easy to be fooled into thinking that these are all the same concept because they define essentially the same operation on $\mathbb{R}$, but 1) this is not obvious and requires proof, and 2) they generalize in different directions, so should be regarded as different in full generality. Note that the condition for $x^n$ to be defined where $x$ is an element of a topological group and $n$ a real number is quite strong and rarely satisfied.
Note also that your proof using logarithms runs into unnecessary subtleties when $u, v$ are taken to be complex numbers, since in this setting the logarithm is not uniquely defined. Nevertheless, the exponential law still holds in this setting; it is one of the basic properties that mathematicians point to to justify calling something an exponential in the first place.
In higher mathematics, the exponential further generalizes to:
So it is good to keep in mind that "exponential" does not just refer to one thing.
-
You've probably lost the reader when talking about group theory and topology - you certainly lost me. Best to stick to the level of the question, in my view. – Noldorin Jan 16 '11 at 16:17
@Noldorin: answers are not just for the questioner. The reason questions stick around after the OP accepts an answer is that questions and answers are supposed to be searchable and preserved for future users, who may have a very different background from the OP but may be wondering the same thing. This is an enormous difference between the internet and your grandmother's living room. Like I said, since I figured several other people would be giving answers at the appropriate level, I thought I would try something different. And if your grandmother asked about the properties of silicon... – Qiaochu Yuan Jan 16 '11 at 16:38
Saying "A,B,C are several concepts in higher mathematics that differ from your usual view of D, so you shouldn't take the properties of D for granted" is not itself an argument in 'higher mathematics' even though it refers to it. If Qiaochu used technical details of the latest set-theoretic constructions of group exponential functions by ultrafilters, I might agree with you; but since he just refers to the idea as an example I'd say it's at the appropriate level. – Michael Burge Jan 16 '11 at 16:48
@Noldorin: I can't help but feel like you're missing the point. The existence of people who would appreciate low-level answers does not say anything about the existence of people who would appreciate high-level answers. I guessed that several other people would provide simple answers, and they did - three of them. This question didn't need a fourth one, at least not until the OP clarified what his definitions were. Meanwhile, the comments in this answer apply regardless of what definition of exponentiation the OP is using. I don't know why you think I wrote this answer because I have a – Qiaochu Yuan Jan 16 '11 at 17:12
This discussion reminds me of Mordell's review of Lang's book on Diophantine geometry (projecteuclid.org/DPubS/Repository/1.0/…) and Lang's later review of Mordell's book on Diophantine equations (projecteuclid.org/DPubS/Repository/1.0/…). – KCd Jan 17 '11 at 4:49
This may seem like a pretty pointless proof, at least on the surface, but I suspect there's some subtlety in the way the author's defined things here. (It may even appear circular at first, considering that the logarithm is often introduced as the inverse of the exponential function. Saying that, it can be derived the other way round, and this can sometimes be enlightening.)
A rigorous proof for integer exponents is very straightforward indeed, and follows simply from the definition of the exponential function (of arbitrary base). For arbitrary exponents, things get slightly more complicated. I present a more complete proof below.
So, let us suppose that the author began by defining the (natural) logarithm function,
$$\ln a = \int_1^a \frac{dx}{x} .$$
We can then prove the addition property of logarithms, $\ln (ab) = \ln a + \ln b$, by considering
$$\ln (ab) = \int_1^{ab} \frac{1}{x} \; dx = \int_1^a \frac{1}{x} \; dx \; + \int_a^{ab} \frac{1}{x} \; dx =\int_1^{a} \frac{1}{x} \; dx \; + \int_1^{b} \frac{1}{t} \; dt = \ln (a) + \ln (b)$$
The exponential function can of course be defined as the inverse of the logarithm, i.e.
$$exp(ln(a)) = a$$
Now, to prove the property of exponentials, $e^{u+v} = e^u e^v$, we start as follows.
$$\text{Let}\ u = \ln a, v = \ln b .$$
Then, using this property of logarithms and the definition of the inverse, consider
$$e^{u+v} = e^{\ln a + \ln b} = e^{\ln (ab)} = ab = e^u e^v .\ \square$$
That should hopefuly be straightforward enough to follow. There are of course other equivalent definitions of $exp$ and $ln$. (You can for example define the Taylor series of $exp$, use the Cauchy product, and then simplify, but that's slightly trickier.)
-
It's also quite common to define $ln(x)$ as the integral $\int_1^a \frac{1}{x} \, dx$, which might be the case here. – Jon Jan 16 '11 at 16:01
@Jon Yes, he did define $ln(x)$ like that previously – bobobobo Jan 16 '11 at 16:07
@Jon: Didn't notice that comment, but I was just writing that in my update as you made it. ;) – Noldorin Jan 16 '11 at 16:12
@bobobobo: Then this is your answer. It would have been great if you had given that definition of $\ln(x)$ in your question after Isaac in his comment asked for it. – Hendrik Vogt Jan 18 '11 at 16:16
The necessity of a proof depends very much on how each of those things is defined and the domain over which the variables may vary. For example, if $e^x$ is defined by the power series $\displaystyle \sum_{n=0}^{\infty} \frac{x^n}{n!}$ for all $x \in \mathbb{C}$, then there is a very real need to prove that $e^{x + y} = e^x e^y$, and the proof is non-trivial. On the other hand, if $e$ is some constant defined elsewhere and $x$ and $y$ are positive integers, then there's essentially nothing to prove.
I would also like to add that the law doesn't hold everywhere. If $A$ and $B$ are square matrices (or endomorphisms of a vector space) that do not commute, then in general $\exp(A + B) \ne \exp(A) \exp(B)$. This fact allows us to study certain non-commutative groups using the tools of analysis and differential geometry — this is the field of study known as Lie theory.
-
@ Zhen:"On the other hand, if e is some constant defined elsewhere and x and y are positive integers, then there's essentially nothing to prove." Why is it there is nothing to prove? How do we know that $x^a x^b = x^{a+b}$ in general $\forall x,a,b$? My line of thought is if I can prove that $e^{a_1 + b_1} = e^{a_1}e^{b_1}$ and $(e^{c})^d = e^{cd}$ then I can choose $a_1 = a \log_e(x)$ and $b_1 = b \log_e(x)$ and prove that $x^ax^b = x^{a+b}$. Or you could prove it for natural numbers, extend it to integers, to rationals to reals and to complex. In either case,there is some proof needed right? – user17762 Jan 17 '11 at 16:31
@Sivaram: If $x^a x^b = x^{a+b}$ for any number $x$ and any positive integers $a$ and $b$ essentially by definition of the notation. The only assumption is that multiplication is associative, so in fact, as Qiaochu alludes to, this is true for $x$ in any arbitrary monoid. – Zhen Lin Jan 17 '11 at 17:22
One advanced reason for giving a proof is that in some related contexts such a formula breaks down. On the 2-adic numbers, the power series $A(x) = \exp(2x^2 - 2x)$ when expanded out and written in standard form turns out to converge on ${\mathbf Z}_2$, so in particular $A(1)$ is defined. One can prove $A(1)^2 = 1$, but although naively one may expect $A(1) = 1$ by just plugging 1 directly into the original definition I gave for $A(x)$, in fact $A(1) = -1$. (The point here is that the naive calculation $A(1) = \exp(2 - 2) = \exp(2)\exp(-2) = 1$ is wrong since $\exp(2)$ doesn't make sense 2-adically.)
-
One definition of $e^x$ is $\displaystyle \lim_{n \rightarrow \infty}(1 + \frac{x}{n})^n$. From this definition, it doesn't automatically follow that $e^x e^y = e^{x+y}$.
In fact, it doesn't even follow immediately that $e^x = \displaystyle \lim_{n \rightarrow \infty}(1 + \frac{x}{n})^n = (\displaystyle \lim_{n \rightarrow \infty}(1 + \frac{1}{n})^n)^x = (e)^x$. What this means is $e^x$ is just a short hand notation for the limit which after some analysis we realize it as $(e)^x$.
By limit arguments, we can now show that $\displaystyle \lim_{n \rightarrow \infty}(1 + \frac{x}{n})^n = 1 + \sum_{k=1}^{\infty} \frac{x^k}{k!}$, $\forall x \in \mathbb{R}$.
Now $e^x \times e^y = (1 + \sum_{k=1}^{\infty} \frac{x^k}{k!}) \times (1 + \sum_{k=1}^{\infty} \frac{y^k}{k!})$.
Now we need to realize that we can rearrange the terms in the series and multiply terms of the two series since both of them converge absolutely.
Hence $e^x \times e^y = (1 + \sum_{k=1}^{\infty} \frac{x^k}{k!}) \times (1 + \sum_{k=1}^{\infty} \frac{y^k}{k!}) = 1 + (x+y) + (\frac{x^2 + 2xy + y^2}{2!}) + (\frac{x^3 + 3x^2y + 3xy^2 + y^3}{3!}) + \cdots$
Now make use of the binomial theorem to get
$$e^{x} e^{y} = e^{x+y}$$
PS: Though I have taken care to make sure the line of thought is right, you need to be careful when writing down the argument as to when you can interchange terms in an infinite series, multiply out two infinite series etc etc.
-
One definition of e^x is what you have written. – Qiaochu Yuan Jan 17 '11 at 17:57
@Qiaochu: I prefer to define $e^x$ as $\displaystyle \lim_{n \rightarrow \infty}(1 + \frac{x}{n})^n$ (at least for complex numbers) and then derive all the other formulas like the infinite series etc from this definition. – user17762 Jan 17 '11 at 18:53
yes, but your use of the word "the" is misleading. The crux of this question is that it is not trivial to show that multiple definitions are equivalent. Your second argument doesn't work, either; like I mentioned in the comments, you still have to show that this function exists. – Qiaochu Yuan Jan 17 '11 at 19:05
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2015-04-21 21:14:11
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https://proofwiki.org/wiki/274
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# 274
Jump to navigation Jump to search
Previous ... Next
## Number
$274$ (two hundred and seventy-four) is:
$2 \times 137$
The $11$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$, $121$, $166$, $202$, $265$:
$2 + 7 + 4 = 2 + 1 + 3 + 7 = 13$
The $27$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $244$, $260$, $266$, $268$:
$\nexists m \in \Z_{>0}: m - \map \phi m = 274$
where $\map \phi m$ denotes the Euler $\phi$ function
The $43$rd nontotient:
$\nexists m \in \Z_{>0}: \map \phi m = 274$
where $\map \phi m$ denotes the Euler $\phi$ function
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2022-10-04 23:03:31
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https://mailman.mit.edu/pipermail/galib/2001-June/000423.html
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EASEA v0.6b compiler
Pierre COLLET Pierre.Collet at Polytechnique.fr
Tue Jun 19 21:32:30 EDT 2001
Hello everybody,
I have been working for the past two years on a language specifically
designed to easily specify evolutionary algorithms.
A compiler comes along with the language that takes .ez files as input, to
produce pure C++ with calls to existing evolutionary libraries.
Two libraries are supported, one of which is GALib, hence my mail to this
list.
With this concept, the end-user only needs to program his fitness function
in C, as well as his specific mutation and crossover functions and EASEA
does the rest, that is: building C++ classes around the defined genome and
getting everything together to create a complete GALib or EO C++ source
file, ready to be compiled by a C++ compiler (I appended a onemax written in
EASEA at the end of this mail for reference).
EASEA Millennium Edition v0.6b was just released, and even though it is
still a prototype, it proved to be stable enough for me to send a mail on
this list.
Up to now, a parallel version of EASEA-GALib (with MPI) has been used to
optimise airfoil shapes (with FORTRAN fitness functions), and the standard
EASEA-GALib compiler is currently being used for several real-world
applications. The EASEA language is taught with success in six French
universities, and several research papers were published, which used EASEA
as an implementation language for the research projects.
EASEA can be used in several ways: the programmer can exclusively use EASEA
to write his project, or he can use EASEA as a primer, allowing him to very
rapidly write a first version of his C++ program using GALib or EO that he
can refine and tune later on directly in C++.
Finally, as EASEA can generate C++ code for EO or GALib out of the same .ez
file, it makes sense to compare both libraries on the same evolutionary
algorithm. Moreover, a .ez file can be seen as an independent way to
describe an EA. One can also take benefit of the fact that it is
recompilable on any environment to use it for comparison and exchange
purposes (EASEA runs indifferently under Windows or UNIX).
The EASEA language and compiler is now at the point where it needs feedback
from many users to evolve towards an optimal language (a meta-evolution ?
:-). Therefore, EASEA is available on the
http://sourceforge.net/projects/easea/ repository, and two mailing lists
were created a couple of days ago so that users can share their experience
with the language, like you share your experience on GALib on this mailing
list.
The first one (http://lists.sourceforge.net/lists/listinfo/easea-help)
should obviously be used whenever help is required, while the second one
(http://lists.sourceforge.net/lists/listinfo/easea-dev-issues) is dedicated
to discussions on missing features, future developments, ...
I would very much like to hear from you on those mailing lists so that you
could tell me which features you are missing most, whether you had
difficulties with installing the software, ...
Well, I need you all to guide me towards the good direction.
I append at the end of this mail an example of a onemax coded with EASEA, so
that you can have a feeling for what the language looks like.
If you download the EASEA compiler, you will be able to compile this basic
example (provided with EASEA) with the "easea onemax -galib -v" command.
Looking forward to hearing from you on the mailing lists,
Tell me what you think...
Pierre COLLET
___________________________________________________________________________
Pierre COLLET -- DREAM European Project
Centre de Mathematiques Appliquees -- Ecole Polytechnique
91128 Palaiseau cedex -- FRANCE
Tel. +33 (0)3 85.57.16.46 / +33 (0)1 69.33.46.19
URL: http://www.cmap.polytechnique.fr/~collet/
____________________________________________________________________________
EVOLUTION ARTIFICIELLE 2001
5th International Conference on Evolutionary Algorithms
October 29-31 2001 - Le Creusot - France
Submission date may 11th 2001
URL: http://www.cmap.polytechnique.fr/~ea01
____________________________________________________________________________
Take it EASEA
http://www-rocq.inria.fr/EASEA/
____________________________________________________________________________
/*_________________________________________________________
onemax.ez
EASEA implementation of the ONEMAX problem
Pierre COLLET (Pierre.Collet at Polytechnique.Fr)
17/01/01
__________________________________________________________*/
\User declarations :
#define SIZE 10
float pMutPerGene=0.1;
inline void swap(bool& a, bool& b)
{bool c=a; a=b; b=c;}
\end
\User classes :
GenomeClass { bool x[SIZE]; }
\end
\GenomeClass::initialiser : // "initializer" is also accepted
for (int i=0;i<SIZE;i++) Genome.x[i]=tossCoin(.5)?1:0;
\end
\GenomeClass::crossover :
int CrossoverPosition=random(0,SIZE);
for(int i=0;i<CrossoverPosition+1;i++)
swap(child1.x[i],child2.x[i]);
\end
\GenomeClass::mutator : // Must return the number of mutations
int NbMut=0;
for (int i=0;i<SIZE;i++)
if (tossCoin(pMutPerGene)){
NbMut++;
Genome.x[i]=Genome.x[i]?0:1;
}
return NbMut;
\end
\GenomeClass::evaluator : // Returns the score
int Score=0;
for (int i=0; i<SIZE;i++)
Score+=(int)Genome.x[i];
return Score;
\end
\Default run parameters : // Please let the parameters appear in this
order
Number of generations : 15 // NB_GEN
Mutation probability : 1 // MUT_PROB
Crossover probability : 1 // XOVER_PROB
Population size : 30 // POP_SIZE
Selection operator : Tournament // RouletteWheel, Deterministic, Ranking,
Random
Offspring population size : 12 // 40%
Replacement strategy : plus // Comma, SteadyState, Generational
Discarding operator : Worst // Best, Tournament, Parent, Random
Evaluator goal : Maximise // Minimise
Elitism : On // Off
\end
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2022-01-25 17:48:36
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http://tug.org/pipermail/macostex-archives/2008-May/035154.html
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# [OS X TeX] Fine point of mathematical typing?
Michael Sharpe msharpe at ucsd.edu
Tue May 13 00:52:27 CEST 2008
A minute or two with a text edit can change it to
\begin{verbatim}
2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131,
173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443,
491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911,
953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451,
1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973,
2003, 2039, 2063, 2069, 2129, 2141, 2273, 2339, 2351, 2393, 2399, 2459,
2543, 2549, 2693, 2699, 2741, 2753, 2819, 2903, 2939, 2963, 2969, 3023,
3299, 3329, 3359, 3389, 3413, 3449, 3491, 3539, 3593, 3623, 3761, 3779,
3803, 3821, 3851, 3863, 3911, 4019, 4073, 4211, 4271, 4349, 4373, 4391,
4409, 4481, 4733, 4793, 4871, 4919, 4943, 5003, 5039, 5051, 5081, 5171,
5231, 5279, 5303, 5333, 5399, 5441, 5501, 5639, 5711, 5741, 5849, 5903,
6053, 6101, 6113, 6131, 6173, 6263, 6269, 6323, 6329, 6449, 6491, 6521,
6551, 6563, 6581, 6761, 6899, 6983, 7043, 7079, 7103, 7121, 7151, 7193,
7211, 7349, 7433, 7541, 7643, 7649, 7691, 7823, 7841, 7883, 7901, 8069,
8093, 8111, 8243, 8273, 8513, 8663, 8693, 8741, 8951, 8969, 9029, 9059,
9221, 9293, 9371, 9419, 9473, 9479, 9539, 9629, 9689, 9791,...
\end{verbatim}
which aligns quite well.
On May 12, 2008, at 12:57 PM, Roberto Avanzi wrote:
> I have the following problem. I want to typeset a list of numbers,
> like the following short file
>
>
> \documentclass[12pt]{article}
>
> \begin{document}
>
> \begin{quote}
> \noindent%
> $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$,
> $173$, $179$, $191$, $233$, $239$, $251$, $281$, $293$, $359$,
> $419$, $431$, $443$,
> $491$, $509$, $593$, $641$, $653$, $659$, $683$, $719$, $743$,
> $761$, $809$, $911$,
> $953$, $1013$, $1019$, $1031$, $1049$, $1103$, $1223$, $1229$,
> $1289$, $1409$, $1439$, $1451$,
> $1481$, $1499$, $1511$, $1559$, $1583$, $1601$, $1733$, $1811$,
> $1889$, $1901$, $1931$, $1973$,
> $2003$, $2039$, $2063$, $2069$, $2129$, $2141$, $2273$, $2339$,
> $2351$, $2393$, $2399$, $2459$,
> $2543$, $2549$, $2693$, $2699$, $2741$, $2753$, $2819$, $2903$,
> $2939$, $2963$, $2969$, $3023$,
> $3299$, $3329$, $3359$, $3389$, $3413$, $3449$, $3491$, $3539$,
> $3593$, $3623$, $3761$, $3779$,
> $3803$, $3821$, $3851$, $3863$, $3911$, $4019$, $4073$, $4211$,
> $4271$, $4349$, $4373$, $4391$,
> $4409$, $4481$, $4733$, $4793$, $4871$, $4919$, $4943$, $5003$,
> $5039$, $5051$, $5081$, $5171$,
> $5231$, $5279$, $5303$, $5333$, $5399$, $5441$, $5501$, $5639$,
> $5711$, $5741$, $5849$, $5903$,
> $6053$, $6101$, $6113$, $6131$, $6173$, $6263$, $6269$, $6323$,
> $6329$, $6449$, $6491$, $6521$,
> $6551$, $6563$, $6581$, $6761$, $6899$, $6983$, $7043$, $7079$,
> $7103$, $7121$, $7151$, $7193$,
> $7211$, $7349$, $7433$, $7541$, $7643$, $7649$, $7691$, $7823$,
> $7841$, $7883$, $7901$, $8069$,
> $8093$, $8111$, $8243$, $8273$, $8513$, $8663$, $8693$, $8741$,
> $8951$, $8969$, $9029$, $9059$,
> $9221$, $9293$, $9371$, $9419$, $9473$, $9479$, $9539$, $9629$,
> $9689$, $9791$, $\ldots$
> \end{quote}
>
> \end{document}
>
>
> BUT, the spacing in the last line is not the same as in the line
> above. How can I force LaTeX to typeset the numbers in the last line
> alined to the numbers in the line above? (yes, it is for number
> theory course notes :-)
>
> Roberto
>
>
> TeX FAQ: http://www.tex.ac.uk/faq
> List Reminders and Etiquette: http://www.esm.psu.edu/mac-tex/list/
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https://www.nature.com/articles/s41598-017-16254-z?error=cookies_not_supported&code=4dea5b64-23db-4be6-bc35-ede5087034ac
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Article | Open | Published:
# Discrimination of CRISPR/Cas9-induced mutants of rice seeds using near-infrared hyperspectral imaging
Scientific Reportsvolume 7, Article number: 15934 (2017) | Download Citation
## Abstract
Identifying individuals with target mutant phenotypes is a significant procedure in mutant exploitation for implementing genome editing technology in a crop breeding programme. In the present study, a rapid and non-invasive method was proposed to identify CRISPR/Cas9-induced rice mutants from their acceptor lines (huaidao-1 and nanjing46) using hyperspectral imaging in the near-infrared (NIR) range (874.41–1733.91 nm) combined with chemometric analysis. The hyperspectral imaging data were analysed using principal component analysis (PCA) for exploratory purposes, and a support vector machine (SVM) and an extreme learning machine (ELM) were applied to build discrimination models for classification. Meanwhile, PCA loadings and a successive projections algorithm (SPA) were used for extracting optimal spectral wavelengths. The SVM-SPA model achieved best performance, with classification accuracies of 93% and 92.75% being observed for calibration and prediction sets for huaidao-1 and 91.25% and 89.50% for nanjing46, respectively. Furthermore, the classification of mutant seeds was visualized on prediction maps by predicting the features of each pixel on individual hyperspectral images based on the SPA-SVM model. The above results indicated that NIR hyperspectral imaging together with chemometric data analysis could be a reliable tool for identifying CRISPR/Cas9-induced rice mutants, which would help to accelerate selection and crop breeding processes.
## Introduction
Crop breeding has come to the stages of targeted genome editing that has benefited from the development of the CRISPR/Cas9 (clustered regularly interspaced short palindromic repeats/CRISPR-associated nuclease 9, Cas9) gene editing system. CRISPR/Cas9-mediated targeted gene modification was first introduced for plant genome editing and crop breeding purposes in 20131. This revolutionary technology allows plant breeders to target very specific pieces of DNA without introduction of foreign DNA and to influence key traits accurately and quickly2. Although the CRISPR/Cas9 system is an excellent tool for genome editing, the next major challenge is to identify the resulting mutants due to the extent of off-target mutations and the differences in cleavage specificity among species3. Several approaches have been developed for identifying the mutants produced by the CRISPR/Cas9 system, such as T7 endonuclease I (T7EI) assay, high-resolution melting curve analysis (HRAM), restriction fragment length polymorphism (RFLP) and polymerase chain reaction (PCR)/restriction enzyme (RE) assay4,5,6,7,8. However, these DNA- and protein-based techniques need the complex process of extractions are time- and labour-intensive, costly, and require expensive capital equipment. Moreover, some precious samples will be destroyed by these techniques.
Therefore, a method for the selection of mutants introduced by the CRISPR/Cas9 system without requiring any wet chemistry, particularly extraction protocols, would be advantageous where large numbers of samples must be analysed. In contrast to biochemical assays, near-infrared (NIR) spectroscopy does not require technical expertise or complex techniques, and the spectrophotometer can be installed anywhere with no requirements for reagents or complicated protocols9. The radiation from light sources in the NIR region (750–2500 nm) provides a unique spectral signature in the form of a spectrum corresponding to the energy absorption and consequent vibrations of organic molecules based on relative overtones and combinations of O-H, N-H, and C-H functional groups10. Previous research in the transgenic field on the application of NIR spectroscopy and chemometrics has demonstrated this technique’s capacity to identify mutants mediated by transgenic technology11. García-Molina et al.9 applied the NIR spectroscopy combined with partial least square (PLS) analysis to discriminate low gliadin wheat grain from non-transgenic wheat lines with excellent classification accuracy (as high as 96%). Luna et al.12 used NIR and support vector machine-discriminant analysis (SVM-DA) to discriminate completely transgenic soybean oils from non-transgenic ones. Additionally, NIRs combined with chemometrics have also been employed successfully in the identification of transgenic rice13, corn14 or mutant of barley15. The basis of this technology for the detection of mutants generated using transgenic technology is that it could be used to identify phenotypic changes caused by genotypic changes that ultimately bring about changes in organic molecular bonds11. However, until now, no report in the literature has described the application of NIR hyperspectral imaging in CRISPR/Cas9-induced mutant recognition in the efficiency of the plant breeding process. In contrast to conventional NIR spectroscopy, NIR hyperspectral imaging has been used to combine traditional optical imaging and a spectral method and is capable of capturing images over broad contiguous wavelengths in the NIR region and has received much attention in cereal science16,17,18.
Therefore, the main objectives of this study were (1) to study the feasibility of screening CRISPR/Cas9-induced mutant rice seeds using NIR hyperspectral imaging and chemometrics analysis; (2) to identify important wavelengths that can be attributed to the differences between wild-type (WT) and mutant rice strains; (3) to build an optimal discrimination model based on important wavelengths to simplify the prediction model and to speed up the operation; and (4) to visualize the number and locations of mutants by developing imaging processing algorithms.
## Results and Discussion
### The Morphological characterization and thousand-grain weight
According to Ishimaru et al.19, a deletion TGW6 mutant had greater grain length and higher thousand-grain weight (TGW) but did not influence grain width compared with its transformation receptors. Table 1 shows the mean results of morphological features and TGW. The mean values for area, perimeter, length in pixels and TGW were significantly different between mutants and the WT at the 0.05 level when using Duncan’s multiple range test. Area, perimeter and length in pixels of seeds were increased after the TGW6 editing for both huaidao-1 and Nanjing 46. No significant difference of width in pixels was observed between the mutants and the WT (P < 0.05). Increases in TGW of 5.8% and 2.3% were detected for mutants from huaidao-1 and nanjing46, respectively. The bright appearance of seeds is a stable genetic characteristic and was used as one of the assessments for classification. Grain phenotypic changes after gene editing could be acquired by calculating the morphological features from the hyperspectral image of each rice seed in this study. However, the phenotypic changes were slight, and it was difficult to identify mutants by appearances from the images alone.
### Overview of spectra
Only the spectrum range of 975–1646 nm was used for analysis due to the back-end noise caused by the detection instrument. Figure 1(A,C) shows the extracted spectra of the ROI, and Fig. 1(B,D) shows the mean and standard deviation spectra of the WT and mutant samples. The patterns of the reflectance curves were similar to those of other crop seeds, such grape seeds20, oat and groat samples21, and Jatropha curcas L seeds22, although the position and magnitude of the valleys were specific. The NIR spectra information mainly described the shell chemical compositions as the rice seeds were unshelled. There was a peak at approximately 1122.81 nm and valleys at approximately 1200.19 and 1483.46 nm. Two absorption bands at approximately 1122 and 1200 nm contributed to the second overtone of C-H stretching vibrations of carbohydrates21,23, and the band at 1483 nm was related to the N-H stretch first overtone from CONH2 24. The graphic also shows that the curves of the spectra from the different genotypes show consistent trends, and they have similar peaks and valleys positions. Slight differences could were noted from the average spectra of WT and CRISPR/Cas9-induced mutant rice seeds of these two varieties. The mutant rice seeds exhibited higher spectral reflectances throughout the entire NIR spectrum than WT for both huaidao-1 and nanjing46. Therefore, the differences in the extents of magnitude of NIR reflectance might result from the phonotypic changes that result from the CRISPR/Cas9-medited genome editing.
### Classification analysis by PCA
PCA is an effective algorithm for reducing the dimensionality of data into a set of principal components (PCs), solving the problem of multicollinearity and handling potential co-linearity between variables. In this case, PCA programmes developed on all of the average wavelengths of rice samples were first applied to visualize the possible clusters and trends in a PCA score plot. The three-dimensional (3D) principal component score plot of the first three PCs of rice samples is illustrated in Fig. 2. The first three PCs explained the most spectral variations at 99.64% (93.95%, 5.31%, 0.38% for PC1, PC2, PC3, respectively) for huaidao-1 and 98.61% (94.35%, 4.93%, 0.33% for PC1, PC2, PC3, respectively) for nanjing46. As can been seen from Fig. 2, mutants were clustered together from their acceptor lines, but there were still several overlapping places in the PCA score plot. The discrimination between WT and mutants was not clear for huaidao-1 and nanjing46. It is worth mentioning that the target gene edited by the CRISPR/Cas9 system, TGW6, resulted in a slight phenotypic change (TGW and enhanced grain length) without a change in grain quality19. As consequence, the discriminant analysis established by the PCA was not sufficiently effective to select the CRISPR/Cas9-induced mutants. As none of the PCs alone contained sufficient information to fully segregate whole rice seeds from their acceptor lines, SVM and ELM discrimination models were run in the following study.
### Selection of optimal wavelengths
Hyperspectral imaging data contain redundant information, which affects the prediction performance of the model. Variable selection was performed using PCA-loadings and SPA wavelength selection-based techniques to facilitate and speed up the classification. Table 2 shows the effective wavelengths that were identified using the SPA and PCA-loadings method. By applying the optimal wavelength selection method, these two rice varieties showed mainly similar effective wavelengths at 1122.81, 1200.19, 1227.12, 1314.72, 1402.42, and 1581.51 nm. It is said that NIR spectroscopy is susceptible to the chemical bonding of organic matter molecules, such as C-N, C-H and N-H stretching vibrations, which could be caused by genotypic changes that result in phenotypic changes11. Absorption bands at approximately 1122, 1200 and 1314 nm were related to the second overtone of C-H stretching vibrations21,25,26,27. The wavelength of 1402 nm presents the O-H stretch from carboxyl acids28. A peak in the vicinity of 1580 nm was determined to be associated with the first overtone of O-H stretching vibrations29. These wavelengths carrying the discriminant information are believed to correspond to the NIR spectral bands relevant to changes in a mutant’s properties caused by TGW6 gene editing by the CRISPR/Cas9 system.
### Classification analysis using a discrimination model
PCA is a powerful method for finding the possible clusters of trends among samples; nevertheless, it cannot be applied for building predictive models for classification. Therefore, discriminant models built on both full and effective spectra were proposed to highlight the chemical differences between samples in the latter step (Table 3). The accuracies (in percentages) obtained for the calibration and prediction sets were summarized. The SVM and ELM models all achieved good recognition results in cases of large samples. For samples from Huaidao-1, better discrimination models using full NIR spectra were obtained from SVM, showing excellent accuracies of 92.38% on the calibration set and 92.50% on the prediction set. The classification capacity of the SVM model was also acceptable for Nanjing46, with an accuracy close to 90%.
Normally, hyperspectral images contain high-dimensional data with redundancy among contiguous wavelengths. According to Wold et al. (1996) discriminant models based on optimal wavelengths might be equally or more efficient than full spectra30. Judicious selection of spectra could discard uninformative wavelengths and decrease sensitivity to non-linearity, which would improve a model’s robustness and improve the model’s operating speed. As can been seen from Table 3, optimal wavelength algorithms affected the performance of mutant detection for both of these rice varieties. The SPA optimal wavelength algorithm improved the performance of the two models, but the PCA loading lines algorithm reduced the models’ recognition capacities. The numbers of effective wavelengths decreased to only 6% for nanjing46 and 5.5% for huandao-1 after using SPA method. The SPA-SVM was suitable for CRISPR/Cas9-induced mutant identification from both rice varieties, as good results were obtained in both the calibration and predication sets. For samples from huaidao-1, correct classifications of more than 90% were obtained in the calculation and predication sets, indicating that these selected spectra had reliable discrimination power for classification. SPA-ELM was only slightly worse than the SPA-SVM model for huaidao-1. For samples from nanjing46, SPA-SVM also improved the classification capacity. The discrimination capacity of the SPA-ELM model was slightly worse than that obtained from all of the wavelengths but was still rated as acceptable. Nearly all discrimination models built on optimal wavelengths selected by the PCA loading lines method yielded worse results than models based on full wavelengths. The overall results indicated that it was feasible to discriminate CRISPR/Cas9-induced mutant seeds using hyperspectral imaging and that the SPA-SVM recognition model was a reliable and robust model. Therefore, only the SPA-SVM discrimination model was used in the further study.
### CRISPR/Cas9-induced mutant visualization
NIR hyperspectral imaging, known as a chemical imaging system, can collect both spatial and spectral information related to chemical constituents of a sample. As described in the Materials and Methods section, discrimination models on the full spectra of all pixels lead to heavy computational burdens and require high-profile computer hardware. In such cases, SVM models developed on optimal wavelengths selected by SPA were applied to the objects’ pixels to predict classes from different genotypes of rice. A very uniform set of seeds (size and morphology) were inspected in this step. The mean pixels of each object were calculated using the object-wise approach from the outset31. Figure 3 shows some example of classification maps for identifying mutants. It is impossible to distinguish WT and CRISPR/Cas9-induced mutant seeds from both genotypes of rice by the naked eye; the differences within the same sample could be easily discerned from the final visualization distribution maps. It was noted that some seeds on the classification map were fallible. For samples from huidao-1, the misclassified number were 20 (90.50% accurate classification) for WT and 23 (89.04%) for mutant, respectively. And the predictive accuracies of nanjing46 were 87% and 91.42% for WT and mutant, respectively. That is to say, the SPA-SVM model produced a satisfied classification. The morphological characters of seeds in classification map had some changes due to the low resolution of the NIR imaging system and the image segmentation algorithm. However, the main shapes of the seeds and their locations were clearly shown on the prediction map. The results indicated that NIR hyperspectral imaging together with chemometric analysis was a promising technique to identify and locate single CRISPR/Cas9-induced mutant rice seeds, which has the potential to be a powerful tool for evaluating large numbers of samples from CRISPR/Cas9 gene editing performance trials for breeding programmes.
## Conclusions
CRISPR/Cas9 genome editing technology has shown great potential for targeting gene editing and facilitating crop breeding programmes. However, a time-consuming screening of large populations is required to identify mutants because of extensive potential off-target effects and the targeting specificity of the technology. This paper focused on the use of NIR hyperspectral imaging (975–1646 nm) combined with discrimination model (SVM and ELM) and optimal wavelength selection methods (PCA loadings and SPA) to identify and visualize CRISPR/Cas9-induced mutant rice seeds. PCA was first conducted to find identify clusters between WT and mutants. The score images acquired from PCA on the hyperspectral data reveal accredited discriminations between WT and mutants from two types of rice varieties. Moreover, a PCA loadings method and SPA were applied to extract effective features that were valuable for discrimination. After the appropriate data pretreatment, the SPA-SVM model had robust and valuable calibration and predication capacities, with a classification accuracy of approximately 90%. Finally, the classification of mutant seeds could be visualized on prediction maps by predicting the features of each pixel on individual hyperspectral images. The use of NIR hyperspectral imaging combined with chemometrics and image processing technology for screening CRISPR/Cas9-induced mutants is a very attractive platform and has the potential to be widely used in rapid plant breeding programmes for screening, as it is non-invasive and cost-effective and its does not require any pretreatments.
## Materials and Methods
### CRISPR/Cas9-induced rice samples
The gene-edited mutant rice seeds and their acceptor lines were provided by Zhejiang Academy of Agriculture Science, China. Rice varieties huaidao-1 and nanjing46 were used as the genetic transformation receptors. We designed a vector to target the gene THOUSAND-GRAIN WEIGHT 6 (TGW6)19 for both huaidao-1 and nanjing46. TGW6, the major QTL for thousand-grain weight, encodes a protein with indole-3-acetic acid (IAA)-glucose hydrolase activity. Loss of function of the TGW6 allele enhances grain weight through pleiotropic effects on source organs and leads to significant yield increases19. The gRNA target sequence was inserted into the Cas9/gRNA plasmid. The constructed plasmid and target sites are shown in Fig. 4(A,B). There were no other differences in the seeds between their transformation receptors and the corresponding mutants.
To detect the effect of the gene-editing, the genome DNA was extracted from each edited plant using the SDS method, and the target gene was sequenced. Each PCR mixture (30 μl volume) consisted of 10× PCR buffer (Takara Biotechnology Co.), 200 mM dNTP (Takara Biotechnology Co.), 0.5 mM of each primer, 1.25 U Taq DNA polymerase (Takara Biotechnology Co.), and template. The PCR programme consisted of a 95 °C/3 min denaturation, followed by 35 cycles of 94 °C/30s, 55 °C/30s and 72 °C/30s, with a final elongation step of 72 °C/7 min. The resulting amplicons were separated by electrophoresis through 2% agarose gels in 0.5× TBE with GelRed staining. As Fig. 4C showed that there were 2 bp and 3 bp missing in the TGW6 gene-edited mutants of huaidao-1 and nanjing46, respectively.
### Near-infrared hyperspectral image acquisition and calibration
Rice seeds were placed on a controlled conveyer belt and scanned using the NIR hyperspectral imaging system (ImSpector N17E; Spectral Imaging Ltd., Oulu, Finland). The devices of this instrument include a high-performance CCD camera (Hamamatsu, Hamamatsu City, Japan), an imaging spectrograph (ImSpector N17E; Spectral Imaging Ltd., Oulu, Finland) with a wavelength range from 874.41 to 1733.9100 nm, two 150 W tungsten halogen lamps (Fiber-Lite DC950 Illuminator; Dolan Jenner Industries Inc., Boxborough, MA, USA) for illumination, a moving platform controlled by a stepper motor (Isuzu Optics Corp., Taiwan, China) for sample motion, and a computer.
Before acquiring nondeformable and clear NIR images, the exposure time of camera, the distance between the CCD camera and conveyor, and the speed of the conveyor moment were adjusted to 3.2 ms, 28.7 cm and 23 mm/s, respectively. NIR hyperspectral images form a three-dimensional structure (x, y, λ) of multivariate data for processing and analysis, where x and y are the spatial dimensions (numbers of rows and columns in pixels), and λ represents the number of wavebands. Hyperspectral images of the rice seeds had a spectral resolution of 5 nm and 256 spectral channels.
For correcting the effects of light source and obtaining data reflection percentage (Rho), raw hyperspectral images (Iraw) were calibrated with two reference standards using the following equation:
$$Rho=\frac{{I}_{raw}-{I}_{dark}}{{I}_{white}-{I}_{dark}}$$
(1)
where a dark reference image (Idark) was acquired by turning off the light source together with covering the camera lens completely with its opaque cap to remove the influence of dark current in the camera. The white reference image (Iwhite) was acquired using a standard white Teflon tile with nearly 100% reflectance. All of the corrected hyperspectral images were then used for spectral information extraction, principal component images acquisition, discrimination purposes, and image visualization. The key steps for discrimination of the CRISPR/Cas9-induced mutants and extracting the morphological characterization of rice are presented in Fig. 5.
### Morphological characterization and thousand-grain weight analysis
The TGW6 gene is believed to change rice seed shape and organ size32, therefore, the morphological characterization from the sample was extracted from the hyperspectral image using an image processing technique. A total of four features (area, perimeter, length, width) from 130 seeds of each rice varieties were extracted. The hyperspectral image at a wavelength of 1139.26 nm was transformed to a greyscale image for later morphological characterization. The four morphological characterizations are as follows:
Perimeter: the number of contour pixels for each seed;
Area: the total number of pixels inside the seed contour pixels;
Length: the number of pixels that specifies the major axis length of the seed;
Width: the number of pixels that specifies the minor axis length of the seed.
Five independent samples of 100 seeds were measured, and the means were converted to thousand-grain weight (TGW).
### Spectral data extraction and pretreatment
Before spectral data extraction, the whole rice seed was segmented from the background and defined as the region of interest (ROI). The reflectance of the rice samples in the ROI was averaged and was taken as the mean spectrum of the relative sample. Spectra were obtained from all hyperspectral images of seeds and were saved in a spectral matrix (X). To improve the signal-to-noise ratio and the distinguishing capacity of the models, raw spectra were subjected to noise suppression by wavelet transformation33 using Daubechies 8 with decomposition scale 3.
A total of 660 intact samples of each variety were selected for image acquisition. The dataset of each variety was divided into a calibration set and a predication set at a ratio of 2:1 using the Kennard-Stone algorithm34. Thus, there were 440 seeds used as the calibration set, and the remaining 220 seeds were used as the prediction set for each variety. All rice seeds were given a category assignment; the wild-type seeds were assigned 1, and the mutants were assigned 2.
### Chemometrics and data analysis
Because there is a large amount of hyperspectral image data containing hidden information, a reliable method is needed to process and extract the features of the spectra. The first step involving analysis was performed using an exploratory principal components analysis (PCA)35. The PCA was applied to reduce the spectral dimensions of the hyperspectral image data by projecting the data into a low dimensional subspace, and each of the spectrums was projected in an alternative set of coordinates called the principal components (PCs). The scores of the most significant PCs corresponding to each NIR spectrum were used. The number of PCs was less than or equal to the number of original variables. From the PCA score plot, it is possible to observe the clusters and trends from the different samples.
Variable (wavelength) selection was performed using wavelength selection-based techniques to facilitate and speed up the classification in the next stage of this study. The spectral bands in hyperspectral images are highly correlated; thus, contiguous spectral bands may contain redundant information. Therefore, it is advantageous to extract feature components to perform better predictions and a simpler process. However, inspection of characteristic spectra allows the analysis of similarities and differences between WT and mutants. The PCA loading lines method and a successive projections algorithm (SPA)36 were applied to select optimal wavelengths. Loading values resulting from the PCA of the raw spectral data represent the regression coefficient and indicate the most dominant wavelength. Therefore, peaks and valleys based on the wavelength-loading map were selected as important wavelengths. SPA is a forward variable selection algorithm that is designed to select optimal wavelengths with minimal redundancy to address collinearity problems. The optimal wavelengths are finally determined according to the minimum root mean square error of validation (RMSEV) in the validation set of an MLR calibration. The SPA was described in detail in the literature37.
In successive stages, support vector machine (SVM) and extreme learning machine (ELM) discrimination models were applied on the raw spectral data, and optimal wavelengths were then selected to screen and distinguish the WT and mutant strains. SVM is a supervised learning model that has been widely used for classification38. Compared with other machine learning methods, this method can develop models with fewer training samples and can overcome the local minimum in a neural network. Detailed information on this popular model can be found in the literature38. An SVM model with the radial basis function (RBF) as a kernel function was used in this case, and different penalty parameters (c) and kernel function parameters (g) were chosen to achieve the highest recognition rate. ELM, a feedforward neural network, can also be applied in binary classification applications39. ELM operates with a single layer of hidden nodes, where the weights connecting inputs to hidden nodes are randomly assigned and never updated. In this paper, optimal numbers of hidden nodes for training were determined from 1 to 80 in steps of 1 to have the best discrimination capability. The performances of the discrimination models were evaluated based on the classification accuracies of the calibration set and the prediction set.
### CRISPR/Cas9-induced mutant visualization
Finally, a very uniform set of seeds from different genotype were presented to the optimal classification model and tested in order to inspect the robustness of the model. To acquire these validation data, the exposure time of camera, the distance between the CCD camera and conveyor, and the speed of the conveyor moment of the NIR hyperspectral imaging system were adjusted to 4 ms, 15.1 cm and 16 mm/s, respectively. The major advantage of hyperspectral imaging is to provide spatial and spectral information simultaneously, which facilitates visualizing the distribution of the features in the tested sample. In the current study, an optimal classification model combined with hyperspectral image processing was used to map and visualize each pixel of the hyperspectral images to demonstrate mutants. This visualization model was run by calculating the dot product between the spectrum of each pixel in the image and the accuracy rate obtained from the optimal model base on particular wavelengths. The classification prediction map is displayed in colours, where each colour represents the corresponding rice varieties.
### Software tools
NIR hyperspectral images were analysed using the ENVI version 4.6 Hyperspectral image analysis soft package (ITT, Visual Information Solutions, Boulder, CO, USA). Shape features extraction, NIR spectral extraction, multivariate data analysis and all stages involved in image visualization purposes were applied using MATLAB version R2010b (The Math-Works, Natick, MA, USA). Statistical comparisons were made using a one-way analysis of variance (ANOVA). The differences between means were established using the Student-Newman-Keuls tests (p < 0.05) in the SPSS version 18.0 statistical programme (SPSS Inc., Chicago, IL, USA). In addition, graphs were prepared using origin Pro 8.5 SR0 (Origin Lab Corporation, Northampton, MA, USA) software.
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## Acknowledgements
This work was supported financially by the 863 National High-Tech Research and Development Plan (Project No: 2013AA102301), National Keypoint Research and Invention Programme of the Thirteenth State Key Laboratory Breeding Base for Zhejiang Sustainable Pest and Disease Control (No. 2010DS700124-KF1712) and the Chinese Postdoctoral Science Foundation (2006M601940).
## Author information
### Author notes
1. Xuping Feng and Cheng Peng contributed equally to this work.
### Affiliations
1. #### Key Laboratory of Spectroscopy, Ministry of Agriculture, College of Biosystems Engineering and Food Science, Zhejiang University, Hangzhou, 310058, China
• Xuping Feng
• , Xiaodan Liu
• & Yong He
2. #### Institute of Quality and Standard for Agro-products, Zhejiang Academy of Agricultural Sciences, Hangzhou, 310021, China
• Cheng Peng
• & Xujun Feng
• Yue Chen
### Contributions
X.F. and X.L. performed the measurements. X.F. and P.C. wrote the manuscript. X.F., and Y.C. designed the experiment and reviewed the manuscript. Y.H. and X.F. reviewed the initial design of the experiments and provided guidance for the writing of the manuscript. All authors reviewed the manuscript.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding authors
Correspondence to Xujun Feng or Yong He.
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http://holani.net/error-estandar/error-estandar-formula-wiki.php
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# holani.net
Home > Error Estandar > Error Estandar Formula Wiki
# Error Estandar Formula Wiki
## Contents
M x = ( σ 1 2 COV 12 COV 13 … COV 12 σ 2 2 COV 23 … COV 13 COV 23 σ 3 2 … ⋮ ⋮ ⋮ Answer this question Flag as... Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, the positive square root of variance, σ2. For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation: σ ^ = 1 n − 1.5 have a peek here
The minimum excess kurtosis is γ 2 = − 2 {\displaystyle \gamma _{2}=-2} ,[a] which is achieved by a Bernoulli distribution with p=1/2 (a coin flip), and the MSE is minimized and Keeping, E.S. (1963) Mathematics of Statistics, van Nostrand, p. 187 ^ Zwillinger D. (1995), Standard Mathematical Tables and Formulae, Chapman&Hall/CRC. Answer this question Flag as... Esto es coherente, ya que las mediciones caen fuera del rango de valores en el cual sería razonable esperar que ocurrieran si el modelo teórico fuera correcto. https://en.wikipedia.org/wiki/Standard_error
## Error Estandar Formula Excel
A natural way to describe the variation of these sample means around the true population mean is the standard deviation of the distribution of the sample means. doi:10.2307/2682923. About this wikiHow 413reviews Click a star to vote Click a star to vote Thanks for voting!
• Add up all the numbers and divide by the population size: Mean (μ) = ΣX/N, where Σ is the summation (addition) sign, xi is each individual number, and N is the
• This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall
• When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a "finite population correction"[9]
• One standard deviation about the central tendency covers approximately 68 percent of the data, 2 standard deviation 95 percent of the data, and 3 standard deviation 99.7 percent of the data.
• For various values of z, the percentage of values expected to lie in and outside the symmetric interval, CI=(−zσ,zσ), are as follows: Percentage within(z) z(Percentage within) Confidence interval Proportion within Proportion
• So it is not unreasonable to assume that the standard deviation is related to the distance of P to L.
• When considering more extreme possible returns or outcomes in future, an investor should expect results of as much as 10 percent plus or minus 60 pp, or a range from 70
• s 2 = ∑ i = 1 n ( x i − x ¯ ) 2 n − 1 {\displaystyle s^{2}={\frac {\displaystyle \sum _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)^{2}}{n-1}}} Ejemplo[editar] Aquí se muestra cómo calcular
• The distribution of these 20,000 sample means indicate how far the mean of a sample may be from the true population mean.
Journal of Sound and Vibrations. 332 (11): 2750–2776. El ECM de un estimador θ ^ {\displaystyle {\hat {\theta }}} con respecto al parámetro desconocido θ {\displaystyle \theta } se define como ECM ( θ ^ ) = E Como el tamaño de la muestra tiende a infinito, el teorema del límite central garantiza que la distribución de la media muestral es asintóticamente la distribución normal. Error Estandar De Estimacion Formula En el 95% de los casos μ estará entre los límites calculados a partir de la media, pero en el 5% de los casos no lo estará.
Because these 16 runners are a sample from the population of 9,732 runners, 37.25 is the sample mean, and 10.23 is the sample standard deviation, s. Formula Del Error Estandar L.; Casella, George (1998). Risk is an important factor in determining how to efficiently manage a portfolio of investments because it determines the variation in returns on the asset and/or portfolio and gives investors a https://en.wikipedia.org/wiki/Standard_deviation In this case, expressions for more complicated functions can be derived by combining simpler functions.
The next graph shows the sampling distribution of the mean (the distribution of the 20,000 sample means) superimposed on the distribution of ages for the 9,732 women. Standard Error Formula Keith (2002), Data Reduction and Error Analysis for the Physical Sciences (3rd ed.), McGraw-Hill, ISBN0-07-119926-8 Meyer, Stuart L. (1975), Data Analysis for Scientists and Engineers, Wiley, ISBN0-471-59995-6 Taylor, J. JSTOR2281592. ^ Ochoa1,Benjamin; Belongie, Serge "Covariance Propagation for Guided Matching" ^ Ku, H. For example, in the case of the log-normal distribution with parameters μ and σ2, the standard deviation is [(exp(σ2)−1)exp(2μ+σ2)]1/2.
## Formula Del Error Estandar
Did this article help you? have a peek at these guys Si no se conoce y n es grande (habitualmente se toma n ≥ 30):[5] Aproximaciones para el valor z α / 2 {\displaystyle z_{\alpha /2}} para los niveles de confianza estándar Error Estandar Formula Excel ISBN970686136X. Formula De Error Estandar Particle physics conventionally uses a standard of "5 sigma" for the declaration of a discovery.[6][not in citation given] A five-sigma level translates to one chance in 3.5 million that a random
A square with sides equal to the difference of each value from the mean is formed for each value. http://holani.net/error-estandar/error-estandar-de-estimacion-wiki.php This level of certainty was required in order to assert that a particle consistent with the Higgs boson had been discovered in two independent experiments at CERN,[7] and this was also Retrieved 22 April 2016. ^ a b Goodman, Leo (1960). "On the Exact Variance of Products". By using this site, you agree to the Terms of Use and Privacy Policy. Formula Para Error Estandar
The calculation of the sum of squared deviations can be related to moments calculated directly from the data. How do I find the mean of one group using just the standard deviation and a total number of two groups? La figura ilustra 50 realizaciones de un intervalo de confianza para una población media dada μ. Check This Out Journal of the Royal Statistical Society.
To show how a larger sample will make the confidence interval narrower, consider the following examples: A small population of N = 2 has only 1 degree of freedom for estimating Standard Error Formula Excel Los datos representan la edad de los miembros de un grupo de niños: {4, 1, 11, 13, 2, 7} 1. Retrieved 13 February 2013.
## p.37.
Como el valor deseado 250 de μ está dentro del intervalo de confianza resultante no hay razón para creer que la máquina no está correctamente calibrada. Al usar este sitio, usted acepta nuestros términos de uso y nuestra política de privacidad. Esta medida es más estable que el recorrido y toma en consideración el valor de cada dato. Standard Error Vs Standard Deviation This is known as Bessel's correction.[5] As a slightly more complicated real-life example, the average height for adult men in the United States is about 70inches, with a standard deviation of
In the following formula, the letter E is interpreted to mean expected value, i.e., mean. σ ( X ) = E [ ( X − E ( X ) ) 2 doi:10.2307/2281592. ↑ Jack Hayya, Donald Armstrong y Nicolas Gressis (julio de 1975). «A Note on the Ratio of Two Normally Distributed Variables». Repeating the sampling procedure as for the Cherry Blossom runners, take 20,000 samples of size n=16 from the age at first marriage population. this contact form Retrieved 17 July 2014.
If the uncertainties are correlated then covariance must be taken into account. p.1. Introduction to Statistical Inference. Distribución de probabilidad continua[editar] Es posible calcular la desviación estándar de una variable aleatoria continua como la raíz cuadrada de la integral σ 2 = ∫ ( x − μ )
A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean. f k = ∑ i n A k i x i or f = A x {\displaystyle f_ ρ 5=\sum _ ρ 4^ ρ 3A_ ρ 2x_ ρ 1{\text{ or }}\mathrm doi:10.1016/j.jsv.2012.12.009. ^ "A Summary of Error Propagation" (PDF). H., Principles and Procedures of Statistics with Special Reference to the Biological Sciences., McGraw Hill, 1960, page 288. ^ Mood, A.; Graybill, F.; Boes, D. (1974).
For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7.
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2018-09-19 13:13:33
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https://atsa-es.github.io/atsa-labs/sec-mssmiss-overview.html
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## 11.1 Covariates with missing values or observation error
The specific formulation of Equation (8.1) creates restrictions on the assumptions regarding the covariate data. You have to assume that your covariate data has no error, which is probably not true. You cannot have missing values in your covariate data, again unlikely. You cannot combine instrument time series; for example, if you have two temperature recorders with different error rates and biases. Also, what if you have one noisy temperature sensor in the first part of your time series and then you switch to a much better sensor in the second half of your time series? All these problems require pre-analysis massaging of the covariate data, leaving out noisy and gappy covariate data, and making what can feel like arbitrary choices about which covariate time series to include.
To circumvent these potential problems and allow more flexibility in how we incorporate covariate data, one can instead treat the covariates as components of an auto-regressive process by including them in both the process and observation models. Beginning with the process equation, we can write $$$\begin{gathered} \begin{bmatrix}\mathbf{x}^{(v)} \\ \mathbf{x}^{(c)}\end{bmatrix}_t = \begin{bmatrix}\mathbf{B}^{(v)} & \mathbf{C} \\ 0 & \mathbf{B}^{(c)}\end{bmatrix} \begin{bmatrix}\mathbf{x}^{(v)} \\ \mathbf{x}^{(c)}\end{bmatrix}_{t-1} + \begin{bmatrix}\mathbf{u}^{(v)} \\ \mathbf{u}^{(c)} \end{bmatrix} + \mathbf{w}_t,\\ \mathbf{w}_t \sim \,\text{MVN}\begin{pmatrix}0,\begin{bmatrix}\mathbf{Q}^{(v)} & 0 \\ 0 & \mathbf{Q}^{(c)} \end{bmatrix} \end{pmatrix} \end{gathered} \tag{11.1}$$$ The elements with superscript $${(v)}$$ are for the $$k$$ variate states and those with superscript $${(c)}$$ are for the $$q$$ covariate states. The dimension of $$\mathbf{x}^{(c)}$$ is $$q \times 1$$ and $$q$$ is not necessarily equal to $$p$$, the number of covariate observation time series in your dataset. Imagine, for example, that you have two temperature sensors and you are combining these data. Then you have two covariate observation time series ($$p=2$$) but only one underlying covariate state time series ($$q=1$$). The matrix $$\mathbf{C}$$ is dimension $$k \times q$$, and $$\mathbf{B}^{(c)}$$ and $$\mathbf{Q}^{(c)}$$ are dimension $$q \times q$$. The dimension of $$\mathbf{x}^{(v)}$$ is $$k \times 1$$, and $$\mathbf{B}^{(v)}$$ and $$\mathbf{Q}^{(v)}$$ are dimension $$k \times k$$. The dimension of $$\mathbf{x}$$ is always denoted $$m$$. If your process model includes only variates, then $$k=m$$, but now your process model includes $$k$$ variates and $$q$$ covariate states so $$m=k+q$$.
Next, we can write the observation equation in an analogous manner, such that $$$\begin{gathered} \begin{bmatrix} \mathbf{y}^{(v)} \\ \mathbf{y}^{(c)} \end{bmatrix}_t = \begin{bmatrix}\mathbf{Z}^{(v)} & \mathbf{D} \\ 0 & \mathbf{Z}^{(c)} \end{bmatrix} \begin{bmatrix}\mathbf{x}^{(v)} \\ \mathbf{x}^{(c)} \end{bmatrix}_t + \begin{bmatrix} \mathbf{a}^{(v)} \\ \mathbf{a}^{(c)} \end{bmatrix} + \mathbf{v}_t,\\ \mathbf{v}_t \sim \,\text{MVN}\begin{pmatrix}0,\begin{bmatrix}\mathbf{R}^{(v)} & 0 \\ 0 & \mathbf{R}^{(c)} \end{bmatrix} \end{pmatrix} \end{gathered} \tag{11.2}$$$ The dimension of $$\mathbf{y}^{(c)}$$ is $$p \times 1$$, where $$p$$ is the number of covariate observation time series in your dataset. The dimension of $$\mathbf{y}^{(v)}$$ is $$l \times 1$$, where $$l$$ is the number of variate observation time series in your dataset. The total dimension of $$\mathbf{y}$$ is $$l+p$$. The matrix $$\mathbf{D}$$ is dimension $$l \times q$$, $$\mathbf{Z}^{(c)}$$ is dimension $$p \times q$$, and $$\mathbf{R}^{(c)}$$ are dimension $$p \times p$$. The dimension of $$\mathbf{Z}^{(v)}$$ is dimension $$l \times k$$, and $$\mathbf{R}^{(v)}$$ are dimension $$l \times l$$.
The $$\mathbf{D}$$ matrix would presumably have a number of all zero rows in it, as would the $$\mathbf{C}$$ matrix. The covariates that affect the states would often be different than the covariates that affect the observations. For example, mean annual temperature might affect population growth rates for many species while having little or no affect on observability, and turbidity might strongly affect observability in many types of aquatic surveys but have little affect on population growth rate.
Our MARSS model with covariates now looks on the surface like a regular MARSS model: $$$\begin{gathered} \mathbf{x}_t = \mathbf{B}\mathbf{x}_{t-1} + \mathbf{u} + \mathbf{w}_t, \text{ where } \mathbf{w}_t \sim \,\text{MVN}(0,\mathbf{Q}) \\ \mathbf{y}_t = \mathbf{Z}\mathbf{x}_t + \mathbf{a} + \mathbf{v}_t, \text{ where } \mathbf{v}_t \sim \,\text{MVN}(0,\mathbf{R}) \end{gathered}$$$ with the $$\mathbf{x}_t$$, $$\mathbf{y}_t$$ and parameter matrices redefined as in Equations (11.1) and (11.2): $$$\begin{gathered} \mathbf{x}=\begin{bmatrix}\mathbf{x}^{(v)}\\ \mathbf{x}^{(c)}\end{bmatrix} \quad \mathbf{B}=\begin{bmatrix}\mathbf{B}^{(v)} & \mathbf{C} \\ 0 & \mathbf{B}^{(c)}\end{bmatrix} \quad \mathbf{u}=\begin{bmatrix}\mathbf{u}^{(v)}\\ \mathbf{u}^{(c)}\end{bmatrix} \quad \mathbf{Q}=\begin{bmatrix}\mathbf{Q}^{(v)} & 0 \\ 0 & \mathbf{Q}^{(c)}\end{bmatrix} \\ \mathbf{y}=\begin{bmatrix}\mathbf{y}^{(v)}\\ \mathbf{y}^{(c)}\end{bmatrix} \quad \mathbf{Z}=\begin{bmatrix}\mathbf{Z}^{(v)} & \mathbf{D} \\ 0 & \mathbf{Z}^{(c)}\end{bmatrix} \quad \mathbf{a}=\begin{bmatrix}\mathbf{a}^{(v)}\\ \mathbf{a}^{(c)}\end{bmatrix} \quad \mathbf{R}=\begin{bmatrix}\mathbf{R}^{(v)} & 0 \\ 0 & \mathbf{R}^{(c)}\end{bmatrix} \end{gathered} \tag{11.3}$$$ Note $$\mathbf{Q}$$ and $$\mathbf{R}$$ are written as block diagonal matrices, but you could allow covariances if that made sense. $$\mathbf{u}$$ and $$\mathbf{a}$$ are column vectors here. We can fit the model (Equation (11.3)) as usual using the MARSS() function.
The log-likelihood that is returned by MARSS will include the log-likelihood of the covariates under the covariate state model. If you want only the the log-likelihood of the non-covariate data, you will need to subtract off the log-likelihood of the covariate model: $$$\begin{gathered} \mathbf{x}^{(c)}_t = \mathbf{B}^{(c)}\mathbf{x}_{t-1}^{(c)} + \mathbf{u}^{(c)} + \mathbf{w}_t, \text{ where } \mathbf{w}_t \sim \,\text{MVN}(0,\mathbf{Q}^{(c)}) \\ \mathbf{y}^{(c)}_t = \mathbf{Z}^{(c)}\mathbf{x}_t^{(c)} + \mathbf{a}^{(c)} + \mathbf{v}_t, \text{ where } \mathbf{v}_t \sim \,\text{MVN}(0,\mathbf{R}^{(c)}) \end{gathered} \tag{11.4}$$$ An easy way to get this log-likelihood for the covariate data only is use the augmented model (Equation (11.2) with terms defined as in Equation (11.3) but pass in missing values for the non-covariate data. The following code shows how to do this.
y.aug = rbind(data, covariates)
fit.aug = MARSS(y.aug, model = model.aug)
fit.aug is the MLE object that can be passed to MARSSkf(). You need to make a version of this MLE object with the non-covariate data filled with NAs so that you can compute the log-likelihood without the covariates. This needs to be done in the marss element since that is what is used by MARSSkf(). Below is code to do this.
fit.cov = fit.aug
fit.cov$marss$data[1:dim(data)[1], ] = NA
extra.LL = MARSSkf(fit.cov)\$logLik
Note that when you fit the augmented model, the estimates of $$\mathbf{C}$$ and $$\mathbf{B}^{(c)}$$ are affected by the non-covariate data since the model for both the non-covariate and covariate data are estimated simultaneously and are not independent (since the covariate states affect the non-covariates states). If you want the covariate model to be unaffected by the non-covariate data, you can fit the covariate model separately and use the estimates for $$\mathbf{B}^{(c)}$$ and $$\mathbf{Q}^{(c)}$$ as fixed values in your augmented model.
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2022-11-27 17:55:53
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https://hrj.episciences.org/volume/view/id/35
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# Volume 24
### 1. On some Ramanujan's P-Q Identities
In this paper, we obtain some P-Q eta-function identities of Ramanujan on employing some modular equations in Ramanujan's alternative theory of elliptic functions of signature 4.
### 2. On the values of the Riemann zeta-function at rational arguments
In a companion paper, On multi Hurwitz-zeta function values at rational arguments, Acta Arith. {\bf 107} (2003), 45-67'', we obtained a closed form evaluation of Ramanujan's type of the values of the (multiple) Hurwitz zeta-function at rational arguments (with denominator even and numerator odd), which was in turn a vast generalization of D. Klusch's and M. Katsurada's generalization of Ramanujan's formula. In this paper we shall continue our pursuit, specializing to the Riemann zeta-function, and obtain a closed form evaluation thereof at all rational arguments, with no restriction to the form of the rationals, in the critical strip. This is a complete generalization of the results of the aforementioned two authors. We shall obtain as a byproduct some curious identities among the Riemann zeta-values.
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2019-05-26 02:01:15
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http://en.wikipedia.org/wiki/Vectorization_(mathematics)
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# Vectorization (mathematics)
In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization of an m×n matrix A, denoted by vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of one another:
$\mathrm{vec}(A) = [a_{1,1}, \ldots, a_{m,1}, a_{1,2}, \ldots, a_{m,2}, \ldots, a_{1,n}, \ldots, a_{m,n}]^T$
Here $a_{i,j}$ represents the $(i,j)$-th element of matrix $A$ and the superscript $^T$ denotes the transpose. Vectorization expresses the isomorphism $\mathbf{R}^{m \times n} := \mathbf{R}^m \otimes \mathbf{R}^n \cong \mathbf{R}^{mn}$ between these vector spaces (of matrices and vectors) in coordinates.
For example, for the 2×2 matrix $A$ = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the vectorization is $\mathrm{vec}(A) = \begin{bmatrix} a \\ c \\ b \\ d \end{bmatrix}$.
## Compatibility with Kronecker products
The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular,
$\mbox{vec}(ABC)=(C^{T}\otimes A)\mbox{vec}(B)$
for matrices A, B, and C of dimensions k×l, l×m, and m×n. For example, if $\mbox{ad}_A(X) = AX-XA$ (the adjoint endomorphism of the Lie algebra gl(n,C) of all n×n matrices with complex entries), then $\mbox{vec}(\mbox{ad}_A(X)) = (I_n\otimes A - A^T \otimes I_n ) \mbox{vec}(X)$, where $I_n$ is the n×n identity matrix.
There are two other useful formulations:
$\mbox{vec}(ABC)=(I_n\otimes AB)\mbox{vec}(C) =(C^{T}B^{T}\otimes I_k)\mbox{vec}(A)$
$\mbox{vec}(AB)=(I_m\otimes A)\mbox{vec}(B) =(B^{T}\otimes I_k)\mbox{vec}(A)$
Vectorization is an algebra homomorphism from the space of n×n matrices with the Hadamard (entrywise) product to Cn with its Hadamard product[disambiguation needed]:
vec(A $\circ$ B) = vec(A) $\circ$ vec(B).
## Compatibility with inner products
Vectorization is a unitary transformation from the space of n×n matrices with the Frobenius (or Hilbert–Schmidt) inner product to Cn :
tr(A* B) = vec(A)* vec(B)
where the superscript * denotes the conjugate transpose.
## Half-vectorization
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the vectorization. The half-vectorization, vech(A), of a symmetric n × n matrix A is the n(n + 1)/2 × 1 column vector obtained by vectorizing only the lower triangular part of A:
vech(A) = [ A1,1, ..., An,1, A2,2, ..., An,2, ..., An−1,n−1,An−1,n, An,n ]T.
For example, for the 2×2 matrix A = $\begin{bmatrix} a & b \\ b & d \end{bmatrix}$, the half-vectorization is vech(A) = $\begin{bmatrix} a \\ b \\ d \end{bmatrix}$.
There exist unique matrices transforming the half-vectorization of a matrix to its vectorization and vice versa called, respectively, the duplication matrix and the elimination matrix.
## Programming language
Programming languages that implement matrices may have easy means for vectorization. In Matlab/GNU Octave a matrix A can be vectorized by A(:). In Python NumPy arrays implement the 'flatten' method, while in R the desired effect can be achieved via the 'c()' or 'as.vector()' functions.
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2015-01-29 01:20:09
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https://ltwork.net/whatencion-is-the-difference-between-your-hometowns-location--12080318
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# Whatención is the difference between your hometowns location and your hometown location as place
###### Question:
Whatención is the difference between your hometowns location and your hometown location as place
### Which expression is equivalent 18 1/5 - (-22 2/5) - ( - 40 1/5)
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### 1. what was the Rowlatt act?
1. what was the Rowlatt act?...
### A car is filled up with 20 gallons of gas. The car uses .25 gallons per minute. How much time will the car travel ?
A car is filled up with 20 gallons of gas. The car uses .25 gallons per minute. How much time will the car travel ?...
### Use one of the triangles to approximate EF in the triangle below.
Use one of the triangles to approximate EF in the triangle below. $Use one of the triangles to approximate EF in the triangle below.$...
### What doctrine states that if a property owner is lax in protecting his or her rights, the property owner
What doctrine states that if a property owner is lax in protecting his or her rights, the property owner may lose those rights?...
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Deoxygenated blood obtains oxygen from the...
### The diversity seen in darwin’s finches is a good example of a. adaptive radiation c. convergent evolution
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### Determine the amounts American Eagle reports for net cash flows from operating activities, investing
Determine the amounts American Eagle reports for net cash flows from operating activities, investing activities, and financing activities in its statement of cash flows for the most recent year. What are total cash flows for the year f...
### What are your characteristics as a student who does projects given by your teacher?
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### EARLY RACIAL TENSIONS: What are the similarities between the treatment of someimmigrants, women, Native Americans, and African Americans? *(For my
EARLY RACIAL TENSIONS: What are the similarities between the treatment of some immigrants, women, Native Americans, and African Americans? * (For my finals )...
### Is the quality of a person's voice that reveals his or her attitude toward the subject of the speech.
Is the quality of a person's voice that reveals his or her attitude toward the subject of the speech. speed tone volume pitch...
### CAN SOMEONE HELP IM IN A TEST Simplify the expression (10 + 3)(10 - 31).
CAN SOMEONE HELP IM IN A TEST Simplify the expression (10 + 3)(10 - 31)....
### Biography of Thomas Chippendale and an explanation of his style and three pieces of his work
Biography of Thomas Chippendale and an explanation of his style and three pieces of his work...
### 3 possitive aspects and 3 negative ones about the book: the red house mystery
3 possitive aspects and 3 negative ones about the book: the red house mystery...
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2023-02-04 18:22:32
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http://www.maplesoft.com/support/help/Maple/view.aspx?path=Task/AngleBetweenTwoVectors
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Angle between Two Vectors - Maple Programming Help
# Online Help
###### All Products Maple MapleSim
Home : Support : Online Help : Tasks : Linear Algebra : Vector Manipulations : Task/AngleBetweenTwoVectors
Angle between Two Vectors
Description This template calculates the angle between two vectors.
Angle between Two Vectors
Enter first vector:
> $⟨1,2⟩$
$\left[\begin{array}{r}{1}\\ {2}\end{array}\right]$ (1.1)
Enter second vector:
> $⟨-1,3⟩$
$\left[\begin{array}{r}{-}{1}\\ {3}\end{array}\right]$ (1.2)
Angle between vectors:
> $\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\left[\mathrm{VectorAngle}\right]\left(,\right)$
${\mathrm{arccos}}{}\left(\frac{{1}}{{10}}{}\sqrt{{5}}{}\sqrt{{10}}\right)$ (1.3)
Commands Used
See Also
## Was this information helpful?
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2017-03-25 01:59:15
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http://www.maths.usyd.edu.au/ut/pub-seek.py?au1=Lai+KF&pgsz=1000
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# Publication Search Results
Matches for:
• Author=Lai KF
1. Ke WF, Lai KF, Zhang RB. Quantum codes from Hadamard matrices, Linear and Multilinear Algebra, 58 (2010), no. 7, 847–854.
2. Kueh KL, Lai KF, Tan KS. Stickelberger elements for $$\Z_p^d$$ extensions of function fields, Journal of Number Theory, 128 (2008), 2776–2783. MR2441076
3. Lai KF, Vostokov SV. The Kneser relation and the Hilbert pairing in multidimensional local field, Mathematische Nachrichten, 280 (2007), no. 16, 1–18.
4. Lai KF, Chen ZJ, Zhao CL. Algebraic Groups, Foundation of Modern Mathematics, Science Press, Beijing, China, (2006), 453. ISBN 7-03-017861-0
5. Lai KF, Zhao CL. Overconvergent $$p$$-adic Siegel modular forms, Algebra and Number Theory, Algebra and Number Theory Silver Jubilee Conference, R. Tandon (ed.), Hindustan Book Agency, New Delhi, (2005), 250–257. ISBN 81-85931-57-7 MR2193357
6. Kisin M, Lai KF. Overconvergent Hilbert modular forms, American Journal of Mathematics, 127 (2005), no. 4, 735–783. MR2154369
7. Chi WC, Lai KF, Tan KS. Integer points on elliptic curves, Pacific Journal of Mathematics, 222 (2005), no. 2, 237–252. MR2225071
8. Lai KF. $$C_2$$ building and projective space, Journal of the Australian Mathematical Society, 76 (2004), 383–402. MR2053511
9. Chi WC, Lai KF, Tan KS. Integral points on elliptic curves over function fields, Journal of the Australian Mathematical Society, 77 (2004), 197–208. MR2083745
10. Lai KF, Zhang RB. Multiplicity free actions of quantum groups and generalized Howe duality, Letters in Mathematical Physics, 64 (2003), 255–272. MR2009264
11. Lai KF, Yeung KM. Rational Points in Flag Varieties, Journal of Number Theory, 95 (2002), 142–149. 2003i:11089
12. Lai KF, Mok KP. On the differential geometry of the (1, 1) tensor bundle, Tensor. (new series), 63 (2002), 15–27. MR2009423
13. Lai KF, Zhao CL. Modular Curves, Peking University Graduate Text, Peking University Press, Beijing China, (2002), 243. ISBN 7-301-05483-1
14. Lai KF, Voskuil H. $$p$$-adic automorphic functions for the unitary group in three variables, Algebra Colloquium, 7:3 (2000), 335–360. 2002c:11059
15. Gérardin P, Lai KF. Asymptotic behavior of eigenfunctions for the Hecke algebra on homogeneous trees, Special Functions: Proceedings of the international workshop, International Workshop on Special Functions- Asymptotics, Harmonic Analysis and Mathematical Physics, Charles Dunkl, Mourad Ismail, Roderick Wong (eds.), Special functions, World Scientific Publishing Co. Pte. Ltd., Singapore, (2000), 114–117. ISBN 981-02-4393-6 2001m:31020
16. Gérardin P, Lai KF. Asymptotic behaviour of eigenfunctions on semi-homogeneous tree, Pacific Journal of Mathematics, 196 (2000), no. 2, 415–427. 2001m:31020
17. Lai KF, Vostokov SV. Explicit pairing and class field theory of multidimensional complete fields, St. Petersburg Mathematical Journal, 11 (2000), no. 4, 611–624. MR1713933
18. Gérardin P, Lai KF. Opérateurs invariants sur les immeubles affines de type $$A$$, Comptes Rendus de l'Académie des Sciences, 329 (1999), 1–4. 2000i:20048
19. Lai KF, Chan WK, Castillo R. Foliations in supergravity, Journal of the Australian Mathematical Society. (Series B), 41 (1999), 161–166. 2000i:83073
20. Lai KF, Chan WK, Simpson K. Analysis of software regression test, Proceedings of the 3rd Annual IASTED International Conference, Software Engineering and Applications, N.C. Debnath (ed.), IASTED/ACTA PRESS, Zurich, (1999), 295–301. ISBN 0-88986-273-7
21. Lai KF, Bondarko M, Vostokov SV. Galois structure for abelian $$p$$-extensions of Dedekind domains, Journal für die Reine und Angewandte Mathematik, 517 (1999), 51–59. 2000j:11180
22. Eie M, Lai KF. On Bernoulli identities and applications, Revista Matem\'tica Iberoamericana, 14 (1998), no. 1, 167–213. 99h:11017
23. Lai KF, Lee R. Finite group actions on Siegel modular spaces, Transactions of the American Mathematical Society, 345 (1994), 37–45.
24. Jacquet H, Lai KF, Rallis S. A trace formula for symmetric spaces, Duke Mathematical Journal, 70 (1993), no. 2, 305–372. 94d:11033
25. Chan WK, Lai KF, Castillo R. Riemannian foliation in $$N=1$$, $$D=11$$ supergravity, Il Nuovo Cimento, 108 B (1993), no. 7, 739–752. 95g:83101
26. Lai KF. Regular elliptic classes, Bulletin of the Canadian Mathematical Society, 35 (1992), no. 2, 230–236. 93i:11061
27. Lai KF. On Arthur's invariant trace formula, Proceedings of the Special Programme, Algebraic Geometry and Algebraic Number Theory, Nankai Institute of Mathematics, (1992), 35–63. 96g:11053
28. Lai KF, Fen S. Topological Groups, Science Press, Academica Sinica, Beijing, (1991),
29. Lai KF. Orbital integrals on symmetric spaces, Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 312 (1991), 913–917. 93b:22015
30. Lai KF. Lefschetz numbers and unitary groups, Bulletin of the Australian Mathematical Society, 43 (1991), 193–209. 92a:11063
31. Lai KF. On Arthur's class expansion of the relative trace formula, Duke Mathematical Journal, 64 (1991), 111–117. 92k:22031
32. Lai KF, Lan YZ. Representation Theory of $$\mathrm{GL}(2)$$ and theory of automorphic forms, Peking University Press, (1990),
33. Lai KF, Mok N. On a vanishing theorem on irreducible quotients of finite volume of polydiscs, , Lecture Notes in Mathematics, 1198 Springer Verlag, (1986), 163–171. 88h:32031
34. Lai KF. Functional equation of Dirichlet series, Advances in Math (Peking), 14 (1985), 263–266.
35. Jaquet H, Lai KF. On a relative trace formula, Compositio Mathematica, 54 (1985), 243–310. 86j:11059
36. Lai KF. Algebraic cycles on compact Shimura surface, Mathematische Zeitschrift, 189 (1985), 593–602. 87a:11057
37. Chan WL, Lai KF. Dual optimal distributed systems with non-negative controls, Journal of Mathematical Analysis and Applications, 104 (1984), 143–154. 86m:49035
38. Jacquet H, Lai KF. Sur une formule des traces relatives, Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 296 (1983), 959–963. 86k:11029
39. Lai KF. On the cohomology of congruence subgroups of symplectic groups, Nagoya Mathematics Journal, 85 (1982), 155–174. 83g:10024
40. Lai KF. Hilbert's Twelfth Problem — the Reciprocity Law and Langlands' Conjecture, Acta Scientiarum Naturalium Universitatis Sunyatseni, 65 (1981), 104–114.
41. Lai KF. Orders of finite algebraic groups, Pacific Journal of Mathematics, 97 (1981), 83g:20048
42. Lai KF. Conjugation of Canonical Models, Proceedings of the SEAMS Conference, (1980), 24–25.
43. Lai KF. On a conjecture on elliptic curve, Applied Mathematics and Computation, 1 (1980), 13–27.
44. Lai KF. Linear algebraic groups, Acta Scientiarum Naturalium Universitatis Sunyatseni, 60 (1980), 94–110.
45. Lai KF. Compactification of homogeneous space, Acta Hangzhou Normal College, (1980), 32–35.
46. Lai KF. Tamagawa number of reductive algebraic groups, Compositio Mathematica, 41 (1980), 153–188. 82d:20043
47. Lai KF. Ringed space, Univ. Amoiensis Acta Sien. Naturalium, 18 (1979), no. 3, 27–38.
48. Lai KF. Introduction to algebraic varieties, Univ. Amoiensis Acta Sien. Naturalium, 18 (1979), no. 4, 27–28.
49. Lai KF. On the Tamagawa number of quasi-split groups, Bulletin of the American Mathematical Society, 82 (1976), 300–302. 53:5483
Number of matches: 49 Page 1 of 1 Select page: 1
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2013-05-23 12:50:17
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https://physicshelpforum.com/threads/calculating-wavelength-of-a-2d-grating-different-distances-between-slits.15607/
|
# Calculating wavelength of a 2D grating (different distances between slits)
#### Femke
For my report I need a formula to calculate the wavelength with a multiple slit experiment. I have found: d sin(\theta) = m \lambda.
However, for the experiment a two dimensional grating is used. The distance between the vertical slits are not the same as the distance between the horizontal slits. What formula can I use to calculate the wavelength?
I added an example to hopefully make it more clear.
#### Woody
The asymmetric grating should not matter,
it will just give you an asymmetric interference pattern.
Note that rather than parallel lines of constructive and destructive interference
you will get a grid of constructive and destructive interference.
the (different) horizontal and vertical spacing of the interference will mirror the (different) horizontal and vertical spacing of the grid
and (of course) the wavelength (frequency) of the light.
Note that you actually get two equations from a single grid.
so you get:
$$\displaystyle d_{H} sin(\theta_{Hm}) = m \lambda$$
and:
$$\displaystyle d_{V} sin(\theta_{Vm}) = m \lambda$$
Where the V and H subscripts indicate the vertical and horizontal grids respectively.
donglebox
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2020-02-24 11:45:31
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https://www.gradesaver.com/textbooks/math/applied-mathematics/elementary-technical-mathematics/chapter-7-section-7-1-ratio-exercises-page-271/52
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## Elementary Technical Mathematics
Always express the numeric ratio in the same order as given in the text version. 6011 to SS 100 = $32\ to\ 60=\frac{32}{60}=\frac{32\div4}{60\div4}=\frac{8}{15}$
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2021-04-20 12:47:50
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https://mathspace.co/textbooks/syllabuses/Syllabus-830/topics/Topic-18391/subtopics/Subtopic-250185/?textbookIntroActiveTab=guide
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# 1.05 Review: Literal equations
Lesson
Solving for a quantity of interest is an important skill to learn. It can come in very handy when you know the value of one algebraic symbol but not another.
For example, in the formula $A=pb+y$A=pb+y, the value $A$A is by itself on the left-hand side of the equals sign. In common language, we might say that it has been "solved for" because it is by itself, even though we do not yet know its value.
When we previously tried to solve equations, we took steps to get the variable by itself. When solving for a quantity of interest, we might have more than one variable, but we still use a similar process:
• Group any like terms
• Simplify using the inverse of addition or subtraction.
• Simplify further by using the inverse of multiplication or division.
#### Practice questions
##### Question 1
Solve for $x$x in the following equation:
$y=\frac{x}{4}$y=x4
##### Question 2
Solve for $R$R in the following equation:
$V=IR-E$V=IRE
##### Question 3
Solve for $x$x in the following equation:
$\frac{x}{9}+\frac{n}{2}=5$x9+n2=5
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2021-12-03 10:56:21
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https://homework.cpm.org/category/CCI_CT/textbook/int3/chapter/11/lesson/11.1.1/problem/11-8
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### Home > INT3 > Chapter 11 > Lesson 11.1.1 > Problem11-8
11-8.
In Lesson 11.1.2 you will focus on multiplying and dividing rational expressions. Recall what you learned about multiplying and dividing fractions in a previous course as you answer the questions below. To help you, the following examples have been provided.
$\frac { 9 } { 16 } \cdot \frac { 4 } { 6 } = \frac { 36 } { 96 } = \frac { 3 } { 8 } \quad \text { and } \quad \frac { 5 } { 6 } \div \frac { 20 } { 12 } = \frac { 5 } { 6 } \cdot \frac { 12 } { 20 } = \frac { 60 } { 120 } = \frac { 1 } { 2 }$
1. Without a calculator, multiply $\frac { 2 } { 3 } \cdot \frac { 9 } { 14 }$ and reduce the result. Describe your method for multiplying fractions.
Multiply the numerators together to find the numerator of the final answer, and multiply the denominators together to find the denominator of the final answer.
Do $2$ and $14$ have a common factor? Do $3$ and $9$ have a common factor? It is possible to reduce before multiplying across? Use whichever method is easiest for you.
$\frac{3}{7}$
2. Without a calculator, divide $\frac { 3 } { 5 } \div \frac { 12 } { 25 }$ and reduce the result. Then use a calculator to check your answer. Describe your method for dividing fractions.
Take the reciprocal of the divisor, and multiply the two resulting fractions.
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2020-08-08 11:32:29
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https://www.projecteuclid.org/euclid.ojm/1315318354
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## Osaka Journal of Mathematics
### Complement to explicit description of Hopf surfaces and their automorphism groups
#### Abstract
In the previous paper we determined $\widetilde{\Aut}(X)$ of each Hopf surface $X = W/G$ with $W=\mathbf{C}^{2} - (0,0)$ so that its holomorphic automorphism group is given by $\Aut(X) = \widetilde{\Aut}(X)/G$. We calculate the group of connected components $\pi_{0}(\Aut(X))$ by reviewing the classification.
#### Article information
Source
Osaka J. Math., Volume 48, Number 2 (2011), 583-588.
Dates
First available in Project Euclid: 6 September 2011
https://projecteuclid.org/euclid.ojm/1315318354
Mathematical Reviews number (MathSciNet)
MR1772841
Zentralblatt MATH identifier
1234.32004
Subjects
Primary: 32J15: Compact surfaces
#### Citation
Matumoto, Takao; Nakagawa, Noriaki. Complement to explicit description of Hopf surfaces and their automorphism groups. Osaka J. Math. 48 (2011), no. 2, 583--588. https://projecteuclid.org/euclid.ojm/1315318354
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2019-10-22 00:03:49
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https://brilliant.org/discussions/thread/good-proof-problems/
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×
# Good proof problems
1) if S is the circumcentre of a $$\triangle ABC$$ Then prove that $$\angle BSD = \angle BAC$$ given that D is the mid point of BC.
2) Let P be any point inside a regular polygon. If $$d_i$$ is the distance of P from the $$i^{th}$$ side, prove that $$d_1+d_2+d_3+....+d_n$$ is constant.
3) I is the incentre of Triangle ABC.X and Y are the feet of the perpendiculars from A to BI and CI. Prove XY || BC
I liked these problems, so i thought that I should share them with the community! :D
Edit: Xuming Liang is forbidden to comment.
Note by Mehul Arora
2 years, 1 month ago
Sort by:
Question 3: Denote the point where $$BI$$ hits $$AC$$ as $$E$$. Denote the angle measures as $$A, B, C$$ according to the vertices of the triangle. All angles in this solution are in degrees.
By Exterior Angle Theorem, we have that $$\angle BEA = C + \frac{B}{2}$$. Since $$\angle AYE = 90$$, we have that $$\angle YAE = 90 - C - \frac{B}{2}$$.
Connecting $$A$$ to $$I$$, we know that $$AI$$ is the angle bisector of $$\angle A$$, which implies that $$\angle IAE = 90 - \frac{B}{2} - \frac{C}{2}$$. This implies that $$\angle IAY = \frac{C}{2}$$.
Now notice since $$\angle AXI = \angle AYI = 90$$, we have that quadrilateral $$AXIY$$ is cyclic. This means that if we connect $$X$$ and $$Y$$, we have that $$\angle IAY = \angle IXY = \frac{C}{2}$$. However we also know that $$\angle XCB = \frac{C}{2}$$. Therefore by alternate interior angles, we know that $$XY || BC$$.
- 2 years, 1 month ago
@Alan Yan How u so genius?
- 2 years, 1 month ago
Question 1. $$\angle BSD = \frac{1}{2}\angle BSC$$ (Perpendicular Bisector.)
$$\angle BAC = \frac{1}{2}\angle BSC$$ (Central angle and inscribed angle substending the same arc.)
Therefore $$\angle BSD = \angle BAC$$.
- 2 years, 1 month ago
Question 2. Partition the polygon into triangles. Let $$m$$ be the side length and let $$A$$ be the total area. The area of the triangles will be $$\frac{1}{2}md_1 , \frac{1}{2}md_2 , \frac{1}{2}md_3 , ...$$. Adding these areas yield the total area.
This implies that $\frac{1}{2}m(d_1+d_2+...+d_n) = A \implies \sum{d_i} = \frac{2A}{m}$ which means the sum of the distances is constant.
- 2 years, 1 month ago
in triangle BDC and CDS,
BS = SC
BD = DC
and angle d is 90
so both triangles are congruent and then angle BSD = angle CSD
also 2.angleBSD = 2.angleBAC
hence proved
- 2 years, 1 month ago
Hints: 1. Property of circumcenter 2. Area
- 2 years, 1 month ago
You are not allowed to comment on such posts :) :P
- 2 years, 1 month ago
Exactly :P
- 2 years, 1 month ago
@Xuming Liang Geom god! _/_
The first one is quite simple. The second one is interesting :)
But you man! These probs are a left hand's play for you, provided you are right handed. :3
- 2 years, 1 month ago
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2017-10-18 04:10:10
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https://quantumcomputing.stackexchange.com/questions/8945/density-matrix-of-a-product-of-bell-states
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# Density matrix of a product of Bell states
Suppose I share two Bell states among two participants Alice and Bob in the following manner : $$|\psi\rangle=\left(\dfrac{|0\rangle_1|0\rangle_2+ |1\rangle_1|1\rangle_2}{\sqrt{2}}\right)\left(\dfrac{|0\rangle_3|0\rangle_4+ |1\rangle_3|1\rangle_4}{\sqrt{2}}\right)$$ Now suppose Alice has qubits $$(1,4)$$ and Bob has $$(2,3)$$. I want to find out the density matrices corresponding to Alice, Bob, and combined.
For the first case should I calculate $$|\psi\rangle\langle\psi|$$, what should be done, in case there was only one Bell pair shared between Alice and Bob, I would have done $$\rho_A = \mathrm{Trace}_B(\rho)$$ can this be generalized when there are more than one Bell pair shared in the sense that I have shared? Can somebody help?
Yes, the overall density matrix shared between Alice and Bob is $$|\psi\rangle\langle\psi|$$. To get the desnity matrix of either Alice or Bob, you should calculate $$\text{Tr}_B|\psi\rangle\langle\psi|\qquad\text{Tr}_A|\psi\rangle\langle\psi|$$ respectively.
However, in this particular case, the calculation is much simply. Let $$|\phi\rangle$$ be the Bell pair such that $$|\psi\rangle=|\phi_{12}\rangle|\phi_{34}\rangle.$$ Because there's a separable partition between (1,2) and (3,4), this is not changed by the partial trace. Thus $$\text{Tr}_B|\psi\rangle\langle\psi|=\left(\text{Tr}_2|\phi\rangle\langle\phi|\right)\otimes \left(\text{Tr}_3|\phi\rangle\langle\phi|\right).$$
You imply that you know how to do the partial trace for a single Bell state. The answer is $$I/2$$. So, we have $$\text{Tr}_B|\psi\rangle\langle\psi|=\frac{1}{4}I\otimes I,$$ the maximally mixed state of two qubits. Similarly, $$\text{Tr}_A|\psi\rangle\langle\psi|=\left(\text{Tr}_1|\phi\rangle\langle\phi|\right)\otimes \left(\text{Tr}_4|\phi\rangle\langle\phi|\right)=\frac{1}{4}I\otimes I$$
• What if we had another bell state $\dfrac{|0\rangle_3|0\rangle_4+|1\rangle_3|1\rangle_4}{\sqrt{2}}$ with the previous ones. That should change the combined density matrix, but the individual matrices would be $\dfrac{I\otimes I\otimes I}{8}$? – Upstart Nov 26 '19 at 10:54
• @Upstart Assuming the subscripts are actually 5 and 6, then yes. – DaftWullie Nov 26 '19 at 11:52
• But, when we have the third state $\dfrac{|0\rangle_5|0\rangle_6+ |1\rangle_5|1\rangle_6}{\sqrt{2}}$ with the $5$th and $6th$ qubit with say Charlie, then if we wanted $\rho_A$, then that would imply partial trace over $2, 3, 5,6$ – Upstart Nov 26 '19 at 12:22
• @Upstart True. This is why we don't do follow-up questions in comments - there's not enough space to be explicit enough about the assumptions. I was assuming you meant that Alice would have one of 5 or 6 and Bob would have the other. – DaftWullie Nov 26 '19 at 12:29
• Should I ask it as a separate question? – Upstart Nov 26 '19 at 12:59
One can perhaps guess the answer without full calculation. Noting that "tracing" intuitively means losing information, then, if you A is maximally entangled with B, then you lose information about B (or A) then you end up with no information about A. That is basically how qubits lose information to the environment (here B is like an environment for A).
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2020-01-25 08:37:36
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http://mathhelpforum.com/calculators/3544-program-ti-84-a.html
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# Math Help - program on TI-84
1. ## program on TI-84
Does anyone know the program to solve the polynomial on graph calculator. Ti-84.
For example: y = x^3 + 3x^2+ 5x + 6
or y = x^4....ans so on.
with this, when doing a problem, if you don't remember or take time to factor it out. So the program might also help you save a lot of time during exams or homework.
if anyone know this program, could you please give it to people, so people can share it. Thanks a lot for your generous.
2. ## TI-84 solve function
when you say solve y = x^3 + 3x^2+ 5x + 6 etc.
what do you mean? do you mean solve for x in terms of y?
or solve for y = 0?
or what?
3. i mean, factor it out, that function is quite easy to factor because when pull the x out. x(x^2 + .............). However, if i have a function , f(x) = x^3 - 3x^2 - 5. how do you factor it on TI 84 calculator ? Thanks.
4. Originally Posted by distance
i mean, factor it out, that function is quite easy to factor because when pull the x out. x(x^2 + .............). However, if i have a function , f(x) = x^3 - 3x^2 - 5. how do you factor it on TI 84 calculator ? Thanks.
It depends on the polynomial. Sometimes the command "simplify" will work. I don't recall if there is a "factor" function on the 84.
If the polynomial has integer zeros (possibly rational zeros) it may factor the expression. However, if some of the zeros are complex it won't do anything.
As far as writing a program to do the trick, you can do it for polynomials up to and including 4th degree (if you can find the formulae for solving cubics and quartics) but I can tell you from experience programming for all eventualities is a pain!
-Dan
5. Originally Posted by distance
i mean, factor it out, that function is quite easy to factor because when pull the x out. x(x^2 + .............). However, if i have a function , f(x) = x^3 - 3x^2 - 5. how do you factor it on TI 84 calculator ? Thanks.
I don't have a TI 84, but I believe that the CAS is based on DERIVE
and in DERIVE the factor function will do the job (with the correct options
set anyway) see attachment.
RonL
6. Originally Posted by distance
i mean, factor it out, ...
Hi,
I'm sorry to say but you can't do that with a TI84.
The TI84 calculates only with numbers. The letter keys which you hit when typing on a TI84 are only addresses of storage places. (For instance: You can't type commands letter by letter but you have to pick them out of a menu list)
The first TI which do symbolic calculation is the TI89. There a small version of Derive is implemented, and there you'll find the factor-command in the menu F2:Algebra.
Bye
EB
7. Yes I know the thread is old but just in case anyone is interested there is a program that does this....
A program called PolySmelt or otherwise known as Poly Root Finder and Simultaneous Equation solver will do the trick for you, well it works on my TI-83 and works on other people's that have a TI-84. Not sure where you can download it, TI website should help. Its a simple program to use and a handy one for finding roots of polynomials.
8. Originally Posted by Random333
Yes I know the thread is old but just in case anyone is interested there is a program that does this....
A program called PolySmelt or otherwise known as Poly Root Finder and Simultaneous Equation solver will do the trick for you, well it works on my TI-83 and works on other people's that have a TI-84. Not sure where you can download it, TI website should help. Its a simple program to use and a handy one for finding roots of polynomials.
I do not own a TI but when I used someone elses and found a way to solve simulatenous equations. You need to use the matrix feature they have in the calculuator.
1)Line up your equations. Varaibles with variables.
2)Go to the Matrix section.
3)Make a new matrix "A" of dimensions n x n where n is the number of variables (or the number of equations).
4)Type in the representative coefficients in the matrix.
5)Make a new matrix "B" of dimension n x 1 where.
6)Type in the representative constant coefficients (the numbers in right of the equal sign).
7)Leave and enter the main screen.
8)Type in
Code:
A^(-1)B
9)The new matrix with have representative solutions to this linear system.
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2014-03-14 15:03:28
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http://ianfinlayson.net/class/cpsc240/labs/04-class
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# Lab 4: Writing a Class
### Objective
To work on coding a class based off a UML diagram
For this lab, you will be writing the Java code for the class that you made the UML diagram for last lab. Pull up your diagram and base the Java class off of that.
A few notes on the implementation of the methods:
• The constructor should set the student's name based on the parameters. The credits and GPA should be set to 0. The ID should be set to the next available ID. The next available ID should then be incremented by one. Because it's static, each new student object shares this value, so they should all get unique IDs. You should initialize the next ID to 1 where it's declared (not in the constructor).
• The method to add a course should factor the grade into their GPA. To do that, weight the student's current GPA by their old number of credits, and the grade from the new course by the number of credits. For example, imagine a student's current GPA is 3.2, and they have 60 credits. If they get an A- in a 3 credit course, then their new GPA is calculated as:
$\Large \frac{3.2 \times 60 + 3.7 \times 3}{60 + 3}$
The 3.7 comes from the number equivalent to an A-. You can get the list of these values from this page.
It should then add the courses credits into the student's total. If the student has 60 credits and adds a 3 credit course, then they now have 63.
• The method to print a report should print all of the student's details to the screen in a readable way. The format is up to you.
• The method which determines if the student can graduate should check that the student has at least 120 credits and has a GPA of at least 2.0. If so, they can graduate; if not, not.
### Testing
You can use the Main.java file to test your program. You might have to change your method names slightly to work with this (but please do so, cause that's how I'll grade your lab).
Here is an example run:
\$ java Main
Alice, Johnson 1 0.00 0
Bob, Williams 2 0.00 0
Claire, Smith 3 0.00 0
Alice, Johnson 1 2.28 124
Bob, Williams 2 2.36 122
Claire, Smith 3 2.16 117
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2020-01-23 21:26:29
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https://mathinfocusanswerkey.com/math-in-focus-grade-5-chapter-13-practice-5-answer-key/
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# Math in Focus Grade 5 Chapter 13 Practice 5 Answer Key Parallelogram, Rhombus, and Trapezoid
This handy Math in Focus Grade 5 Workbook Answer Key Chapter 13 Practice 5 Parallelogram, Rhombus, and Trapezoid provides detailed solutions for the textbook questions.
## Math in Focus Grade 5 Chapter 13 Practice 5 Answer Key Parallelogram, Rhombus, and Trapezoid
Complete. Figure ABCD is a parallelogram. Measure the sides and angles of the figure.
Explanation:
Question 1.
AD = ___ cm
Explanation:
Property of a Parallelogram:
• Opposite sides are equal and parallel.
Question 2.
AB _______ cm
AB = 7 cm
Explanation:
Property of a Parallelogram:
• Opposite sides are equal and parallel.
Question 3.
BC = __ cm
BC = 6 cm
Explanation:
Property of a Parallelogram:
• Opposite sides are equal and parallel.
Question 4.
DC = ______ cm
DC = 7 cm
Explanation:
Property of a Parallelogram:
• Opposite sides are equal and parallel.
Question 5.
m∠A = ____
m∠A = 55°
Explanation:
Property of a Parallelogram:
• Opposite angles are equal.
Question 6.
m∠B = ___
m∠B = 120°
Explanation:
Property of a Parallelogram:
• Opposite angles are equal.
Question 7.
m∠C = ___
m∠C = 55°
Explanation:
Property of a Parallelogram:
• Opposite angles are equal.
Question 8.
m∠D = ___
m∠D = 120°
Explanation:
Property of a Parallelogram:
• Opposite angles are equal.
Question 9.
Name the parallel sides of the figure. ______
$$\overline{A B}$$ || $$\overline{D C}$$
$$\overline{A D}$$ || $$\overline{B C}$$
Question 10.
Name the opposite angles that are equal. ______
m∠A = m∠C
m∠B = m∠D
This parallelogram is not drawn to scale. Fill in the blanks.
Question 11.
m∠Q = m∠___
= _____
m∠Q = m∠S = 75°
Explanation:
Property of a Parallelogram:
• Opposite angles are equal.
Question 12.
m∠P = 180° – ___
= ____
m∠P = 180° – m∠S =180° – 75° = 105°
Explanation:
Property of a Parallelogram:
• Sum of any two adjacent angles is 180°
Question 13.
m∠R = m∠___
= ____
m∠R = m∠P = 105°
Explanation:
Property of a Parallelogram:
• Opposite angles are equal.
These parallelograms are not drawn to scale. Find the unknown angle measures.
Question 14.
m∠a = 56°
Explanation:
Property of a Parallelogram:
• Opposite angles are equal.
m∠a = 56°
Question 15.
m∠b = 45°
Explanation:
Property of a Parallelogram:
• Sum of any two adjacent angles is 180°
m∠b = 180-135 =45°
Question 16.
m∠f = 46°
Explanation:
Property of a Parallelogram:
• Sum of any two adjacent angles is 180°
180-73=107°
m∠f = 107 – 61 =46°
Question 17.
m∠g = 81°
Explanation:
Property of a Parallelogram:
• Opposite angles are equal.
So opposite angle is 139°
m∠g = 139 -58 = 81°
Question 18.
m∠x = 90°
m∠y = 90°
m∠z = 90°
Property of a Parallelogram:
• Opposite angles are equal.
• Sum of any two adjacent angles is 180°
m∠y = 90°
m∠x = 180 -90° =90°
m∠z = m∠x = 90°
Question 19.
m∠p = 90°
Property of a Parallelogram:
• Sum of any two adjacent angles is 180°
m∠p = 180-62+28= 90°
Complete. Write the name of another side or angle of each rhombus.
Question 20.
AB = BC
= ___ = ____
AB = BC= CD= DA
Explanation:
property of a Rhombus:
• All sides are equal and, opposite sides are parallel to each other.
AB = BC= CD= DA
Question 21.
m∠B = m∠______
m∠B = m∠D
Explanation:
property of a Rhombus:
• Opposite angles are equal.
m∠B = m∠D
Question 22.
m∠A = m∠____
m∠A = m∠C
Explanation:
property of a Rhombus:
• Opposite angles are equal.
m∠A = m∠C
Question 23.
UV = _______
= ___ = ___
UV = VS= ST = TU
Explanation:
property of a Rhombus:
• All sides are equal and, opposite sides are parallel to each other.
UV = VS= ST = TU
Question 24.
m∠S = m∠____
m∠S = m∠U
Explanation:
property of a Rhombus:
• Opposite angles are equal.
m∠S = m∠U
Question 25.
m∠T = m∠______
m∠T = m∠V
Explanation:
property of a Rhombus:
• Opposite angles are equal.
m∠T = m∠V
This rhombus is not drawn to scale. Fill in the blanks.
Question 26.
m∠X = m∠__ = ____
m∠X = m∠Z= 53°
Explanation:
property of a Rhombus:
• Opposite angles are equal.
m∠X = m∠Z= 53°
Question 27.
m∠W = ____ – ___ = ____
m∠W = 180 – 53° = 127°
Explanation:
property of a Rhombus:
• Sum of any two adjacent angles is 180°
m∠W = 180 – m∠Z = 180 – 53° = 127°
Question 28.
m∠Y = m∠___ = ____
m∠Y = m∠W = 127°
Explanation:
property of a Rhombus:
• Opposite angles are equal.
m∠Y = m∠W = 127°
These rhombuses are not drawn to scale. Find the unknown angle measures.
Question 29.
m∠p = 125°
Explanation:
property of a Rhombus:
• Opposite angles are equal.
m∠p= 125°
Question 30.
m∠q = 180 – 57° = 123°
Explanation:
property of a Rhombus:
• Sum of any two adjacent angles is 180°
m∠q = 180 – 57° = 123°
Question 31.
m∠r = 180 – 129° = 51°
Explanation:
property of a Rhombus:
• Sum of any two adjacent angles is 180°
m∠r = 180 – 129° = 51°
Question 32.
m∠s = 52°
Explanation:
property of a Rhombus:
• All sides are equal and, opposite sides are parallel to each other.
AB=BC=CD=DA
then in Triangle ABC AB= BC. then it is isosceles Triangle .
The isosceles triangle property states that when two sides are equal, the base angles are also equal.
So, m∠s = 52°
Question 33.
m∠t = 90°
Explanation:
property of a Rhombus:
• All sides are equal and, opposite sides are parallel to each other.
AB=BC=CD=DA
then in Triangle ABC AB= BC. then it is isosceles Triangle .
The isosceles triangle property states that when two sides are equal, the base angles are also equal.
So, m∠s = 45°
in triangle sum of three angles is 180°
So, m∠t = 180 -45+ 45° = 90°
Question 34.
m∠s = 37°
Explanation:
in triangle ABC sum of three angles is 180°
m∠s + m∠v = 180°- 106°= 74°
property of a Rhombus:
• All sides are equal and, opposite sides are parallel to each other.
AB=BC=CD=DA
then in Triangle ABC AB= BC. then it is isosceles Triangle .
The isosceles triangle property states that when two sides are equal, the base angles are also equal.
So, m∠s =m∠v = 74° divided by 2 = 37°
Measure the unknown angles. Then fill in the blanks. ABCD is a trapezoid where $$\overline{A B}$$ || $$\overline{D C}$$.
Question 35.
m∠A = ____
Question 36.
m∠B = ___
m∠B = 110
Question 37.
m∠C = ____
m∠C = 70
Question 38.
m∠D = ___
m∠D = 80
Question 39.
m∠A + m∠D = m∠___ + m∠___ = ____
m∠A + m∠D = m∠100+ m∠80= 180
Supplementary angles = 180°
These trapezoids are not drawn to scale. Find the unknown angle measures.
Question 40.
m∠A = m∠116°
m∠D = m∠64°
m∠D = 180 – 116
= 64°
Explanation:
Supplementary angles = 180°
m∠64° + m∠116° = 180°
Trapezium and Its Properties
• Angle: The sum of angles in a trapezoid-like other quadrilateral is 360°. …
• Two angles on the same side are supplementary, that is the sum of the angles of two adjacent sides is equal to 180°.
Question 41.
m∠G = m∠58°
m∠F = m∠122°
m∠F = 180 – 58 = 122
Explanation:
Supplementary angles = 180°
m∠58° + m∠122° = 180°
Question 42.
m∠L = m∠109°
m∠K = m∠71°
m∠K = 180 – 109 = 71
Explanation:
Supplementary angles = 180°
m∠71° + m∠109° = 180°
Question 43.
m∠P = m∠97°
m∠Q = m∠83°
m∠Q = 180 – 97 = 83°
Explanation:
Supplementary angles = 180°
m∠83° + m∠97° = 180°
Explanation:
m∠S = m∠81°
m∠R = m∠99°
m∠R = 180 – 81 = 99°
Explanation:
Supplementary angles = 180°
m∠81° + m∠99° = 180°
These trapezoids are not drawn to scale. Find the unknown angle measures.
Question 44.
m∠a = 60°
Explanation:
m∠w = 30°
m∠x = 90
90 + 30 = 120
m∠a = 180 – 120 = 60°
Sum of the angles of a triangle = 180°
m∠b = 110°
m∠u = 70° given
m∠u + m∠b = 180°
180 – 70 = 110°
110 + 70 = 180°
Supplementary angles = 180°
Question 45.
∠d = 55°
∠c = 90°
Explanation:
In quadrilateral the sum of angles = 360°
∠w = 90°
∠Y = 85°
YX = 180°
85° + 40° = 125°
180° – 125° = 55°
∠d = 55°
125° + 55° + 90° = 270°
360° – 270° = 90°
∠c = 90°
Question 46.
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2023-02-02 07:48:32
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https://www.coursehero.com/sg/managerial-accounting/cost-of-goods-sold-cogs-budget/
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# Cost of Goods Sold (COGS) Budget
The cost of goods sold budget shows the expenses a company incurs for producing a product.
The next step in the master budget process is to determine its cost of goods sold budget. The cost of goods sold (COGS) is a manufacturing expenses that a company incurs when finished inventory is sold.
To prepare the cost of goods sold budget, an accountant follows these steps:
1. Enter the beginning work in process inventory balance (from the beginning balance sheet).
2. Add the budgeted direct materials, budgeted direct labor, and budgeted manufacturing overhead used in production.
3. Subtract the budgeted ending work in process inventory balance to determine the budgeted cost of goods manufactured (COGM).
4. Add the beginning finished goods inventory balance (from the beginning balance sheet).
5. Subtract the budgeted ending finished goods inventory balance.
#### Cost of Goods Sold Budget Steps
The result of these calculations will be the budgeted cost of goods sold, which will appear on the pro forma income statement.
To complete the cost of goods sold (COGS) budget, budgeted ending inventory is needed. Heavenly Sleep Systems has chosen to maintain a stable ending inventory of 1,000 units. Another option some companies use is to maintain ending inventory equal to a percentage of the following month's sales. This helps reduce the amount of cash tied up in inventory and helps decrease inventory carrying costs.
Heavenly Sleep Systems uses information from its previous budgets to create its cost of goods sold budget. For Quarter 1, budgeted production is 11,500 units. The cost of cotton per unit is $5. The cost of labor per unit is$10. Management would multiply the number of units produced times the cost per unit to arrive at the total cost of cotton and the total cost for direct labor. The overhead for Quarter 1 is $574,250. To find the total cost of goods manufactured in Quarter 1, management would add the total cost of cotton$57,500, total cost of direct labor $115,000, and total manufacturing overhead$574,250.
Next, the change in inventory levels must be accounted for in order to arrive at cost of goods sold. What management just calculated was cost of goods manufactured. However, the number of units manufactured does not always equal the number of units sold.
The budgeted number of units in beginning and ending inventory is multiplied by the cost per unit to find the total value of beginning and ending inventory. The cost per unit is calculated by dividing the total cost of goods manufactured by the number of units manufactured $\65.72=\3{,}124{,}750/47{,}550$. There are a variety of ways to track the value of inventory. For simplicity, Heavenly Sleep Systems has used the total cost of manufacturing for the entire year and divided by the total number of production units for the year to arrive at \$65.72.
Finally, cost of goods sold is calculated by adding the value of beginning inventory and subtracting the value of ending inventory from the cost of goods manufactured.
Heavenly Sleep Systems
Cost of Goods Sold Budget
For Year Ended December 31, 2019
Quarter 1 Quarter 2 Quarter 3 Quarter 4
Total Cost of Cotton $\57{,}500$ $\50{,}000$ $\60{,}250$ $\75{,}000$
Direct Labor Budget $\115{,}000$ $\100{,}000$ $\120{,}500$ $\140{,}000$
Total Manufacturing Overhead $\574{,}250$ $\574{,}250$ $\577{,}750$ $\685{,}250$
= Budgeted Cost of Goods Manufactured $\746{,}750$ $\724{,}250$ $\758{,}500$ $\900{,}250$
$+\;\text{Beginning Inventory}\;(\text{Units}\times\text{Cost})$ $\98{,}580\;(1{,}500\times\65.72)$ $\65{,}720\;(1{,}000\times\65.72)$ $\65{,}720\;(1{,}000\times\65.72)$ $\65{,}720\;(1{,}000\times\65.72)$
$-\;\text{Budgeted Ending Inventory } (\text{Units}\times\text{Cost})$ $\65{,}720\;(1{,}000\times\65.72)$ $\65{,}720\;(1{,}000\times\65.72)$ $\65{,}720\;(1{,}000\times\65.72)$ $\65{,}720\;(1{,}000\times\65.72)$
= Budgeted Cost of Goods Sold $\779{,}610$ $\724{,}250$ $\758{,}500$ $\900{,}250$
The cost of goods sold budget considers all costs associated with the production of a product and is usually one of the largest expenses on a company's income statement.
After the COGS is determined, an accountant can calculate gross profit. An accountant figures out this preliminary profit number on the income statement by taking total net sales revenues and subtracting the cost of goods sold. The gross profit amount helps managers figure out production efficiency and estimate the overall profitability of a company.
After gross profit is determined, an accountant can layer in other items of the pro forma income statement, such as selling and administrative expenses, operating profit, and interest income and expense. This leads to the calculation of pro forma pretax income.
After this, the accountant estimates income taxes and subtracts them from pretax income. The result is the net income, or "bottom line," of the pro forma income statement.
Now that the operating budget is complete, the financial budget is created to finalize the master budget.
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2019-04-26 02:39:57
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http://libros.duhnnae.com/2017/jun9/149859543927-Continuum-limits-of-random-matrices-and-the-Brownian-carousel-Mathematics-Probability.php
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# Continuum limits of random matrices and the Brownian carousel - Mathematics > Probability
Continuum limits of random matrices and the Brownian carousel - Mathematics > Probability - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.
Abstract: We show that at any location away from the spectral edge, the eigenvalues ofthe Gaussian unitary ensemble and its general beta siblings converge toSine beta, a translation invariant point process. This process has a geometricdescription in term of the Brownian carousel, a deterministic function ofBrownian motion in the hyperbolic plane.The Brownian carousel, a description of the a continuum limit of randommatrices, provides a convenient way to analyze the limiting point processes. Weshow that the gap probability of Sine beta is continuous in the gap size and$\beta$, and compute its asymptotics for large gaps. Moreover, the stochasticdifferential equation version of the Brownian carousel exhibits a phasetransition at beta=2.
Autor: Benedek Valko, Balint Virag
Fuente: https://arxiv.org/
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2018-11-18 07:45:08
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http://project-navel.com/navel/news/magazines/2006-01.html
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2017-10-21 23:12:45
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https://math.stackexchange.com/questions/922470/an-algorithm-to-obtain-square-of-any-number
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# An algorithm to obtain square of any number.
I have discovered a new way to obtain square of any number. I think its a new algorithm to find square of a number than just multiplying it with itself; if its not new then let me know. Algorithm is as follows:
1. Divide the number in two parts with one part containing only the number at unit's place say part 'A', and other part say 'B', containing the remaining number.
2. Now square the number at unit's place. The square will be one of these; {0,1,4,9,16,25,36,49,64,81}. The unit's place digit in this square is the unit's place digit in actual final answer.Write it in the answer. If the square of digit at unit's place is a two digit no like from 16 to 81 in above set; write only the digit at unit's place from this square in the final answer and carry the remaining digit.
3. Multiply the actual number to be squared by part 'B'(the remaining part than the number at unit's place as described in step 1 ).
4. Multiply the parts 'A' and 'B'.
5. Add results of step 3 with results of step 4.
6. Add the carried digit from step 2 to the sum in prior step, that is step 5.
7. Now write this sum before the number we wrote at unit's place of final answer in step 2.
8. This number we now obtain from step 7, is the square of our number.
Example:
Lets find square of 127 by above algorithm;
1. A = 7 and B = 12 here...
2. A^2 = 49 thus final answer will have 9 at units place and 4 is carried to add later.
3. Given number multiplied by 'B' => 127*12 = 1524
4. A*B => 12*7 = 84
5. 1524+84 = 1608 ..... ( step 3 + step 4 )
6. 1608+4 =1612 .... (4 is carried as stated in step 2)
7. Now, 16129 is the answer... (From step 2 and 6)
• You have some number $N = A+10B$. Then $N^2 = N\left( A + 10B\right) = NB + \left( A+10B\right)A = 10NB + 10AB + A^2$. You're performing this third calculation. You seem to have a firm grasp of arithmetic, at least. ;) – COTO Sep 7 '14 at 14:07
Ok so let's try to square $10b+a$ and we let $a^2=10c+d$ where it is possible that $c=0$.
Step $1$ tells us to identify $a$ and $b$
For step $2$ we have $a^2=10c+d$ and we write $d$ in the final place and remember $c$
Step $3$ gives us $b\cdot (10b+a) =10b^2+ab$
Step $4$ gives us $ab$
Step $5$ gives us $10b^2+2ab$
Step $6$ gives us $10b^2+2ab+c$
In step $7$ "putting the number before" is the same as multiplying by $10$ and adding so we get $10\cdot(10b^2+2ab+c)+d$
To check at step 8 we have $100b^2+20ab+10c+d=100b^2+20ab+b^2=(10b+a)^2$
So your method works. I couldn't say whether it is new, but it requires two multiplications at steps $3$ and $4$, plus the squaring step.
There is an interestingly similar method for speeding up the multiplication of two large numbers. Let $M$ be a large power of $10$, say, so that the numbers $a, b, c, d$ are of comparable size.
Multiplying $(aM+b)(cM+d)=acM^2+(ad+bc)M+bd$ looks as though it requires the computation of four products $ac, ad, bc, bd$, but if we compute $ac, bd$ and $(a+b)(c+d)$ (three products) we find that $ad+bc=(a+b)(c+d)-ac-bd$. This turns out to be computationally more efficient.
The key thing about a method like yours is not so much whether it works (all sorts of methods work) or whether it is new, but whether it is a practical improvement on existing methods.
$$\Large (10X+Y)^2=100X^2+20XY+Y^2$$
• Although the OP used $A$ and $B$ in the other way ($10B+A$). – Clement C. Sep 7 '14 at 14:01
• @ClementC. see the edit? – RE60K Sep 7 '14 at 14:03
• (yes... sorry for nitpicking.) – Clement C. Sep 7 '14 at 14:22
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2020-02-25 01:38:18
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https://mersenneforum.org/showthread.php?s=967d6b9f0e2215fa2574fd67f9d94d31&p=553607
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mersenneforum.org Jason Zimba vs the Creature From the Dozenal Abyss
Register FAQ Search Today's Posts Mark Forums Read
2020-08-13, 20:57 #100 sweety439 Nov 2016 1000101101112 Posts https://dozenal.fandom.com/wiki/10: the properties of the number 10 (dozen, not ten) https://vignette.wikia.nocookie.net/...20200613134857: the dozenal digits which I use (dozenal is the base (radix) which I use) https://dozenal.fandom.com/wiki/Divisibility_rule: Dozenal divisibility rules https://dozenal.fandom.com/wiki/Prime_number: Dozenal properties of the prime numbers https://dozenal.fandom.com/wiki/Form...s_exactly_once: some formulas like ddd * ddd = dddddd and dd * dd * dd = dddddd, which use all digits (form 0 to E) exactly once https://dozenal.fandom.com/wiki/Harshad_number: Dozenal harshad numbers https://dozenal.fandom.com/wiki/Kaprekar%27s_routine: Kaprekar's routine in dozenal https://dozenal.fandom.com/wiki/Narcissistic_number: Dozenal Narcissistic numbers https://dozenal.fandom.com/wiki/Polydivisible_number: Dozenal polydivisible numbers https://dozenal.fandom.com/wiki/Table_of_divisors: table of divisors for numbers 1 to 1000 (decimal 1728), written in dozenal https://dozenal.fandom.com/wiki/Table_of_prime_factors: table of prime factors for numbers 1 to 1000 (decimal 1728), written in dozenal https://dozenal.fandom.com/wiki/Repeating_dozenal: repeating dozenal, like repeating decimal Last fiddled with by sweety439 on 2020-08-13 at 20:58
2020-08-13, 21:06 #101 sweety439 Nov 2016 23·97 Posts https://dozenal.fandom.com/wiki/Number_classes, this is a very long article, listed many classes of numbers (e.g. highly composite numbers, generalized pentagonal numbers, Catalan numbers, happy numbers, Frugal numbers, Lychrel numbers, Mersenne primes, Sophie Germain primes, irregular primes, palindromic primes, permutable primes, left-truncatable primes) in dozenal
2020-08-13, 21:08 #102
sweety439
Nov 2016
223110 Posts
Quote:
Originally Posted by tuckerkao The main reason it's because hex is not the default math base that everyone has to use on the daily routines. Of course, Sexigesimal won't be an ideal default base anyway. Let the numbers speak for themselves, people will eventually realize decimal isn't the best base for the default. For Dozenal multiplication tables: 2, 3, 4, 6 are easy. 8, 9, Ɛ are medium, only 5, 7 and Ӿ are hard. Dozenal 7 times table is actually easier than Decimal 6 times(on the odds) table if you know both bases equally well. Like what retina said, Dozenal as the default math base won't leave behind the kids behind.
A dozenal 100*100 multiplication table
2020-08-14, 01:59 #103
tuckerkao
Jan 2020
23×7 Posts
Quote:
Originally Posted by Uncwilly So, by your logic English letters B, C, D, E, G, P, T, and Z should have different names because people can mis-hear them.
I've posted my playing dozenal cards on page 4 of this thread.
If I apply your units, I'll have "2, 3, 4, 5, 6, 7, 8, 9, A, B, J, Q, K, A", then I cannot properly play it anymore.
Last fiddled with by tuckerkao on 2020-08-14 at 02:00
2020-08-14, 02:02 #104
Uncwilly
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2×4,297 Posts
Quote:
Originally Posted by tuckerkao I've posted my playing dozenal cards on page 4 of this thread. If I apply your units, I'll have "2, 3, 4, 5, 6, 7, 8, 9, A, B, J, Q, K, A", then I cannot properly play it anymore.
I think that this is a fine place to pause this thread.
2020-08-14, 02:13 #105 Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 22·2,281 Posts ...Aw... and just when two kindred spirits finally found each other on the waves of ether.
2020-08-14, 02:16 #106
LaurV
Romulan Interpreter
Jun 2011
Thailand
2·11·397 Posts
Quote:
Originally Posted by xilman Here is the start of the table in cuneiform: 𒐕𒐖𒐗𒐘𒐙𒐚𒐛𒐜𒐝
[offtopic]
what trick do I need to do to firefox to see those? (like installing fonts, etc?)
[/offtopic]
2020-08-14, 06:38 #107
xilman
Bamboozled!
"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across
240028 Posts
Quote:
Originally Posted by LaurV [offtopic] what trick do I need to do to firefox to see those? (like installing fonts, etc?) [/offtopic]
You will certainly need to have fonts installed. I use the noto series on my Linux machines but I can't advise on Windows or Mac systems.
That may be sufficient. We can't know unless the fonts are installed.
2020-08-14, 12:29 #108
Xyzzy
"Mike"
Aug 2002
7,691 Posts
Quote:
Originally Posted by LaurV [offtopic] what trick do I need to do to firefox to see those? (like installing fonts, etc?) [/offtopic]
Quote:
Originally Posted by xilman You will certainly need to have fonts installed. I use the noto series on my Linux machines but I can't advise on Windows or Mac systems. That may be sufficient. We can't know unless the fonts are installed.
For Linux (Fedora 32) this package worked for us:
Code:
\$ sudo dnf info google-noto-sans-cuneiform-fonts-20181223-7.fc32.noarch
Last metadata expiration check: 2:35:08 ago on Fri 14 Aug 2020 04:38:07 AM CDT.
Available Packages
Version : 20181223
Release : 7.fc32
Architecture : noarch
Size : 352 k
Repository : fedora
Summary : Sans Cuneiform font
Description : Noto fonts aims to remove tofu from web by providing fonts for all
: Unicode supported scripts. Its design goal is to achieve visual harmonization
: between multiple scripts. Noto family supports almost all scripts available
: in Unicode.
:
: Noto Sans font for Cuneiform.
Similar Threads Thread Thread Starter Forum Replies Last Post sweety439 And now for something completely different 22 2020-07-21 14:49 sweety439 sweety439 0 2020-06-11 06:36 tuckerkao Lounge 7 2020-02-11 04:44 jasong Sierpinski/Riesel Base 5 8 2005-04-29 05:13
All times are UTC. The time now is 08:54.
Sun Sep 20 08:54:58 UTC 2020 up 10 days, 6:05, 0 users, load averages: 2.00, 1.52, 1.49
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2020-09-20 08:54:59
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http://mathoverflow.net/questions/118530/not-isomorphic-varieties-with-isomorphic-tilting-algebras
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# Not isomorphic varieties with isomorphic tilting algebras
Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies condition $Ext^{i}(T, T)=0$ for $i \neq 0$. Tilting algebra $A=End(T)$ is a finite dimensional algebra of finite global dimension. Main result of such geometric tilting theory is an equivalence of triangulated categories $D(X)$ and $D(A)$.
Are there examples of two not isomorphic smooth projective varieties with tilting objects $(X, T)$ and $(X', T')$ such that $End_X(T) \cong End_{X'}(T')$?
In particular this would imply that $\operatorname{D^b}(X) \simeq \operatorname{D^b}(X')$, so by results of Bondal and Orlov $\omega_X$ can't be (anti-)ample, because in this case $X \cong X'$. Moreover, I suppose that if $\omega_X$ is ample than $X$ could not admit a tilting object, but I don't know how to prove this.
-
Related to your last paragraph: there are examples of non-isomorphic varieties which are derived equivalent. The derived equivalences are then called Fourier-Mukai transforms. For example, in dimension 3, all crepant resolutions of a variety with terminal singularities are derived equivalent. There is lots of information in the chapter "Derived categories of coherent sheaves on algebraic varieties" by Yukinobu Toda in Triangulated Categories, edited by Holm, Joergensen, and Rouquier, including some discussion of tilting objects, but an answer to your question didn't obviously follow for me. – Hugh Thomas Jan 10 '13 at 23:48
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2015-02-27 21:34:03
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https://www.khanacademy.org/test-prep/gmat/problem-solving/v/gmat-math-12
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We're on problem 65. They wrote 2, 4, 6, 8, n, 3, 5, 7, and 9. And they tell us, in the list above, if n is an integer between 1 and 10, inclusive-- so that means n could be 1, and it could be 10, or any number inbetween, and it's an integer-- then the median must be-- the median, right, not the mean, the median. So the median means the middle number, right? Let's put all the other numbers in order. So we have 2, 3, 4, 5, 6, 7, 8, 9. And they're essentially saying that n could be anywhere in this. It could be a 1 here, it could be right here, it could be another 2, it doesn't tell us. It could be any number, it could be 2, could be another 3, could be 10. We don't know. But before we stick n in here, let's figure out what the middle is. Right here we have eight numbers, so the middle of the range is actually-- there is no middle number-- but when we add n, there will be a middle number. So let's just think about it for a little bit. If n is 5 or less, so n goes into this bucket, n goes someplace here, then what happens? Then what becomes the middle number? If n is 5 or less, then we'll have n here. And n might be 5, n might go right here. But what becomes the middle number? Then we have 1, 2, 3, 4 on this side of the middle number, and we have 1, 2, 3, 4 on that side of the number. So then, the median, which is the middle number, becomes 5. That's if n is 5 or less. If n is less than or equal to 5, then we know that the median will be equal to 5. Now let's do the opposite thought experiment. What if n is 6 or more? What if n is greater than or equal to 6? So it goes someplace over here, so n will go someplace here. Could be 6, I might have to put it right there. But wherever I put it on the side, I'll have 4 to the right of 6, and I'll have 4 to the left of 6. n isn't on this side anymore, 4 to the left of 6. So then the median would be 6. And either of these cases have to be true for n. So the median is either 5 or 6, and that is choice B. Problem 66. I like this brownish color. Reminds me of when I was in middle school math competitions. 4u7, n23, 162. And you add them all together, you get one 1,222. And what they want to know is, if n and u represent single digits in the correctly worked computation above, what is the value of n plus u? So the best thing to do is to try to work these out, and see what happens. So 7 plus 3 is 10, plus 2 is 12. So you would write a 2 here, which they already wrote, and carry a 1. And now we have 1 plus u-- let me write this down-- 1 plus u plus 2 plus 6 is equal to, well, it's either going to be equal to 2, or some digit and a 2. So let's just think about it a little bit. 1 plus u plus 2 plus 6, that's equal to u plus 9. u plus 9 is going to have to equal something. So if u is 9, and it has to be something that ends with a 2. And remember, u can only be between 0 and 9. It's a digit, it has to be an integer, it has to be a digit. So let's think about it a little bit. It has to be something that ends in a 2. So the only next thing above 9 that ends as a 2 is 12. Because you can't get to 22. To get to 22, you would have to add 13, and you can't be 13. So this has to be a 12, so u has to be equal to 3. That's the only possibility that'll give you something that ends in a 2. So if we assume that u is 3, then we have 1 plus 3 plus 2 plus 6, which is 12, carry the 1, you get 1 plus 4 plus n plus 1 is equal to 12. So you get 1 plus 4 plus 1. That's 6, plus n is equal to 12, and n would be equal to 6. And they want to know what n plus u is. So n plus u, 6 plus 3. That's equal to 9. And that is choice B, 9. 67. Look at that, this is a dense looking one, but let me write down the little equation they wrote on top. r is equal to 400 times d plus s minus p, all of that over p. If stock is sold three months after it is purchased, the formula above relates p, d, s, and r, where p is the purchase price of the stock. That's the purchase price. d is the amount of any dividend received. Fair enough. s is the selling price of the stock, and r is the yield of the investment as a percent. Fair enough. If Rose purchased $400 worth of stock-- so p is equal to$400-- received a $5 dividend-- so d is equal to$5-- and sold the stock for \$420, for three months after purchasing-- so this formula applies, because this is the formula for selling after three months-- what was the yield of her investment, according to the formula? Assuming she paid no commissions. We just substitute in. So let's see. The yield would be equal to 400 times 5 plus 420 minus 400, p is 400, all of that over 420. That is equal to 420 minus 400 is 20, plus 5 is 25. 400 times 25 over-- this shouldn't be 420, this is p. The price you paid was 400. 25 over 400. I just substituted these values into this equation. This cancels out, and I'm just left with a yield of 25, and it's probably in terms of percent. Yep, 25%. E. All right, problem 68. Let's switch colors. 68. The temperatures in degree Celsius recorded at 6:00 in the morning in various parts of a certain country were 10 degrees, I'm not going to write all of it, 5, minus 2, minus 1, minus 5, and 15. What is the median of these temperatures? The median just means the middle. Don't confuse that with the average, which is the mean. Or the mean, which is the average. Median means middle, so let's just put them in order and figure out the middle. So the smallest of these numbers is minus 5, cross it out. Then we have minus 2, cross it out. Then we have minus 1, cross it out. Then we have 5, cross it out, then we have 10, cross it out. Then we have 15. So this is interesting about median. If there is no true middle number, you have three on this side of negative 1, or negative 1 and less, and you have three on that side, so there's no middle number, because we have an even number of numbers. So what you want to do is, you take the two middle numbers, which are negative 1 and 5, and you average them. So it's negative 1 plus 5 over 2, which equals 4 over 2, which is equal to 2. 2 degrees Celsius, which is choice C. Problem 69. If y times three 3x minus 5 over 2 is equal to y, and y does not equal 0, then x is equal to what? Well the simplest thing, since y doesn't equal 0, we can divide both sides of this equation by y. You divide both sides by y. y divided by y is 1, y divided by y is 1. So then we're left with 3x minus 5 over 2 is equal to 1. Multiply both sides by 2, you get 3x minus 5 is equal to 2. Add 5 to both sides, 3x is equal to 7. And you get x is equal to 7 over 3, which is choice C. And I'm out of time, so I'll see you in the next video.
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2017-05-27 21:16:23
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https://answers.ros.org/question/355980/ament_target_dependencies-vs-target_link_libraries/
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What are the situations where I should use ament_target_dependencies and when should I use target_link_libraries? I also notice that there are occasions where I need both for the dependencies to be correctly added.
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2020-07-16 15:58:13
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http://www.ck12.org/tebook/Texas-Instruments-Algebra-I-Teacher%2527s-Edition/r1/section/11.4/
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<meta http-equiv="refresh" content="1; url=/nojavascript/"> Manual Fit | CK-12 Foundation
You are reading an older version of this FlexBook® textbook: CK-12 Texas Instruments Algebra I Teacher's Edition Go to the latest version.
# 11.4: Manual Fit
Created by: CK-12
0 0 0
This activity is intended to supplement Algebra I, Chapter 10, Lesson 7.
ID: 12274
Time required: 20 minutes
## Activity Overview
In this activity, students will manipulate parabolas in vertex form so that the curve matches a set of data points graphed as a scatter plot. This activity will serve to reinforce understanding of the vertex form for a parabola. In the extension, students will find an equation in vertex form and standard form that matches points from the Gateway Arch in St. Louis.
• Graph the parabola so that its vertex and shape match a set of plotted points.
• Understand the value of a and its contribution to shape and direction of opening.
• Apply knowledge of parabolas to parabolic shapes in real world problems.
Teacher Preparation and Notes
• This activity is intended for an Algebra 1 or Algebra 2 class.
• Students will need to be able enter data into lists and graph as a scatter plot. They will also need to be able to graph a function and adjust window settings.
• Students will answer questions about the vertex, direction of opening, and the relative width of opening for a particular shape.
Associated Materials
## Problem 1 – Match the graph, Part 1
Vertex form for the equation of a parabola is shown on the worksheet. Students will use this information in the next few questions to help them answer questions.
Instruct students that to change the graph, they can substitute different values into the $Y=$ equation. Students should enter initial values (non-zero) into the vertex form of the equation to begin their exploration.
Notice that the students may not get the “exact” answer that they wish. Tell them that they will find more exact methods of finding matching equations for data in later classes.
Acceptable equation: $y = 2(x - 1)^2 + 3$
You may also want to cover the effects of changing the window on the appearance of the graph. Stress the importance of knowing the minimum, maximum, and scale to determine the equation.
## Problem 2 – Match the graph, Part 2
Students are given another set of plotted points and asked to grab and move the parabola to match the graph. Encourage discussion of the placement of the vertex, and the relative width of the curve. This time, a negative value for $a$ is required. $y=-0.25x^2$
## Problem 3 – Match the Double Arches
After matching the data well, the $“Mâ€$ double arches appear quite nicely. Discussion could follow about reflections, symmetry, and the design of company logos using mathematical or geometric figures that are pleasing to the eye.
Note: Students can enter the less than or equal to $(\le)$ and the greater than or equal to $(\ge)$ symbols by pressing $2^{nd} \ [0]$, the Catalog menu, and selecting from the list.
To enter the word and, students can press $2^{nd}$ [MATH] and move to the LOGIC menu.
## Problem 4 – The Main Cables of a Suspension Bridge
Several loops of cable are represented here. Students will be matching an equation to a particular piece of the graph. What the students have learned about vertex form should be of help in this problem.
$\text{Section A:} \ y = 0.2(x + 4)^2\\\text{Section B:} \ y = 0.2(x - 4)^2$
To graph the given screen, see the equations to the right. Conditional statements are used to limit the domain of the function.
## Extension – The Gateway Arch in St. Louis
The Gateway Arch in St. Louis, the “Gateway” to America, is a shape that looks like a parabola to the casual observer (It is actually called a catenary curve.).
Students will create an equation in vertex form to match the data given in $L1$ and $L2$.
Using the same data, students are asked to match the graph in standard form. Important things to remember are; what does the value of a do to the graph, and what would your $y-$intercept be ($c$ in the equation)?
Discussion that follows includes how the equations are the same, and different. Assist the students in expanding the vertex form so that a direct comparison can be made for the two equations.
Other Arches
This section gives students a few real-world situations where they can find parabolas. Students can find the equations that model these situations.
• Hang a chain (or necklace) against a piece of graph paper and trace its graph (or take a digital photo). Write an equation in vertex form to match the shape of the curve.
• Place a laminated piece of graph paper behind a drinking fountain and take a digital photo. Write an equation to match the shape of the curve.
Feb 22, 2012
Aug 19, 2014
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2014-10-01 08:13:54
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https://bsahely.com/2018/12/09/a-uni%EF%AC%81ed-mechanistic-framework-for-developmental-and-evolutionary-change-enrico-borriello-sara-i-walker-and-manfred-d-laubichler/
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# A unified, mechanistic framework for developmental and evolutionary change | Enrico Borriello, Sara I. Walker and Manfred D. Laubichler
Reproduced from: https://arxiv.org/abs/1809.02331
# A unified, mechanistic framework for developmental and evolutionary change
Enrico Borriello∗1,2, Sara I. Walker†1,2,5,7 and Manfred D. Laubichler‡1,3,4,6
1ASU-SFI Center for Biosocial Complex Systems, Arizona State University,Tempe, AZ, USA
2Beyond Center for Fundamental Concepts in Science, Arizona State University, Tempe AZ
3Santa Fe Institute, Santa Fe, NM, USA
4Marine Biological Laboratory, Woods Hole, MA, USA
5School of Earth and Space Exploration, Arizona State University, Tempe AZ
6School of Life Sciences, Arizona State University, Tempe AZ
7Blue Marble Space Institute of Science
September 10, 2018
## Abstract
The two most fundamental processes describing change in biology – development and evolution – occur over drastically different timescales, difficult to reconcile within a unified framework. Development involves a temporal sequence of cell states controlled by a hierarchy of regulatory structures. It occurs over the lifetime of a single individual, and is associated to the gene expression level change of a given genotype. Evolution, by contrast entails genotypic change through the acquisition or loss of genes, and involves the emergence of new, environmentally selected phenotypes over the lifetimes of many individuals. Here we present a model of regulatory network evolution that accounts for both timescales. We extend the framework of boolean models of gene regulatory network (GRN) – currently only applicable to describing development – to include evolutionary processes. As opposed to one-to-one maps to specific attractors, we identify the phenotypes of the cells as the relevant macrostates of the GRN. A phenotype may now correspond to multiple attractors, and its formal definition no longer require a fixed size for the genotype. This opens the possibility for a quantitative study of the phenotypic change of a genotype, which is itself changing over evolutionary timescales. We show how the realization of specific phenotypes can be controlled by gene duplication events, and how successive events of gene duplication lead to new regulatory structures via selection. It is these structures that enable control of macroscale patterning, as in development. The proposed framework therefore provides a mechanistic explanation for the emergence of regulatory structures controlling development over evolutionary time.
Keywords: gene regulatory network, boolean network, evolution, development, control.
## 1 Introduction
Understanding the mechanisms underlying the emergence and persistence of new cell types is a central problem in the evolution and development of multicellular organisms. Whereas all cell types can in principle access the same genetic information, in practice, regulation of gene expression restricts this, such that only a subset of an organism’s total genomic information content is accessible to a given cell type at a given time, permitting differentiation of many phenotypes from a single genotype [1]. Regulation of gene expression therefore plays a dominant role in establishing cell types. From a formal point of view, the question of how new cell types emerge, therefore, reduces to the problem of understanding how new regulatory structures emerge to specify and control the expression of novel phenotypes.
The interplay among these regulatory genes, and their interaction with the other components of the cell governs the expression levels of both mRNA and proteins, where the set of interactions is described as a “gene regulatory network” (GRN). Understanding the topology of GRNs controlling body form is a central problem in developmental biology. Understanding their change over evolutionary time is an equally important problem in morphogenesis, as it provides a mechanistic explanation for the differentiation of body structures.
Nonetheless, any attempt at trying to predict the steady states of the regulatory process – i.e. the cell types – in terms of a dynamical model (coupled ordinary differential equations, boolean network models, stochastic gene networks, to name a few common approaches) faces the difficulty of having to reconcile the fixed number of genes in these models, whose expression level is representative of a given cell type [2–4], with the possibility for the size of the genotype to change over evolutionary timescales. Chromosome loss and gain, single gene and whole genome duplication, as well as horizontal gene transfer all alter the number of genes participating in the dynamics of a GRN. In doing so, these processes deprive the mapping of cell types to gene expression patterns of its original meaning. Therefore, even when successful in explaining developmental change, current GRN models must be redefined after each modification of the genotype for their use in evolutionary biology.
In this manuscript, we generalize the well established framework of boolean, dynamical models of GRNs as proposed by Kauffman [5], to include features of evolutionary biology. In Boolean models, the attractors of the network dynamics encode different, stable cellular phenotypes, permitting a model for how multiple cell type identities can be encoded in the same regulatory structure. The novelty of our approach consists in relaxing the restrictive one-to-one mapping between network attractors and cell phenotypes by redefining phenotypes as collections of gene expression patterns with a given subset of genes sharing the same pattern (section 2). While the traditional definition, assuming a one-to-one map between phenotype and genotype, yields increasingly fine-tuned specifications for the phenotype for progressively larger genotypes, our novel definition permits greater evolutionary plasticity by identifying the phenotypes with a macrostate, as opposed to individual (micro)states, of a dynamical system.
We will show that, under the relaxed assumption of identifying phenotypes as macrostates of the underlying Boolean GRN, a fixed genotypic size is not necessary for specifying or retaining phenotypes through evolutionary processes. We will exploit this possibility to study the emergence of new cell types, as well as the consolidation or loss of old types, as a consequence of the changing size and topology of the GRN over evolutionary timescales, and of shifting environmental conditions. As such, our model also addresses an inconsistency arising from considering concepts belonging to different levels of description of a well conceived ontology of biological objects [6] as being modeled as ’same-level’ processes: for example, gene expression levels and phenotypes, as they were interchangeable. Our approach will instead assume gene expression to be at a lower level of the ontology than phenotype, while phenotype and environment will belong to the same, higher ontological level.
In what follows, we focus on the case study of gene duplication, and use it as an example of genotype-changing evolutionary process. Gene duplication has occurred in all three domains of life [7], and is an ancient mechanism dating to before the last universal common ancestor [8]. Features of genome evolution and function, not somehow related to gene duplication events, seem to be the exception rather than the rule. Gene duplication is, by far, the dominant force in creating new genes, and at least 50% of genes in prokaryots [9, 10] and over 90% of those in eukaryotes [11] are the result of gene duplication. Nonetheless, with the exception of a few papers [12–14], it has rarely been discussed as a mechanism for evolving new regulatory patterns for cell type identity.
We propose the interplay between gene duplication and natural selection as the driving force responsible for the assemblage of genetic “modules”, or “core sets of regulatory genes” [15–21], in charge of specific cell functions. Evidence in favor of functional modularity in biology has steadily increased over the last three decades, and, as of today, modules have been both reconstituted in vitro [22], and transplanted from one cell type to another [23]. It is now well accepted that biological functions are only rarely attributed to individual molecules – the role of hemoglobin in transporting oxygen along the bloodstream being among the best examples. Far more often, a biological function results instead from the interaction among many different proteins, like in the transduction process converting pheromone detection into the act of mating in yeast [24–26]. Network controllability is usually reinterpreted in GRN dynamical models as the mathematical counterpart of extracellular signaling. Our macrostate interpretation of the phenotypes still allows us to adopt and assign biological meaning to network controllability techniques. For example, the control kernel of a GRN is defined in [27] as the minimum number of genes/nodes whose expression it is necessary to control, in order to steer the dynamics of the rest of the network toward a desired attractor (phenotype). Focusing on the developmental timescale we show it is still possible to easily rephrase, generalize, and adapt the notion of control kernel within our new framework. Therefore, our unified framework, while being suitable to mechanistic studies of GRN mutations over the evolutionary timescale, is still able to describe developmental change.
As a final application of our method, we will show how our definition gives rise to a nested hierarchy of phenotypes in a GRN (section 5), arising through evolutionary accumulation of new phenotypic states. As a consequence of this nested structure, the older a phenotypic trait is, the less likely it is to be disrupted by mutations. Our model can therefore add further theoretical justification for the preservation of early assembled modules of a GRN, like the ones underling development of phyletic body plans, suggested by Davidson and Erwin as having being preserved since the Early Cambrian. [28]
The manuscript is structured as follows: The next section contains a brief review of dynamical boolean models of GRNs sufficient to orient those not familiar with the general theory. It reviews the main features of Kauffman’s seminal theory, and of the identification between cell types and attractors. We then expose our core idea that the cell types are more properly described by collections of attractors, i.e. by macrostates of the dynamics. We then conclude the section by drawing an example network that we will use in the sections 3 and 4 as a toy model to explore the consequences of our hypothesis. Section 3 is devoted to the main consequence of our approach. We show how gene duplication and mutation events alter the nature of the cell types expressed as a consequence of selective pressure induced by a shifting environment. Section 4 shows how our generalized approach does not inhibit the possibility of adopting network controllability methods to describe epigenetically induced developmental change. We again use the same toy model to illustrate key concepts, and show extracellular control is now more easily achieved than in Kauffman’s original framework. Before concluding, we show in section 5 that our framework naturally explains the progressive acquisition of compatible phenotypic traits in terms of a nested phenotypic structure that shelters earlier traits from deleterious mutations.
## 2 Boolean Models
This section describes the mathematical details of the boolean, dynamical model we will assume in the rest of this manuscript. In living tissues, the intrinsic patterns of gene expression coupled with signaling input dictates cell fate. Both of these processes can readily be modelled by a boolean network. Boolean networks were originally proposed by Kauffman [5] as a viable mathematical model of GRNs. They permit exploration of the complex steady-state dynamics of GRNs, where the attractors of the dynamics can be identified with different cell types/fates, e.g. quiescence, proliferation, apoptosis, differentiation, etc. The interested reader can check [29–41] only to cite some remarkable examples of the vast literature corroborating the success of the boolean approach.
### 2.1 Review of Boolean Models
At a given time t, the state of the boolean model of a GRN is known when the state xi(t) of every gene i is known. The Boolean nature of the model assumes two possible states for gene i: active, which corresponds to xi(t) = 1, or inactive, with xi(t) = 0. For a network describing the interactions among n genes, the state at time t is specified by an n−dimensional boolean array x1(t), . . . , xn(t). The dynamics of the GRN is then described by n boolean functions f1, . . . , fn which provide the state of the network at time t + 1, given its state at time t (synchronous update, but asynchronous models are also possible [42]):
For n genes, 2n possible states (gene expression patterns) exist, and the dynamics of the network is represented by a trajectory (a time series) in the discrete space containing the totality of these states. For deterministic functions fi, and because of the finite size of this state space, these trajectories will eventually converge to either a fixed state or a cycle of states. These special activation patterns take the name of attractors of the boolean network, and were identified by Kauffman as corresponding to the stable phenotypes of the GRN.
Nothing has been said so far about the functional form of the boolean functions fi, and readers interested in an insightful analysis of how constrained such a specification is from experimental data are redirected to [43,44]. For reasons that will become clearer in section 3, and given the conceptual nature of this manuscript, we will restrict ourselves to the case of boolean network with thresholds:
where aji is the relative weight of the regulatory signal from gene j to gene i (activation when aji is positive, inhibition otherwise), bi is the activation threshold of gene i, and sgn(x) is a unitary step function, defined by sgn(x) = 0 if x ≤ 0 but sgn(x) = 1 if x > 0. In a more compact notation, we can write
where we have introduced the adjacency matrix A = (aij), and the columns B = (bi) and X = (xi). A useful feature of these models is that they convey the topology of the network, implicit in the definition of generic boolean functions fi, in a very explicit way, with the edges of the network representing non-null entries of the adjacency matrix.
Now that we have laid the groundwork for the numerical models we will be using in the rest of this manuscript, let us go back to the interpretation of the attractors of a boolean network as the phenotypes of a GRN. Given this identification, robustness and evolvability of GRNs can be mapped directly to the dynamics of the attractor landscape of the corresponding boolean model [13, 45, 46]. In this framework, the emergence of new phenotypes has a precise mathematical meaning as the acquisition of new attractors, which can arise due to mutations in the network structure.
Many steady-state attractors are permitted in a boolean network, making it an ideal model for describing how the genomic information contained within a single initial fertilized egg is differentially expressed in so many distinct cell types in response to different regional specification and morphogenetic histories [47].
In the absence of external regulation, the likelihood of a given cell type, or phenotype, is quantified in terms of the number of initial configurations of the network converging on the attractor state encoding that cell type. The higher the number of initial configurations leading to the same equilibrium dynamics, i.e. the production of a well defined set of proteins, the more likely the cell type that set of proteins represents will be. We will refer to these probabilities as the basins of attraction of the possible cell types encoded in the GRN.
On the other hand, the effect that external regulation, in the form of extracellular signaling during development, might have in determining the expression of a specific subset of genes, is even more dramatic, as it might force the development toward cell types that were not even initially accessible to the unperturbed GRN. The minimum number of genes whose expression needs to be controlled, for the cell fate to be determined, is the control kernel associated to that cell type [27].
### 2.2 GRNs as Evolvable Systems
Most of the predictions derived within the framework we have just reviewed rely on the assumption that different cell phenotypes correspond to different attractors of the GRN. While the connection between cell types and attractors is both numerically and experimentally well-motivated, their one-to-one correspondence might be an artifice due to the diminutive size of the mathematical models that were actually solvable in the not-so-distant past, when a small number of attractors were naturally identified with different phenotypes. Recent advances in computing power are finally making the study of much larger networks possible. One interesting example is the recent dynamical model developed by Fumiã and Martins [48] for the integration of the main signaling pathways involved in cancer. The signaling among almost 100 different genes is responsible for 63 different attractors, that eventually correspond to only three phenotypically distinct and incompatible cell fates: apoptotic, quiescent, and proliferative. An important observation is that these phenotypes are determined by just a small subset of values, e.g. the constant activation of the effector caspases in apoptotic cells, within a much larger gene activation pattern.
A second example of several attractors all sharing the same phenotypic identity is already present in the much smaller GRN describing cell-fate determination during Arabidopsis thaliana flower development [33]. Four among ten possible attractors all represent the same inflorescence meristematic cell type.
Therefore, while most of the literature on network robustness and evolvability focuses on individual attractors, the concept of phenotype – that we redefine here as a macro-state of the cell characterized by the activation values (fixed or cycling) of only a subset of gene nodes – seems to be better suited to the idealization of biological systems. While an attractor can represent a phenotype by itself, a phenotype can be compatible with multiple attractors.
Our definition is exemplified in the following diagram, showing the case of a phenotype Φ defined by the expression of the two genes represented by dark gray boxes, and the inaction of the two genes shown in light gray tone:
The expression of any other gene (boxes with a question mark) can take any value, as long as the four genes entering the definition of Φ exhibit the right expression pattern.
A critical motivation for this redefinition of the mathematical identity of a phenotype is a conceptual difficulty which naturally arises in Kauffman’s original framework [5]. Kauffman’s definition loses its strict meaning in light of evolutionary biology, for the identification of the attractors of a boolean network with the phenotypes of a GRN would require – after the regulatory network has lost or acquired genes – the comparison between genotypes of different lengths.
An immediate, advantageous consequence of our definition is that it keeps its meaning even after changes occur in the genotype. As an example, the following diagram, shows the case of a single gene duplication event enlarging the genome expressing Φ:
As this mutation is not affecting any of the defining genes, this change in the size of the genome does not alter our definition of Φ. Therefore, it makes sense to study whether Φ is still expressed by the new GRN, and whether the gene replica is favoring or disfavoring Φ’s basin. As we will show in section 3, these questions can be addressed in quantitative terms. The inclusion of different gene replicas will induces different changes in the relative size of the basin of attraction of Φ, and a good dynamical model of the GRN will be enough to determine the duplication events that increase the basin of Φ. When Φ is selected, these are likely to be the duplication events that are fixated more often.
A second, less relevant difference between our approach and those outlined in previous literature on GRNs, is that we will not require the basin of attraction of a specific cell type to be exactly 100% for the network to be representative of that specific type. Instead, we will only require one basin to be much larger than the remaining others. In the following example, aimed at making these ideas more concrete, we will assume 80% as the threshold a basin needs to exceed for the remaining attractors to be treated as negligible.
### 2.3 A Sample Network
In the next two sections, we will explore the consequences of our approach in studying evolutionary and developmental processes withing the boolean model framework. To anchor our discussion to a concrete example, we will introduce a small, tractable network that we will adopt as a toy model, and modify as needed.
The example is shown in figure 1. It represents the GRN expressing a phenotype that we will call cell type 1. The subset of genes defining the phenotype are represented by nodes 1, 2, and 3. Inheriting the terminology adopted in [49], we will refer to them as the effector genes. The characteristic pattern of cell type 1 is assumed to have effector genes 1 and 3 expressed, and inactive gene 2 (blue = active, white = inactive in figure 1). We will refer to this by saying that cell type 1 has expression pattern 101. In determining the basin of attraction of cell type 1, we will sum the basins of every attractor of the GRN which is compatible with the expression pattern 101, i.e. every pattern of the form 1, 0, 1, x4, . . . , x8, where x4x8 (gray nodes) can take any possible (binary) value representing the expression/suppression of the remaining genes labeled from 4 to 8. We will refer to these other genes as terminal selectors (transcription factors that control cell type-specific gene expression in differentiated cells [50, 51]).
The GRN of figure 1 actually encodes two possible cell types, one being type 1, as we said, the other having the activation pattern 110. We will refer to this second phenotype as cell type 2. Despite encoding both phenotypes, the basin of attraction of cell type 1 includes more than 80% of the possible initial activation patterns, all leading to the equilibrium dynamics characterized by the expression of effector genes 1 and 3, and the suppression of gene 2. This is why we assume this network to be a viable description of cell type 1.
For the reader interested in reproducing our example: The GRN dynamics is modeled using a boolean threshold network (as described in section 2.1), with equal weight edges aij = ±1 for activating/inhibiting links (continuous/dashed line in figure 1), non-zero thresholds b1 = −1 and b5,6,8 = 1. But it is important to remark that none of the conclusions we will draw in the next sections depend on the mathematical details of the example we are using.
## 3 Evolution
We have seen how our re-definition of the phenotypes of a GRN in the framework of boolean, dynamical models is not affected by mutations changing the size of the genome. In this section, we seek to show an explicit example of how this allows the prediction of the genes whose duplication will reinforce a phenotype favored by natural selection and the consequent appearance of regulatory modules over evolutionary time. The underlying idea is exemplified in the following diagram (same example of phenotype Φ adopted in subsection 2.2), with arrows showing the regulation induced by an ideally connected terminal selector which activates exactly the effectors that are supposedly expressed, and suppresses just the genes that are inactive in our former definition of Φ:
It is easy to foresee that the duplication of this ideal terminal selector would positively impact the change in the basin of attraction of Φ. In a more realistic scenario, duplication would be followed by divergence, represented in the following diagram by a difference in one of the effector genes regulated by the replica:
In this section we will consider duplication events like the one just described. For each terminal selector in the GRN, we will consider the entire spectrum of possible events of duplication + divergence, and show ideal pathways leading to cell type changes in a shifting environment. With reference to figure 2, we will describe a simple evolutionary model for the differentiation of progenitor cells of type 1, as encoded in our network from subsection 2.3, into cells of type 2 (already encoded, but unlikely), and the newly born cell type 3, a novel phenotype induced by the modifications the network is going through.
We are assuming here that the knowledge of the environment is equivalent to the knowledge of the selected cell type: they are on the same ontological level, and share the same mathematical definition in our theoretical model. The environmental pressure changes in both space and time, and selects cells that undergo mutation events that favor the emergence of the newly preferred cell type. In this example, sister cells [52] of type 1 start being selected for cell type 2, and undergo mutation events $\textcircled{A}$ and $\textcircled{B}$ until cell type 2 represents the dominant phenotype. In our model, the mutation events are represented by gene duplication and divergence of one of the terminal selectors (nodes 4–8 in fig. 1), during which the network acquires a non-mutated replica of a preexistent terminal selector, and then a mutation in the way the replica regulates the remaining genes and itself.
Let us first provide a mathematical description of what we mean by a perfect replica of a preexistent gene. We will then introduce divergence in the form of a mutation affecting the replica’s downstream signaling.
Non-mutated gene replica: Given a boolean network N with n nodes, we want to study the dynamics of the n + 1-node network N’ obtained by the inclusion of the perfect replica of a node already present in N.
If the dynamics of N is governed by the set of boolean equations
then the updating rules of N’ will have the form
In the previous equations, xi is the value (0 or 1) of node i at time t, while x’i is the simplified notation for the value of the same node at time t+1. The φi functions are new functions that we want to determine, given our knowledge of the functions fi. Without loss of generality, we can assume node n + 1 to be a perfect replica of node 1. Let us then consider the requirements we impose on φi(x1, . . . , xn, xn+1), after we include node n + 1:
1. The assertion that node n + 1 is a perfect replica of node 1 translates into the assumption thatHere we are just stating that node replicas obey the same updating rules as the original nodes.
2. We also want the new node not to affect the system, when not expressed. Therefore,
3. Lastly, we want nodes 2, . . . , n to see node 1 and n + 1 as indistinguishable:
The previous conditions are enough to determine the values taken by the functions φi for any dynamical state of N’ with the only exception being those cases where x1 and xn+1 are both 1. This is the only genuinely new scenario whose output cannot be predicted in terms of an equivalent configuration of N. The problem is easily solved in the case of boolean networks with thresholds (subsection 2.1), as we know the prescription that gives the updating rule of a node in terms of the state of the network, eq. (2). Building the functions φi in a similar fashion, we deduce that it is both natural, and enough, to impose aj+n,i = aji and bi+n = bi for the previous three conditions to be satisfied.
Mutation: The signaling of gene replica i is represented by the string of numbers aij = 0, ±1, with j = 1, . . . , n (n = current number of genes in the network). A mutation is introduced by changing one of these numbers to another possible value. For example, changing aij from 0 to ±1 corresponds to the creation of a new edge. Changing it from ±1 to 0 means deleting an edge which is instead departing from the original node. Changing it from ±1 to ∓1 changes an activating link into an inhibiting one or vice-versa. Different models of mutation could be assumed, which in turn determine the minimum numbers of replicas the cell needs to acquire in order to change from one type to another. For simplicity, our examples show only the sequence of mutations that minimizes the number of steps needed to complete the phenotypic shift. Therefore, we consider all possible mutations of the kind previously described, and then select the one that maximizes the basin of attraction of the preferred cell type.
We can now return to the example shown in fig. 2 and deduce that at least two mutation events are needed to convert sister cells of type 1 to type 2 (activation pattern: 110). By acquiring a mutated replica of terminal selector 7, event $\textcircled{A}$, that inactivates effector gene 3 instead of activating it, the dynamics of the GRN is equally likely to converge toward cell type 1 or 2 (figure 3). The subsequent acquisition of a mutated replica of gene 6, event $\textcircled{B}$, that, differently from the original one, does not activate gene 7, determines the complete shift toward cell type 2 (88% of the configuration space). The highly connected subset of nodes that appear in the B-panel of figure 3 is responsible for the convergence of the dynamics toward the characteristic activation pattern which defines cell type 2. Whatever the function associated to this cell type is, the newly-created module is in charge of it.
The assumption that sister cells of cell type 2 start being selected for cell type 3 (activation pattern: 100), determines the selection of mutation events like $\textcircled{C}$, where the GRN acquires a mutated replica of terminal selector 8 that suppresses gene 3. One mutation event is now enough to determine the complete shift toward cell type 3.
The last example we will consider here is the convergence of cell type 2 back to cell type 1. After the acquisition of mutations $\textcircled{A}$ and $\textcircled{B}$, the preferred cell type is again type 1. This can be achieved (complete shift from less than 12% to more than 96%) with a single mutation event, marked $\textcircled{D}$ in figure 3, consisting of a mutated replica of terminal selector 8 that activates effector gene 3. This last evolution of the GRN expresses the same phenotype as the initial network in figure 1, but carries memory of the evolutionary path that led it through its type 2 period in the form of the module visible in the D-panel of figure 3.
We hope this example is enough to convince the reader that, as long as a viable mathematical description of the GRN is available, as well as the knowledge of the preferred phenotypes for a shifting environment, our approach provides the necessary selected gene duplication and mutation event(s) with mechanistic meaning. The possibility to retain cell type mathematical identities for evolving GRNs is the key features enabling it.
We have also shown that our method allows the identification of GRNs that differ for both genotype and network topology (like the GRN in figure 1 and the evolved network in the D-panel of figure 3) as encoding the same cell type (type 1 in our example). Therefore, apparently redundant topological features in the structure of a GRN might carry the imprint of their evolutionary history, and of the environment that induced them.
## 4 Development
Next we want to show how external control of the expression of accurately chosen terminal selectors – e.g. as under the effect of drug therapies, or because of extracellular signalling during development – can determine cell differentiation, and the expression of not only cell type 2, but also novel cell types that were not initially encoded in our sample network.
For an explanation of cell differentiation not invoking extracellular signaling, and relying instead on cellular noise, the reader is redirected to [53]. We will focus here on the possibility of epigenetic signaling to induce the appearance of new cell types in a Lamarckian fashion [54]. This approach has been extensively investigated as an alternative explanation [55,56] to the current paradigm of treatment-selected, drug-resistant clones in tumor progression [59, 60]. We generalize it here to the broader problem of cell differentiation over a developmental timescale. We also show the role played by our generalized definition of phenotype in reducing the size of the control kernel, as defined in [27].
The original approach developed in [27], consists of taking every possible combination of any subset of genes and their possible expression patterns. For each combination, and each initial state, time series of the transient expression pattern of the remaining genes are generated, while the selected genes are pinned to the specific selected values. When by pinning the smallest number of nodes this operation moves all possible initial conditions toward the basin of attraction of a specific attractor, Kim et al. define that smallest subset of nodes as the control kernel of the attractor state the system is converging to.
For consistency with our less constrained definition of what a phenotype is (subsection 2.2), we need to re-define the notion of control kernel accordingly. The generalized definition of control kernel we adopt here does not assume the forced terminal state to be a specific attractor, just a phenotype (i.e. any possible attractor compatible with the definition of that phenotype). As a result, our control kernels are significantly smaller than those found in [27], even when restricting ourselves to controlling only terminal selectors. To show this explicitly, we have considered again our sample network from subsection 2.3, and the effect of pinning just one terminal selector at a time.
The result is shown in table 1. Forcing the expression of terminal selector 7 (referred to as “TS7 is ON” in the table) is enough to enlarge the basin of attraction of cell type 2 from less than 20% to exactly 100%. Likewise, forcing the expression of TS8 makes the network shift toward a phenotype characterized by the expression pattern 111, that was not even encoded in the network. Eventually, we might assume 80% (or some other high percentage value) to be enough for the network to be considered locked in a certain cell phenotype. In this case TS4 would be an alternative control kernel for the cell type defined by the 111 pattern. This new definition of control kernel, focused on forcing the GRN into a less restrictive activation pattern than a single, specific attractor, seems to be much easier to achieve experimentally, better suited for large/realistic networks, and therefore relevant to drug treatments.
Cell type differentiation in response to extracellular signaling, is still a key aspect of our generalized approach to dynamical GRN models. Even if mainly aimed at incorporating evolutionary features (section 3), our framework entirely preserves (and sometimes facilitates) the applicability of network control theory to GRN models in explaining both developmental or drug-induced change.
## 5 Phenotype Preservation
In their seminal paper on the GRN structures devoted to the development of animal body plans [28], Davidson and Erwin postulated the existence of a modular structure for these networks: more ancient gene modules, assemble early, and are strongly selected for, because of the vital nature of the function they perform. A similar, modular approach can be recognized in [18], and our redefinitions of the phenotypes (section 2) naturally reflects Hartwell et al.’s point of view: “If the function of a protein were to directly affect all properties of the cell, it would be hard to change that protein, because an improvement in one function would probably be offset by impairments in others.”
We have already shown that our many-to-one activation patterns to phenotype map favors modularity. Our aim in this section is to show that it also gives rise to a nested phenotypic structure which makes functions less likely to be disrupted the earlier it gets selected.
We have seen how a phenotype is defined in terms of the activation values of just a subset of nodes in the gene expression pattern. If, for example, Φ1,1 is defined by the activation of just n1,1 out of n nodes in our boolean model, then there will be 2n−n1,1 possible microstates still compatible with it, for expressing Φ1,1 is not constraining the entire activation pattern. This also means that, after the selection of Φ1,1, there will be still room for the selection of additional functions. Let Φ2,1 and Φ2,2 be two of them (see figure 4), characterized by the expression pattern of n2,1 and n2,2 additional effector genes, respectively, and let us assume they both get selected. Let us refer to these two novel functions as generation 2 phenotypes, while Φ1,1 will belong to first generation. We can expect this process to continue until the present time, with more recent phenotypes defined by more and more detailed activation patterns of an increasing number of effector genes, as required by the fulfillment of the definitions of all the phenotypes selected over the previous generations. It is important to keep in mind that such extrapolation is possible because we can retain the same definitions of the phenotypes Φi,j over successive generations, even if we know the genotype has very likely changed in the meantime.
With reference to figure 4, while phenotype Φ1,1 is present as long as n1,1 effector genes exhibit the right expression pattern, phenotype Φ5,4 will rely on the right activation of
effector genes.
This also means that more recent traits are a more likely target of mutations and, therefore, more easily disrupted, while older ones, on top of being favored by natural selection, are naturally sheltered from deleterious mutations.
## 6 Conclusions
This manuscript represents a synthesis of mathematical models of gene regulatory networks within theoretical evolutionary biology, which also accounts for development.
Despite of their predictive power, and their possibility to mathematically reinterpret cell fates as the steady states of the abstract, non-linear dynamics of the regulatory processes they describe, dynamical models (boolean models in this manuscript) of GRNs have been limited by this identification between mathematical attractors and biological phenotypes. Evolutionary processes, like single gene and whole genome duplication, both chromosome gain and loss, horizontal gene transfer, etc., alter the size of the genome, and deprive the exact attractor-phenotype identification as a one-to-one map of its original meaning. The leading role played in evolution and speciation by the interplay between these processes and natural selection, makes current dynamical models of GRNs unfit to describe genetic network evolution.
Motivated by this limitation, we have explored the consequences of relaxing this one-to-one identification between cell types and attractors. We have shown with a simple numerical model that, redefining a cell type as a collection of attractors of a GRN, all sharing a common, characterizing property, enables retaining the main features of Kauffman’s original theory, and permitting easy generalization using network control theory. It is now no longer needed for the genome to retain a fixed size for the cell type type definition to preserve its meaning.
This opens the possibility to quantitative studies of the effect that gene duplication, mutation, and natural selection have on cell types, by virtue of allowing the study of evolutionary processes within the well-established framework of dynamical systems theory.
More specifically, we have shown the emergence of elementary GRN modules responsible for cell fate determination as an immediate consequence of the selection processes of optimal evolutionary mutations and gene duplication events. We have incorporated the formalism of network controllability, preserving its biological interpretation as a viable mathematical description of extracellular/epigenetic control of gene expression in an evolutionary model. Finally, we have shown that the remaining degrees of freedom arising from the identification of the phenotypes of the GRN with the macrostates, as opposed to the states, of the dynamical network gives rise to a nested phenotypic structure, and that this structure naturally makes more ancient phenotypes less likely to be disrupted by mutations.
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# Below is an excerpt from a letter that was sent by the
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Manager
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Below is an excerpt from a letter that was sent by the [#permalink] 02 May 2005, 00:59
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Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporation’s unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
(C) The chairman is innocent of any criminal offense.
(D) The corporation has expanded steadily over the past six years.
(E) Any legal proceedings against the chairman have resulted in his acquittal.
Source: Arco or 1000 Series
Director
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C and E are nowhere close.
A is out because the word "all" is used.
B is out because the paragraph does not say the misdeeds or charges are business or personal
I would go with D. The paragraph says six year of unbroken growth. Expanded means enhance bulk or volume and is directly related to growth while unbroken is directly related to steady as it mean uninterrupted or continuous.
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Below is an excerpt from a letter that was sent by the [#permalink] 05 Oct 2007, 11:09
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporations unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
A. The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
B. Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
C. The chairman is innocent of any criminal offense.
D. The corporation has expanded steadily over the past six years.
E. Any legal proceedings against the chairman have resulted in his acquittal.
Director
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Re: CR- Chairman [#permalink] 05 Oct 2007, 12:37
GMATBLACKBELT wrote:
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporations unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
A. The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
B. Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
C. The chairman is innocent of any criminal offense.
D. The corporation has expanded steadily over the past six years.
E. Any legal proceedings against the chairman have resulted in his acquittal.
A) too strongly worded - "all" we do not know for sure if this is so
B) this is beyond the scope of the argument, we do not know if the company's record growth was the result of any misdeeds of the chairman
C) the chairman may or may not be, but hasn't been found guilty (maybe later)
D) correct as found in the stem: "as the corporations unbroken six-year record of growth will show"
E) nothing in the stem notes this, we have to assume beyond the scope of what we are told
Manager
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Re: CR- Chairman [#permalink] 05 Oct 2007, 13:51
I will go with A.
Individuals seeking to control.....have demanded my resignation.
So chairman thinks all those who demand his regination are motivated to ......
In D , I feel growth does not mean expansion.
Whats the OA?
Manager
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i like E [#permalink] 05 Oct 2007, 14:09
E seems to me to be the answer pretty clearly. D is meant to be a popular wrong answer. problem with d is 'expansion' vs. growth. growth he infers in the passage means a monetary growth. expansion means size. the corporaton may have grown (in terms of where it reaches customers) and sold more product, but not necessarily expanded.
this question is real tough. But I think the distinction is but slight, but there. and E is correct.
Manager
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also` [#permalink] 05 Oct 2007, 14:17
i got a hunch oa might be d, and i would disagree. consider it says 'grown steadily'. that means at the same rate each year. the passage does not infer that. just that it has grown each year. each year could have been quite different in its rate of growth.
very difficult question. i do like E.
i could see e being wrong though because it does not infer his acquittal just that he has not been found guilty - meaning he could still be in the process of several court cases.
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CR - letter from a chairman [#permalink] 09 Dec 2007, 17:36
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporation’s unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
(C) The chairman is innocent of any criminal offense.
(D) The corporation has expanded steadily over the past six years.
(E) Any legal proceedings against the chairman have resulted in his acquittal.
Current Student
Joined: 03 Jan 2007
Posts: 42
Followers: 0
Kudos [?]: 1 [0], given: 0
Re: CR - letter from a chairman [#permalink] 09 Dec 2007, 21:47
bulls wrote:
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporation’s unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
(C) The chairman is innocent of any criminal offense.
(D) The corporation has expanded steadily over the past six years.
(E) Any legal proceedings against the chairman have resulted in his acquittal.
"A" -> "Individuals seeking to control the corporation for their own purposes have demanded my resignation."
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
-> Can't be inferred
(C) The chairman is innocent of any criminal offense.
-> Who knows?
(D) The corporation has expanded steadily over the past six years.
-> Steadily? Expanded? <-> unbroken six-year record of growth
(E) Can't be inferred.
Current Student
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Re: CR - letter from a chairman [#permalink] 09 Dec 2007, 23:22
GMAT TIGER wrote:
alnasl wrote:
bulls wrote:
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporation’s unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
(C) The chairman is innocent of any criminal offense.
(D) The corporation has expanded steadily over the past six years.
(E) Any legal proceedings against the chairman have resulted in his acquittal.
"A" -> "Individuals seeking to control the corporation for their own purposes have demanded my resignation."
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
-> Can't be inferred
(C) The chairman is innocent of any criminal offense.
-> Who knows?
(D) The corporation has expanded steadily over the past six years.
-> Steadily? Expanded? <-> unbroken six-year record of growth
(E) Can't be inferred.
Hard to tell whether it has been expanded steadily.
-6 yr 0.0001% Growth
-5 yr 0.0001% Growth
-4 yr 0.0001% Growth
-3 yr 50% Growth
-2 yr 0.0001% Growth
-1 yr 0.0001% Growth
= unbroken six-year record of growth
=! has expanded steadily over the past six years
BTW, what's the OA?
Manager
Joined: 11 Aug 2007
Posts: 64
Followers: 1
Kudos [?]: 10 [0], given: 0
Re: CR - letter from a chairman [#permalink] 10 Dec 2007, 04:38
bulls wrote:
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporation’s unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
(C) The chairman is innocent of any criminal offense.
(D) The corporation has expanded steadily over the past six years.
(E) Any legal proceedings against the chairman have resulted in his acquittal.
It's D by POE:
(a) "all those" is to extreme and never mentioned. Statement says only about "Individuals". Probably, somebody wants his resignation for other reasons. We don't know
(b) again, nothing helps us conclude that "any misdeeds". "Any" is too extreme
(c) the same: "innocent of any criminal offense"? We don't know. Maybe he stole a pen from his classmate back in college:)
(d) exactly! "expanded steadily over the past six years" = "corporation’s unbroken six-year record of growth"
(e) again, "any" is too extreme. Probably, he was adjudged guilty for stealing the pen:) We don't know.
Manager
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Below is an excerpt from a letter that was sent by the [#permalink] 07 Jan 2008, 15:16
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporation’s unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
(C) The chairman is innocent of any criminal offense.
(D) The corporation has expanded steadily over the past six years.
(E) Any legal proceedings against the chairman have resulted in his acquittal
Director
Joined: 12 Jul 2007
Posts: 865
Followers: 13
Kudos [?]: 221 [0], given: 0
Re: CR - Chairman of corporation [#permalink] 10 Jan 2008, 07:26
Quote:
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
Individuals seeking to control the corporation for their own purposes have demanded my resignation
Assumption A can be properly inferred from the passage.
Quote:
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
He doesn't say anything that leads us to believe he even committed any misdeed, much less that he committed them for the good of the company (we don't even know what he's accused of doing! maybe it's embezzling company funds)
Quote:
(C) The chairman is innocent of any criminal offense.
We don't know whether or not he's innocent, we just know that he hasn't been convicted. He could be guilty of Murder 1 for all we know.
Quote:
(D) The corporation has expanded steadily over the past six years.
the corporation’s unbroken six-year record of growth
We don't know if it's expanded steadily over the past six years or has had great fluctuations in how much it's grown each year.
Quote:
(E) Any legal proceedings against the chairman have resulted in his acquittal
no court of law in any state has found me guilty of any criminal offense whatsoever
Have they? We know none of them have resulted in his conviction, but perhaps they resulted in mistrials or something of that nature. There are more than two outcomes of a trial, we can't assume that because he wasn't convicted he was acquitted.
I'd love to get an OA ruling on this one though.
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Re: CR - Chairman of corporation [#permalink] 16 Jan 2008, 22:49
Inference questions are special in that the solutions to these questions directly follow from the passage and generally won’t include any extreme words.
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes. [Intriguing - but goes overboard by stating extreme – “all those who demanded his resignation …” – Eliminate it]
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation. [Irrelevant to the argument - eliminate it]
(C) The chairman is innocent of any criminal offense. [Hold it]
(D) The corporation has expanded steadily over the past six years. [Irrelevant to the argument - eliminate it]
(E) Any legal proceedings against the chairman have resulted in his acquittal [Irrelevant to the argument - eliminate it]
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Below is an excerpt from a letter that was sent by the [#permalink] 28 Jan 2008, 05:26
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporation’s unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
(C) The chairman is innocent of any criminal offense.
(D) The corporation has expanded steadily over the past six years.
(E) Any legal proceedings against the chairman have resulted in his acquittal.
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Re: cr -1000 [#permalink] 28 Jan 2008, 13:38
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.(Stated in the passge)
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.(Hold It)
(C) The chairman is innocent of any criminal offense.(Not a good inference)
(D) The corporation has expanded steadily over the past six years.(Stated in he passage
(E) Any legal proceedings against the chairman have resulted in his acquittal.(Out of scope)
left with B
OA?
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Below is an excerpt from a letter that was sent by the [#permalink] 03 Jun 2008, 11:03
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporations unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
(C) The chairman is innocent of any criminal offense.
(D) The corporation has expanded steadily over the past six years.
(E) Any legal proceedings against the chairman have resulted in his acquittal.
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Re: CR-Excerpt from a letter [#permalink] 03 Jun 2008, 13:10
B.
Here is my reasoning in red next to each answer.
What's the OA?
apurva1985 wrote:
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporations unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
The question states as much so it can't be an inference.
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
Even though this statement starts outwith "any", it later says "may have committed". The entire answer seems to only make one step from the passage. He passage doesn't state what his motivations were, but it is implied by his reference to the benefit to the company. It's not necessarily true, but it is inferred (in my opinion).
(C) The chairman is innocent of any criminal offense.
This answer uses the word "any". Generally speaking, when an answer uses such an absolute word, it's a red flag that it is wrong.
(D) The corporation has expanded steadily over the past six years.
Again, the passage actually states as much, so this isn't really an inference.
(E) Any legal proceedings against the chairman have resulted in his acquittal.
This goes one step (or more) too far. The passage says nothing about legal proceedings against the chairman, so the next step would be he was acquitted. When thinking of inferences, find something that goes just 1 step further. This goes 1 step to legal proceedings and then a 2nd step to an acquittal from those proceedings.
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Re: CR-Excerpt from a letter [#permalink] 03 Jun 2008, 19:35
Actually I think that D makes sense as I was torn betwen A and D. The CR is asking what can be inferred from the passage:
Below is an excerpt from a letter that was sent by the chairman of a corporation to the stockholders.
A number of charges have been raised against me, some serious, some trivial. Individuals seeking to control the corporation for their own purposes have demanded my resignation. Remember that no court of law in any state has found me guilty of any criminal offense whatsoever. In the American tradition, as you know, an individual is considered innocent until proven guilty. Furthermore, as the corporations unbroken six-year record of growth will show, my conduct of my official duties as chairman has only helped enhance the success of the corporation, and so benefited every stockholder.
Which of the following can be properly inferred from the excerpt?
(A) The chairman believes that all those who have demanded his resignation are motivated by desire to control the corporation for their own purposes.
This comes close in my mind as he states that" Individuals seeking to control the corporation for their own purposes have demanded my resignation" however my reasoning here is that there is no way to infer this includes EVERYBODY...he mentions individuals which could be a few, some or many but by no means all
(B) Any misdeeds that the chairman may have committed were motivated by his desire to enhance the success of the corporation.
Absolutely not! There is no way to infer this, the chairman mentions them as two separate reasons
(C) The chairman is innocent of any criminal offense.
He could be lying his ass off and be behind the next Enron scandal. We can't say he's not guilty just because he says so!
(D) The corporation has expanded steadily over the past six years.
He says "as the corporations unbroken six-year record of growth will show". CLEARLY you dont have to infer much to believe this right? Bingo!
(E) Any legal proceedings against the chairman have resulted in his acquittal.
Just remember, at times, as is the case with RC, an inference question only goes as far as you can throw a stick and is very close to something already stated in the passage maybe with a different play on words
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Re: CR-Excerpt from a letter [#permalink] 03 Jun 2008, 19:43
actually i disagree with D..
if anything the corp has grown..by record numbers..that is hardly any indication of steady growth..it has grown explosively..
i think A is best..
what is the source of this question.?
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Re: CR-Excerpt from a letter [#permalink] 03 Jun 2008, 19:51
That's a good point, however I think a "six year record of growth" could also mean that yeah the company has grown by 2% or 5% etc a year which even by a compounded effect is not something to write home about.
I think it really comes down to picking between the "lesser evil" here. My main issue with A is the use of ALL which is a big leap of faith from "individuals". One that may be bigger than assuming that a "six year record of growth" equates to "expanded steadily over the past six years"...
Not a fun question that's for sure....
Re: CR-Excerpt from a letter [#permalink] 03 Jun 2008, 19:51
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2015-05-26 00:22:43
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https://gamedev.stackexchange.com/questions/71103/how-to-draw-non-triangulated-mesh/71104
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# How to draw non-triangulated mesh?
I am working with DirectX (C#/C++). I am wondering is it possible to do not triangulate meshes and what the difference between cases (in loading and rendering code)? If so, how to do this? I know DrawIndexed requires primitive_topology to be set - which value should I use then? I always used triangulated mesh before with no problems but it is interesting..
• Look up Geometry Shaders. There are the only effective way to draw non-triangulated meshes. Then again, you're more of generating them on the GPU side rather than "drawing" a mesh that you've previously created. – Thebluefish Mar 1 '14 at 17:30
• Thanks! Ill try them after finishing with learning PS/VS. – Croll Mar 1 '14 at 23:22
Using directx you must draw triangle meshes. If your mesh data contains quads, then create an index buffer that indexes each quad as two triangles and draw by calling DrawIndexed. For example if you have vertex1,vertex2,vertex3,vertex4 which describes a quad then index them by 0,1,2,0,2,3 or according to the desired winding order.
• So i have to triangulate vertices anyway. Nice. – Croll Feb 27 '14 at 19:48
• @DmitrijA - That's the way GPU hardware works, and DX doesn't pretend that other primitive types exist, so yes, you're stuck with triangles. – Maximus Minimus Feb 27 '14 at 19:58
Do you mean draw stuff like cubes, rectangles.. etc?
It´s easy!
Per example for a rectangle (square or something with 4 vertices) :
CUSTOMVERTEX Dirt[] =
{
{ 0.0f, 600.0f, 0.0f, 1.0f, D3DCOLOR_XRGB(221, 216, 148), },
{ 700.0f, 600.0f, 0.0f, 1.0f, D3DCOLOR_XRGB(221, 216, 148), },
{ 0.0f, 700.0f, 0.0f, 1.0f, D3DCOLOR_XRGB(221, 216, 148), },
{ 700.0f, 700.0f, 0.0f, 1.0f, D3DCOLOR_XRGB(221, 216, 148), },
};
instead of having only 3 vertices
then d3ddev->CreateVertexBuffer(4 * sizeof(CUSTOMVERTEX), 0, CUSTOMFVF, D3DPOOL_MANAGED, &v_buffer, NULL); instead of 3 * sizeof....
and atlast : d3ddev->DrawPrimitive(D3DPT_TRIANGLESTRIP, 0, 2); instead of D3DPT_TRIANGLELIST and change the uiPrimitiveCount to 2...
It also explains cubes and stuff...
Sorry if I didnt helped much
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2019-09-23 01:24:01
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https://www.cheenta.com/smo-senior-2014-problem-4-number-theory/
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Select Page
# Understand the problem
For each positive integer $n$ let $$x_n=p_1+\cdots+p_n$$ where $p_1,\ldots,p_n$ are the first $n$ primes. Prove that for each positive integer $n$, there is an integer $k_n$ such that $x_n
##### Source of the problem
SMO (senior)-2014 stage 2 problem 4
Number Theory
Medium
##### Suggested Book
Excursion in Mathematics
Do you really need a hint? Try it first!
We have $P_1 = 2 , P_=3 , P_3=5 , P_4 =7 , P_5 = 11 \ and \ so \ on ….$. Now to understand the expression $x_n , observe . $For \ n=1 \ , \ 2 < 2^2 < 2+3$ $For \ n=2 \ , \ 2+3 < 3^2 < 2+3+5$ $For \ n=3 \ , \ 2+3+5 < 4^2 < 2+3 +5+7$ $For \ n=4 \ , \ 2+3 +5+7 < 5^2 <2+3 +5+7 +11$ Now proceed to prove $\forall n \geq 5$ .
Observe $\forall n \geq 5$ we have $P_n > (2n-1)$. [where $n \in Z^+$] Then try to use $x_n = P_1 + P_2 + …+P_5+….. +P_n > 1 +3 + ….+ 9 +… (2n-1) = n^2 \\ \Rightarrow x_n > n^2 , \forall n \geq 5[where \ n \in Z^+]$ .
Think if $x_n=P_1 + P_2 + …. + P_5 +…P_n = b^2 for \ some \ n , b \in Z^+$ , then we are done . If not so , then think $m$ be the largest non negative integer such that $(n+m)^2 < x_n$ . Now note that the next perfect square is $(n+m+1)^2$ . Observe that if we can prove that $(n+m+1)^2 – (n+m)^2 = (2n+ 2m +1) \geq P_{n+1}$ , then we are done . Now try to verify this claim .
Suppose our claim is not true i.e. $P_n < 2n + 2m +1$ . So, $P_n < 2n + 2m +1 \\ \Rightarrow 2n+ 2m \geq P_n , \forall n \in Z^+ \\ \Rightarrow (2n +2m-2)+(2n+ 2m -4)+…..2m \geq P_n + P_{n-1}+……+P_1 \\ \Rightarrow n^2 + 2mn -n \geq P_n + P_{n-1}+……+P_1 \\ \Rightarrow n^2 + 2mn -n \geq x_n \\ \Rightarrow n^2 + 2mn +m^2 > n^2 + 2mn -n\geq x_n \\ \Rightarrow (n+m)^2 > x_n$ . Contradiction! since we have assumed $x_n = P_1 + P_2+……+P_{n-1} > (n+m)^2$ . Thus ,$(n+m+1)^2 \in (x_n , x_{n+1})$ .
# Connected Program at Cheenta
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2020-07-05 11:07:45
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https://socratic.org/questions/what-is-the-slope-of-a-line-with-the-equation-y-3-5-x-2-1
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# What is the slope of a line with the equation (y + 3) = 5(x - 2)?
$\left(y + 3\right) = 5 \left(x - 2\right)$ has a slope of $5$
The general slope point form of a line through a point $\left(\textcolor{red}{\overline{x}} , \textcolor{red}{\overline{y}}\right)$ with slope $m$ is
$\textcolor{w h i t e}{\text{XXX}} y - \textcolor{red}{\overline{y}} = \textcolor{b l u e}{m} \left(x - \textcolor{red}{\overline{x}}\right)$
From this general form we can see that the given equation has a slope of $\textcolor{b l u e}{5}$
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2022-10-06 10:22:39
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http://clay6.com/qa/43277/for-a-crystal-system-a-b-c-and-alpha-beta-gamma-neq-90-
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Browse Questions
# For a crystal system a = b = c and $\alpha=\beta=\gamma\neq 90^{\large\circ}$
rhombohedral
Hence (C) is the correct answer.
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2017-03-30 04:56:06
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https://developers.here.com/documentation/isoline-routing-api/dev_guide/topics/use-cases/reverse-time-isoline.html
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# Calculate reverse direction isoline routing
This tutorial calculates an isoline in the reverse direction. The result is the answer to the question: from where can I get to the destination within a given range? This can be useful when determining which vehicles from a fleet of vehicles, such as a taxi service, can get to the destination within a given time. To trigger calculation in reverse direction, use the destination parameter instead of origin.
curl -X GET \
'https://isoline.router.hereapi.com/v8/isolines?transportMode=car&destination=52.51578,13.37749&range[type]=time&range[values]=600'
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2022-09-30 22:51:36
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http://openstudy.com/updates/513e977ae4b029b0182c362b
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## mathishard2 Group Title A man has five pairs of socks (no two pairs are the same color). If he randomly selects two socks from a drawer, what is the probability that he gets a matching pair? one year ago one year ago
• This Question is Open
1. izzy56658989 Group Title
1/2 chances or 5/10
2. mathishard2 Group Title
The answer in the book says 1/9...
3. izzy56658989 Group Title
i never said it was gonna be right there was a half chance i was wrong
4. mathishard2 Group Title
I'm not trying to show that you're wrong... I just wanted to see if someone could explain this question to me. Because math is hard. And I have a test on probablility tomorrow.
5. izzy56658989 Group Title
k there's 5 there it's an uneven number which means if u tryed to pair them all up 1 would be left behind
6. izzy56658989 Group Title
lik if u got rid of that 1 sock and did 4x2 =8+1=9
7. izzy56658989 Group Title
there's ur denominator
8. satellite73 Group Title
he picks one sock it is some color
9. satellite73 Group Title
then there are 9 socks left in the drawer, and only one has the same color as the sock he picked originally
10. izzy56658989 Group Title
exactly it never said they were the same colors
11. satellite73 Group Title
so the probability that the next sock he picks has the same color is $$\frac{1}{9}$$
12. satellite73 Group Title
it is easier to think about it if you imagine picking one at a time instead of two at once
13. satellite73 Group Title
we can also solve with thinking of picking two at once the number of ways he can pick the two socks out of the total of ten socks is $\dbinom{10}{2}=\frac{10\times 9}{2}=5\times 9=45$ of those 45 ways, 5 of them are pairs that match, so you get the probability as $\frac{5}{45}=\frac{1}{9}$
14. satellite73 Group Title
i think the first way is easier to envision, but it is a matter of taste i suppose
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2014-08-01 15:59:31
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https://agua.tidymodels.org/reference/h2o_tune.html
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Control model tuning via h2o::h2o.grid()
Usage
agua_backend_options(parallelism = 1)
Arguments
parallelism
Level of Parallelism during grid model building. 1 = sequential building (default). Use the value of 0 for adaptive parallelism - decided by H2O. Any number > 1 sets the exact number of models built in parallel.
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2023-04-01 06:46:28
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http://www.physicsforums.com/showthread.php?t=522775
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## Pressure and Ionisation
How does the pressure change the amount by which air ionizes? I know that high pressure suppresses ionization. But why?
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2013-06-19 02:48:19
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https://www.gamedev.net/forums/topic/281363-static_cast/
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# static_cast
## Recommended Posts
Does static_cast cause the argument to be converted right there on the spot, or does it only store the cast into the LHS? For example, this code is throwing an access violation error when it gets to the first delete[] statement:
// real_out and im_out are double** arrays
for (int i = 0; i < height; i++)
for (int j = 0; j < width; j++)
{
fftData[i][k] = static_cast<float>(real_out[i][j]);
fftData[i][k + 1] = static_cast<float>(im_out[i][j]);
}
for (int i = 0; i < MRCHeight; i++)
{
delete[] real_out[i]; <=== ERROR HERE! ACCESS VIOLATION!
delete[] im_out[i];
}
delete[] real_out;
delete[] im_out;
However, if I comment out the static_cast statements, no problems. Is this always the case? For some reason I thought the cast was only applied to a return value...not the actual value itself. If this really is what is causing the problem, what cast should I use to make sure the fftData value is properly assigned as a float?
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cant use my hands ;-) .. post deleted
for (int j = 0; j < width; j++)
{
fftData[i][k] = (float)(real_out[i][j]);
fftData[i][k + 1] = (float)(im_out[i][j]);
}
[Edited by - hoLogramm on November 8, 2004 4:48:57 AM]
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Quote:
Original post by StonieJFor some reason I thought the cast was only applied to a return value...not the actual value itself.
That is correct. A cast actually creates a temporary rvalue of the target type. The original lvalue is not modified.
I suspect you are writing past the end of your fftData array, trashing the bookkeeping headers for the real_out and im_out arrays.
i.e. it's the assignment, not the cast which is at fault.
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Quote:
Original post by hoLogrammtry this instead:for (int j = 0; j < width; j++) { fftData[i][k] = (float)(real_out[i][j]); fftData[i][k + 1] = (float)(im_out[i][j]); }
If the jist of your post is that old C casts are preferrable over the new C++ cast operators then I disagree very strongly. The C++ cast operators are much more explicit and you actually have to stop and think a second what kind of cast you really need/want rather than just casting away for the heck of it.
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There's almost certainly a bug in that original code because the first nested loops are equivalent to:
for (int i = 0; i < height; i++){ fftData[i][k] = static_cast<float>(real_out[i][width - 1]); fftData[i][k + 1] = static_cast<float>(im_out[i][width - 1]);}
The way it is at the moment you just keep overwriting the values width times.
Enigma
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no, it doesn't look like you're overwriting values... except for possibly i. Your indentation makes it look like the deletion loop happens inside the first for(i = 0; i < width...). However, I suspect that's just a cut-and-paste artifact.
Is MRCHeight == height ?
what type is real_out to begin with that you would want to cast?
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The code posted most certainly does overwrite values for width > 1. If you unroll the inner loop of the first double-loop:
for (int i = 0; i < height; i++){ /* unroll this loop for (int j = 0; j < width; j++) { fftData[i][k] = static_cast<float>(real_out[i][j]); fftData[i][k + 1] = static_cast<float>(im_out[i][j]); }*/ fftData[i][k] = static_cast<float>(real_out[i][0]); fftData[i][k + 1] = static_cast<float>(im_out[i][0]); fftData[i][k] = static_cast<float>(real_out[i][1]); fftData[i][k + 1] = static_cast<float>(im_out[i][1]); fftData[i][k] = static_cast<float>(real_out[i][2]); fftData[i][k + 1] = static_cast<float>(im_out[i][2]); // ...}
k never changes, so the values of fftData[i][k] and fftData[i][k + 1] are written multiple times.
Enigma
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I agree that using C++ style casts' is almost always preferable to C cast.
Just for the simple fact that you can much easier grep for them. ie finding all static_cast<float>(..) is alot easier to find that (float).
The latter would produce alot of false positives. for things like funciton prototypes and definitions with no parameters.
Cheers
Chris
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Crap, perhaps I should have posted the whole algorithm. I left out the k terms because I just got lazy, but I will include them now. Ultimately, this is what I'm trying to do:
1) Read in two arrays of data.
2) Merge them into a single array of data.
Shouldn't be too hard I'd think. Here is the complete, relevant code segment with my error code (i.e. I left out some stuff that simply wasn't vital):
void CEMData::FFT(){ double **real_in = new double*[MRCHeight]; double **im_in = new double*[MRCHeight]; double **real_out = new double*[MRCHeight]; double **im_out = new double*[MRCHeight]; try { if (!real_in) throw Exception("Possibly out of memory."); if (!im_in) throw Exception("Possibly out of memory."); if (!real_out) throw Exception("Possibly out of memory."); if (!im_out) throw Exception("Possibly out of memory."); for (int i = 0; i < MRCHeight; i++) { real_in[i] = new double[MRCWidth]; im_in[i] = new double[MRCWidth]; real_out[i] = new double[MRCWidth]; im_out[i] = new double[MRCWidth]; } for (int i = 0; i < MRCHeight; i++) { if (!real_in[i] || !im_in[i] || !real_out[i] || !im_out[i]) throw Exception("Possibly out of memory."); } // the imaginary input array should be all zeros for (int i = 0; i < MRCHeight; i++) ZeroMemory(im_in[i], sizeof(double) * MRCWidth); // POPULATE INPUT ARRAYS (not shown) // do the one-dimensional transform on the rows for (int i = 0; i < MRCHeight; i++) fft(MRCWidth, real_in[i], im_in[i], real_out[i], im_out[i]); // permute the rows and store them in the input variables (since we will be inputting them again) for (int i = 0; i < MRCHeight; i++) { for (int j = 0; j < MRCWidth; j++) { real_in[i][j] = real_out[j][i]; im_in[i][j] = im_out[j][i]; } } // do the one-dimensional transform on the rows again (which used to be the columns) for (int i = 0; i < MRCHeight; i++) fft(MRCWidth, real_in[i], im_in[i], real_out[i], im_out[i]); // delete the input data since we're done with it for (int i = 0; i < MRCHeight; i++) { delete[] real_in[i]; delete[] im_in[i]; } delete[] real_in; delete[] im_in; // convert the transform data from double to float // HERE IS WHERE THE PROBLEMS START OCCURRING fftData = new float *[MRCHeight]; if (!fftData) throw Exception("Possibly out of memory."); for (int i = 0; i < MRCHeight; i++) fftData[i] = new float[MRCWidth]; /* NOTE FOR THE INNER LOOP...YOU ONLY NEED THE FIRST W/2 + 1 COLUMS OF THE DATA...NOT THE ENTIRE WIDTH */ for (int i = 0; i < MRCHeight; i++) { int k = 0; for (int j = 0; j < MRCWidth / 2 + 1; j++) { fftData[i][k] = static_cast<float>(real_out[i][j]); fftData[i][k + 1] = static_cast<float>(im_out[i][j]); k += 2; } } // delete the original output data since we're done with it for (int i = 0; i < MRCHeight; i++) { delete[] real_out[i]; <=== ERROR HERE!!! delete[] im_out[i]; } delete[] real_out; delete[] im_out;}
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Two things that look suspicious:
1.
for (int i = 0; i < MRCHeight; i++) { for (int j = 0; j < MRCWidth; j++) { real_in[i][j] = real_out[j][i]; im_in[i][j] = im_out[j][i]; } }
This will only work if MRCHeight == MRCWidth otherwise you are reading out of bounds of real_out and im_out.
2.
int k = 0; for (int j = 0; j < MRCWidth / 2 + 1; j++) { fftData[i][k] = static_cast<float>(real_out[i][j]); fftData[i][k + 1] = static_cast<float>(im_out[i][j]); k += 2; }
fftData[i] is defined to be an array of size MRCWidth but you are writing ((MRCWidth / 2) + 1) * 2 items to it, or MRCWidth + 2 items, so you are overwriting the bounds of the array.
Enigma
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Quote:
Original post by EnigmaThere's almost certainly a bug in that original code because the first nested loops are equivalent to:for (int i = 0; i < height; i++){ fftData[i][k] = static_cast(real_out[i][width - 1]); fftData[i][k + 1] = static_cast(im_out[i][width - 1]);}
I don't see how this code overwrites any data.
And, while the new listing writes past the end of the array, it's very odd that it doesn't crash without the cast.
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The code I posted and you quoted does not overwrite values. The original code was:
for (int i = 0; i < height; i++)for (int j = 0; j < width; j++){ fftData[i][k] = static_cast<float>(real_out[i][j]); fftData[i][k + 1] = static_cast<float>(im_out[i][j]);}
Which simply writes real_out[i][0] to fftData[i][k] and then immediately overwrites that by writing real_out[i][1] to fftData[i][k] on the next iteration of the loop. The code I posted and you quoted is functionally equivalent but rather that overwriting each time it simply writes the final value directly.
The point is rather moot now that StonieJ has posted the real code he was using. While attempting to cut down code for forum posting is a Good Thing™ you have to be careful to retain the real problem. I originally highlighted the loop because I assumed he was simply using the wrong variable to index into fftData.
Enigma
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Use asserts!!!
If your program causes an access violation when calling delete then you HAVE written to memory that you shouldn't have. In ~95% of cases it's past the end of an array.
The only question is where. If you comment out a line of code and it works, then that line of code is probably where it was overwriting an array.
Make sure you always use asserts to verify that you are within the bounds or your array. Add them now and it should show you where problem is.
Notice how no one ever posts code that causes an access violation on delete, where they did actually use asserts? That's because the asserts found the problem for those people already.
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2017-11-21 10:35:53
|
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https://svhol.pbmichel.com/exam
|
exam
# Exam Questions
The idea of this page is to collect questions which might be asked in the exams. Whenever you review the slides or work on exercises, just put anything here which might be asked.
I guess it will be most easy for the moderating group to put questions about the topic they present here, too.
Please try to maintain categories of useful size, so each gets its own edit button.
## General
Please give a short overview over the lecture.
• Intorduction: What is Verification and Specification? What is a Logic? –> Propositional logic
• Functional programming and specification (ML & Isabelle/HOL)
• HOL: language and semantic aspects (e.g. types, terms, theory, cons. ext., …)
• Proofs in HOL
• Specific things in HOL: sets, functions, relations, and fixpoints
• Verifying functions
• Inductively defined sets
• Specifiaction of PL semantics (operational-, denotational- & axiomatical semantics)
• –> programm verification and programm logic (Hoare Logic)
What is Verification?
• argument for building the system right (NOT building the right system –> validation)
What is Specification?
• defining all possible behaviours of the specified system
What is the difference between model- and property-oriented specifications?
• Model-oriented is based on well-defined mathematical objects like sets and functions, which are used to construct a representation of the system state, as well as the operations on these states. Thus, such specifications reason about a transition system.
• Property-oriented is purely declarative. It uses some logic language to express the properties of the functions in the system.
What is a Model?
## Core HOL
What's the most important thing that needs to be proven when defining types in a HOL theory $T = (\chi, \Sigma, A)$?
• Proof that the newly introduced term $S$ of type $r \Rightarrow bool$ has at least one element inside. Formally $T \vdash \exists x.\; S\; x$
Why is it important that the new types defined in HOL are non-empty?
• (see FAQ for hints)
How are the elements True and False introduced in HOL?
• Since there should always be a distinguished infinite set in U whose elements and subsets are also fully contained in U, there's a guarantee that there will be a set with two elements (say the subset {0,1} of the infinite set) and that one is going to be mapped further on to the values True and False 1)
Why must all functions be total in Isabelle/HOL?
How does the automatic proof of termination work for definition, primrec, and fun?
What is the general procedure for proving termination manually in function?
## Calculi
Why do we need calculi.
• Basically, calculi are the tools we use to derive (and prove) theorems. Given a set of valid formulas and a sound calculus, new valid formulas (tautologies) can be derived. As far as I know, the semantic notion of “validity” corresponds to “provability” in the calculus, since it does not consider any notion of “truth”.
Please explain the following properties: Soundness (Correctness), Completeness, …
• A deductive system is sound if all provable formulas are valid. It is complete if all valid formulas are provable.
• Validity is a semantic property ($A \models B$), while provability a syntactic one ($A \vdash B$). So, in other words we have:
• Soundness: $A \vdash B \longrightarrow A \models B$
• Completeness: $A \models B \longrightarrow A \vdash B$
What are the advantages/disadvantages of the Hilbert calculus compared to the Gentzen calculus.
How can we prove formulas without a calculus?
• Calculi are based purely on syntax. We can always rely on semantic proofs, e.g. truth table or tableaux methods.
1)
Did I miss something?
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2019-06-16 17:04:40
|
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https://alecospapadopoulos.wordpress.com/category/uncategorized/
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Evaluating the CDF of the Skew Normal distribution
Posted: April 4, 2020 in Research papers, Uncategorized
Tags: , , ,
A paper I wrote together with Christine Amsler and Peter Schmidt (yes, I cannot resist to say, the Peter Schmidt of the KPSS time series stationarity test, and one of the founders of Stochastic Frontier Analysis), has just been approved for publication in a special issue of Empirical Economics that will be dedicated to efficiency and productivity analysis. The paper is
Amsler C, A Papadopoulos and P Schmidt (2020). “Evaluating the CDF of the Skew Normal distribution.” Forthcoming in Empirical Economics. Download the full paper incl. the supplementary file.
ABSTRACT. In this paper we consider various methods of evaluating the cdf of the Skew Normal distribution. This distribution arises in the stochastic frontier model because it is the distribution of the composed error, which is the sum (or difference) of a Normal and a Half Normal random variable. The cdf must be evaluated in models in which
the composed error is linked to other errors using a Copula, in some methods of goodness of fit testing, or in the likelihood of models with sample selection bias. We investigate the accuracy of the evaluation of the cdf using expressions based on the bivariate Normal distribution, and also using simulation methods and some approximations. We find that the expressions based on the bivariate Normal distribution are quite accurate in the central portion of the distribution, and we propose several new approximations that are accurate in the extreme tails. By a simulated example we show that the use of approximations instead of the theoretical exact expressions may be critical in obtaining meaningful and valid estimation results.
The paper computes values of the Skew Normal distribution using 17 different mathematical formulas (approximations or exact), and/or algorithms and different software. with particular focus on the accuracy of computation of the Skew Normal CDF by the use of the Bivariate standard Normal CDF, since the latter is readily available, but also on what happens deep into the tails. There, the CDF values as so close to zero or unity that it would appear it wouldn’t matter for empirical studies, if one simply imposed a non-zero floor and a non-unity ceiling, and be ok. It is not ok. In Section 7 of the paper we show by a simulated example, that using the Bivariate standard Normal CDF only (with or without floor/ceiling) may lead to failed estimation, while inserting an approximate expression in its place for the left tail solves the problem. This is a result we did not anticipate: it says that approximate mathematical expressions may perform better than exact formulas due to computational limitations related to the latter.
Ignorability and estimator consistency in binary Logistic regression
Posted: May 10, 2019 in Technical Reports, Uncategorized
Tags: , , , ,
In (counterfactual) Treatment Effects Analysis, we learn that a fundamental condition in order to be able to estimate treatment effects reliably is that the treatment variable is “ignorable conditional on the control variables” (see Rosenbaum and Rubin 1983). When ignorability does not hold, as it happens with most cases of observational, non-randomized data, various methods have been developed to obtain ignorability, or in more precise words, to construct a sample (through “risk adjustment”, “balancing on propensity scores”, etc) that “imitates” a randomized one.
We are also told that ignorability is analogous to regressor exogeneity in the linear regression setup, and so that when ignorability does not hold, essentially we have endogeneity and the estimation will produce inconsistent and so unreliable estimates, see e.g. Imbens (2004), or Guo and Fraser “Propensity Score Anaysis” (2010), 1st ed., pp 30-35.
This is simply wrong. The treatment variable may not be ignorable and yet the estimator can be consistent. This means that we can estimate consistently the treatment effect even if the treatment is non-ignorable. We illustrate that non-ignorability does not necessarily imply inconsistency of the estimator, through the widely used Binary Logistic Regression model (BLR).
The BLR model starts properly with a latent-variable regression, usually linear,
$y^{\ell}_i = \beta_0 + \beta_1T_i + \mathbf z'_i \gamma + u_i,\;\;\; i=1,...,n \;\;\;\;(1)$
Where $y^{\ell}_i$ is the unobservable (latent) variable, $T_i$ is the treatment variable, $\mathbf z_i$ is the vector of controls and $u_i$ is the error term. We obtain the BLR model if we assume that the error term follows the standard Logistic distribution conditional on the regressors, $u_i | \{T_i, \mathbf z_i\} \sim \Lambda (0, \pi^2/3)$. Then we define the indicator variable $y_i \equiv I\{y^{\ell}_i >0\}$, which is observable, and we wonder what is the probability distribution of $y_i$ conditional on the regressors. We obtain
$\Pr\left (y_i = 1 | \{T_i, \mathbf z_i\}\right) = \Lambda\left (\beta_0 + \beta_1T_i + \mathbf z'_i \gamma\right)\;\;\;\;(2)$
and in general,
$\Pr\left (y_i | \{T_i, \mathbf z_i\}\right) = \left[\Lambda\left (\beta_0 + \beta_1T_i + \mathbf z'_i \gamma\right)\right]^{y_i}\cdot \left[1-\Lambda\left (\beta_0 + \beta_1T_i + \mathbf z'_i \gamma\right)\right]^{1-y_i}\;\;\;\;(3)$
This likelihood is estimated by the maximum likelihood estimator (MLE).
Turning to ignorability, it can be expressed as
$\Pr \left (y_i | \{T_i, \mathbf z_i\}\right) = \Pr \left (y_i |\mathbf z_i\right)\;\;\;\;(4)$
Essentially ignorability means that the treatment variable is totally determined by the controls, or maybe, that if it is only partly determined by them, its other “part” is independent from the dependent variable/outcome.
Comparing $(4)$ with $(3)$ we see that ignorability of treatment in the context of the BLR model, is equivalent to the assumption $\beta_1=0$.
“Great”, you could say. “So run the model and let the data decide whether ignorability holds or not”. Well, the issue is whether, when ignorability does not hold, the MLE remains a consistent estimator so that we can have confidence in the estimates that we will obtain. And the assertion that we find in the literature, is that non-ignorability destroys consistency.
Does it? Let’s see: in order for the MLE to be inconsistent, it must be the case that the regressors in the latent-variabe regression (eq. 1), are correlated with the error term. The controls are assumed independent from the error term from the outset. What is argued, is that if $T_i$ is non-ignorable, then it is associated with $u_i$.
We just have seen that ignorability implies that $\beta_1 =0$. So if non-ignorability is the case, we have that $\beta_1 \neq 0$. How does this imply the inconsistency condition “$T_i$ is not independent from $u_i$“?
It doesn’t. The (informal) argument is that if the treatment variable is not fully determined by the controls, it “must” be statistically associated with the unmodeled/random factors represented by $u_i$. But there is nothing here to support a priori this assertion. Whether the treatment variable is endogenous or not, must be argued per case, with respect to the actual situation that we analyze and model. Certainly, if the argument is that the treatment is ignorable, then, if the controls are exogenous to the error term (which is the maintained assumption), so will be the treatment variable also. But if it is non-ignorable, it does not follow automatically that it is endogenous.
Therefore, depending on the real-world phenomenon under study and the available sample, we may very well have a consistent MLE in the BLR model, and so
a) be able to test validly the ignorability assumption, and
b) estimate treatment effects reliably even if the treatment is non-ignorable.–
At the request of a comment, here is a quick Gretl code to simulate a situation where the Treatment is not ignorable, but it is independent from the error term and so it can be consistently estimated. Play around with the sample size (now $n=5000$) , or embed the script into a simple index loop (with matrices to hold the estimates for each run, then fill a series with the estimates from the matrix, then take basic statistics to see that the estimator is consistent).
<hansl>
nulldata 5000
set hc_version 2 #uses HC2 robust standard errors
#Data generation
genr U1 = randgen(U,0,1) #auxialiary variable
genr Er = -log((1-U1)/U1) #Logistic error term Λ(0,1)
genr X1 = randgen(G,1,2) # continuous regressor following Exponential
genr N1 = randgen (N,0,1) # codetermines the assignment of treatment
genr T = (X1+N1 >0) #Bernoulli treatment
genr yL = -0.5 + 0.5*T + X1 + Er # latent dependent variable
#The Treatment is not ignorable because it influences directly the latent dependent variable.
genr Depvar = (yL >0) #obseravble dependent variable
#Estimation
list Reglist = const T X1 #OLS estimation for starting values
ols Depvar Reglist –quiet
matrix bcoeff = \$coeff #starting value scale parameter of the error term
#This so that the names of the variables appear in the estimation output
string varblnames = varname(Reglist)
string allcoefnames = varblnames
#command for maximum likelihood estimation
catch mle logl = Depvar*log(CDF) + (1-Depvar)*log(1-CDF)
series g = lincomb(Reglist,bcoeff)
series CDF = 1/(1+ exp(-g)) #correct specification of the distribution of the error term
params bcoeff
param_names allcoefnames
end mle –hessian
</hansl>
Changes
Posted: October 18, 2017 in Uncategorized
It just has been made official: my PhD focus has changed, and it will now be about the Two-tier Stochastic Frontier model, on which I have already published a paper. The thesis will contain new distributional specifications for the model, among them one that allows for statistical dependence, and two applications where I apply the model to new situations that the existing literature has not touched upon. The projection is that the whole thing will be finished in the next 6 months, since most of the work has been done in the past years, as a … recreational break.
Δύο Ασκήσεις για τη Διαχρονική Προσέγγιση στο Ισοζύγιο Τρεχουσών Συναλλαγών
Posted: May 5, 2015 in Uncategorized
Economics.SE has gone into Public Beta (and CES into Leontief)
Posted: December 2, 2014 in Uncategorized
Tags: , , ,
Economics.SE has gone into Public Beta and now everybody can participate there. Of course the content is still not much, no one expects a lot of content during the private Beta. But it already appears that the Economics.SE can eventually have the right mix and balance regarding style, level and focus of questions and answers.
For example an answer proves that the C.E.S production function converges to the Leontief technology (and to Cobb-Douglas) –but the answer also treats the case where the CES is not homogeneous of degree one, it does not exhibit constant returns to scalewhat happens then?
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2021-07-28 05:23:01
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https://stanford.edu/~hamidi/talk/2019-01-informs/
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# INFORMS 2019
Date
Oct 22, 2019
We consider the $k$-armed stochastic contextual bandit problem with $d$ dimensional features, when both $k$ and $d$ can be large. To the best of our knowledge, all existing algorithm for this problem have a regret bound that scale as polynomials of degree at least two in $k$ and $d$. The main contribution of this paper is to introduce and theoretically analyse a new algorithm (REAL-bandit) with a regret that scales by $r^2(k+d)$ when $r$ is rank of the $k\times d$ matrix of unknown parameters. REAL-bandit relies on ideas from low-rank matrix estimation literature and a new row-enhancement subroutine that yields sharper bounds for estimating each row of the parameter matrix that may be of independent interest.
##### Nima Hamidi
###### Ph.D. Student in Statistics
I’m a Ph.D. student at Stanford and my area of research includes the theory of bandit problems.
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2021-06-21 23:24:52
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http://ndl.iitkgp.ac.in/document/ejUvTDBjTTFyMVNhZU1IZ3lwV3ptSXk0dHJDanRqVXdvQ0pxUHAyNzJmYz0
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### On the average communication complexity of asynchronous distributed algorithmsOn the average communication complexity of asynchronous distributed algorithms
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Author Tsitsiklis, John N. ♦ Stamoulis, George D. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1995 Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Asynchronous distributed algorithms Abstract We study the communication complexity of asynchronous distributed algorithms. Such algorithms can generate excessively many messages in the worst case. Nevertheless, we show that, under certain probabilistic assumptions, the expected number of messages generated per time unit is bounded by a polynomial function of the number of processors under a very general model of distributed computation. Furthermore, for constant-degree processor graphs, the expected number of generated messages is only $\textit{O(nT)},$ where $\textit{n}$ is the number of processors and $\textit{T}$ is the running time. We conclude that (under our model) any asynchronous algorithm with good time complexity will also have good communication complexity, on the average. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1995-03-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 42 Issue Number 2 Page Count 19 Starting Page 382 Ending Page 400
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Source: ACM Digital Library
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2020-08-06 07:43:31
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https://ask.sagemath.org/answers/26876/revisions/
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Since $\cos(t)$ is negative for $\pi/2 < t \le \pi$, computing with $(\cos(t))^{3/2}$ is an issue. I do not think numerical_integral will handle complex values, but I could be wrong.
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2021-05-10 01:57:09
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https://plato.stanford.edu/entries/causation-backwards/
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# Backward Causation
First published Mon Aug 27, 2001; substantive revision Fri Feb 26, 2021
Sometimes also called retro-causation. A common feature of our world seems to be that in all cases of causation, the cause and the effect are placed in time so that the cause precedes its effect temporally. Our normal understanding of causation assumes this feature to such a degree that we intuitively have great difficulty imagining things differently. The notion of backward causation, however, stands for the idea that the temporal order of cause and effect is a mere contingent feature and that there may be cases where the cause is causally prior to its effect but where the temporal order of the cause and effect is reversed with respect to normal causation, i.e., there may be cases where the effect temporally, but not causally, precedes its cause.
The idea of backward causation should not be confused with that of time travel. These two notions are related to the extent that both agree that it is possible to causally affect the past. The difference, however, is that time travel involves a causal loop whereas backward causation does not. Causal loops for their part can only occur in a universe in which one has closed time-like curves. In contrast, backward causation may take place in a world where there are no such closed time-like curves. In other words, an ordinary system $$S$$ taking part in time travel would preserve the temporal order of its proper time during its travel, it would keep the same time sense during its entire flight (a watch measuring $$S$$’s proper time would keep moving clockwise); but if the same system $$S$$ were to become involved in a process of backward causation, the order of its proper time would have to reverse in the sense that the time sense of the system would become opposite of what it was before its back-in-time travel (the watch will start to move counter-clockwise). So neither backward causation nor time travel logically entails each other and time travel is distinct from back-in-time travel.
## 1. History
The philosophical debate about backward causation is relatively new. Only little consideration of the problem can be found in the philosophical literature before Michael Dummett and Anthony Flew initiated their discussion in the mid 1950s. The reason for this is twofold. No empirical phenomena seem to demand a notion of backward causation for our understanding of them. And for a long time it was thought that such a notion involved either a contradiction in terms or a conceptual impossibility. David Hume’s definition of the cause as the one of two events that happens before the other thus rules out that the cause can happen after its effect. Moreover, according to Kant’s idea of synthetic a priori truth the claim that the cause temporally precedes its effect was considered to state such a truth. In 1954 Michael Dummett and Anthony Flew had a discussion about whether an effect can precede its cause. Dummett defended the idea whereas Flew argued that it involved contradictions in terms.
Two years later, Max Black (1956) presented an argument against backward causation, which became known as the bilking argument, and later attempts to meet the argument seemed to generate all kinds of paradoxes. Imagine $$B$$ to be earlier than $$A$$, and let $$B$$ be the alleged effect of $$A$$. Thus, we assume that $$A$$ causes $$B$$, even though $$A$$ is later than $$B$$. The idea behind the bilking argument is that whenever $$B$$ has occurred, it is possible, in principle, to intervene in the course of events and prohibit $$A$$ from occurring. But if this is the case, $$A$$ cannot be the cause of $$B$$; hence, we cannot have backward causation. Since then philosophers have debated the effectiveness of the bilking argument in particular and, in general, the validity and the soundness of the concept of backward causation.
In the 1960s and 1970s, physicists began to discuss the possibilities of particles travelling with a speed greater than light, the so-called tachyons, and as a consequence a similar debate about paradoxes involving backward causation arose among them. In case superluminal particles, like tachyons, exist and could be used to generate signals, it seemed possible to communicate with the past because tachyons going forward in time with respect to one set of reference frames would always be seen as travelling backwards in time from another set of reference frames.
Now and then physicists and philosophers also invoke backward causation in order to explain some experimental and theoretical results within quantum mechanics. For instance, this might be in connection with understanding quantum entanglement after several experiments have proven the violation of Bell’s inequalities.
## 2. Philosophy
A general notion of backward causation raises two sets of questions: those concerning conceptual problems and those that relate to empirical or physical matters. Among the first sets of questions that require a satisfactory answer are the following:
### 2.1 Time and Backward Causation
Can metaphysics provide a notion of time that allows that the effect precede its cause? Answering this question, one may argue, on the one hand, as Maudlin (2002, p. 184) does, that in case we allow backward and forward causation to be parts of our description of the world, the “metaphysical picture of the past generating the future must be abandoned, and along with it the mathematical tractability of local theories.” On the other hand, one may argue, as Evans (2015) among others does, that backward causation is not “precluded by the known structure of reality.” Indeed, Evans’s claim presupposes that the future is ontologically on par with the past and that basic physical laws are time symmetric such that the influence of a physical process may under certain special conditions reach backwards in time.
A proper notion of backward causation requires that the future is just as real as the present and the past. It is common among metaphysicians to distinguish between three different views on time. The first is presentism. This view claims that only events that exist now really exist. Past or future events do not exist. Past events have ceased to exist, whereas future events are yet to become real. So only statements about the present or related to the present have a definite truth value. The second view is possibilism according to which both past and present events exist, but future events are still only possible or non-existing. This view is sometimes called the growing block universe. As a consequence, the view holds that only statements about past and present events have a definite truth-value, but statements about the future are either probably true or may completely lack any truth-value. Finally, the third view is called eternalism, also named the block universe. This position maintains that every past, present and future event tenselessly exists at a certain time and that statements about these events therefore have a definite truth-value at every other time.
Usually, presentism and the growing block universe are associated with the dynamic view of becoming. The transient now plays an ontological role as the ever changing time in which things become real or perhaps, if you are a presentist, cease to exist. One reason to prefer the growing block universe from presentism may be the analysis of forward causation. In order for a present event to be caused by a past event, the past event must exist. Nothing, which does not exist in the past, can cause something that presently exists. By the same token, if the backward causation is a conceptual possibility, something must exist in the future in order for it to cause something in the present.
Thus, backward causation demands eternalism or a static account of time in the sense that there is no objective becoming, no coming into being such that future events exist on the par with present and past events. It means that the future is real, the future does not merely consist of unrealised possibilities or even nothing at all. Ordinarily we may think of the past as a nothing that once was a something. But when asked what makes sentences about the past true or false, we would probably also say that it is the facts of the past that make present sentences about the past either true or false. The fact that I went to the cinema yesterday makes it true today when I say that I went to the cinema yesterday. This view is a realist one with respect to the past. If backward causation is to be conceptually possible, it forces us to be realists with respect to the future. The future must contain facts, events with certain properties, and these facts can make sentences about the past/future true or false. Such a realist account is provided by static and tenseless theories of time. Eternalism is such a theory and holds that the participation of time into the past, the present and the future depends on the perspective we human beings put on the world. The attribution of pastness, presentness and futureness to events is determined by what we take to exist at times earlier than and times later than the time of our experience. (For further discussion, see the entry on being and becoming in modern physics.)
### 2.2 Affecting the Past?
Does backward causation mean that a future cause is changing something in the past? Talking about forward causation we think of the cause as the event that produces its effect or brings it about. It is not part of our notion of forward causation that the cause changes anything in the future. A cause determines what the effect will be. Regardless of whether one is an advocate for presentism, the growing block universe, or eternalism, one never considers the cause as an event that will make the future different from what it will be. Indeed, without the forward-directed cause the future would have been different from what it is going to be.
Even most protagonists consider it an unwarranted consequence that the notion of backward causation, if consistent, involves the idea that the future is able to change the past. Their answer has therefore usually been that if we have the power to bring something about in the past, what came about really already existed when the past was present. We have to make a distinction between changing the past so it becomes different from what it was and influencing the past so it becomes what it was. A coherent notion of backward causation only requires that the future is able to have an influence on what happens in the past. Nonetheless, we can say, quite parallel to forward causation, that the past would have been different if the backward-directed cause had not made the past as it turned out to be.
### 2.3 Distinguishing Cause from Effect
Can the cause be distinguished from its effect so that the distinction does not depend on a temporal ordering of the events? For a long time the distinction between cause and effect was their temporal order. This view goes at least back to Hume who claimed, “We may define a cause to be an object followed by another, and where all the objects, similar to the first, are followed by objects similar to the second,” which is to say that “[…] if the first object had not been, the second never had existed” (Hume, [1748] 2007, 56). Although Hume did not explicitly say so, the expression “followed by” has always been read as “temporally followed by” and not “causally followed by”. So the adherents of this definition have usually tried to give an account of causation in which the cause and the effect are seen as temporal regularities between types of events. But we can also see that Hume himself added a counterfactual statement about causation which cannot be derived from his temporal definition. Apparently, he noticed that if we have a cause and an effect, there exists a relationship between them that is not given by the temporal order.
What is required for backward causation to be possible is some account of the direction of causation which does not rely on the direction of time. Various alternative proposals refer to counterfactuals, probabilities, agency, manipulation and intervention, common cause or causal forks. Among these it seems that only a Humean notion of causation explicitly makes a temporal identification of the cause and the effect. But there are also problems with some of the other accounts.
For example, it is quite common to follow David Lewis and define causation in terms of counterfactuals (Lewis, 1973). Assume event $$c$$ causes event $$e$$, then we have a situation in which both $$c$$ and $$e$$ occur, and in which the counterfactual statement “If $$c$$ had not occurred, then $$e$$ would not have occurred” is true. Thus, in Lewis’s view, $$e$$ is causally dependent on $$c$$ if, and only if, $$e$$ is counterfactually dependent on $$c$$. According to the traditional theory, formulated by Lewis and Stalnaker, any counterfactual statement is true if the consequent is true in the closest possible world to the actual world in which the antecedent is true. Apparently, Lewis’ definition delivers a non-temporal causal asymmetry, since the effect $$e$$ is counterfactually dependent on the cause $$c$$.
As we saw above, Hume also believed that causal statements entail counterfactuals, but the question is whether causal statements are definable in terms of counterfactuals. Here is an objection against such an attempt: Consider the following example. Because of severe frost during the night, ice covers the lake this morning. So, given the circumstances, if it had not been severe frost, there would not have been any ice on the lake. However, in these circumstances, the frost as the cause of the ice seems to be not only causally necessary for the ice, but the frost also seems to be causally sufficient for the ice cover. In other words, given the actual circumstances, it seems correct to say that the frost is causally sufficient as well as necessary for the ice. If we are going to represent causally sufficiency in terms of a counterfactual statement, we might then say that, given the actual circumstances, if no ice had covered the lake this morning, there would not have been a severe frost during the night. But if one accepts this objection, it shows that causal statements cannot be defined in terms of counterfactuals because such a definition does not give us the wanted asymmetry between cause and effect since each is counterfactually dependent on the other.
Sometimes this problem is called the Problem of Effects. According to this problem, the effect does not distinguish itself other than temporally from the cause because both the cause and the effect are counterfactually dependent on each other. Over the years several attempts have provided solutions to this problem. Lewis himself proposed some criteria consisting of a hierarchy of possible worlds such that a world in which $$c$$ occurs but $$e$$ does not occur is closer to the actual world than a world in which $$c$$ does not occur due to the absence of $$e$$. But a number of philosophers have challenged the aptness of these criteria, among others Bennett (1974); Faye (1989); Horwich (1993); Baker (2003); Choi (2007); Wasserman (2015), and Seli (2020).
However, and more importantly in the present context, is that Stalnaker-Lewis’ theory of counterfactuals has difficulties with backtracking counterfactuals and backward causation because if $$c$$ occurs later than $$e$$, the proposed method of truth evaluation assumes that $$e$$ occurs in the relevant possible worlds in which $$c$$ does not occur. In general, the assessment of a counterfactual conditional is carried out by assuming that the possible world must be identical with the actual world up to $$c$$. Therefore, it is stipulated that the closest possible world is one in which everything happens just as in the actual world up to the time of $$c$$’s occurrence, which means, given $$e$$ occurs before $$c$$, that the possible world will include the occurrence of $$e$$. But then it is necessarily true that there is never a possible world closer to the actual world which includes $$c$$ but not $$e$$. This creates a problem because we consider any causal connection between $$c$$ and $$e$$ as contingent. Rather we would expect that if we face a case of backward causation where both a present event $$c$$ and a past event$$e$$ occur, the following counterfactual would be true: “If $$c$$ had not occurred, then $$e$$ would not have occurred.” The truth evaluation of this counterfactual, if it were to represent backward causation, would require that the closest world without $$c$$ is also one without $$e$$. Nevertheless, the traditional theory does not allow such an evaluation.
How else can causality be specified so that the order of the cause-effect relationship is not time-dependent? Several possibilities seem open for such an account. Here we shall just mentioned some. One could hold that the cause makes the effect more probable in the circumstances; one could argue that causation can be understood in terms of manipulation and intervention; one could argue that we can specify causation in terms of transmission of information or conserved physical quantities. Or, finally, one could argue that causality is a primitive notion that may imply, or be used to explain, regularities, counterfactuals, probabilities, transmission of signals, or manipulation and intervention, but it cannot be completely analyzed in any of these terms. If we take into consideration that higher animals show a sense of causation, it may be a good indication that the origin of this notion goes back in history and that it stems from the cognitive evolution of grasping in which ways one’s environment is stable and in which ways one’s behavior can interact and change the environment.
This observation may come close to Lewis much later analysis of causation as influence (Lewis, 2000). Thus, it is through our ability to influence our environment that we improve our ability to manipulate and intervene in the cause of events and through which we receive knowledge of causal relationships and the causal order of this relationship (Faye, 1989; Woodward, 2003). However, such an understanding of causation is already rather complex, building on our capacity of foreseeing the effect of our own actions, on our capacity of foreseeing the effects of others’ actions, on our understanding the cause of others actions, and on our understanding the cause of physical events (Gärdenfors, 2006, p. 41).
### 2.4 The Bilking Argument
Can the bilking argument be challenged in such a way that the mere possibility of intervention does not generate any serious paradoxes? The bilking argument is due to Max Black (1956) who assumed the following scenario. Suppose Houdini makes a prediction about the outcome of, say, a coin about to be flipped $$B$$ before someone actually does the flipping $$A$$. We may also assume that in the past Houdini rarely failed in his predictions. In this case we might be tempted to say that the Houdini’s answer is caused by the later flipping. But, as Black argued, after Houdini’s prediction of $$B$$, we can always intervene such that the coin is not flipped or arranged opposite to Houdini’s prediction. The implication is that $$A$$ seems both to be the cause and not to be the cause of $$B$$. Black also argued that if $$A$$ is the cause of $$B$$, then the causal antecedents of $$A$$ are independent of $$B$$. Hence, if we cannot, after $$B$$ has occurred, prevent $$A$$ from happening, then $$A$$ cannot be said to be causally independent of $$B$$. But since it is in human power to intervene with respect to $$A$$, Black concluded that $$A$$ cannot be the cause of $$B$$
Since manipulation and intervention is so central for our knowledge of causal relations, the image of our capability of intervening in the course of backward causation, after the alleged effect has occurred, seems to violate the whole idea that the notion of backward causation is coherent. However, the force of the bilking argument can, it seems, be weakened in various ways.
First, one may hold that it is not a problem for our notion of backward causation that we can in principle intervene in the course of the events. Usually, we find out whether two events are causally connected trough manipulation and intervention. In case we can produce one event $$Q$$ by bringing about another $$P$$, or we can obstruct one event $$Q$$ by preventing another $$P$$, we think that $$P$$ is the cause of ($$Q$$. Likewise, we believe that if $$P$$ causes $$Q$$ in the relevant circumstances, we may be able to prevent $$Q$$ from happening if we intervene in the causal order after $$P$$ occurs by changing some of these circumstances, which make $$P$$ causing $$Q$$. Should $$Q$$ occurs in spite of this intervention, there must under the new circumstances be another event $$X$$ rather than $$P$$ that causes $$Q$$.
Now, the bilking argument holds that backward causation is impossible because we can always intervene after we have observe that the alleged effect occurs and obstruct the alleged cause from occurring. Since nothing prohibits us from doing this whenever we want, it demonstrates that backward causation does not take place. Indeed, if we actually intervene and prevent $$A$$ after $$B$$ has occurred, then of course a particular later $$A$$ (which does not exist) cannot be the cause of a particular earlier $$B$$ (which exists). But in all those cases where nobody actually intervenes, events of the same type as $$A$$ may be the cause of events of the same type as $$B$$. This situation is not different from what may happen in some cases of forward causation. Assume that $$P$$ causes $$Q$$ in the relevant circumstances. We may still prevent a particular $$P$$ from happening, but at the same time a particular $$Q$$ may nevertheless occur because in the given circumstances it is caused by another event than $$P$$.
Second, if a later event $$A$$ really causes an earlier one $$B$$, then it would be impossible to intervene into the cause of the event after $$B$$ has happened and therefore impossible to prevent $$A$$ from happening. If someone tries, she will by all means fail. It may intuitively sound strange as long as we think of backward causation as consisting of something we can control directly by our everyday actions. But if backward causation is a notion that is applicable only to processes that human beings are unable to control in any foreseeable way the notion would not provoke our intuitions so much.
But even if we would have full control of backward causal processes and able to intervene in their cause of events, we may not, in such cases where we actually intervene, exclude backward causation from taking place. Assume someone tosses a coin a minute after the magician Houdini has predicted whether it would be heads or tails. Furthermore, assume that Houdini’s predictions are highly correlated with the outcomes of the tossed coin. Presumably, no such high correlation would exist unless we were facing a case of backward causation. However, it turns out that the construction of a correlation may behave differently depending on whether we consider Houdini’s predictions to be purely physical events or to have a semantic content.
In an example like the one mentioned it is reasonable to think that we discover the correlation between Houdini’s predictions and the subsequent outcomes of the tossed coin by comparing the semantic content of the predictions with one of the two iconic sides of the coin that define the semantic content. The side facing up determines whether the prediction is true or not. First, in order to establish a causal correlation between the content of the predictions and the outcome of the tosses we would look for a high percentage of correct statements made by Houdini. This is an epistemically necessary condition for establishing a causal correlation. Second, there cannot ontologically be a causal correlation between the semantic content of Houdini’s prediction and the (reading of) outcome of the head or tail if no flipping of the coin takes place. His predictions would not carry any truth-value and we would think of them as pure guesses. Hence, when someone attempts to bilk this experiment, he or she will either arrange the outcome opposite to the content of Houdini’s prediction or abstain from flipping the coin.
However, one might suggest that a possible reply to Black’s bilking argument is to say that both tossing the coin and preventing the toss may backwardly cause Houdini’s behavior. That the preclusion of the coin from being tossed may be a cause of Houdini’s response has also been proposed by Brian Garrett (2020). But he argues, in contrast to the scenarios described below, that the lack of flipping the coin may be the direct cause of the Houdini’s earlier prediction (because of backward causal preemption) and not only the cause of an earlier lack of prediction. Thus, it seems at least consistent to argue that both the toss and the prevention could cause Houdini’s earlier behavior regardless of whether this behavior is a prediction or the lack of a prediction, but only as long as we consider both actions to be physical events. Indeed, if we solely consider Houdini’s prediction (regardless of its semantic content) or his lack of prediction as purely physical events and do the same with the actions “tossing the coin” and “preventing tossing the coin,” we may have a high correlation of these physical events.
Thus, it seems that we have three scenarios in which we can observe backward causation even though in two of them one may attempt to bilk the cause after the alleged effect has occurred.
Scenario 1: The experimenter asks Houdini to predict the outcome of a tossed coin a minute later. It turns out that there is a high positive correlation between Houdini’s answers and the actual heads or tails.
Scenario 2: The experimenter now asks Houdini to make his predictions, but arranges the coin such that it shows heads or tails opposite to Houdini’s pronouncements. In this case, there will be a negative correlation between the semantic content of his answers and the outcome of the tosses. (Indeed, you could also arrange the outcome so there would be no correlation at all.) But one could still argue that the very fact that Houdini did not abstain from responding was because the physical act of handling the coin automatically prompted him to produce an answer. One will observe a high positive correlation between Houdini making physical pronouncements and the physical handlings of the coin, although there exists a negative or no correlation between the semantic content of his answers and the outcomes in the form of heads and tails.
Scenario 3: The experimenter asks Houdini to make his predictions. However, he abstains in situations where someone subsequently prevents the coin from flipping, but in situations where no one intervenes; there is a high correlation between his predictions and the outcomes. In those cases where someone prevents the coin from being flipped, and Houdini therefore abstains from saying anything, there cannot be any correlation between the truth-value of Houdini’s answers (since he does not produce any). However, there still seems to be a high correlation between his behaviors (i.e. the physical lack of responses) and the subsequent prevention of any outcome.
For those reasons one may argue that the bilking argument is not as powerful as Max Black might have thought.
### 2.5 Free will
Does backward caution imply fatalism? An often advanced objection against backward causation is that if eternalism and backward causation are possible, then the future is already determined. And if it is already determined now what the future is going to be, then it does not matter what an agent will do, since everything in the future is set and done. An agent cannot do other than what the future is determined to be. Although this argument seems very appealing, it may not hold for a closer scrutiny.
Being an eternalist one may distinguish between a determined future and a determinate future. Also this pair of concepts have different names in the literature. Sometimes determinism is called physical determinism, causal determinism or nomological determinism, whereas determinateness is named logical determinism, temporal determinism, or block determinism. A future event is determined now if and only if a present event is causally or nomologically sufficient for it to happen. In contrast, a future event is determinate with respect to the present if and only if this event occurs tenselessly at a future time. Thus, eternalists will say that an agent still has a choice to make tomorrow about whether or not she will go and see her parents, because no present event causally determined her choice tomorrow. Her choice is nevertheless determinate, and therefore it is true today what she will do tomorrow.
The discussion of free will has a long history and the overwhelming parts of this discussion has been dedicated to the problem that if the world is completely governed by deterministic laws, does it then make sense to talk about free will? Indeed, the outcome of this discussion very much depends on how we understand the notion of free will. Usually the concept of free will connects to whether or not a human agent could have done otherwise. Some philosophers, the compatibilists, believe that as long a human agent is not subject to any external or internal forces, that individual is free to do whatever he or she wants. But then the compatibilists have to dismiss some part of the following argument of consequence.
1. We cannot change the past.
2. We cannot change the laws of nature.
3. Hence, we cannot change either the past or the laws of nature (by 1 and 2).
4. If determinism is true, then our present actions are necessary consequences of the past and the laws of nature.
5. Hence, if determinism is true, then we can do nothing now to change the fact that our present actions are necessary consequences of the past and the laws of nature (by 4).
6. Hence, if determinism is true, then we cannot change that our current actions occur (by 3 and 5).
7. Hence, if determinism is true, then we have no power to do things differently than what we do (by 6).
8. Free will requires a power to do things differently.
9. Hence, if determinism is true, then we do not have free will (by 7 and 8).
The non-compatibilist will argue that we have free will in the sense that we could have done otherwise. Hence determinism cannot be true.
Indeed, the compatibilist will accept (1), because even an advocate of backward causation does not claim that the past can be changed. Moreover, the compatibilist does not have to accept backward causation. Most likely the compatibilist will not challenge the non-compatibilist by saying that we may be able to influence the past. It is much more likely that the compatibilist will object to the argument either by saying that not all laws of nature are deterministic or by saying that the brain operates based on ceteris paribus laws including a clause of the absence of external and internal forces. For the compatibilist this suffices to say that the agent could have done otherwise. In other words, the compatibilist’s objection would usually not concern the determinateness of the past.
As we can see, the above argument of consequence combines determinateness of the past and determinism of laws. Whatever the compatibilist would object in order to reject the argument of consequence, the argument cannot be used mutatis mutandis to argue against eternalism or the block universe, because in contrast to the discussion of free will and the past that focuses on determinism, the discussion of free will and the future focuses on determinateness. For even if indeterminism is true, i.e. if some processes of nature are indeterministic, one could still argue in favor of the block universe as some of its advocates pointed out long time ago (See Grünbaum, 1967, 28–35). Therefore, the argument against backward causation based on free will has to be different from the one posed against traditional fatalism. It has to look something like this:
1. If backward causation is possible, then the future has to be determinate.
2. If the future is determinate, it is now the case that a wanted action occurs at a later time t or it is now the case that it does not occur at this later time t(by 1).
3. If it is now the case that a wanted action occurs at a later time t or it is now the case that it does not occur at this later time t, I cannot perform a wanted action that does not occur at this later time t.
4. Hence if I cannot form a wanted action at a later time t, then I cannot be free to do otherwise (2 and 3).
5. Now, assuming that the future is determinate.
6. Hence, I am not free to do otherwise (by 4 and 5).
7. Hence, I have no free will.
8. However, I know from experience that I could have done otherwise.
9. Hence, the future is not determinate.
10. Hence, backward causation is not possible.
A common criticism has been that if the agent’s actions tomorrow are determinate, and it is therefore true today that the agent, say, will visit her parents, then the agent cannot do anything other than visit her parents. However, this conclusion seems to be a non sequitur. In order to reach such a conclusion, one must tacitly assume that the agent’s action causally determines the outcome of her choice, whereas the argument were believed to show that backward causation violates the notion of free will.
In contrast, the eternalist could argue that the reason why it is true today that an agent will visit her parents tomorrow is because the agent makes a decision that causally determines her visit before she goes to see them. If the agent makes the opposite decision tomorrow, it will be true today that she will not visit her parents. Whatever she chooses tomorrow, it will be her decision which makes it true today that she is going to see her parents. The outcome of her decision tomorrow is determinate not because the present truth value fixes her decision but because her future outcome fixes the present truth value. (Nor is it the case that the outcome of her decision causally determines her previous decision.) Thus, the eternalist may argue that even though the future is determinate, it does not exclude people from having free will. If people have free will, the argument goes, the fact that the outcome of their future decision is determinate with respect to the past does not affect their ability to choose freely.
Of all the philosophical problems to which backward causation (and time travel) gives rise, the paradoxes are those that have generated the most heat in both physics and philosophy because, if they are valid, they exclude backward causation from being both metaphysically and logically possible. The paradoxes can grossly be divided into three kinds: (1) Bootstrap paradoxes involve a causal or information loop; (2) Consistency paradoxes involve generating a possible inconsistency; and (3)Newcomb’s paradox seems to foreclose free will. So if backward causation (and time travel) should be logically possible, one has to show that the paradoxes can be resolved and that therefore arguments based on them are invalid.
The bootstrap paradoxes arise in cases where you have a causal chain consisting of particular events in which $$a$$ causes $$b, b$$ causes $$c$$, and $$c$$ causes $$a$$. The problem here is that the occurrence of $$a$$ presupposes the occurrence of $$c$$; in other words, the cause presupposes its effect. But how can something be required of what itself requires? Indeed this seems paradoxical. Some philosophers therefore think that this makes the idea of causal loops incoherent. Hugh Mellor even believes that
the possibility of causal loops can be excluded a priori, and so therefore can the closed timelike paths entailed by closed time, backward time-travel and all kinds of backward causation. (1991: 191).
His proof goes like this. Take four chains of events. Each of them consists of three particular events $$a, b$$, and $$c$$, all different tokens of the same kind of events $$A, B$$ and $$C$$. We then construct the chain such that
1. $$b \Rightarrow c \Rightarrow a$$
2. $${\sim}b \Rightarrow{\sim}c \Rightarrow{\sim}a$$
3. $$b \Rightarrow c \Rightarrow{\sim}a$$
4. $${\sim}b \Rightarrow{\sim}c \Rightarrow a$$
The first two sequences may be called G-chains and the other two H-chains. Moreover, Mellor assumes that all tokens of $$A, B$$ and $$C$$ are distributed among the four chains so that the number of chains is exactly the same, namely one fourth of the sequences. Mellor then defines a causal relation between two singular events $$a$$ and $$b$$ in terms of a situation $$k$$ which makes $$b$$ more likely to occur given $$a$$ than without $$a$$, i.e., $$\rP(b\mid a) \gt \rP(b\mid {\sim}a)$$. But we can see that the number of chains in which $$b$$ is combined with $$a$$ is equal to the number of chains in which $$b$$ is not combined with $$a$$. In fact we have that $$\rP(b\mid a) =$$ $$\rP({\sim}b\mid {\sim}a) =$$ $$\rP(b\mid {\sim}a) =$$ $$\rP({\sim}b\mid a)$$. From this it follows that a particular $$b$$’s chance in $$k$$ cannot increase with respect to $$a$$ compared to its chance without $$a$$. Hence $$a$$ cannot affect $$b$$, and therefore causal loops are impossible.
Some philosophers have not found this argument very convincing. Faye (1994) has pointed to the following problematic issues. First, Mellor measures the probability of singular events (propensities) instead of the probability of certain kinds of events. Second, he does not differentiate between circumstances in which a $$B$$ is followed by an $$A$$ and those in which a $$B$$ is not followed by an $$A$$. The argument is valid only if it can be proved, and not be stipulated, that (1) and (3) happen surrounded by the same facts. Many people would say that in a world of (1) must be different from a world of (3) in some other important respects than merely containing $$a$$ or $${\sim}a$$, especially since Mellor claims that the argument is valid for deterministic situations as well. Third, the equal distribution of the various chains seems quite selective. In Mellor’s G&H world, in which the number of the four chains is equal, and therefore in which the probabilities are equal, there cannot be any causal relationship between the individual $$b$$ and the individual $$a$$ due to the fact that the occurrence of $$a$$ or $${\sim}a$$ happens under exactly the same circumstances given $$b$$. Finally, fourth, it seems appropriate to claim that any negative argument, like Mellor’s, should be able to show that what holds true of one world can be proved to hold true of every other world similar in all relevant respects, but in which G-chains and H-chains are not equally distributed.
It is clear that any world which contains G-chains rather than H-chains does not show the same inconsistency as Mellor’s G&H-world does. If it can be proved that causal loops in such worlds are consistent with the adopted definition, then causal loops are possible. In other words, if we set up a consistent model in which $$A$$ increases the probability of $$B$$, and $$B$$ increases the probability of $$A$$, we have then proven that causal loops are possible and that Mellor’s argument is invalid. The claim is therefore that both
1. $$\rP(A\mid B) \gt \rP(A\mid {\sim}B)$$
2. $$\rP(B\mid A) \gt \rP(B\mid {\sim}A),$$
can be shown to be true with respect to a world containing $$A$$s and $$B$$s. Assume the following probabilities, which hold for the distributions among $$A, {\sim}A, B$$, and $${\sim}B$$, are
\begin{align} \rP(A \amp B) &= 0.7 \\ \rP(A \amp{\sim}B) &= 0.1 \\ \rP({\sim}A \amp B) &= 0.1 \\ \rP({\sim}A \amp{\sim}B) &= 0.1. \end{align}
On the basis of the definition of the conditional probability, we get
\begin{align} \rP(A\mid B) = \frac{\rP(A \amp B)}{\rP(B)} &= 7/8 \\ \rP(A\mid {\sim}B) = \frac{\rP(A \amp{\sim}B)}{\rP({\sim}B)} &= 1/2 \\ \rP(B\mid A) = \frac{\rP(A \amp B)}{\rP(A)} &= 7/8; \\ \rP(B\mid {\sim}A) = \frac{\rP({\sim}A \amp B)}{\rP({\sim}A)} &= 1/2. \end{align}
Thus (i) and (ii) are both true with respect to the stated world; hence we have proven, according to Mellor’s own definition of causality, that it is consistent to talk about causal loops. Mellor has not been able to establish any satisfying a priori argument against causal loops or backwards causation.
Moreover, even if one assumes that Mellor were correct in ruling out causal loops a priori, he may be wrong in holding that this impossibility entails the impossibility of time travel as well as backward causation. Mellor’s argument presupposes that it is the same kind of processes obeying the same kind of macroscopic physical laws which enters into both the forward and backward part of the causal loop. This assumption may hold for time travel but not for backward causation.
The consistency paradoxes arise when you, for instance, try to kill your younger self by a backward causal process but evidently have to fail. The reason why you must fail is quite obvious. Your younger self belongs to the past and therefore, since you cannot change the past, you cannot commit retro-suicide. This answer tacitly assumes that resurrection is impossible. You may, of course, kill your younger self in the past without changing the past if you have come alive again later on. This is not what is paradoxical. What is paradoxical is the fact that you are assumed to be able to kill your younger self in the sense that you are well-equipped to make these kinds of retro-killings, you may even be targeting your younger self, but you must always miss. The same holds, indeed, for all those people who stay alive into the present. You cannot retro-kill somebody yesterday who is alive today. There must be certain constraints which prohibit you from making retro-suicide or retro-killing, and these constraints may be very local, changing from case to case, or they may be universal in nature depending on some physical laws. So, on the one hand, the assumption is that it physically possible for you to kill somebody in the past; but, on the other hand, it is physically impossible for you to do what is physically possible. This is the paradox.
A way out of the paradox was suggested by David Lewis (1976) who argued that the ability of killing somebody should be understood as a possibility compossible with the relevant fact. As an opera singer, for example, you are able to sing operas, since you have the physical capacity and training to do so, but because of a temporary loss of voice, you cannot hum a single tune. What you can do relative to one set of facts, is something you cannot do relative to another set of facts. This contextual solution explains why you are able to retro-kill your younger self, given the fact that your gun is in proper working-order, you have a good aim at your target, and no one forces you to abstain from taking action. But it also explains why you are unable to retro-kill anybody who is alive today because you cannot change the past. The consistency paradox exists only in virtue of an equivocation of a context-sensitive ‘can’, and if we notice that, we see that the paradox vanishes like dew before the sun.
Some may reply that we are still talking about different abilities. In contrast to the case in which the opera singer sometimes cannot sing, your attempt to carry out retro-suicide inevitably fails. The opera singer is able to sing operas because he has shown it before and can demonstrate it again, but the attempted retro-killer has not proved and can never prove his ability. Therefore you are never in a situation where you can kill your younger self. If we accept this objection, we may reformulate the solution by saying that the contextual solution explains why you should be able to retro-kill your younger self under the appropriate circumstances. But, again, how can we talk about the ability to make retro-suicide relative to certain facts at all, if there are no possible worlds in which you carry out your deed. It seems reasonable to say that you have the ability to do something if there is a possible world in which you carry out this action. This is true of the opera singer. He can sing operas because he does it in a possible world in which he has not lost his voice. But you cannot make retro-suicide because there is no possible world in which you kill your younger self. You are unable, even in principle, to do so.
In sum, the consistency paradox is no paradox as long as you do not insist on changing the past. You are unable to change the past, and therefore you are unable to retro-kill anybody who is alive when you try to kill them. The paradox seems to arise only because you wrongly believe that you are able to do something you are unable to do.
Now if there is no paradox on the conceptual level, what then is it that makes retro-suicide physically impossible? It could be either local facts or global facts. Local facts that could constrain your action of retro-killing are many. Your hand was shaking while firing your gun, you got a fly in your eye, you were disturbed by a cat, you just fainted, etc. These constraining facts seem reasonable by themselves; they could have happened independently of your overall capacity of killing somebody in the past, but also in the actual situation interact with your ability and turn the action into an unsuccessful event. The problem is merely that such an explanation looks suspicious. It is a general fact that we cannot retro-kill anybody who is alive after the time the death purportedly took place. Likewise it is a general fact, assuming that backward causation (or time travel) is physically possible, that we can retro-kill anybody who is not alive after the time the purported death took place. But the explanation of a general fact requires an appeal to a general fact or a law of nature. Thus a reference to a singular contingent fact to explain why you never succeed in killing your younger self seems not to fulfil the requirement of being an explanation.
The problem may be better understood in following way: each time you try to retro-kill somebody who is not alive after the time the purported effect of killing took place, your assassination may still fail because of your hand was shaking, etc. Such particular facts explain why you actually missed the target which you in principle were able to hit. But to say that you are in principle able to perform retro-killing means that there are laws of nature that normally allow you to perform such an action in the appropriate circumstances. Similarly, each time you try to retro-kill somebody who is alive after the purported death took place, you may fail for one reason or another. But you must always fail to retro-kill somebody who is alive after you did your action, i.e., you are in principle unable to retro-kill such a person. In those cases it is physically impossible for you to kill, say, your younger self. It seems, therefore, that there should be some laws of nature, working on either a local level or a global level, which violate such an action and makes it physically impossible.
A possible solution may be found in a recent result which shows that the most basic features of quantum mechanics may ensure that we could never alter the past, even if it should be possible to interact with the past. The two physicists, Daniel Greenberger and Karl Svozil (2005 in Other Internet Resources), imagine some form of quantum mechanical feedback by introducing figurative beam splitters which are unitary, i.e., the splitters allow the feedback loop to be reversed because they have the same number of entry ports and exit ports. From quantum mechanics we know that an object may behave like a wave and that some unitary operator describes the propagation of a physical system. The system is represented by a wave function, also referred to as a path, and the time evolution of the system is calculated as a sum over all possible paths from the initial state to the final state. This calculation is usually restricted to the forward direction of time. Now, if we think of some of the paths as unfolding backwards in time, Greenberger and Svozil are able to prove that either the forward and the backward component paths of the loop cancel out, or that the propagator, which establishes the feedback in time, “wipes out the alternative possible futures, thus guaranteeing the future that has already happened”. Thus, if you could aim at something in the past, the laws of nature prohibit you to act in ways that are in conflict with what makes the future what it is (what it already turned out to be). The authors’ conclusion is that if you go back in time or effect “the past quantum mechanically, you would only see those alternatives consistent with the world you left behind you”.
This thought experiment involves a player playing a game against a fortune teller. In this game there are two boxes of which the player may select one or both. One of the two boxes is transparent, let’s call it $$A$$, the other is opaque, call it $$B$$. Before the player makes her choice the fortune teller, based on her prediction of the player’s choice, puts a certain amount of money in one or both boxes. When the player makes her choice, the following information is available to her: 1) Up to now the fortune teller has been able to predict the future with absolute certainty and has foreseen what other players have chosen. Moreover: 2) sometimes the fortune teller puts 1,000,000 dollars in box $$B$$, but only if the player selects box $$B$$ and does not take both boxes, whereas she puts 1,000 dollars in box $$A$$, regardless of whether the player chooses box $$A$$ or both boxes. So when the player has to choose between the two boxes, it is already determined whether box $$B$$ contains 1,000,000 or nothing. If the fortune teller is correct in her prediction, the player will get 1,000,000 dollars, only if she selects box $$B$$; however, if the player wants to earn 1,001,000 dollars and selects both boxes, she will miss most of the fortune and only receive 1,000 dollars, which she can already see in box $$A$$.
This puzzle was originally proposed by William Newcomb but never published. After its first publication by Robert Nozick (1969) it was much discussed within decision theory. But it has also been debated in the context of backward causation, because it gives a nice illustration of some of the philosophical problems that arise in relation to backward causation. In this debate it has been used to demonstrate various claims: that backward causation is impossible or that it implies fatalism or determinism. The earliest discussion of some of the consequences for backward causation and free will can be found in George Schlesinger (1974) and in an exchange between Don Locke (1978, 1979) and André Gallois (1979). Here Schlesinger argued that there are good arguments for taking both boxes, $$A$$ and $$B$$, and equally good arguments for taking only box B, whereas Locke held that backward causation is irrelevant and the player ought to take both boxes. However, Gallois believed that Locke’s arguments for taking both boxes are misconceived.
Two opposite arguments for what the player should decide can be given. Assuming the player has a free will it seems rational to say that the fortune teller cannot in principle predict what the player is going to do. When the player makes her choice, it is already determinate whether box $$B$$ contains 1,000,000 dollars or nothing. Therefore her choice cannot be affected by the fortune teller’s prediction, and she would be better off by selecting both boxes, hoping that there already is 1,000,000 dollars in box $$B$$. The other argument maintains that based on previous experience—which tells us that the fortune teller has always been correct about her predictions—it would be most rational of the player to take only $$B$$. From here philosophers have disagreed about the strength of the argument. One could insist that fortune teller can predict the player’s choice only based on relevant knowledge of that person’s past or present behavior or state of mind (compatibilism). Alternatively one could argue that the fortune teller could have such a perfect knowledge only from information about the player’s choice in the future.
Nevertheless, it has been argued that the Newcomb paradox demonstrates that backward causation is impossible. George Schlesinger (1980:75 ff.), for instance, imagines a perfect judge who is allowed to check the two boxes after the fortune teller has placed the money in the boxes, but before the player has made her choice. It is clear that if the perfect judge then informs the player about the content of the two boxes, we have a case where the fortune teller’s prediction causes the player to select both boxes as she is informed about the 1,000,000 dollars in box $$B$$. Indeed this would be a clear example of Black’s case of bilking. But Schlesinger also argues that the same holds true, if the perfect judge merely knows it without informing the player. However, it is difficult to understand why this should be the conclusion. If we assume that the player has a free will and that the fortune teller is able to predict the result of the player’s choice, it seems most rational for the player to learn from experience and opt for box $$B$$ only rather than being greedy and take both boxes. As long as the perfect judge is silent, his knowledge about the content of the two boxes seems not to be able to influence the player’s decision. Nor does the player’s possible knowledge of the existence of a perfect judge seem to have any effect on her decision, because this information does not add anything to the information she already has; namely, that the fortune teller has placed the money in the boxes before she makes her decision and that the fortune teller has never failed to predict the outcome in advance.
Another problem is that the paradox seems to indicate that backward causation implies fatalism and determinism. If backward causation is possible we cannot have a free will. If the result of the player’s action can be predicted, because it is already true today what the player will choose tomorrow, she cannot be free to pick. If the fortune teller already knows that the player will choose both boxes, this has to be true, and the player cannot do otherwise. And if the fortune teller already knows that the player will select only box $$B$$, this has to be true, and the player cannot but do it. However, this argument seems to be misguided.
The intuitive strength of the argument stems from the general assumption that backward causation presupposes an ontologically closed future—a metaphysical position about time usually named eternalism. Hence it is not an argument only against backward causation but against eternalism as well. Suppose it is now already true or false what is going to happen tomorrow; then there must be some future truth-makers that determine that it is now already true or false what is going to happen. The consequence seems to be that what the player is going to do tomorrow is already determined today; hence the player cannot do otherwise tomorrow than what is true today. Therefore the player cannot have a free choice, and it seems futile of her to make any decision.
However, it can be maintained that the player still has her free choice. Nothing around the fortune teller causes the player to make a particular decision. The player is as free as she would be with no prediction. What the fortune teller is able to predict is the result of the player’s choice. The choice itself may nevertheless be free. Moreover, unless one replaces forward causation with backward causation, one cannot argue that it is the outcome of the player’s choice that causes her decision. The statement about the outcome of her decision is true, according to the fortune teller, because of the particular decision the player makes; a statement about the player’s decision is not true because of the outcome of her decision. Consequently, even with respect to the fortune teller, the player’s decision may be regarded as free in the sense that nothing in its past nor in its future determines what the decision actually becomes. The reason why the fortune teller is able to make her prediction could be that the player’s choice instigates an information channel backwards in time.
Philosophers who reject eternalism may not leave the discussion here. They could argue that it may very well be the case that the player’s choice is neither causally determined by past events nor by future events. Yet, eternalism implies that the player’s decision is ontologically determinate before it is made, since it is true today what she will decide tomorrow. Therefore she is not as free as she would be if the future is ontologically open, i.e., it is not yet ontologically determinate what the future is going to be, including the player’s choice. The eternalist may attempt to rebut these critics by arguing that as long as the player’s decision is not causally determined, it is free in any possible sense.
## 4. Physics
The notion of backward causation raises a very different set of questions that need to be answered before a physically adequate notion has been developed.
1. What, if anything, would in physical terms characterize backward causation?
One has to remember that causality as such is an everyday notion that has no natural application in physics. How we could physically identify backward causal processes depends very much on which feature we take our ordinary notion of causation to apply to a physical process. In physics we may be tempted to associate it with different physical notions of processes. Four suggestions have been put forward: (a) the causal link can be identified with the transference of energy; (b) it can be identified with the conservation of physical quantities like charge, linear and angular momentum; (c) it can be identified with interaction of forces; or (d) it can be identified with the microscopic notion of interaction. It appears with respect to all four suggestions, however, that the involved descriptions are invariant under the time reversal operation.
The most fundamental laws of nature are time reversal invariant in the sense that our physical theories allow description of the fundamental reactions and processes in terms of the time reversed order. Such processes are said to be reversible in time. Maxwell’s theory of electromagnetism, for instance, admits two kinds of mathematical solutions for the equations describing the radiation of energy in an electromagnetic field. One is called the retarded solution where radiation appears as outgoing concentric waves, the other is named the advanced solution according to which radiation appears as incoming concentric waves. Apparently the advanced solution describes the temporal inverse phenomena of the retarded solution so that these two solutions are usually regarded as the time reverse solution of the other. Nevertheless, retarded waves, like the increase of entropy in quasi-closed systems, appear to be de facto irreversible although they are described in terms of time invariant laws. Nature seems to prefer certain processes rather than their temporally inversed counterparts in spite of the fact that the laws of nature do not show such a preference. Light, radiation and ripples on a pond always spread outwards from their source rather than inwards just like entropy of a quasi-closed system is always moving from lower to higher states.
### 4.1 The Wheeler-Feynman Absorber Theory
The Wheeler-Feynman theory takes for granted that outgoing, expanding waves are identical with retarded radiation and incoming, contracting waves with advanced radiation. But is such identification without any problems? Not quite. An example with retarded and advanced emitters illustrates clearly why. Think of a stone being thrown directly into the middle of a circular pond. The ripples move outwards from the point where the stone hits the water (the source) in a coherent, organized wave front and eventually reach the edges (the absorber). Moreover, the source acts earlier than the absorber. What will the inverse process look like? It depends on how we understand such a process, whether or not we consider a case that includes a reversed source and a reversed absorber. (A) If they are included, the edges of the pond will now act as the source and the converging waves will eventually reach the middle of the pond. We may create something like this if we dropped a big ring horizontally into the pond. Inside the ring the waves would move inwards in an organized wave front towards the centre. In this case the source (the drop of the ring) would still act earlier than the absorber (the ripples meeting at the middle of the pond from all sides). (B) But if our understanding of the inverse process does not include an exchange of the source with the absorber and vice versa, then the ripples reach the edges of the pond (the absorber) earlier than the stone plunges into the water (the source). This is definitely not a state of affairs we could bring about. Furthermore, if we were to observe such a process, the ripples would seem to move inwards as contracting waves. The problem is that both kinds of inverse processes would seem to appear to us as organized incoming waves but one would be a case of retarded radiation and the other of advanced radiation.
This may not be the only problematic assumption of the Wheeler and Feynman theory. Huw Price (1996) has singled out other problems. Among them is the question of how we may experience the difference between retarded and advanced waves. When Wheeler and Feynman attributed to the source a field of half retarded and half advanced waves, they assumed that the field actually consists of retarded as well as an advanced component. Price objects, however, that there is no measurable difference between the two kinds of waves, and we cannot justify such a distinction by an appeal to the nature of the source because both emitters and absorbers can be associated with retarded as well as advanced waves. Instead he believes that these components are fictitious and that Wheeler and Feynman’s formalism merely offer two different descriptions of the same wave. The problem of the asymmetry, as he sees it, has nothing to do with the fact that transmitters are associated with outgoing radiation rather than incoming radiation but that transmitters are centered on organized outgoing wave fronts whereas receivers are not centered on similar organized incoming wave fronts.
### 4.2 Tachyons
When the discussion of tachyons began to appear in physics in the 1960s, it was soon noticed that such particles according to some frames of reference were associated with negative energies going backwards in time. To understand how, consider the trajectory of the same tachyon in relation of three different reference frames, $$S, S^*$$, and $$S^{**}$$ in the Minkowski-space. Now assume that $$A$$ is, in relation to $$S$$, the emission of a tachyon at $$t_{1}$$ and $$B$$ is the absorption of the tachyon at $$t_{2}$$. According to an observer in $$S, A$$ will be earlier than $$B$$ and the tachyon will carry positive energy forward in time. Nevertheless it is always possible to select a reference frame $$S^*$$ in relation to which an observer will see $$A$$ happen simultaneously with $$B$$ and yet another reference frame $$S^{**}$$ in relation to which an observer sees $$A$$ happens at $$t_{2}^{**}$$ whereas $$B$$ happens at $$t_{1}^{**}$$. According to the observer in $$S^{*}$$, $$A$$ will take place later than $$B$$ and the tachyon carries negative energy backwards in time (See Figure 1).
Figure 1: Spacetime diagram of tachyon
In Figure 1 the planes represent the hypersurfaces of simultaneity. In relation to frame $$S$$ the tachyon source is at rest, and a tachyon is emitted at event $$A$$, with a superluminal but finite velocity. The absorption of the tachyon, event $$B$$, will accordingly occur later than $$A$$ in relation to the observer in $$S$$, and the arrow of trajectory is for that reason pointing into the future above the hypersurface passing through $$A$$ and standing perpendicular to the world-line of the source. But neither with respect to the frame $$S^*$$ nor $$S^{**}$$ is the tachyon source at rest and the hypersurfaces are therefore tilted in relation to the arrow of trajectory. An observer in $$S^*$$ observes the tachyon to have infinite speed, and therefore the hypersurface is tilted so much that it coincides with the arrow. The observer in $$S^{**}$$ is moving so fast with respect to the tachyon source that the hypersurface becomes titled so much that the arrow points into the past below the hypersurface.
E. Recami (1978) tried to avoid the idea that tachyons could move backwards in time by introducing the so-called reinterpretation principle according to which all negative energy tachyons should be interpreted as if they have positive energy and move forward in time. This would mean that the causal order of tachyons should not be regarded objective since both $$A$$ and $$B$$ sometimes denoted the emission and sometimes the absorption depending on the frame of reference. There are, however, good reasons to believe that this suggestion does not solve the problems it was intended to (Faye 1981/1989).
### 4.3 Quantum Mechanics
Other physical candidates for backward causation can be founded in the physics literature. Richard Feynman once came up with the idea that the electron could go backwards in time as a possible interpretation of the positron (Feynman 1949). In fact he imagined the possibility that perhaps there were only one electron in the world zig-zagging back and forth in time. An electron moving backwards in time would carry negative energy whereas it would with respect to our ordinary time sense have positive charge and positive energy. But few consider this as a viable interpretation today (Earman 1967a, 1976).
More recently, the Bell type experiments have been interpreted by some as if quantum events could be connected in such a way that the past light cone might be accessible under non-local interaction; not only in the sense of action at a distance but as backward causation. One of the most enticing experiments of this kind is the Delayed Choice Quantum Eraser designed by Yoon-Ho Kim et al. (2000). It is a rather complicated construction. It is set up to measure correlated pairs of photons, which are in an entangled state, so that one of the two photons is detected 8 nanoseconds before its partner. The results of the experiment are quite amazing. They seem to indicate that the behavior of the photons detected these 8 nanoseconds before their partners is determined by how the partners will be detected. Indeed it might be tempting to interpret these results as an example of the future causing the past. The result is, however, in accordance with the predictions of quantum mechanics.
However, David Ellerman (2015) argues that interpreting delayed-choice experiments or similar experiments as revealing cases of backward causation or retrocausation relies on what he calls the separation fallacy:
We have seen the same fallacy of interpretation in two-slit experiments, which-way interferometer experiments, polarization analyzers, and Stern-Gerlach experiments. The common element in all the cases is that there is some separation apparatus that puts a particle into a certain superposition of spatially “entangled” or correlated eigenstates in such a manner that when an appropriately spatially-positioned detector induces a collapse to an eigenstate, then the detector will only register one of the eigenstates. The separation fallacy is that this is misinterpreted as showing that the particle was already in that eigenstate in that position as a result of the previous “separation.” In fact the superposition evolves until some distinction is made that constitutes a measurement, and only then is the state reduced to an eigenstate. The quantum erasers are more elaborate versions of these simpler experiments, and a similar separation fallacy arises in that context.
Thus, Ellerman argues that when one describes a system in a superposition of certain eigenstates, it does not mean that the system is in any of these eigenstates before some measurement is carried out. Therefore, it is wrong to interpret delayed-choice experiments, and their like, as if the future measurement determines some past eigenstates, which all were parts of a superposition.
In his discussion of the experimental violation of Bell’s inequalities Don Howard (1989) distinguishes, based on an earlier work done by Jon Jarrett, between locality and separability. The locality condition states that a measurement of a pair of objects emerging from a singlet is statistically independent of the setting of the apparatus used to measure its counterpart. However, the separability condition is defined as the joint probability is equal to the the product of the probability of each state. We know that the experimental violation of Bell’s inequalities involves the invalidation of one of these conditions. Moreover, if we take Ellerman’s argument into consideration, the two entangled particles, although separated, are still in a state of superposition until the measurement takes place.
If we consider the notion of the entangled state in quantum mechanics, we find that it is characterized as a unified, non-separable state due to the help of the notion of superposition of possible eigenstates represented by one common wave function for the correlated pair. Such a superposition is neither distance-dependent nor time-dependent. Therefore it is not surprising that based on the correct predictions of quantum mechanics it is impossible to find support of the violation of normal causation within this kind of experiment. With reference to the philosophical discussion about quantum mechanical entanglement, we can conclude that the experimental results of this sort violate the principle of separability rather than the principle of locality.
Phillippe Eberhard and Ronald R. Roos (1989) have established a theorem which says that if quantum mechanics is correct, it is impossible to use quantum effects to generate a break in the chain of normal causation. Quantum field theory does not allow any superluminal communication between different observers. Indeed, this is not so strange, since quantum field theory is relativistically invariant whereas superluminal frames of reference are not. But Eberhard and Roos’ theorem does not rule out all forms of backward causation. Two possible scenarios are still open: (1) entangled pairs exchange some form of superluminal information (and energy) below the limits of Heisenberg’s uncertainty relations; or (2) causation may be symmetrical so that the direction of causation in a physical system is determined by its boundary conditions.
Costa de Beauregard (1977, 1979), for instance, has suggested that when a system of two photons in a singlet state is measured by two observers in two regions separated by a space-like distance, then it is precisely the act of observation that produces the past of the measuring process in the sense that it influences the source that emitted the two photons. de Beauregard’s idea is that the element of reality being revealed in the formulation of the EPR paradox is real only because it was created by actually performed acts of observation that was propagated backwards in time with one of the two correlated quantum objects from the measuring device to the source of the photons.
Some physicists, like Elitzur et al. (2016), suggest a form of too-late-choice experiment that supports such a time-symmetric interpretations of quantum mechanics according to which backwards causation plays a significant role. In the normal EPR experiment, each measurement determines, say, the spin value of two separated, say, electrons in a singlet state along a certain orientation of the apparatus. The outcome then proves that the spin value of one electron has been affected by the distant experimenter’s choice of spin orientation of the other electron. However, what Elitzur et al. now imagine is a reversal setting in which a chosen spin value determines the corresponding orientation. Based on their analysis, they conclude that “it turns out that the orientation is similarly subject to nonlocal effects [as the spin value.]”
Several other philosophers and physicists have come forward with similar ideas. Aharonov and Vaidman (1997) have formulated a two vector approach to quantum mechanics “in which a quantum system is described, at a given time, by two (instead of one) quantum states: the usual one evolving toward the future and the second evolving backwards in time from a future measurement.” Also Cramer’s transactional interpretation of quantum mechanics involves the idea of a second wave travelling backwards in time (Cramer, 1986). The basic assumption behind all of them is that in the micro-world we find only causal symmetry, and this fact together with proper boundary conditions can be used to give an explanation of outcomes that seem otherwise paradoxical. Such quantum correlation experiments can, however, be interpreted in many other ways. In the end it seems as if it all depends one whether one assumes that Bell experiments break with either the locality condition or the separability condition. A time-symmetric interpretation of quantum mechanics is required only if one believes that it is the locality condition that is violated in order to avoid being in conflict with relativity theory.
### 4.4 Two alternatives
These alleged examples of backward causation have one thing in common. They are all based on the idea that fundamental physical processes are by themselves symmetric in nature. Our ordinary notion of causation does not track any nomological feature of the world. What counts as the cause and the effect depends on the observer’s projection of his or her temporal sense onto the world. So it is still an open question how a coherent notion of backward causation can fit into this general understanding of nature. The question we therefore have to answer is the following:
1. How can we distinguish between forward causation and backward causation if all basic physical processes are time symmetric according to our description of nature?
Two very different reactions to this problem seem possible.
#### 4.4.1 Boundary Conditions
One proposal is to say that if we came across reversed cases of de facto irreversible processes, such as running a film backwards in which the cream converged in a coffee cup, such cases should be interpreted as examples of backward causation (Price 1996). Such a claim build on a common interpretation of time reversal invariance of processes according to which this descriptive feature of the dynamical equations of physics makes reversal processes symmetric in time. Many philosophers have defended such an interpretation, in particular Hans Reichenbach (1956, 1929 [1958]) and Adolf Grünbaum (1963). This also led them to argued that only de facto irreversible processes, such as those described by statistical thermodynamics, could be used to defined a physical orientation of time. Thus, it seems to be the case that the actual world consisting of mostly de facto irreversible processes on the macroscopic level due to a prevalent set of boundary conditions is temporally and causally symmetric on the microscopic level. On the macroscopic level, de facto irreversible processes emerge, because the boundary conditions are a result of the huge degree of freedom among microprocesses or coherence conditions forced upon the underlying microscopic processes.
The point is here to argue that it is the absence of the right initial or boundary conditions on the macroscopic level that makes backward causation so rare or nearly empirically impossible. This suggestion is based on three basic assumptions: (i) there is no objective asymmetry in the world, causal processes are intrinsically symmetric in nature, or causation is bidirectional, and therefore the fundamental processes of the micro-world are temporally symmetric; (ii) causal asymmetry is subjective in the sense that any attribution of an asymmetry between cause and effect depends on our use of counterfactuals and our own temporal orientation; (iii) backward causation, or advanced action, is nonetheless possible because sometimes the correlation of certain past events depends on the existence of causally symmetric processes and some future boundary conditions. For instance, advanced actions in electrodynamics require that the existence of transmitters in the future are centered on organized incoming wave fronts; and advanced actions in quantum mechanics require that their present states are in part determined by the future conditions (measurements) they are to encounter. This feature is then taken to explain quantum entanglement and the violation of Bell’s inequalities in quantum mechanics.
A simple consideration seems to support this interpretation. Think of a particle travelling between two boxes. The normal observer and the counter-observer who has an inverse time sense will describe the exchange in conflicting terms. To the normal observer Box 1, say, will be considered as the emitter because it loses energy before anything in Box 2 happens. Therefore, Box 2 will be considered as the receiver since it gains energy at a later time. So in relation to the normal observer, the particle travels from Box 1 to Box 2. The counter-observer, however, sees the situation with opposite eyes. In relation to him, Box 2 loses energy and not until thereafter does Box 1 gain a similar amount of energy. Accordingly, in relation to the counter-observer, the particle moves from Box 2 to Box 1. In other words whether a box is considered to be an emitter or a receiver depends on the observer’s time sense.
#### 4.4.2 Nomic conditions
The other proposal denies that basic physical processes are time symmetric and argues, in contrast, that the causal asymmetry is objective and therefore that there exists an intrinsic difference between the cause and the effect of all physical processes. John Earman (1967b and 1969) may be the first who argued against the interpretation that time reversal invariance of processes is identical with invariance under the exchange of the temporal order between earlier and later. Two reversed tokens of the same type of physical process do not developed in the opposite direction of time.
Hence, backward causation should not be considered as a notion about boundary conditions but as a notion concerned with processes that nomically distinguish themselves from forward causal processes. Thus, if there are processes in the world that might be seen as a manifestation of backward causation, these are not to be depicted by a description that leaves them to be time reversed cases of ordinary forward causal processes (Faye 1981/1989, 1997, 2002). This alternative interpretation rests on a basic claim and four assumptions.
The fundamental claim is that for any observer it is possible to identify experimentally the cause and the effect so that these remain the same even in relation to counter-observers, i.e., observers having the opposite time sense of ours. In support of this claim consider the following thought experiment. Two boxes, each having a shutter, are facing each other. Assume, ex hypothesis, that Box 1 is the particle source and Box 2 is the particle receiver. The question is how a normal observer and a counter-observer can come to agreement that particles move from Box 1 to Box 2. The answer can be found through a series of manipulations with the shutters, I would say. There are four possible combinations of the two shutters: open-open, close-close, open-close, close-open. Let us call any change of energy in Box 1, regardless of whether it emits or receives a particle, $$A$$ and, similarly, any change of energy in Box $$2 B$$. Whether $$A$$ or $$B$$ stand for a gain or a loss of energy can be determined by weighing the two boxes. (i) In case both boxes are closed, no particle will leave Box 1 and no particle is received by Box 2, thus no gain or loss of energy occurs, and both the normal observer and the counter-observer see a situation of not-$$A$$, not-$$B$$. (ii) In case both boxes are open a particle leaves Box 1 and is received by Box 2. Again this can be observed by measuring the change of energy in the two boxes. Thus the observers will see a situation of both $$A$$ and $$B$$. (iii) In case Box 1 is closed and Box 2 is open, they will observe no change of energy in Box 1 (because it is closed) and, since no particle is leaving Box 1, no particle will reach Box 2 although its shutter is open. Hence the observers measure no energy change in this box. Thus they see not-$$A$$ and not-$$B$$. (iv) Finally, if Box 1 is open and Box 2 is closed, a particle leaves Box 1, but none is received by Box 2. In other words, there is a loss or a gain of energy in Box 1, but no loss or gain of energy in Box 2. So the observers see $$A$$ and not-$$B$$. The upshot of this toy experiment is that the normal observer as well as the counter-observer experience two $$A$$s but only one $$B$$, and one not-$$A$$ but two not-$$B$$s; therefore both will agree that the particles move from Box 1 to Box 2.
This means that what a normal observer identifies as a forward causal process will be regarded as a backward causal process in relation to the counter-observer in the sense that the very same event acting as a past cause for the normal observer will act as a future cause for the counter-observer. This indicates, too, that in relation to a normal observer forward causation and backward causation cannot be regarded as two different manifestations of nomologically reversible (but de facto irreversible) processes since both manifestations—the common process and the very improbable reversed process—would develop forward in time. If this claim is true, it implies that the description of physical processes should reflect such an intrinsic asymmetry in a way that the nomic description varies according to whether the process in question goes forward or backwards in time. Moreover, we must also be able to distinguish theoretically (and not only experimentally) between the normal observer’s report and the counter-observer’s report of the same process by a separate convention in respect to whether the process is forward moving or backward moving. What we want is a characterization of every physical process so that the invariance of cause and effect corresponds to nomological irreversibility.
In order to establish a nomic, intrinsic distinction between forward causal processes and backward causal processes one has to take departure in four assumptions. (i) Process tokens and process types are distinct in the sense that only process types are reversible, process tokens are not. (ii) A normal observer will describe causal processes propagating forward in time in terms of positive mass and positive energy states pointing into her future whereas she will describe the same tokens in terms of negative mass and energy states pointing into her past. This reflects two possible solutions of the four-momentum vector in the theory of relativity. (iii)Thus, one must distinguish between a passive time reversal operation and an active time reversal operation. The passive transformation is applied to the same process token by describing it in terms of opposite coordinates and opposite energy states. The active transformation, in contrast, brings about another token of the same process type in virtue of some physical translation or rotation of the system itself, both tokens having the same energy sign pointing in the same direction of time. (iv) The description in terms of positive mass and the positive energy flow corresponds to the intrinsic order of the propagation.
Now, let us try to apply the nomic interpretation to the above consideration concerning the exchange of a particle between two boxes. In relation to the normal observer who describes the particle in terms of its positive energy component, it travels from Box 1 to Box 2 because Box 1 looses energy at an earlier time and Box 2 gains energy at a later time. The same situation is by the counter-observer described in terms of the particle’s negative energy component as a situation where something happens in Box 2 before it happens in Box 1. In relation to the counter-observer, Box 2 would not, as the boundary interpretation suggests, loose energy. On the contrary, Box 2 would seem to gain energy, but the counter-observer would describe the particle as a series of negative energy states reaching into his future supposing the particle to be moving from Box 2 to Box 1 carrying negative energy. But, as we have just argued, the particle really moves from Box 1 to Box 2, from the counter-observer’s future into his past carrying positive energy.
Consequently, the nomic interpretation holds that in relation to our normal time sense the causal direction of ordinary processes is identical with that of their reversed processes. In other words, take two tokens of a nomologically reversible process type, say $$A$$ and $$B$$, and let $$B$$ be the actively time reversed process of $$A$$, then this interpretation claims that $$A$$ and $$B$$ causally develop in the same direction of time. So, according to this view, neither incoming, contracting electromagnetic waves nor the decrease of entropy would count as examples of backward causation as long as such processes involve ordinary types of matter, i.e., matter that possesses positive mass and/or energy pointing, in relation to our normal time sense, towards the future. The notion of backward causation should instead be applied to matter of a different type, particles that appear to have, according to usual conventions, negative mass and/or energy pointing, in relation to our normal time sense, towards the future but positive mass and/or energy pointing towards the past. Such advanced matter, if it exists, should be distinguished from both ordinary retarded matter as well as tachyons by always being described with respect to our time sense in terms of negative mass and energy stretching forward in time. A consequence is that a world in which advanced matter exists together with retarded matter, and where advanced matter is able to interact directly with the same amount of retarded matter, both would, in case they actually did interact, annihilate without leaving any trace of energy.
How and whether the notion of backward causation has a role to play in physics has yet to be seen. But as long as no common agreement exists among philosophers and physicists about what in the physical description of the world corresponds to our everyday notion of causation, it would still be a matter of theoretical dispute what counts as empirical examples of backward causation.
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• Gorovitz, G., 1964, “Leaving the Past Alone”, Philosophical Review, 73: 360–371.
• Grünbaum, A., 1963, Philosophical Problems of Space and Time, New York: A.A. Knopf; 2nd expanded edition, 1973, published in Boston Studies in Philosophy of Science.
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• Gärdenfors, P., 2006, How Homo Became Sapiens, Oxford: Oxford University Press.
• Horwich, P., 1987, Asymmetries in Time, Cambridge, MA: MIT Press.
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• Howard, D., 1989, “Holism, Separability, and the Metaphysical Implication of Bell Experiments”, in J. Cushing and E. McMullin (eds.), Philosophical Consequences of Quantum Theory: Reflections on Bell’s Theorem, Notre Dame: The University of Notre Dame Press.
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• Maudlin, J., 2002, Quantum Non-locality and Relativity, Oxford: Blackwell Publishing.
• Mellor, D.H., 1981, Real Time, Cambridge: Cambridge University Press.
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• Nozick, R., 1969, “Newcomb’s Problem and Two Principles of Choice”, in R. Rescher et al. (eds.) Essays in Honor of Carl. F. Hempel, Dordrecht: Reidel, pp. 114–146.
• Price, H., 1996, Time’s Arrow and Archimedes’ Point, Oxford: Oxford University Press.
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2022-12-05 13:29:31
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https://www.fisicalab.com/en/section/vertical-launch
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Among all the constant acceleration motions, or uniformly accelerated rectilinear movements (u.a.r.m.), there are two of particular interest: free fall and vertical launch. In this section, we will study vertical launch. Both are governed by the equations of the uniformly accelerated rectilinear motion (u.a.r.m.):
$y={y}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$
$v={v}_{0}+a\cdot t$
$a=\text{cte}$
Vertical launch
In vertical launch, an object is launched vertically up or down from a height H without taking into consideration any kind of friction with the air or any other obstacle. It is a uniformly accelerated rectilinear motion (u.a.r.m.) in which the acceleration is gravity. On the surface of the Earth, the acceleration of gravity can be considered constant and directed downward. It is designated by the letter g, and its value is 9.8 m/s2.
To study vertical launch motion, normally, we will use a system of reference whose origin of coordinates is located at the base of the positive y-axis, as can be seen in the picture.
Vertical launch upward
The body is launched upward from a height H, with a velocity greater than 0. As it ascends, its speed decreases until it reaches 0 (maximum height). From that point onwards, its velocity is negative and it begins to descend.
Vertical launch down
The body is launched downwards from a height H, with a speed smaller than 0 which will remain negative throughout the entire motion.
The vertical launch is a uniformly accelerated rectilinear motion (u.a.r.m.) or constant acceleration motion in which a body is launched vertically with some initial velocity from a certain height and do not find any resistance on its way. We can distinguish two cases based on the system of reference considered:
• We launch the body upward and therefore the initial velocity is positive (v0>0). In this case the equations for the upward vertical launch are:
$y=\mathrm{H}+{v}_{0}t-\frac{1}{2}g{t}^{2}$
$v={v}_{0}-g\cdot t$
$a=-g$
• We launch the body downward and therefore the initial velocity is negative (v0<0). In this case the equations for the downward vertical launch are:
$y=\mathrm{H}-{v}_{0}t-\frac{1}{2}g{t}^{2}$
$v=-{v}_{0}-g\cdot t$
$a=-g$
Where:
• y: Final position of the body. Its unit in the International System (SI) is the meter (m)
• v, v0: The final and initial velocity of the body respectively. Its unit in the International System (SI) is the meter per second (m/s)
• a: Acceleration of the body while in motion. Its unit in the International System (SI) is the meter per second squared (m/s2)
• t: Time spent on the motion. Its unit in the International System (SI) is the second (s)
• H: Height from which the body is launched. It is a measurement of length and therefore is unit is the meter (m)
• g: Value of the gravitational acceleration which on Earth surface can be considered equal to 9.8 m/s2
Experiment and Learn
Data
g = 9.8 m/s2 | |
Vertical launch
The blue ball in the figure represents a body suspended above the ground. Drag it to the initial height H that you want and select the value of the initial velocity ( v) with which the body will be launched vertically. Then press the Play button to drop it.
Notice that, once the simulation is started, you can slide the time t(s) and see how, under the label Data, the corresponding values of position (y) and velocity (v) are calculated, as the body falls to the ground.
Verify that:
• If v0 is positive the body ascend to reach the highest point to then descend
• If v0 is negative the body descends from the start
• If the value of v0 is 0 it is a free fall motion
Solved exercises worksheet
Here you can test what you have learned in this section.
A rookie equilibrist
difficulty
A rookie equilibrist is standing on a platform 12 meters above the ground. While practicing juggling with 2 balls, he stumbles and throws both balls at 9 m/s, however, he throws one up which we will call A and the other one down which we will call B. Considering that gravity is 10 m/s2, calculate
a) The time they are in the air.
b) Their velocity when they hit the ground.
c) The maximum height that ball A reaches.
A launch of negligible mass
difficulty
From a height of 40 meters an object of negligible mass is thrown downward with a velocity of 20 m/s. How long will it take to hit the ground? What will be its velocity on impact?
Vertical launch and free fall
difficulty
A stone is let to free fall to the bottom of a cliff with a height of 80 m. A second later a second stone is thrown downward so that it reaches the bottom at the same time as the first one
1. What was the launch velocity of the second stone?
2. What was the velocity of the first stone when they both hit the bottom?
3. How long was the second stone in the air?
Formulas worksheet
Here is a full list of formulas for the section Vertical Launch. By understanding each equation, you will be able to solve any problem that you may encounter at this level.
Click on the icon to export them to any compatible external program.
Equation of position of downward vertical launch
$y=\mathrm{H}-{v}_{0}t-\frac{1}{2}g{t}^{2}$
Equation of position in upward vertical launch
$y=\mathrm{H}+{v}_{0}t-\frac{1}{2}g{t}^{2}$
Position equation of uniformly accelerated rectilinear motion - y-axis
$y={y}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$
Equation of velocity of downward vertical launch vertical
$v=-{v}_{0}-g\cdot t$
Equation of velocity of the upward vertical launch
$v={v}_{0}-g\cdot t$
Equation of acceleration on the Earth surface
$a=-g$
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2019-08-22 20:41:22
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https://socratic.org/questions/if-the-length-of-a-21-cm-spring-increases-to-57-cm-when-a-5-kg-weight-is-hanging
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# If the length of a 21 cm spring increases to 57 cm when a 5 kg weight is hanging from it, what is the spring's constant?
Dec 26, 2015
We should expect $k$ to be somewhere in the hundreds for typical springs.
This is simply looking at the equation
$\setminus m a t h b f \left(F = - k \Delta y\right)$
...for when an object is hanging off a spring.
So, the only force acting on the spring is the force $\setminus m a t h b f \left({F}_{g}\right)$ due to gravity $\setminus m a t h b f \left(g\right)$. The generic force $F$, then, is ${F}_{g}$. Thus, with $g < 0$ (where down is negative):
${F}_{g} = - k \Delta y$
$m g = - k \left({y}_{f} - {y}_{i}\right)$
$\frac{m g}{{y}_{i} - {y}_{f}} = k$
color(blue)(k) = (("5 kg")(-"9.80665" cancel"m""/s"^2))/("0.21" cancel"m" - "0.57" cancel"m")
$= \textcolor{b l u e}{{\text{136.20 kg/s}}^{2}}$ or $\textcolor{b l u e}{\text{N/m}}$
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2020-06-05 19:49:14
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https://math.stackexchange.com/questions/4068448/uniform-continuity-on-0-infty-and-limits-of-the-function-at-lim-at-0
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# Uniform continuity on $(0, \infty)$ and limits of the function at $\lim$ at $0$ and $\infty$
Are the following statements True or False?
1. If $$f : (0, \infty) \rightarrow \mathbb{R}$$ is uniformly continuous then $$\lim_{x \rightarrow 0} f(x)$$ exists.
2. If $$f : (0, \infty) \rightarrow \mathbb{R}$$ is uniformly continuous then $$\lim_{x \rightarrow \infty} f(x)$$ exists.
For part 1, this is what I tried : consider $$\{x_n\}$$, $$\{y_n\}$$ be two Cauchy sequences converging to 0. Then by uniform continuity $$\{f(x_n)\}$$ and $$\{f(y_n)\}$$ are Cauchy and convergent to the same point $$l = \lim_{n \rightarrow \infty} f(x_n)$$. To show $$l = f(\lim_{n \rightarrow \infty} x_n) = f(0)$$, which is satisfied as f is continuous on $$(0, \infty)$$ hence the limits can be exchanged. Hence $$\lim_{x \rightarrow 0} f(x)$$ exists.
For the second part, by uniform continuity on $$[b, \infty)$$, $$b$$ sufficiently large, given $$\epsilon > 0$$, there exists $$\delta > 0$$ such that whenever $$|x-y| < \delta$$, $$|f(x)- f(y)| < \epsilon$$. Consider a sequence $$x_n \rightarrow \infty$$. I am stuck here.
Could you please check my ideas and help me finish the problems? Thank you.
• It's actually possible to have $f(\{M,\infty)\})=\mathbb R$ for every $M>0.$
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2022-01-21 16:12:18
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http://mathhelpforum.com/advanced-algebra/194490-x-3-1-0-soutions-really-cyclic-group.html
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# Thread: x^3-1=0, soutions really a cyclic group?
1. ## x^3-1=0, soutions really a cyclic group?
"Prove that the cube roots of unity (ie the roots of $x^3-1=0$) form a cyclic group of order three."
I'm assuming that the operation is multiplication, but then that would mean that the set of solutions to the doesn't contain the identity element. Furthermore, it wouldn't be closed under multiplication, since multiplying the two complex roots gives a number outside of the set of solutions. Is there something to this exercise that I'm not getting?
EDIT: Nevermind, I just realized I had a dumb algebra error.
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2017-12-17 16:33:57
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http://forum.allaboutcircuits.com/threads/emitter-follower-analysis.83287/
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# Emitter-follower analysis
Discussion in 'General Electronics Chat' started by gerases, Apr 2, 2013.
1. ### gerases Thread Starter Member
Oct 29, 2012
177
2
Hi all,
This circuit is driving me crazy and I'm not smart enough to figure it out on my own.
My question is: how do we get 6.42V on the base? Usually, there's a voltage divider at the base and then it's a little more simple. But here, I don't know where to start.
The whole emitter follower arrangement has a few mysteries for me. When the emitter goes straight to ground, it's clear, the base current flows, the CE path starts to pass current depending on how much current flows into the base, which in turn is determined by the base resistor.
With the emitter-follower, it's more complicated because the base current depends on Vload and the Vload depends on VCE, which in turn depends on Ib. It's like a chicken and egg problem.
Thank you!
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2. ### Jony130 AAC Fanatic!
Feb 17, 2009
3,993
1,116
In this circuit you post the BJT is in a saturation.
So I don't know what you want to do with this circuit?
screen1988 likes this.
3. ### gerases Thread Starter Member
Oct 29, 2012
177
2
Well, can you tell me how to calculate the drop over the Rin in general in a configuration like this?
4. ### Jony130 AAC Fanatic!
Feb 17, 2009
3,993
1,116
Emitter follower is a very simply circuit. The output voltage is always 0.6V (Vbe) lower the the input voltage.
See some examples
Also notice that the base current is (β+1) smaller then emitter current (load current).
So our base current source B1 see our load ( Re resistor) not as 100Ω resistor. But B1 see (β+1)*Re load.
Here you have anther example.
Spouse we have a 1K resistor voltage divider supply from 10V battery.
Without any load connect to the output terminal of our voltage divider the output voltage is equal 5V. Now we connect a 100Ω load resistor across the output terminal.
And now our voltage divider output voltage drops from 5V to 0.83V. So we ruin our circuit.
To fix this issue we add a buffer (emitter follower) see the diagram
Now I hope that you see why we say that emitter follower has a high input impedance and low output impedance. It's all thanks to BJT and his current gain.
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• ###### 12a.PNG
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screen1988 and absf like this.
Feb 17, 2009
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6. ### gerases Thread Starter Member
Oct 29, 2012
177
2
By Rin I meant the 1K resistor on the base. Sorry.
I tried to apply that method, but there, we have a voltage divider and everything is a bit simpler because we know the base voltage straight away. Here, I don't see where to begin. How come the base voltage in this case is not simply 9V?
Last edited: Apr 2, 2013
7. ### Jony130 AAC Fanatic!
Feb 17, 2009
3,993
1,116
As usually we start our analysis by applying Kirchhoff's law.
Vcc = IB*Rb + Vbe + Ie*Re
And for the BJT
Ie = Ib *(β + 1)
Ib = (Vcc - Vbe)/( Rb + Re*(β+1))
and Voltage at base
Vb = Vcc - Ib*Rb or Vb = Vbe + Ie*Re
So if we assume β = 100; and Vbe = 0.65V we have
Ib = (9V - 0.65)/(1K+ 1K*101) = 8.35V/102k = 81.87μA
and
Ie = 101 * 81.87μA = 8.26mA
Now we can check whether BJT is on active region or in saturation
Collector current cannot be greater than Ic_max = (Vcc - Vce_sat)/(Rc + Re) ≈ 9V/2K = 4.5mA.
But we calculate 8.26mA.
So if Ic >> Ic_max BJT is in saturation region. And in saturation Ic = β*IB don't hold anymore.
All we can do next is to write this equation
Ie = Ib + Ic
So we have
Ie = Ve/Re
Ib = (Vcc - Vbe - Ve)/Rb
Ic = (Vcc - Vce_sat - Ve)/Rc
And using these three equations we can solve for Ve
$Ve = ( \frac{Vcc - Vbe}{RB} + \frac{Vcc - Vce_{sat}}{Rc}) * (RB||RC||RE) = 5.75V$
So voltage at base is equal to
Vb = Vbe + Ve = 6.4V
I assume Vbe = 0.65V and Vce_sat = 0.1V.
This is all we need to do to find DC bias current in this circuit.
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8. ### gerases Thread Starter Member
Oct 29, 2012
177
2
Very cool. What are the vertical bars in (RB||RC||RE)?
9. ### Jony130 AAC Fanatic!
Feb 17, 2009
3,993
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"||" - parallel connection.
In our case we have RC = RB = RE = 1K so equivalent resistance is equal
Req = 1K/3 = 333R
Last edited: Apr 2, 2013
screen1988 and gerases like this.
Oct 29, 2012
177
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Oh, I see!
11. ### wmodavis Well-Known Member
Oct 23, 2010
737
150
Actually the circuit you refer to is not an emitter follower. It is a common emitter BJT circuit with the emitter resistor not bypassed.
12. ### Jony130 AAC Fanatic!
Feb 17, 2009
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How can you know that? There is no input/output terminal on the diagram.
And what if I connect "the load" directly to Re resistor?
13. ### crutschow Expert
Mar 14, 2008
13,502
3,376
Yes, if you take the output from Re it will be acting as an emitter follower. But a normal emitter follower circuit has no collector resistor since it serves no purpose and can actually slow down the stage response due to Miller negative AC feedback through the base-collector capacitance. So when a BJT stage is seen with both collector and emitter resistors, the logical assumption is that it is a CE circuit with an emitter bias stabilization resistor (which can be AC bypassed or not as required by the circuit requirements).
14. ### wmodavis Well-Known Member
Oct 23, 2010
737
150
I can know that because your original post was sooo incomplete wrt providing detail as to what you were asking. See Cruts post.
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2017-01-22 04:13:46
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http://boazspot.blogspot.com/2007/11/
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## Monday, November 19, 2007
### client/server vs. p2p
I realized that in some ways, what I'm been struggling with lately is a matter of architecture.
In a client/server approach, the resources are collected together in centralized places and server programs are your interface to those resources. In a peer to peer approach, there are many sources of information. In fact each node is on the same level. I feel like in each of the projects I was working on, I was being pushed to either be a client or a server, and I didn't particularly want to be either.
I know that hierarchy has its place, but on the whole, I'm much more comfortable with a p2p approach to many aspects of work and life.
## Monday, November 12, 2007
### back to physics
Ok, I need to do some integrals. I'm trying to find the number of electrons scattering per unit time into a given longitudinal velocity.
I need to do the integral:
$\int_0^\1 du\int_0^\infty dq {1\over (qu)^3}(2-u^2)\delta(p_z - qu)f(q)$
where $f(q)$ is typically a Gaussian.
I can do the $u$ integral first which seems like the right thing to do. But then to check it, I can integrate over $p_z$ from the momentum acceptance to infinity and get the Touschek lifetime. But I can't see how it gives the right answer. Anyway, its fun to get back to this stuff.
## Saturday, November 10, 2007
### bridge
I've been trying to understand a certain piece of technical history that is important for accelerator physics. It involves people at each stage and their own skill sets. I'll leave it somewhat vague for now, because I don't understand it well enough. On one end we have various math ideas which from a certain perspective join numerical and analytical approaches, but from another, its just some math. These go under the names differential algebra, non-standard analysis and truncated power series algebra. At the other end of the bridge we have collaborative work on designing a particle accelerator.
The thing is that in some ways this is really just a personal bridge. I am comfortable reading math. And I am comfortable working in a team with an open environment. But in between has been a huge mess.
I was just trying to find references to Foucault's The Order of Things. I found this essay.
What the author says is that Foucault's grand schemes aren't particularly new, and his referencing is pretty poor, but when discussing particulars, he adds new depth.
Continuing my free association: I was just listening to this radio show starring Richard Stallman!
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2018-07-18 12:44:18
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https://zbmath.org/?q=an:0910.35145
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zbMATH — the first resource for mathematics
Regularity of the free boundary for the porous medium equation. (English) Zbl 0910.35145
This paper deals with the initial value problem for the porous medium equation $$u_t=\Delta u^m$$ with compactly supported initial data, written for pressure $$f=mu^{m-1}$$. The authors prove that, under rather general assumptions on the initial data, the free boundary is a smooth surface locally in time. First, a model linear degenerate equation is studied on the half-space $$x\geq 0$$. The basic idea is to establish Schauder type coercive estimates for solutions of the model equation. Then the result is extended to a certain class of quasilinear degenerate evolution equations. Then the proof of regularity of the free boundary is given. Using a global change of coordinates, the authors transform the free boundary problem to a fixed boundary problem for a degenerate quasilinear equation, which can be solved in appropriately defined Hölder spaces.
MSC:
35R35 Free boundary problems for PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B65 Smoothness and regularity of solutions to PDEs
Full Text:
References:
[1] Sigurd Angenent, Analyticity of the interface of the porous media equation after the waiting time, Proc. Amer. Math. Soc. 102 (1988), no. 2, 329 – 336. · Zbl 0653.35040 [2] D. G. Aronson, Regularity propeties of flows through porous media, SIAM J. Appl. Math. 17 (1969), 461 – 467. · Zbl 0187.03401 [3] D. G. Aronson, Regularity properties of flows through porous media: A counterexample., SIAM J. Appl. Math. 19 (1970), 299 – 307. · Zbl 0255.76099 [4] D. G. Aronson, Regularity properties of flows through porous media: The interface., Arch. Rational Mech. Anal. 37 (1970), 1 – 10. · Zbl 0202.37901 [5] D. G. Aronson, L. A. Caffarelli, and Juan Luis Vázquez, Interfaces with a corner point in one-dimensional porous medium flow, Comm. Pure Appl. Math. 38 (1985), no. 4, 375 – 404. · Zbl 0544.35058 [6] D. G. Aronson and J. L. Vázquez, Eventual \?^{\infty }-regularity and concavity for flows in one-dimensional porous media, Arch. Rational Mech. Anal. 99 (1987), no. 4, 329 – 348. · Zbl 0642.76108 [7] Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189 – 213. · Zbl 0692.35017 [8] Luis A. Caffarelli and Avner Friedman, Regularity of the free boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math. 101 (1979), no. 6, 1193 – 1218. · Zbl 0439.76084 [9] Luis A. Caffarelli and Avner Friedman, Regularity of the free boundary of a gas flow in an \?-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), no. 3, 361 – 391. · Zbl 0439.76085 [10] L. A. Caffarelli, J. L. Vázquez, and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the \?-dimensional porous medium equation, Indiana Univ. Math. J. 36 (1987), no. 2, 373 – 401. · Zbl 0644.35058 [11] Luis A. Caffarelli and NoemíI. Wolanski, \?^{1,\?} regularity of the free boundary for the \?-dimensional porous media equation, Comm. Pure Appl. Math. 43 (1990), no. 7, 885 – 902. · Zbl 0728.76103 [12] K. Höllig, H.O. Kreiss, $$C^{\infty }$$ regularity for the porous medium equation, Univ. of Winsconsin Madison, Computer Scienced Dept., Technical report # 600. · Zbl 0597.35107 [13] Barry F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), no. 2, 381 – 415. · Zbl 0365.35030 [14] J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math. 20 (1967), 797 – 872. · Zbl 0153.14503 [15] M. V. Safonov, The classical solution of the elliptic Bellman equation, Dokl. Akad. Nauk SSSR 278 (1984), no. 4, 810 – 813 (Russian). [16] M. V. Safonov, Classical solution of second-order nonlinear elliptic equations, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1272 – 1287, 1328 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 3, 597 – 612. [17] Lihe Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math. 45 (1992), no. 1, 27 – 76. · Zbl 0832.35025 [18] Lihe Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math. 45 (1992), no. 2, 141 – 178. · Zbl 0774.35042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-12-08 12:57:47
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https://www.hpmuseum.org/forum/thread-2397-post-21271.html
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Entering partial derivatives?
11-06-2014, 01:46 PM
Post: #1
DrD Senior Member Posts: 1,095 Joined: Feb 2014
Entering partial derivatives?
User Interface question: $$f(x,y):=x^2y+xy^2$$
How best to enter this second mixed partial derivative into the entry line(?):
$$\frac{∂^2f(x,y)}{∂x∂y}$$
-Dale-
11-06-2014, 03:11 PM
Post: #2
Han Senior Member Posts: 1,811 Joined: Dec 2013
RE: Entering partial derivatives?
One way is:
f(x,y):=x^2*y+x*y^2;
diff(f(x,y),x,y);
For higher order derivatives, you simply add more commas and variables. For the $$\frac{\partial}{\partial x^2}$$ you can use the shortcut:
diff(f(x,y),x$2); You can also mix and match the two, such as: diff(f(x,y),y$2,x);
Graph 3D | QPI | SolveSys
11-06-2014, 10:06 PM
Post: #3
DrD Senior Member Posts: 1,095 Joined: Feb 2014
RE: Entering partial derivatives?
Is there a short way to test a point (1,2) on the partial with respect to x?
diff((f(x,y),x)|x=1,y=2 doesn't work.
diff((f(x,y),x|x=1,y=2) doesn't work.
diff(f(x,y),x)|assume(x = 1,y = 2) doesn't work.
(diff(f(x,y),x)|(assume(x = 1,y = 2))) doesn't work.
(f(x,y),x)'|x=1,y=2) doesn't work.
11-06-2014, 11:36 PM (This post was last modified: 11-07-2014 01:59 PM by Han.)
Post: #4
Han Senior Member Posts: 1,811 Joined: Dec 2013
RE: Entering partial derivatives?
None of those work because your syntax is ambiguous. There is no way (from a software point of view) to tell if the substitution should be applied prior to executing the command diff() or after. While we all understand your intended meaning, such is the nature of the where function.
As for syntax, it is usually:
expression|x=value
or
expression|{x=value,y=value,…}
It appears that the where function gives precedence to substitution over evaluation of functions. On the other hand, subst() has reverse precedence.
Edit:
diff(x^2*y,x)|{x=r,y=s} produces 2*r*s
subst(diff(x^2*y),{x=1,y=2}) produces 4
Graph 3D | QPI | SolveSys
11-07-2014, 09:04 AM
Post: #5
ww63 Junior Member Posts: 43 Joined: Sep 2014
RE: Entering partial derivatives?
Your last commant will result in an error message, because one bracket ')' is missing! ;-)
11-07-2014, 11:32 AM
Post: #6
DrD Senior Member Posts: 1,095 Joined: Feb 2014
RE: Entering partial derivatives?
Thank you for your very informative response, Han!
(Original example): $$f(x,y):=x^2y+xy^2$$
I seem to have omitted in the group of things I did try, that didn't work, was: diff(f(x,y),x)|{x = 1,y = 2}, which is THE form I expected WOULD work.
The subst() function does work, but somewhat misses the ideal: subst(diff(f(x,y),x),{x = 1,y = 2}) returning 8. Whether that is a "shorthand" means of getting the result, stretches the definition of shorthand a bit.
I'd like to suggest that the authors consider extending the utility value of the "|" where command to include applications like: diff(f(x,y),x)|{x = 1,y = 2}. The context is for the substitution to be applied AFTER the differentiation, as would probably be obvious on inspection in handwritten form.
Again, thanks, I learn a great deal from these responses, and hopefully can share accordingly, as time goes on.
-Dale-
11-07-2014, 11:46 AM
Post: #7
parisse Senior Member Posts: 1,027 Joined: Dec 2013
RE: Entering partial derivatives?
This is not possible, because sometimes you want to eval the argument first (like here) and sometimes you want to eval it after the substitution/assumption is done (for example solve(x^2=4)|x>1). That's precisely why you have 2 instructions for subtitution.
If you want to use |, you must compute the derivative first, for example
fx:=diff(f(x,y),x);
fx|[x=1,y=2]
11-07-2014, 12:14 PM
Post: #8
DrD Senior Member Posts: 1,095 Joined: Feb 2014
RE: Entering partial derivatives?
I don't want to beat a dead horse, but could the use of an appended flag on the end of the 'where' command resolve the matter? example: "|{x=1,y=2},0" would indicate "eval before" or "|{x=1,y=2},1" "eval after?"
11-07-2014, 02:00 PM
Post: #9
Han Senior Member Posts: 1,811 Joined: Dec 2013
RE: Entering partial derivatives?
(11-07-2014 09:04 AM)ww63 Wrote: Your last commant will result in an error message, because one bracket ')' is missing! ;-)
Fixed! -- Thank you!
Graph 3D | QPI | SolveSys
11-07-2014, 05:32 PM
Post: #10
parisse Senior Member Posts: 1,027 Joined: Dec 2013
RE: Entering partial derivatives?
(11-07-2014 12:14 PM)DrD Wrote: I don't want to beat a dead horse, but could the use of an appended flag on the end of the 'where' command resolve the matter? example: "|{x=1,y=2},0" would indicate "eval before" or "|{x=1,y=2},1" "eval after?"
It's not very pretty and you would have to explain how evaluation is performed anyway, I prefer to stay with subst and |, with appropriate help (and links to | from subst help and conversly).
11-07-2014, 06:19 PM
Post: #11
Han Senior Member Posts: 1,811 Joined: Dec 2013
RE: Entering partial derivatives?
You can always create an "alias" program file that shortens built-in commands to shorter commands.
For example:
EXPORT SS(a,b):=
BEGIN
return subst(a,b);
END;
Place all your "macros" in the same file, and they will also appear in the Toobox (under User, next to the Catalog) so you can get there with just a few screen taps.
Now, you can just type:
SS(diff(x^2*y+x*y^2,x),{x=1,y=2});
Graph 3D | QPI | SolveSys
11-08-2014, 11:28 AM
Post: #12
DrD Senior Member Posts: 1,095 Joined: Feb 2014
RE: Entering partial derivatives?
Backing away from the subject matter to reflect on the intended product purpose, and handheld calculating devices for the classroom market; can it be said that requiring a student to learn machine-dependent mathematics, meets the educational goal of learning subject mathematics? I don't think so in this case.
In a class setting, math topics move forward quite quickly. During lecture, little time is given for a diversion of how "this or that" machine interface must be tweaked to accomplish the subject theme. During recitation, interposing the human machine interface (HMI) demands, adds a layer of confusion to the central goal of learning the topic. Finally, during exam time, trying to recall which of the various machine dependencies is most likely to reach the expectations of examination problems, makes for greater overall difficulty and time demands.
Earlier, the " | where " statement was discussed regarding it's limitations: "that's the way it is" must be accommodated by users of the Prime. Accordingly, many examples exist which don't work using that facility. Yet in a blackboard class setting, that is precisely the form used to pass constraints back to the parent expression. Subst()'s syntax is not part of the lecturer's normal math lexicon.
This discussion emphasizes that hand held technology doesn't cross connect with the classroom, easily. HMI, lecture, recitation, and exam, would be ever so much the better if objectives being taught were, in the same way, met by the tools being marketed for them.
Personally, I feel the Prime fails educators, students, and others, in this important particular scenario, and needs more work.
-Dale-
11-08-2014, 02:16 PM
Post: #13
Gilles Member Posts: 162 Joined: Oct 2014
RE: Entering partial derivatives?
Hi, why don't use the 'template" wich is in my opinion the fastest way ?
returns 8
11-08-2014, 03:48 PM
Post: #14
DrD Senior Member Posts: 1,095 Joined: Feb 2014
RE: Entering partial derivatives?
Isn't that interesting? In the beginning of my work with this subject, I did use the template key for this.
In the first post, I was trying to find a way to enter second mixed partials: $$\Large\frac{∂^2f(x,y)}{∂x∂y}$$
After that, I wanted to find ways to evaluate the expression at a point with respect to x, (or y), and we moved away from the template feature. It migrated to the generic entry line forms because (in other cases) the template resulted in the entry line form, anyway: (f(x,y),x)'|{x=1,y=2}, or diff(f(x,y),x)|{x=1,y=2}
This generated the responses up to now. So, armed with "such is the limitation of the where command," "your syntax is ambiguous," because of not knowing when to apply (before/after) expression evaluation for a given point, etc., has resulted in my, mostly irrelevant, opinion of the status quo.
Ironically, I guess I hadn't put the last expression into the template, as you did, and it DOES the evaluation (as you discovered), in the manner in which, -I would think-, it should!
You just never know,( or try to remember), what you're going to get when the same expression entered in various ways, produces various results.
Every now and then, diff(f(x,y),x)|{x=1,y=2} ---> [0 0].
Most other times it triggers ["Invalid | Error: Bad Argument Value"].
In short, I think this is an area in which the software could be improved. I leave it to those empowered to effect change. Fortunately, in my case, I have time to try lots of things, hopefully finding a workable solution.
I enjoy the Prime, and learn along the way! That's a long way to explain the, "why," but that's it. The template key is normally my first go to.
-Dale-
11-08-2014, 06:17 PM
Post: #15
lrdheat Senior Member Posts: 490 Joined: Feb 2014
RE: Entering partial derivatives?
This is my feeling as well (and similar thoughts in the graphing environment). The capabilities of the Prime is amazing. Parisse is a genius with XCAS and it's implementation. Parrish is a genius period.
DrD's point about how precise one has to be in how an entry is made in Prime is hugely important. For this amazing platform to succeed in the educational market and professional markets, it is critical that the primary mathematically equivalent means of expression be recognized by the Prime in Home, CAS, and graphing modes. The Prime should not be about finding a specific syntax amongst a variety of mathematically correct and equivalent entries.
I hope that as the Prime matures, this problem will be given a high priority. I fear that the market place has this expectation as a starting point. This may make it difficult to get school systems and the professional market that may have rejected the Prime for these reasons to reconsider...
11-08-2014, 08:47 PM (This post was last modified: 11-08-2014 08:48 PM by Gilles.)
Post: #16
Gilles Member Posts: 162 Joined: Oct 2014
RE: Entering partial derivatives?
I agree with lrdheat. The prime's CAS is far more powerfull than the HP50G CAS for example, but for now the HP50G seems more homogenous in my opinion despite the lacks of his CAS.
Here is how to do for this with the 50G in RPL :
Note that the equation writer is unable to display the formula 2 and 4 on the stack in 'text book' (despite the fact that this syntax is well documented) but the way of the 50G to handle the paranthesis is more logical.
11-09-2014, 11:14 AM
Post: #17
parisse Senior Member Posts: 1,027 Joined: Dec 2013
RE: Entering partial derivatives?
The problem with | is an evaluation order problem and in addition a different meaning for the same variable name. At some point, when you are using a CAS, you have to understand that.
If you are using the HP50G in RPN mode, you have full control of evaluation, so it's easier to understand evaluation. But interaction is harder, especially doing again one computation with some modifications.
My point of view is that it's an error to expect students to use a math software without a minimum of explanations of the math teacher : you can sometimes hide difficulties but they are still there and they will bite later.
And it's not a loss of time to explain evaluation when using a CAS, to the contrary, it should be part of the math course!
11-09-2014, 07:43 PM
Post: #18
lrdheat Senior Member Posts: 490 Joined: Feb 2014
RE: Entering partial derivatives?
I agree with Parrise...CAS principles and limitations should be taught in math classes where CAS will be encountered. Ideally, everyone would have to be on the same CAS system. However, I strongly feel that a CAS be designed to accept primary mathematically equivalent entries such as n root, surd, and ^ to be accepted by users, especially your target market. This must also work across the entire platform...home,CAS, graphing.
That said, I'm a huge fan of Prime and it's team!
11-09-2014, 10:14 PM
Post: #19
DrD Senior Member Posts: 1,095 Joined: Feb 2014
RE: Entering partial derivatives?
I'm not too far removed from the point of view expressed by Parisse, except that I've found that no technology (CAS or otherwise) is academically universal.
I have taken math classes where no calcs were allowed, others where computer software Mathcad, Maple, Mathematica, etc. was the recitation support tool, and one trig class (long ago) where my hp48 was coveted by the professor, as he would always ask me for hp 48 solutions to classroom examples, comparing them with his lecture notes.
The most recent class I attended, linear algebra, the prof used only a few mathcad examples, throughout the course. The material covered, if well understood, didn't require hardware. Sometimes problem sets were a little more extensive, but for those, the process of the solution was paramount, with accuracy almost a secondary concern. However, it was always stressed that subject matter was what lecture time was about, unless it was a class with a technology perspective foremost.
In my vocational life, of course, it was accuracy above all, and hp calcs of various models were always near to hand my entire career. Beginning with hp-25, I still use the hp-50g even after retirement!
Lately, I have much more "fun" with the Prime, though! Thanks to Parisse, and Han, I have learned a great deal, possibly the most important being great respect for their efforts, regardless of how fitting the results may be!
-Dale-
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2019-11-22 00:14:04
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https://dml.cz/handle/10338.dmlcz/118459
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# Article
Full entry | PDF (0.1 MB)
Keywords:
dynamical system; universal minimal dynamical system; Abelian group; absolute
Summary:
Let $M(G)$ denote the phase space of the universal minimal dynamical system for a group $G$. Our aim is to show that $M(G)$ is homeomorphic to the absolute of $D^{2^\omega }$, whenever $G$ is a countable Abelian group.
References:
[1] Balcar B., Błaszczyk A.: On minimal dynamical systems on Boolean algebras. Comment. Math. Univ. Carolinae 31 (1990), 7-11. MR 1056164
[2] Comfort W.W.: Topological Groups. Handbook of set-theoretic topology, North-Holland, 1984, 1143-1260. MR 0776643 | Zbl 1071.54019
[3] van Douwen E.K.: The maximal totally bounded group topology on $G$ and the biggest minimal $G$-space, for Abelian groups $G$. Topology and its Appl. 34 (1990), 69-91. MR 1035461 | Zbl 0696.22003
[4] Ellis R.: Lectures on Topological Dynamics. Benjamin, New York, 1969 (). MR 0267561 | Zbl 0193.51502
[5] Hewitt E., Ross K.A.: Abstract Harmonic Analysis I. Springer, Berlin, 1963.
[6] van der Woude J.: Topological Dynamix. Mathematisch Centrum, Amsterdam, 1982. Zbl 0654.54026
Partner of
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2017-12-15 00:55:06
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https://socratic.org/questions/556c92fa581e2a374e0482da#149813
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# Question #482da
Jun 1, 2015
There aren't any right answers in the four choices (although I don't really understand what you mean by "sound level").
Due to the Doppler effect, the frequency that an observer perceives (it's called the apparent frequency) is different from the real frequency, the frequency that a source makes.
The Doppler effect only affects frequency and wavelength.
It doesn't affect either the speed of a sound, the speed of the observer, the speed of the source or the amplitude of a sound.
The apparent frequency is given by :
$f ' = f \cdot \frac{{v}_{w a v e} + {v}_{o b s}}{{v}_{w a v e} - {v}_{s}}$,
where ${v}_{w a v e}$ is the speed of sound in the medium, ${v}_{o b s}$ is the speed of the observer (${v}_{o b s} > 0$ when the source is in front of him) and ${v}_{s}$ is the speed of the source (${v}_{s} > 0$ when the observer is in front of it).
In our case, we don't care about ${v}_{w a v e}$, we suppose that ${v}_{o b s} = 0$, and there is a source approaching the observer so ${v}_{s} > 0$.
Now, we have :
$f ' = f \cdot \frac{{v}_{w a v e}}{{v}_{w a v e} - {v}_{s}} \implies f ' > f$.
The wavelength is given by :
$\lambda = {v}_{w a v e} / f$.
Therefore :
$\lambda = {v}_{w a v e} / f > \lambda ' = {v}_{w a v e} / \left(f '\right)$.
So the frequency that the observer perceives increases and the wavelength decreases.
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2022-12-07 12:57:58
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https://www.gerardpierre.eu/product-page/john-toiletry-bag
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# John Toiletry Bag
€99.00Price
John Toiletry Bag is quite simple with a stylish outlook which makes life easier for men as they can find their toiletries easily in their regular routine.
John Toiletry Bag is free from the confusion of many small pockets as it has got one large compartment for storing all inside with zipper pocket for the smaller items.
Not only all of that, but our toiletry bag also resists against water to safeguard against unwanted water seepage.
Measurement
23.5x 14 x 13 cm
Weight
258 g
SKU: 0008
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2020-07-13 09:19:04
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https://www.quizover.com/algebra2/section/approach-word-problems-with-a-positive-attitude-by-openstax
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# 3.1 Use a problem-solving strategy
Page 1 / 8
By the end of this section, you will be able to:
• Approach word problems with a positive attitude
• Use a problem-solving strategy for word problems
• Solve number problems
Before you get started, take this readiness quiz.
1. Translate “6 less than twice x ” into an algebraic expression.
If you missed this problem, review [link] .
2. Solve: $\frac{2}{3}x=24.$
If you missed this problem, review [link] .
3. Solve: $3x+8=14.$
If you missed this problem, review [link] .
## Approach word problems with a positive attitude
“If you think you can… or think you can’t… you’re right.”—Henry Ford
The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?
How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?
Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?
When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.
Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in [link] and say them out loud.
Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!
## Use a problem-solving strategy for word problems
We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.
Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.
## Use a problem-solving strategy to solve word problems.
1. Read the problem. Make sure all the words and ideas are understood.
2. Identify what we are looking for.
3. Name what we are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Can you all help me I don't get any of this
4^×=9
Did anyone else have trouble getting in quiz link for linear inequalities?
operation of trinomial
y=2×+9
Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns$3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?
Mayra has $124 in her checking account. She writes a check for$152. What is the New Balance in her checking account?
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2018-02-24 19:35:23
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http://stackoverflow.com/questions/10703813/how-to-set-the-unit-length-of-axis-in-matplotlib
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# How to set the unit length of axis in matplotlib?
For example x = [1~180,000] When I plot it, in x axis, it shows: 1, 20,000, 40,000, ... 180,000 these 0s are really annoying
How can I change the unit length of x axis to 1000, so that it show:1, 20, 40, ... 180 and also show that somewhere its unit is in 1000.
I know I can do a linear transformation myself. But isn't there a function doing that in matplotlib?
-
If you are aiming on making publication quality figures, you'll want to a fine control over the axes labels. One way to do this is to extract the label text and apply your own custom formatting:
import pylab as plt
import numpy as np
# Create some random data over a large interval
N = 200
X = np.random.random(N) * 10 ** 6
Y = np.sqrt(X)
# Draw the figure to get the current axes text
fig, ax = plt.subplots()
plt.scatter(X,Y)
ax.axis('tight')
plt.draw()
# Edit the text to your liking
label_text = [r"$%i \cdot 10^4$" % int(loc/10**4) for loc in plt.xticks()[0]]
ax.set_xticklabels(label_text)
# Show the figure
plt.show()
-
You can use pyplot.ticklabel_format to set the label style to scientific notation.
import pylab as plt
import numpy as np
# Create some random data over a large interval
N = 200
X = np.random.random(N) * 10 ** 6
Y = np.sqrt(X)
# Draw the figure to get the current axes text
fig, ax = plt.subplots()
plt.scatter(X,Y)
ax.axis('tight')
plt.draw()
plt.ticklabel_format(style='sci',axis='x',scilimits=(0,0))
# Edit the text to your liking
#label_text = [r"$%i \cdot 10^4$" % int(loc/10**4) for loc in plt.xticks()[0]]
#ax.set_xticklabels(label_text)
# Show the figure
plt.show()
-
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2014-03-15 04:05:30
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https://www.indepetro.com/trading/bitumen/b-pg/
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#### BITUMEN
Performance Grades are the latest standard of the day. This relatively new method classified bitumen is based on varying temperatures. It is a fully scientific method studying the mechanical specifications of bitumen. In this method, a temperature range is defined for bitumen and the consumer can easily choose the desired product.
Today, a PG is defined for polymer modified bitumen and pure bitumen based on environmental conditions and temperature. A wider PG range means higher resistance and more favorable specifications.
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2021-01-18 13:36:43
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http://math.stackexchange.com/questions/393245/proving-that-a-function-is-differentiable-and-equal-to-a-constant-value-for-all
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# Proving that a function is differentiable and equal to a constant value for all x
Let $f(x)$ denote a strictly positive continuous function defined on all real numbers with the property that $f(2012)=2012$ and $f(x)=f(x+f(x))$ for all $x$. Prove that $f(x)=2012$ for all $x$.
I am trying to prove that f is differentiable before I can do $f'(x)=f'(x+f(x))$. How can I do that? Is this the correct approach?
-
"positive" or "not decreasing" ? – nikita2 May 16 '13 at 8:37
The question says positive. – user76836 May 16 '13 at 9:10
@user76836 Next time, please try to get the basic requirements right from the start. Seeing that "continuous" suddely became a requirement just as I'm about to post my answer is frustrating... – fgp May 16 '13 at 10:35
I am guessing that it is a requirement... I typed the question exactly as it is was in the paper. – user76836 May 16 '13 at 10:41
Here's a counter-example to your assertion as it was originally. You've now added the requirement that $f$ is continuous, which invalidates this. But since I had already written this when I saw the change, I've decided to post it nevertheless $$f(x) = \begin{cases} 2012 &\text{for x \in \mathbb{N}} \\ 2102 &\text{otherwise.} \end{cases}$$
Obviously $f(x) > 0$ for all $x \in \mathbb{R}$.
If $x \in \mathbb{N}$ then $$f(x + f(x)) = f(\underbrace{x + 2012}_{\in \,\mathbb{N}}) = 2012 = f(x) \text{.}$$
If $x \in \mathbb{R}\setminus\mathbb{N}$ then $$f(x + f(x)) = f(\underbrace{x + 2102}_{\in \,\mathbb{R}\setminus\mathbb{N}}) = 2102 = f(x) \text{.}$$
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2014-08-28 03:10:48
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https://arxiver.wordpress.com/2017/03/13/low-mass-young-stellar-population-and-star-formation-history-of-the-cluster-ic-1805-in-the-w4-hsc-ii-region-ga/
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# Low-mass young stellar population and star formation history of the cluster IC 1805 in the W4 H{\sc ii} region [GA]
W4 is a giant H{\sc ii} region ionized by the OB stars of the cluster IC~1805. The H{\sc ii} region/cluster complex has been a subject of numerous investigations as it is an excellent laboratory for studying the feedback effect of massive stars on the surrounding region. However, the low-mass stellar content of the cluster IC~1805 remains poorly studied till now. With the aim to unravel the low-mass stellar population of the cluster, we present the results of a multiwavelength study based on deep optical data obtained with the Canada-France-Hawaii Telescope, infrared data from 2MASS, $Spitzer$ Space Telescope and X-ray data from $Chandra$ Space Telescope. The present optical dataset is complete enough to detect stars down to 0.2~M$_\odot$, which is the deepest optical observations so far for the cluster. We identified 384 candidate young stellar objects (YSOs; 101 Class I/II and 283 Class III) within the cluster using various colour-colour and colour-magnitude diagrams. We inferred the mean age of the identified YSOs to be $\sim$ 2.5 Myr and mass in the range 0.3 – 2.5 M$_\odot$. The mass function of our YSO sample has a power law index of -1.23 $\pm$ 0.23, close to the Salpeter value (-1.35), and consistent with those of other star-forming complexes. We explored the disk evolution of the cluster members and found that the diskless sources are relatively older compared to the disk bearing YSO candidates. We examined the effect of high-mass stars on the circumstellar disks and found that within uncertainties, the influence of massive stars on the disk fraction seems to be insignificant. We also studied the spatial correlation of the YSOs with the distribution of gas and dust of the complex to conclude that IC 1805 would have formed in a large filamentary cloud.
N. Panwar, M. Samal, A. Pandey, et. al.
Mon, 13 Mar 17
18/48
Comments: Accepted for publication in MNRAS; 34 pages, 10 figures
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2017-12-13 20:22:49
|
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https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/4%3A_The_Definite_Integral
|
# 4: The Definite Integral
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$
$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$
• 4.1: Determining Distance Traveled from Velocity
f we know the velocity of a moving body at every point in a given interval and the velocity is positive throughout, we can estimate the object’s distance traveled and in some circumstances determine this value exactly. In particular, when velocity is positive on an interval, we can find the total distance traveled by finding the area under the velocity curve and above the t-axis on the given time interval. We may only be able to estimate this area, depending on the shape of the velocity curve.
• 4.2: Riemann Sums
A Riemann sum is simply a sum of products of the form $$f (x^∗_i )\Delta x$$ that estimates the area between a positive function and the horizontal axis over a given interval. If the function is sometimes negative on the interval, the Riemann sum estimates the difference between the areas that lie above the horizontal axis and those that lie below the axis.
• 4.3: The Definite Integral
The Riemann sum of a continuous function provides an estimate of the net signed area bounded by the function and the horizontal axis on the interval. Increasing the number of subintervals in the Riemann sum improves the accuracy of this estimate, and letting the number of subintervals increase without bound results in the values of the corresponding Riemann sums approaching the exact value of the enclosed net signed area.
• 4.4: The Fundamental Theorem of Calculus
We can find the exact value of a definite integral without taking the limit of a Riemann sum or using a familiar area formula by finding the antiderivative of the integrand, and hence applying the Fundamental Theorem of Calculus.
• 4.E: The Definite Integral (Exercises)
These are homework exercises to accompany Chapter 4 of Boelkins et al. "Active Calculus" Textmap.
### Contributors
Matt Boelkins (Grand Valley State University), David Austin (Grand Valley State University), Steve Schlicker (Grand Valley State University)
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2019-02-23 23:58:28
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|
https://tex.stackexchange.com/questions/298778/biblatex-authoryear-icomp-parenthesis-in-footnotes-and-textcite
|
biblatex authoryear-icomp parenthesis in footnotes and textcite
By typing
Some text \autocite[88]{A01}.
Now Textcite \textcite[88]{A01}
I try to achieve the following output:
Using this answer, I do get the footnotes right. Unfortunately, there are brackets when using \textcite, which I consider bad typography.
Using this answer, \textcite works as expected. However, parenthesis in footnotes now span both year and page number.
Using one of the rich examples have not brought me closer. Whats left is my MWE.
\documentclass{article}
\usepackage[style=authoryear-icomp,autocite=footnote]{biblatex}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{A01,
author = {Author, A.},
year = {2001},
title = {Alpha},
journaltitle = {A Journal},
volume = {1},
number = {1},
}
@misc{B02,
author = {Buthor, B.},
year = {2002},
title = {Bravo},
location = {There},
}
\end{filecontents}
\textheight=120pt% only for the example
\begin{document}
Some text \autocite[88]{A01}.
Some text \autocite[88]{B02}.
Some text \autocite[88]{B02}.
Now Textcite \textcite[88]{A01}
\printbibliography
\end{document}
• What about the comp features? What should \autocites[12]{A01}[13]{B02}, \autocite[12]{A01,B02} and \autocite{A01,B02} give? If you can drop the "comp" feature, things might get easier code-length wise. And while we're at it: What about the "ibid"? Do you want it \textcite-style with the name, or as normally without the name. – moewe Mar 13 '16 at 13:53
• But what about "Knuth 1984, Knuth 1986a, Knuth 1986b", would you want "Knuth 1984,1986a,b" there? (I should have probably said \autocite{knuth:ct:b,knuth:ct:c} in my comment.) – moewe Mar 13 '16 at 14:03
• Sorry, I was confused. Both comp and ibid would be nice to have, hence me using icomp. However, citing as stated above should be priority. – lactea Mar 13 '16 at 14:08
• "Nice to have" doesn't actually say what you think about say "Knuth (1986a), p. 12, (b)" which is what could happen with icomp, I imagine. That would probably not be something you find nice, but then do you want "Knuth (1986a,b)" if possible? And what about "ibid" do you want "Knuth (ibid)" or just "ibid"? (I realise that these questions might be annoying, but they are needed to decide which approach to take, the cite macro in authoryear takes about 10 lines in authoryear-icomp it is 2.5 times that + some helper macros) – moewe Mar 13 '16 at 14:13
• I highly appreciate your efforts. Thank you for asking very specific questions. Your assumptions are right. "(1984; 1986a,b)" would be ideal for \textcite{K84}{K86a}{K86b}. As we usually include page numbers, \textcite[12]{K84}[13]{K86a}{K86b} would lead to "(1984, p. 12; 1986a, p. 13; 1986b)". Using \autocite[12]{K86a}[13]{K86b} gives "Knuth (1986a), p. 12; Knuth (1986b), p. 13." in footnotes, which is sufficient. "ibid" instead of "Knuth (ibid)" meets my faculty requirements, too. If taken care of the sorting manually, the comp feature is not necessary. – lactea Mar 13 '16 at 14:37
Modifying authoryear-icomp with regards to brackets around years is not that simple, so may I interest you in the \DeclareInnerCiteDelims feature of my biblatex-ext package? Simply use the style style=ext-<yourstyle> instead of style=<yourstyle>. The ext-... styles are written so that they can be used as a drop-in replacement of the standard styles.
With \DeclareInnerCiteDelims{footcite}{\bibopenparen}{\bibcloseparen} you wrap the year in \footcite (and \autocite if it ends up in as a \footcite with autocite=footnote) in round brackets.
\documentclass{article}
\usepackage[style=ext-authoryear-icomp,autocite=footnote]{biblatex}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{A01,
author = {Author, A.},
year = {2001},
title = {Alpha},
journaltitle = {A Journal},
volume = {1},
number = {1},
}
@misc{B02,
author = {Buthor, B.},
year = {2002},
title = {Bravo},
location = {There},
}
\end{filecontents}
\textheight=120pt% only for the example
\DeclareInnerCiteDelims{footcite}{\bibopenparen}{\bibcloseparen}
\begin{document}
Some text \autocite[88]{A01}.
Some text \autocite[88]{B02}.
Some text \autocite[88]{B02}.
Now Textcite \textcite[88]{A01}
\printbibliography
\end{document}
Edited for name changes in version 0.4 of biblatex-ext.
If you don't want to use an external package, you will have to modify the cite and cite:postnote bibmacros as follows
\makeatletter
\renewbibmacro*{cite}{%
\iffieldundef{shorthand}
{\ifthenelse{\ifciteibid\AND\NOT\iffirstonpage}
{\usebibmacro{cite:ibid}}
{\ifboolexpr{test {\ifnameundef{labelname}}
or test {\iffieldundef{labelyear}}}
{\usebibmacro{cite:label}%
\setunit{%
\global\booltrue{cbx:parens}%
\printdelim{nonameyeardelim}%
\bibopenparen}%
\usebibmacro{cite:reinit}}
{\iffieldequals{namehash}{\cbx@lasthash}
{\ifboolexpr{test {\iffieldequals{labelyear}{\cbx@lastyear}}
and (test {\ifnumequal{\value{multicitecount}}{0}}
or test {\iffieldundef{postnote}})}
{\setunit{\compcitedelim}%
\savefield{labelyear}{\cbx@lastyear}}}
{\printnames{labelname}%
\setunit{%
\global\booltrue{cbx:parens}%
\printdelim{nameyeardelim}%
\bibopenparen}%
\savefield{namehash}{\cbx@lasthash}%
\savefield{labelyear}{\cbx@lastyear}}}}}
{\usebibmacro{cite:shorthand}%
\usebibmacro{cite:reinit}}%
\setunit{%
\ifbool{cbx:parens}
{\bibcloseparen
\global\boolfalse{cbx:parens}}
{}%
\multicitedelim}}
\renewbibmacro*{cite:postnote}{%
\setunit{}%
\printtext{%
\ifbool{cbx:parens}
{\bibcloseparen
\global\boolfalse{cbx:parens}}
{}}%
\ifbool{cbx:loccit}
{}
{\usebibmacro{postnote}}}
\makeatother
|
2020-09-27 01:47:43
|
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|
https://zbmath.org/authors/?q=vassiliev.victor-a
|
# zbMATH — the first resource for mathematics
## Vasil’ev, Viktor Anatol’evich
Compute Distance To:
Author ID: vassiliev.victor-a Published as: Vasil’ev, V. A; Vasil’ev, V. A.; Vassil’ev, V. A.; Vassiliev, V.; Vassiliev, V. A; Vassiliev, V. A.; Vassiliev, Victor; Vassiliev, Victor A. Homepage: http://www.mi.ras.ru/~vva/ External Links: MGP · Math-Net.Ru · Wikidata · ORCID · GND
Documents Indexed: 122 Publications since 1977, including 11 Books Biographic References: 2 Publications
all top 5
all top 5
#### Serials
11 Functional Analysis and its Applications 10 Russian Mathematical Surveys 6 Funktsional’nyĭ Analiz i ego Prilozheniya 6 Journal of Knot Theory and its Ramifications 6 Moscow Mathematical Journal 3 Mathematical Notes 3 Notices of the American Mathematical Society 3 Doklady Mathematics 3 Oberwolfach Reports 3 Translations of Mathematical Monographs 3 Arnold Mathematical Journal 2 Bulletin of the London Mathematical Society 2 Topology and its Applications 2 Sbornik: Mathematics 2 Izvestiya: Mathematics 2 Translations. Series 2. American Mathematical Society 1 Matematicheskie Zametki 1 Uspekhi Matematicheskikh Nauk [N. S.] 1 Journal of Soviet Mathematics 1 Mathematics of the USSR. Izvestiya 1 Publications de l’Institut Mathématique. Nouvelle Série 1 Soviet Mathematics. Doklady 1 Transactions of the Moscow Mathematical Society 1 European Journal of Combinatorics 1 Combinatorica 1 Algebra i Analiz 1 Russian Academy of Sciences. Sbornik. Mathematics 1 Topological Methods in Nonlinear Analysis 1 Journal of Mathematical Sciences (New York) 1 Itogi Nauki i Tekhniki. Seriya Sovremennye Problemy Matematiki. Noveishie Dostizheniya 1 St. Petersburg Mathematical Journal 1 Selecta Mathematica. New Series 1 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 1 Encyclopaedia of Mathematical Sciences 1 Mathematical Surveys and Monographs 1 Proceedings of the Steklov Institute of Mathematics 1 Student Mathematical Library 1 Journal of Singularities
all top 5
#### Fields
52 Manifolds and cell complexes (57-XX) 39 Global analysis, analysis on manifolds (58-XX) 35 Algebraic topology (55-XX) 21 Several complex variables and analytic spaces (32-XX) 16 Algebraic geometry (14-XX) 15 History and biography (01-XX) 8 Partial differential equations (35-XX) 8 Computer science (68-XX) 7 General and overarching topics; collections (00-XX) 6 Group theory and generalizations (20-XX) 5 Combinatorics (05-XX) 4 Special functions (33-XX) 4 Differential geometry (53-XX) 3 Order, lattices, ordered algebraic structures (06-XX) 3 Real functions (26-XX) 3 Operator theory (47-XX) 3 Convex and discrete geometry (52-XX) 2 Field theory and polynomials (12-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Measure and integration (28-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Integral transforms, operational calculus (44-XX) 2 Functional analysis (46-XX) 2 Quantum theory (81-XX) 1 Commutative algebra (13-XX) 1 Associative rings and algebras (16-XX) 1 Nonassociative rings and algebras (17-XX) 1 Category theory; homological algebra (18-XX) 1 Potential theory (31-XX) 1 Approximations and expansions (41-XX) 1 General topology (54-XX) 1 Numerical analysis (65-XX) 1 Mechanics of particles and systems (70-XX) 1 Optics, electromagnetic theory (78-XX) 1 Astronomy and astrophysics (85-XX)
#### Citations contained in zbMATH
68 Publications have been cited 358 times in 268 Documents Cited by Year
Topology of complements to discriminants and loop spaces. Zbl 0731.32017
Vassiliev, V. A.
1990
Combinatorial formulas for cohomology of knot spaces. Zbl 1015.57003
Vassiliev, V. A.
2001
Singularity theory I. Transl. from the Russian by A. Iacob. 2nd printing of the 1st ed. 1993. Zbl 0901.58001
Arnold, V. I.; Goryunov, V. V.; Lyashko, O. V.; Vasil’ev, V. A.
1998
Complements of discriminants of smooth maps: topology and applications. Transl. from the Russian by B. Goldfarb. Transl. ed. by S. Gelfand. Zbl 0762.55001
Vasil’ev, V. A.
1992
Asymptotic exponential integrals. Newton’s diagram, and the classification of minimal points. Zbl 0418.28001
Vasil’ev, V. A.
1978
Newton’s Principia read 300 years later. Zbl 0693.01005
Arnol’d, V. I.; Vasil’ev, V. A.
1989
Complexes of connected graphs. Zbl 0812.55005
Vassiliev, V. A.
1993
General hypergeometric functions on complex Grassmannians. Zbl 0625.33006
Vasil’ev, V. A.; Gel’fand, I. M.; Zelevinskij, A. V.
1987
Dynamical systems VIII. Singularity theory. II: Classification and applications. Transl. from the Russian by J. S. Joel. Zbl 0778.58001
Arnol’d, V. I.; Goryunov, V. V.; Lyashko, O. V.; Vassiliev, Victor A.
1993
Recognizing textile structures by finite type knot invariants. Zbl 1172.57300
Grishanov, S. A.; Meshkov, V. R.; Vassiliev, V. A.
2009
Applied Picard-Lefschetz theory. Zbl 1039.32039
Vassiliev, V. A.
2002
Invariants of ornaments. Zbl 0854.57011
Vassiliev, Victor A.
1994
Newton’s lemma XXVIII on integrable ovals in higher dimensions and reflection groups. Zbl 1315.14015
Vassiliev, V. A.
2015
Fiedler type combinatorial formulas for generalized Fiedler type invariants of knots in $$M^2\times R^1$$. Zbl 1207.57018
Grishanov, S. A.; Vassiliev, V. A.
2009
Resolutions of discriminants and topology of their complements. Zbl 0991.58014
Vassiliev, Victor
2001
Ramified integrals, singularities and lacunas. Zbl 0935.32026
Vassiliev, V. A.
1995
Invariants of knots and complements of discriminants. Zbl 0895.57001
Vassiliev, Victor A.
1993
Topology of plane arrangements and their complements. Zbl 1002.55015
Vasil’ev, V. A.
2001
Introduction to topology. Transl. from the Russian by A. Sossinski. Zbl 0971.57001
Vassiliev, V. A.
2001
Topological order complexes and resolutions of discriminant sets. Zbl 0953.55011
Vassiliev, V. A.
1999
On invariants and homology of spaces of knots in arbitrary manifolds. Zbl 0929.57004
Vassiliev, V. A.
1998
Holonomic links and Smale principles for multisingularities. Zbl 0888.57004
Vassiliev, Victor A.
1997
Topology of discriminants and their complements. Zbl 0852.55003
Vassiliev, Victor A.
1995
On function spaces that are interpolating at any $$k$$ nodes. Zbl 0805.41003
Vasil’ev, V. A.
1992
Topology of spaces of functions without compound singularities. Zbl 0731.58009
Vasil’ev, V. A.
1989
Singularities. I: The local and global theory. Zbl 0691.58002
Arnol’d, V. I.; Vasil’ev, V. A.; Goryunov, V. V.; Lyashko, O. V.
1988
Braid group cohomologies and algorithm complexity. Zbl 0674.68040
Vasil’ev, V. A.
1988
Singularities. Local and global theory. Zbl 0787.58001
Arnol’d, V. I.; Vasil’ev, V. A.; Goryunov, V. V.; Lyashko, O. V.
1988
Homology groups of spaces of non-resultant quadratic polynomial systems in $${\mathbb R}^3$$. Zbl 1375.55011
Vassiliev, Victor A.
2016
Invariants of links in 3-manifolds and splitting problem of textile structures. Zbl 1221.57017
Grishanov, S. A.; Vassiliev, V. A.
2011
Homology of spaces of knots in any dimensions. Zbl 0994.55010
Vassiliev, V. A.
2001
Topology of two-connected graphs and homology of spaces of knots. Zbl 0941.57006
Vassiliev, Victor A.
1999
On $$r$$-neighbourly submanifolds in $$\mathbb{R}^N$$. Zbl 0922.57020
Vassiliev, Victor A.
1998
Cohomology of knot spaces. Zbl 0727.57008
Vasil’ev, V. A.
1990
Rational homology of the order complex of zero sets of homogeneous quadratic polynomial systems in $$\mathbb{R}^3$$. Zbl 1335.55015
Vassiliev, V. A.
2015
Two constructions of weight systems for invariants of knots in non-trivial 3-manifolds. Zbl 1154.57010
Grishanov, S. A.; Vassiliev, V. A.
2008
On finite order invariants of triple point free plane curves. Zbl 0964.57018
Vassiliev, V. A.
1999
Topological complexity of root-finding algorithms. Zbl 0864.57029
Vassiliev, Victor A.
1996
Complements of discriminants of smooth maps: topology and applications. Transl. from the Russian by B. Goldfarb. Transl. ed. by S. Gelfand. Rev. ed. Zbl 0826.55001
Vasil’ev, V. A.
1994
Cohomologies of braid groups and complexity of algorithms. Zbl 0659.68071
Vasil’ev, V. A.
1988
Sharpness and the local Petrovskiĭ condition for strictly hyperbolic operators with constant coefficients. Zbl 0615.35012
Vasil’ev, V. A.
1987
General hypergeometric functions on complex Grassmannians. Zbl 0614.33008
Vasil’ev, V. A.; Gel’fand, I. M.; Zelevinskii, A. V.
1987
Asymptotic behavior of exponential integrals in the complex domain. Zbl 0435.32002
Vasil’ev, V. A.
1980
Lacunas and local algebraicity of volume functions. Zbl 1402.53059
Vassiliev, V. A.
2018
Stable cohomology of spaces of non-resultant systems of homogeneous polynomials in $$\mathbb{R}^n$$. Zbl 1427.55006
Vassiliev, V. A.
2018
Stable cohomology of spaces of non-resultant polynomial systems in $$\mathbb{R}^3$$. Zbl 1386.55019
Vassiliev, V. A.
2017
Local Petrovskii lacunas close to parabolic singular points of the wavefronts of strictly hyperbolic partial differential equations. Zbl 1356.35014
Vassiliev, Victor A.
2016
Homology of spaces of non-resultant homogeneous polynomial systems in $$\mathbb R^2$$ and $$\mathbb C^2$$. Zbl 1408.13076
Vassiliev, V. A.
2015
On topological invariants of real algebraic functions. Zbl 1271.57002
Vassiliev, V. A.
2011
First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in $$\mathbb R^n$$. Zbl 1113.55014
Vassiliev, V. A.
2005
Combinatorial formulas for cohomology of spaces of knots. Zbl 1089.57013
Vassiliev, V. A.
2004
How to calculate homology groups of spaces of nonsingular algebraic projective hypersurfaces. Zbl 0981.55008
Vasil’ev, V. A.
1999
Homology of $$i$$-connected graphs and invariants of knots, plane arrangements, etc. Zbl 0976.57024
Vassiliev, V. A.
1999
New examples of locally algebraically integrable bodies. Zbl 1434.52006
Vassiliev, V. A.
2019
A few problems on monodromy and discriminants. Zbl 1320.14015
Vassiliev, V. A.
2015
Topological complexity and Schwarz genus of general real polynomial equation. Zbl 1279.68109
Vassiliev, V. A.
2011
Spaces of Hermitian operators with simple spectra and their finite-order cohomology. Zbl 1047.47052
Vassiliev, V. A.
2003
Homology of spaces of homogeneous polynomials in $${\mathbb{R}}^2$$ without multiple zeros. Zbl 0945.55009
Vasil’ev, V. A.
1998
Geometry of differential equations. Dedicated to V. I. Arnold on the occasion of his 60th birthday. Zbl 0896.00014
Khovanskij, A. (ed.); Varchenko, A. (ed.); Vassiliev, V. (ed.)
1998
Topological complexity and real roots of polynomials. Zbl 0897.68055
Vassiliev, V. A.
1996
On spaces of polynomial knots. Zbl 0871.57011
Vassiliev, V. A.
1996
Knot invariants and singularity theory. Zbl 0969.58012
Vassiliev, Victor A.
1995
Geometry of local lacunas of hyperbolic operators with constant coefficients. Zbl 0773.35038
Vasil’ev, V. A.
1992
A geometric realization of the homology of classical Lie groups, and complexes S-dual to flag manifolds. Zbl 0791.57024
Vasil’ev, V. A.
1991
Arnol’d, V. I.; Vasil’ev, V. A.
1990
Singularities. II: Classification and applications. Zbl 0707.58005
Arnol’d, V. I.; Vasil’ev, V. A.; Goryunov, V. V.; Lyashko, O. V.
1989
Characteristic classes of Lagrangian and Legendre manifolds dual to singularities of caustics and wave fronts. Zbl 0514.57004
Vasil’ev, V. A.
1982
The asymptotic of exponential integrals in complex space. Zbl 0429.32004
Vasil’ev, V. A.
1979
New examples of locally algebraically integrable bodies. Zbl 1434.52006
Vassiliev, V. A.
2019
Lacunas and local algebraicity of volume functions. Zbl 1402.53059
Vassiliev, V. A.
2018
Stable cohomology of spaces of non-resultant systems of homogeneous polynomials in $$\mathbb{R}^n$$. Zbl 1427.55006
Vassiliev, V. A.
2018
Stable cohomology of spaces of non-resultant polynomial systems in $$\mathbb{R}^3$$. Zbl 1386.55019
Vassiliev, V. A.
2017
Homology groups of spaces of non-resultant quadratic polynomial systems in $${\mathbb R}^3$$. Zbl 1375.55011
Vassiliev, Victor A.
2016
Local Petrovskii lacunas close to parabolic singular points of the wavefronts of strictly hyperbolic partial differential equations. Zbl 1356.35014
Vassiliev, Victor A.
2016
Newton’s lemma XXVIII on integrable ovals in higher dimensions and reflection groups. Zbl 1315.14015
Vassiliev, V. A.
2015
Rational homology of the order complex of zero sets of homogeneous quadratic polynomial systems in $$\mathbb{R}^3$$. Zbl 1335.55015
Vassiliev, V. A.
2015
Homology of spaces of non-resultant homogeneous polynomial systems in $$\mathbb R^2$$ and $$\mathbb C^2$$. Zbl 1408.13076
Vassiliev, V. A.
2015
A few problems on monodromy and discriminants. Zbl 1320.14015
Vassiliev, V. A.
2015
Invariants of links in 3-manifolds and splitting problem of textile structures. Zbl 1221.57017
Grishanov, S. A.; Vassiliev, V. A.
2011
On topological invariants of real algebraic functions. Zbl 1271.57002
Vassiliev, V. A.
2011
Topological complexity and Schwarz genus of general real polynomial equation. Zbl 1279.68109
Vassiliev, V. A.
2011
Recognizing textile structures by finite type knot invariants. Zbl 1172.57300
Grishanov, S. A.; Meshkov, V. R.; Vassiliev, V. A.
2009
Fiedler type combinatorial formulas for generalized Fiedler type invariants of knots in $$M^2\times R^1$$. Zbl 1207.57018
Grishanov, S. A.; Vassiliev, V. A.
2009
Two constructions of weight systems for invariants of knots in non-trivial 3-manifolds. Zbl 1154.57010
Grishanov, S. A.; Vassiliev, V. A.
2008
First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in $$\mathbb R^n$$. Zbl 1113.55014
Vassiliev, V. A.
2005
Combinatorial formulas for cohomology of spaces of knots. Zbl 1089.57013
Vassiliev, V. A.
2004
Spaces of Hermitian operators with simple spectra and their finite-order cohomology. Zbl 1047.47052
Vassiliev, V. A.
2003
Applied Picard-Lefschetz theory. Zbl 1039.32039
Vassiliev, V. A.
2002
Combinatorial formulas for cohomology of knot spaces. Zbl 1015.57003
Vassiliev, V. A.
2001
Resolutions of discriminants and topology of their complements. Zbl 0991.58014
Vassiliev, Victor
2001
Topology of plane arrangements and their complements. Zbl 1002.55015
Vasil’ev, V. A.
2001
Introduction to topology. Transl. from the Russian by A. Sossinski. Zbl 0971.57001
Vassiliev, V. A.
2001
Homology of spaces of knots in any dimensions. Zbl 0994.55010
Vassiliev, V. A.
2001
Topological order complexes and resolutions of discriminant sets. Zbl 0953.55011
Vassiliev, V. A.
1999
Topology of two-connected graphs and homology of spaces of knots. Zbl 0941.57006
Vassiliev, Victor A.
1999
On finite order invariants of triple point free plane curves. Zbl 0964.57018
Vassiliev, V. A.
1999
How to calculate homology groups of spaces of nonsingular algebraic projective hypersurfaces. Zbl 0981.55008
Vasil’ev, V. A.
1999
Homology of $$i$$-connected graphs and invariants of knots, plane arrangements, etc. Zbl 0976.57024
Vassiliev, V. A.
1999
Singularity theory I. Transl. from the Russian by A. Iacob. 2nd printing of the 1st ed. 1993. Zbl 0901.58001
Arnold, V. I.; Goryunov, V. V.; Lyashko, O. V.; Vasil’ev, V. A.
1998
On invariants and homology of spaces of knots in arbitrary manifolds. Zbl 0929.57004
Vassiliev, V. A.
1998
On $$r$$-neighbourly submanifolds in $$\mathbb{R}^N$$. Zbl 0922.57020
Vassiliev, Victor A.
1998
Homology of spaces of homogeneous polynomials in $${\mathbb{R}}^2$$ without multiple zeros. Zbl 0945.55009
Vasil’ev, V. A.
1998
Geometry of differential equations. Dedicated to V. I. Arnold on the occasion of his 60th birthday. Zbl 0896.00014
Khovanskij, A. (ed.); Varchenko, A. (ed.); Vassiliev, V. (ed.)
1998
Holonomic links and Smale principles for multisingularities. Zbl 0888.57004
Vassiliev, Victor A.
1997
Topological complexity of root-finding algorithms. Zbl 0864.57029
Vassiliev, Victor A.
1996
Topological complexity and real roots of polynomials. Zbl 0897.68055
Vassiliev, V. A.
1996
On spaces of polynomial knots. Zbl 0871.57011
Vassiliev, V. A.
1996
Ramified integrals, singularities and lacunas. Zbl 0935.32026
Vassiliev, V. A.
1995
Topology of discriminants and their complements. Zbl 0852.55003
Vassiliev, Victor A.
1995
Knot invariants and singularity theory. Zbl 0969.58012
Vassiliev, Victor A.
1995
Invariants of ornaments. Zbl 0854.57011
Vassiliev, Victor A.
1994
Complements of discriminants of smooth maps: topology and applications. Transl. from the Russian by B. Goldfarb. Transl. ed. by S. Gelfand. Rev. ed. Zbl 0826.55001
Vasil’ev, V. A.
1994
Complexes of connected graphs. Zbl 0812.55005
Vassiliev, V. A.
1993
Dynamical systems VIII. Singularity theory. II: Classification and applications. Transl. from the Russian by J. S. Joel. Zbl 0778.58001
Arnol’d, V. I.; Goryunov, V. V.; Lyashko, O. V.; Vassiliev, Victor A.
1993
Invariants of knots and complements of discriminants. Zbl 0895.57001
Vassiliev, Victor A.
1993
Complements of discriminants of smooth maps: topology and applications. Transl. from the Russian by B. Goldfarb. Transl. ed. by S. Gelfand. Zbl 0762.55001
Vasil’ev, V. A.
1992
On function spaces that are interpolating at any $$k$$ nodes. Zbl 0805.41003
Vasil’ev, V. A.
1992
Geometry of local lacunas of hyperbolic operators with constant coefficients. Zbl 0773.35038
Vasil’ev, V. A.
1992
A geometric realization of the homology of classical Lie groups, and complexes S-dual to flag manifolds. Zbl 0791.57024
Vasil’ev, V. A.
1991
Topology of complements to discriminants and loop spaces. Zbl 0731.32017
Vassiliev, V. A.
1990
Cohomology of knot spaces. Zbl 0727.57008
Vasil’ev, V. A.
1990
Arnol’d, V. I.; Vasil’ev, V. A.
1990
Newton’s Principia read 300 years later. Zbl 0693.01005
Arnol’d, V. I.; Vasil’ev, V. A.
1989
Topology of spaces of functions without compound singularities. Zbl 0731.58009
Vasil’ev, V. A.
1989
Singularities. II: Classification and applications. Zbl 0707.58005
Arnol’d, V. I.; Vasil’ev, V. A.; Goryunov, V. V.; Lyashko, O. V.
1989
Singularities. I: The local and global theory. Zbl 0691.58002
Arnol’d, V. I.; Vasil’ev, V. A.; Goryunov, V. V.; Lyashko, O. V.
1988
Braid group cohomologies and algorithm complexity. Zbl 0674.68040
Vasil’ev, V. A.
1988
Singularities. Local and global theory. Zbl 0787.58001
Arnol’d, V. I.; Vasil’ev, V. A.; Goryunov, V. V.; Lyashko, O. V.
1988
Cohomologies of braid groups and complexity of algorithms. Zbl 0659.68071
Vasil’ev, V. A.
1988
General hypergeometric functions on complex Grassmannians. Zbl 0625.33006
Vasil’ev, V. A.; Gel’fand, I. M.; Zelevinskij, A. V.
1987
Sharpness and the local Petrovskiĭ condition for strictly hyperbolic operators with constant coefficients. Zbl 0615.35012
Vasil’ev, V. A.
1987
General hypergeometric functions on complex Grassmannians. Zbl 0614.33008
Vasil’ev, V. A.; Gel’fand, I. M.; Zelevinskii, A. V.
1987
Characteristic classes of Lagrangian and Legendre manifolds dual to singularities of caustics and wave fronts. Zbl 0514.57004
Vasil’ev, V. A.
1982
Asymptotic behavior of exponential integrals in the complex domain. Zbl 0435.32002
Vasil’ev, V. A.
1980
The asymptotic of exponential integrals in complex space. Zbl 0429.32004
Vasil’ev, V. A.
1979
Asymptotic exponential integrals. Newton’s diagram, and the classification of minimal points. Zbl 0418.28001
Vasil’ev, V. A.
1978
all top 5
#### Cited by 324 Authors
22 Vasil’ev, Viktor Anatol’evich 7 Jeong, Myeong-Ju 5 Grishanov, Sergei A. 5 Koseleff, Pierre-Vincent 5 Pecker, Daniel 4 Park, Chan-Young 4 Turchin, Victor Éduardovich 4 Zenkina, M. V. 3 Allenov, S. V. 3 Blagojević, Pavle V. M. 3 Feichtner-Kozlov, Dimitry N. 3 Habiro, Kazuo 3 Kanenobu, Taizo 3 Karasev, Roman N. 3 Ohyama, Yoshiyuki 3 Ziegler, Günter Matthias 2 Agranovsky, Mark L. 2 Alvarez, Mario M. 2 Antony, Noelle 2 Bérard, Alain 2 Blanc, David Abraham 2 Brini, Andrea 2 Chernov, Vladimir V. 2 Dye, Heather Ann 2 Ekholm, Tobias 2 Farber, Michael S. 2 Gamkrelidze, Alexander 2 Gel’fand, Israil’ Moiseevich 2 Givental, Alexander Borisovich 2 Goryunov, Victor V. 2 Grandati, Yves 2 Greenblatt, Michael 2 Hanßmann, Heinz 2 Jefferies, Brian R. F. 2 Kaiser, Uwe 2 Kalai, Gil 2 Kamimoto, Joe 2 Kohno, Toshitake 2 Kudryavtseva, Elena A. 2 Labastida, José M. F. 2 Lando, Sergei K. 2 Lichtin, Ben 2 Lin, Fang Hua 2 Lück, Wolfgang 2 Manturov, Vassiliĭ Olegovich 2 Matsuo, Atsushi 2 Meilhan, Jean-Baptiste 2 Mohrbach, Hervé 2 Nikkuni, Ryo 2 Nowik, Tahl 2 Pragacz, Piotr 2 Rouillier, Fabrice 2 Rudyak, Yuli B. 2 Shekhtman, Boris 2 Shvalb, Nir 2 Sinha, Dev P. 2 Stoimenow, Alexander 2 Szűcs, András 2 Volovikov, Aleksej Yur’evich 2 Weiss, Michael S. 2 Yang, Yisong 2 Živaljević, Rade T. 1 Abdullaev, Janikul I. 1 Alsaeed, Suliman 1 Arnold, John M. 1 Aroca, Fuensanta 1 Arone, Gregory Z. 1 Baladze, Vladimer 1 Bamberger, Hagay 1 Barnett, Kathryn 1 Belitskii, Genrich R. 1 Bellettini, Giovanni 1 Beorchia, Valentina 1 Bérczi, Gergely 1 Berest, Yuri Yu. 1 Berglund, Per 1 Bergqvist, Magnus 1 Birman, Joan S. 1 Błaszak, Maciej 1 Bogaevskij, Il’ya Aleksandrovich 1 Borcea, Julius 1 Boyle, Michael 1 Brandenbursky, Michael 1 Braun, Andreas P. 1 Breiding, Paul 1 Brochier, Adrien 1 Broer, Henk W. 1 Brugallé, Erwan 1 Budney, Ryan David 1 Buffenoir, E. 1 Capitanio, Gianmarco 1 Carlet, Guido 1 Casonatto, Catiana 1 Cattaneo, Alberto Sergio 1 Cavalieri, Renzo 1 Chen, Bo-Yong 1 Chen, Weiyan 1 Chenciner, Alain 1 Chilikov, A. A. 1 Chmutov, Sergei V. ...and 224 more Authors
all top 5
#### Cited in 104 Serials
32 Topology and its Applications 25 Journal of Knot Theory and its Ramifications 13 Algebraic & Geometric Topology 11 Journal of Mathematical Sciences (New York) 9 Geometry & Topology 8 Functional Analysis and its Applications 7 Advances in Mathematics 6 Nuclear Physics. B 6 Transactions of the American Mathematical Society 5 Communications in Mathematical Physics 5 Journal of Pure and Applied Algebra 5 Arnold Mathematical Journal 4 Annales de l’Institut Fourier 4 Doklady Mathematics 3 Mathematical Notes 3 Duke Mathematical Journal 3 Inventiones Mathematicae 3 Mathematische Zeitschrift 3 Journal of Complexity 3 Proceedings of the Steklov Institute of Mathematics 2 Communications in Algebra 2 Israel Journal of Mathematics 2 Journal d’Analyse Mathématique 2 Journal of Mathematical Analysis and Applications 2 Theoretical and Mathematical Physics 2 Journal of Geometry and Physics 2 Compositio Mathematica 2 Journal of Combinatorial Theory. Series A 2 Journal of Differential Equations 2 Mathematische Annalen 2 European Journal of Combinatorics 2 Journal of Symbolic Computation 2 Annals of Physics 2 Bulletin of the American Mathematical Society. New Series 2 Celestial Mechanics and Dynamical Astronomy 2 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 2 Annales Mathématiques Blaise Pascal 2 Bulletin des Sciences Mathématiques 2 Sbornik: Mathematics 2 Mathematical Problems in Engineering 2 Journal of Physics A: Mathematical and Theoretical 1 International Journal of Modern Physics A 1 Computers & Mathematics with Applications 1 Discrete Mathematics 1 Journal of Mathematical Physics 1 Periodica Mathematica Hungarica 1 Physics Letters. A 1 Arkiv för Matematik 1 Chaos, Solitons and Fractals 1 Annali di Matematica Pura ed Applicata. Serie Quarta 1 Applied Mathematics and Computation 1 Automatica 1 Bulletin of the London Mathematical Society 1 Czechoslovak Mathematical Journal 1 Geometriae Dedicata 1 Publications Mathématiques 1 Journal of Algebra 1 Journal of Approximation Theory 1 Journal of Combinatorial Theory. Series B 1 Journal of Computational and Applied Mathematics 1 Journal of the Mathematical Society of Japan 1 Journal für die Reine und Angewandte Mathematik 1 Michigan Mathematical Journal 1 Proceedings of the American Mathematical Society 1 Proceedings of the London Mathematical Society. Third Series 1 Publications of the Research Institute for Mathematical Sciences, Kyoto University 1 Siberian Mathematical Journal 1 Tokyo Journal of Mathematics 1 Transactions of the Moscow Mathematical Society 1 Bulletin of the Korean Mathematical Society 1 Annals of Pure and Applied Logic 1 Optimization 1 Constructive Approximation 1 Discrete & Computational Geometry 1 Journal of the American Mathematical Society 1 Forum Mathematicum 1 Science in China. Series A 1 International Journal of Mathematics 1 Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences 1 Historia Mathematica 1 Expositiones Mathematicae 1 Indagationes Mathematicae. New Series 1 Applicable Algebra in Engineering, Communication and Computing 1 Journal of Mathematical Imaging and Vision 1 Advances in Applied Clifford Algebras 1 Selecta Mathematica. New Series 1 The Ramanujan Journal 1 European Journal of Mechanics. B. Fluids 1 Regular and Chaotic Dynamics 1 Journal of the European Mathematical Society (JEMS) 1 Journal of High Energy Physics 1 Journal of the Australian Mathematical Society 1 Journal of Applied Mathematics 1 Portugaliae Mathematica. Nova Série 1 Bulletin of the Brazilian Mathematical Society. New Series 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 1 Journal of Function Spaces and Applications 1 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 1 Journal of Fixed Point Theory and Applications 1 Journal of Topology and Analysis ...and 4 more Serials
all top 5
#### Cited in 50 Fields
144 Manifolds and cell complexes (57-XX) 45 Algebraic topology (55-XX) 43 Algebraic geometry (14-XX) 27 Global analysis, analysis on manifolds (58-XX) 22 Several complex variables and analytic spaces (32-XX) 22 Quantum theory (81-XX) 20 Combinatorics (05-XX) 16 Differential geometry (53-XX) 15 Group theory and generalizations (20-XX) 14 Nonassociative rings and algebras (17-XX) 12 Dynamical systems and ergodic theory (37-XX) 11 Number theory (11-XX) 11 Partial differential equations (35-XX) 10 Convex and discrete geometry (52-XX) 8 Computer science (68-XX) 7 Category theory; homological algebra (18-XX) 7 Ordinary differential equations (34-XX) 7 Mechanics of particles and systems (70-XX) 6 Functions of a complex variable (30-XX) 5 Field theory and polynomials (12-XX) 5 Commutative algebra (13-XX) 5 Associative rings and algebras (16-XX) 5 Special functions (33-XX) 5 Approximations and expansions (41-XX) 5 Functional analysis (46-XX) 5 Relativity and gravitational theory (83-XX) 4 Real functions (26-XX) 4 Integral transforms, operational calculus (44-XX) 4 Operator theory (47-XX) 4 Numerical analysis (65-XX) 2 History and biography (01-XX) 2 Mathematical logic and foundations (03-XX) 2 Order, lattices, ordered algebraic structures (06-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Topological groups, Lie groups (22-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 $$K$$-theory (19-XX) 1 Measure and integration (28-XX) 1 Difference and functional equations (39-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Abstract harmonic analysis (43-XX) 1 Geometry (51-XX) 1 General topology (54-XX) 1 Statistics (62-XX) 1 Mechanics of deformable solids (74-XX) 1 Fluid mechanics (76-XX) 1 Optics, electromagnetic theory (78-XX) 1 Operations research, mathematical programming (90-XX) 1 Systems theory; control (93-XX)
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2021-01-18 05:10:25
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https://www.onemathematicalcat.org/algebra_book/online_problems/solve_by_inspec.htm
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SOLVING SIMPLE SENTENCES BY INSPECTION
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
• PRACTICE (online exercises and printable worksheets)
Want more details, more exercises?
In this exercise, you will practice solving sentences ‘by inspection’.
That is, you will look at the sentence, stop and think, and come up with a number that makes the sentence true.
Some sentences have exactly one solution.
Some sentences have many solutions; you will provide only one.
You should do this exercise without using a calculator.
EXAMPLES:
Question: Solve: $x = 5$
Answer: $5$
Question: Solve: $x \gt 5$
Answer: $\,5.1\,$, or $\,6\,$, or $\,1007.4\,$; any number in the interval $\,(5,\infty)\,$
(You may want to review interval notation.)
Question: Solve: $\displaystyle\frac{x}{6} = \frac{1}{2}$
Answer: $3$
Question: Solve: $x + 2 = 9$
Answer: $7$
Master the ideas from this section
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2021-09-16 21:48:09
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https://edurev.in/course/quiz/attempt/-1_WBJEE-Maths-Test-5/8ff7e3cc-ed94-464c-b889-4b7d3e606ec3
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JEE > WBJEE Maths Test - 5
# WBJEE Maths Test - 5
Test Description
## 75 Questions MCQ Test WBJEE Sample Papers, Section Wise & Full Mock Tests | WBJEE Maths Test - 5
WBJEE Maths Test - 5 for JEE 2022 is part of WBJEE Sample Papers, Section Wise & Full Mock Tests preparation. The WBJEE Maths Test - 5 questions and answers have been prepared according to the JEE exam syllabus.The WBJEE Maths Test - 5 MCQs are made for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for WBJEE Maths Test - 5 below.
Solutions of WBJEE Maths Test - 5 questions in English are available as part of our WBJEE Sample Papers, Section Wise & Full Mock Tests for JEE & WBJEE Maths Test - 5 solutions in Hindi for WBJEE Sample Papers, Section Wise & Full Mock Tests course. Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free. Attempt WBJEE Maths Test - 5 | 75 questions in 120 minutes | Mock test for JEE preparation | Free important questions MCQ to study WBJEE Sample Papers, Section Wise & Full Mock Tests for JEE Exam | Download free PDF with solutions
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WBJEE Maths Test - 5 - Question 1
### An approximate value of (7.995)1/3 correct to four decimal places is
WBJEE Maths Test - 5 - Question 2
### In the expansion of (1+x)(2n+2) the maximum coefficient is :
WBJEE Maths Test - 5 - Question 3
### The coefficient of the middle term in the binomial expansion in powers of x of (1+αx)4 and (1-αx)6 is the same, if α equals
WBJEE Maths Test - 5 - Question 4
The length of tangents drawn from the point (5,1) is to the circle x2+y2+6x-4y-3=0
WBJEE Maths Test - 5 - Question 5
WBJEE Maths Test - 5 - Question 6
The lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangent to the same circle. The radius of this circle is
WBJEE Maths Test - 5 - Question 7
If ω is cube root of unity, then (1 + ω3) - (1 + ω2)3 =
WBJEE Maths Test - 5 - Question 8
is equal to
WBJEE Maths Test - 5 - Question 9
WBJEE Maths Test - 5 - Question 10
If is a singular matrix, then x =
WBJEE Maths Test - 5 - Question 11
WBJEE Maths Test - 5 - Question 12
WBJEE Maths Test - 5 - Question 13
Let f(x) be a function satisfying f ′(x)= f x with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = x2, then value of integral
is equal to
WBJEE Maths Test - 5 - Question 14
A singular solution of the differential equation y2[1+(dy/dx)2]=R2 is :
WBJEE Maths Test - 5 - Question 15
The order and degree of the differential equation d2y/dx2 + (dy/dx)1/3 + x1=0 are respectively
WBJEE Maths Test - 5 - Question 16
If y=sin((1+x2)/(1-x2)), (dy/dx)=
WBJEE Maths Test - 5 - Question 17
The latus rectum of the ellipse 5x2 + 9y2 = 45 is
WBJEE Maths Test - 5 - Question 18
Detailed Solution for WBJEE Maths Test - 5 - Question 18
WBJEE Maths Test - 5 - Question 19
The eccentricity of the ellipse 9x2 + 5y2 − 30y = 0 is
WBJEE Maths Test - 5 - Question 20
The locus of the centre of a circle which touches given circles externally is
WBJEE Maths Test - 5 - Question 21
The product of the perpendicular, drawn from any point on a hyperbola to its asymptotes is
WBJEE Maths Test - 5 - Question 22
The value of
WBJEE Maths Test - 5 - Question 23
A and B are square matrices of order n x n, then (A - B)2 is equal to
Detailed Solution for WBJEE Maths Test - 5 - Question 23
(A - B) x (A - B)
(A - B) x A - (A - B) x B
= A2 - AB - BA + B2
WBJEE Maths Test - 5 - Question 24
The coordinates of a point of the parabola y=x2+7x+2 which is the closest to the straight line y=3x-3 is
WBJEE Maths Test - 5 - Question 25
[sin(tan⁻1(3/4))]2=
WBJEE Maths Test - 5 - Question 26
Let a + b = 4, a < 2 and g(x) be a monotonically increasing function of x.
Then,
Detailed Solution for WBJEE Maths Test - 5 - Question 26
We have a + b = 4 ⇒ b = 4 − a and b − a = 4 − 2 a = t (say)
Thus, f(a) is an increasing function of t. Hence, the given expression increases with increase in (b-a).
WBJEE Maths Test - 5 - Question 27
The number (1-i)3/1-i3 is equal to
WBJEE Maths Test - 5 - Question 28
The length of the latus rectum of the parabola x2-4x-8y+12=0 is
WBJEE Maths Test - 5 - Question 29
The length of the latus rectum of the parabola 4y2+2x-20y+17=0 is
WBJEE Maths Test - 5 - Question 30
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of there parallel lines is :
WBJEE Maths Test - 5 - Question 31
If nP4=30.nC5, then n=
WBJEE Maths Test - 5 - Question 32
Five digit number divisible by 3 is formed using the digits 0, 1 , 2, 3, 4 and 5 without repetition. Total number of such numbers is
Detailed Solution for WBJEE Maths Test - 5 - Question 32
A number is divisible by 3 if and only if the sum of its digits are divisible by 3
Notice that 1 + 2 + 3 + 4 + 5 = 15, which is divisible by 3
The only other way we can have a sum of 5 digits divisible by 3 is to replace the 3 by the 0 making the sum 3 less:
1 + 2 + 0 + 4 + 5 = 12, which is divisible by 3
No other choice of 5 digits can have a sum divisible by 3, because there is no other way to make the sum 12 or 15, and we certainly can't have a sum of 9 or 18
So the number of 5-digit numbers that can be formed from the digits {1,2,3,4,5} is
Number of ways = 1 x 2 x 3 x 4 x 5 = 120
And the number of 5-digit numbers that can be formed from the digits {1, 2, 0, 4, 5} is figured this way
Number of ways = 1 x 2 x 3 x 4 x 4 = 96
Total Number of ways = 120 + 96 = 216
WBJEE Maths Test - 5 - Question 33
A person draws a card from a pack of playing cards, replaces it and shuffles the pack. He continues doing this until he draws a spade. The chance that he will fail in first two times is
WBJEE Maths Test - 5 - Question 34
A five digit number is formed by writing the digits 1,2,3,4,5 in a random order without repetitions. Then the probability that the number is divisible by 4, is
WBJEE Maths Test - 5 - Question 35
You are given a box with 20 cards in it. 10 of these cards have the letter 1 printed on them. The other ten have the letter T printed on them. If you pick up 3 cards at random and keep them in the same order, the probability of making the word IIT is
WBJEE Maths Test - 5 - Question 36
In ΔABC , cosA + cosB + cosC =
Detailed Solution for WBJEE Maths Test - 5 - Question 36
cos A + cos B + cos C = 1 + 4 sin
WBJEE Maths Test - 5 - Question 37
In a ΔABC , ∑(b + c) tan
Detailed Solution for WBJEE Maths Test - 5 - Question 37
Σ(b + c) tan A 2 tan ( B - C 2 ) = Σ(b + c) tan
WBJEE Maths Test - 5 - Question 38
If both the roots of the quadratic eqution x2 - 2kx + k2 + k - 5 = 0 are less than 5, then k lies in the interval
WBJEE Maths Test - 5 - Question 39
If f(x) is continuous and differentiable over [−2,5] and −4 ≤ f′ (x) ≤ 3 for all x in (−2,5) then the greatest possible value of f(5) − f(−2), is
Detailed Solution for WBJEE Maths Test - 5 - Question 39
WBJEE Maths Test - 5 - Question 40
The sum of the series 1.32 + 2.52 + 3.72 +.....upto 20 terms is
WBJEE Maths Test - 5 - Question 41
Let α, β be the roots of the equation (x - a)(x - b) + c = 0, c ≠ 0. The roots of the equation (x - α)(x - β)+ c = 0 are
Detailed Solution for WBJEE Maths Test - 5 - Question 41 Since α, β are the roots of equation
(x - a) (x - b) = c
or x2 - (a + b) x + ab - c = 0
Then α + β = a + b
and αβ = ab - c
or ab = αβ + c
Then the roots of equation is
x2 - x(α + β) + αβ + c = 0
⇒ x2 - x(a + b) + ab = 0
⇒ a and b are the roots
WBJEE Maths Test - 5 - Question 42
Every term of a G.P. is positive and also every term is the sum of two preceding terms. Then the common ratio of the G.P. is
WBJEE Maths Test - 5 - Question 43
If a,b,c are in G.P.,then equations ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common root if d/a,e/b,f/c are in
WBJEE Maths Test - 5 - Question 44
The sum of n terms of two A.P.'s are in the ratio of (7n + 1) : (4n + 27). The ratio of their 11 terms is
Detailed Solution for WBJEE Maths Test - 5 - Question 44
Hence the ratio of the 11th terms is 148 : 111
WBJEE Maths Test - 5 - Question 45
4 [cot⁻13 + cosec⁻1 (√5)] is equal to
WBJEE Maths Test - 5 - Question 46
With reference to a universal set, the inclusion of a subset in another, is relation, which is
WBJEE Maths Test - 5 - Question 47
The locus of a point such that the sum of whose distances from two rectangular lines is 2 units is
WBJEE Maths Test - 5 - Question 48
If the intercept on y-axis is double the intercept on x-axis by any line and if the line passes through (1,2), then equation of line is
WBJEE Maths Test - 5 - Question 49
The general solution of the equation tan2θ.tanθ=1 for n∈I is, θ is equal to
WBJEE Maths Test - 5 - Question 50
The condition that the cubic equation x3 - px2 + qx - r = 0 should have roots in G.P. is given by____
Detailed Solution for WBJEE Maths Test - 5 - Question 50
The roots of the cubic equation
WBJEE Maths Test - 5 - Question 51
The equation 4 sin2x + 4 sin x + a2 − 3 = 0 possesses a solution if a belongs to the interval
WBJEE Maths Test - 5 - Question 52
If p , x1 , x2 , … xi … and q , y1 , y2 , … yi … are in A.P., with common difference a and b respectively, then the centre of mean position of the points Ai(xi, yi) where i = 1, 2, ..., n lies on the line
WBJEE Maths Test - 5 - Question 53
If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x. The expression
3b2x2 + 6bc x + 2c2 is
WBJEE Maths Test - 5 - Question 54
The value of is equal to
WBJEE Maths Test - 5 - Question 55
The locus of the orthocentre of the triangle formed by the lines
(1 + p)x - py + p(1 + p) = 0,
(1 + q)x - qy + q(1 + q) = 0,
and y = 0, where p ≠ q , is
WBJEE Maths Test - 5 - Question 56
The differential equation determines a family of circles with
WBJEE Maths Test - 5 - Question 57
The area of the region between the curves bounded by the lines x = 0 and x = is
WBJEE Maths Test - 5 - Question 58
dx equals
WBJEE Maths Test - 5 - Question 59
Let A = N x N, and let ' ∗ ' be a binary operation on A defined by (a,b) ∗ (c,d) = (ad+bc,bd) for all (a,b), (c,d) ∈ N x N. Then find identity element in A.
WBJEE Maths Test - 5 - Question 60
If the curves intersect each other at right angles, then
WBJEE Maths Test - 5 - Question 61
Let A be a non-empty set and S be the set of functions from A to itself. The composition of function 'O' is a.
WBJEE Maths Test - 5 - Question 62
Let A1 , A2 , A3 , … … , An be n skew-symmetric matrix of same order then will be
WBJEE Maths Test - 5 - Question 63
Let a,b,c be any real numbers. Suppose that there are real numbers x,y,z not all zero such that x=cy+bz, y=az+cx and z=bx+ay. Then a2+b2+c2+2abc is equal to
WBJEE Maths Test - 5 - Question 64
The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1,2 and 3 only, is
WBJEE Maths Test - 5 - Question 65
Given that converges, what is the value to which it converges?
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 66
Circles are drawn on chords of the rectangular hyperbola xy = c2 parallel to the line y = x as diameters. All such circles pass through two fixed points whose co-ordinates are
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 67
Let f(x) = 1 + 2 sinx + 2 cos2 x , 0 ≤ x ≤ π/2 . Then
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 68
If sinβ is the G.M. between sinα and cos α , then cos2 β is equal to
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 69
Z1 , Z2 , Zcorrespond to the vertices of an equilateral triangle and |Z1 − 1| = |Z2 − 1| = |Z3 − 1| . Then
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 70
In a certain culture of bacteria, the rate of increase is proportional to the number present. It if be known that the number doubles in 4 hours, then
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 71
The polynomial 1 − x + x2 − x3 + . . . + x16 − x17 may be written in the form where y = x + 1 and are constants . Then a2 equals
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 72
The set of discontinuities of the function f (x) = ( 1/2 − cos2x ) contains the set
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 73
If pth, 2pth, 4pth terms of an A.P. are in G.P. then common ratio of G.P. is
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 74
An integral solution of the equation tan−1 x + tan−1 = tan−1 3 is
*Multiple options can be correct
WBJEE Maths Test - 5 - Question 75
If represent the equations of three planes, then point of intersection of these planes is
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4 videos|10 docs|54 tests
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2022-08-14 23:47:10
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https://wildonblog.wordpress.com/2015/04/03/bmcbamc-tuesday-to-thursday-forbidden-subgraphs-mantels-theorem-and-the-foundations-of-mathematics/
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## BMC/BAMC 2015 Tuesday to Thursday: Mantel’s Theorem and the foundations of mathematics
On Tuesday, John Talbot gave a great short talk on Mantel’s Theorem: analogues and extensions. Mantel’s Theorem states that the maximum number of edges in a graph on $n$ vertices containing no triangles is $\lfloor n^2/4 \rfloor$. Talbot indicated one of many nice proofs: for all vertices $x$ and $y$ such that $\{x,y\}$ is an edge we have
$(\delta(x) - 1) + (\delta(y) - 1) \le n-2.$
So summing over all $m$ edges we get
$\sum_x \delta(x)^2 = \sum_{\{x,y\}} (\delta(x) + \delta(y)) \le nm.$
The left-hand side is maximized when all the degrees are as equal as possible. So we can assume that $\delta(x) \le \sqrt{m}$ for each vertex $x$. By the Handshaking Lemma we get $2m \le n\sqrt{m}$. Hence $4m^2 \le n^2m$ and $m \le n^2/4$. Moreover, it follows that there is a unique extremal graph, namely the complete bipartite graph with parts of sizes $\lfloor n/2 \rfloor$ and $\lceil n/2 \rceil$.
Talbot mentioned a number of other analogous problems for other forbidden subgraphs, and extensions where graphs are generalized to uniform hypergraphs. He ended with some open questions. The most interesting for me was a conjecture of Turan, that the asymptotic density of edges in a $3$-uniform hypergraph free of the tetrahedron $\{ \{1,2,3\}, \{1,2,4\}, \{1,3,4\}, \{2,3,4\} \}$ is $5/9$. Turan gave an explicit construction attaining this bound: Let $X = \{1,\ldots, r\}$, $Y = \{r+1,\ldots, 2r\}$ and $Z = \{2r+1, \ldots, 3r\}$ and let the $3$-edges be all $3$-subsets of $\{1,\ldots, 3r\}$ of the form $\{x,y,z\}$, $\{x,x',y\}$, $\{y,y',z\}$ or $\{z,z',x\}$. Given any $4$ vertices there must be two in the same subset, say $X$, and a tetrahedron is then ruled out because the graph has no edges of the form $\{x,x',z\}$. The number of edges is roughly $r^3 + 3r^2/2 \times r = 5r^3/2$ so the asymptotic density is $5/2 \left/\right. 27/6 = 5/9$.
On Thursday there was another great talk by Mirna Džamonja on Current challenges in foundations of mathematics, logic and set theory. Most of the talk was a fascinating tour through the history of foundations, from Hilbert’s programme to Gödel, and then post-Gödel results such as the independence of the continuum hypothesis from ZFC (giving a surprising answer to Hilbert’s Problem 1) and a brief mention of Voevodsky’s univalent foundations, which seems to have connections to the proof assistant Coq mentioned in Monday’s panel discussion.
She ended with some results on random graphs that suggests that singular cardinals (i.e. cardinals $\kappa$ such that $\kappa$ is the union of fewer than $\kappa$ cardinals each smaller than $\kappa$) have some special ‘rigidity’ properties, in contrast to the free-for-all often permitted by ZFC. Her final result is elegant and even has a simple proof in ZF. To state it we need the following definition: a graph $G$ on a set of cardinality $\kappa$ is universal for a set of graphs $\mathcal{G}$ if every graph in $\mathcal{G}$ is isomorphic to an induced subgraph of $G$.
Theorem. Let $\kappa$ be a singular cardinal of countable cofinality. There is no graph $G$ of cardinality $\kappa$ containing no clique of cardinality $\kappa$, and universal for graphs of cardinality $\le \kappa$ containing no clique of cardinality $\kappa$.
Proof. Suppose $G$ is such a graph. Let $\kappa = \cup_{i < \omega} \lambda_i$ where $\lambda_i < \kappa$ for each $i$. Make a new graph $G^+$ by taking disjoint cliques $H_i$ of cardinality $\lambda_i$ for each $i < \omega$, and joining all vertices in $G$ to all vertices in each $H_i$. Note that $G^+$ has no clique of cardinality $\kappa$. Let $f : G^+ \rightarrow G$ be an injection. If $x \in H_i$ and $y \in H_j$ where $i < j$ then $\{x, f^{j-i}(y)\}$ is an edge of $G^+$, and hence $\{f^{i+1}(x), f^{j+1}(y) \}$ is an edge of $G$. It easily follows that
$\cup_{i < \omega} f^{i+1}(H_i)$
is a clique in $G$ of cardinality $\cup_{i < \omega} \lambda_i = \kappa$. $\Box$
In fact this proof even shows that no such $G$ exists if we omit the word 'induced' in the definition of universal. In contrast the incredible countable random graph contains every finite or countable graph as an induced subgraph, and it is known that under GCH for every cardinal $\kappa$ there exist a graph of cardinality $\kappa$ universal for all graphs of cardinality $\le \kappa$.
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2017-07-23 06:52:53
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http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=DBSHCJ_2005_v20n1_107
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A PICTURE OF KLEINIAN MODULAR GROUP
Title & Authors
A PICTURE OF KLEINIAN MODULAR GROUP
KIM, HONG-CHAN;
Abstract
We show an algorithm to draw the famous picture of Kleinian modular group PSL(2, $\small{\mathbb{Z}}$), which appears in describing the moduli space of elliptic curves.
Keywords
good divisor;algorithm;
Language
English
Cited by
References
1.
D. Farmer, Groups and Symmetry. A guide to discovering mathematics, Amer. Math. Soc., 1996
2.
N. Jacobson, Basic Algebra II. Second edition, W. H. Freeman and Company, 1989
3.
J. P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics, no. 7, Springer-Verlag, 1973
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2016-12-04 04:07:15
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https://physics.stackexchange.com/questions/329659/speed-limitation?noredirect=1
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Speed limitation [duplicate]
I was thinking about something and I wish to share this thought with you in order to understand better something that bothers me.
Relativity theory says that the speed of light is the highest possible speed that can be developed. My question is: what is physical (and simplified) explanation for this? In other words, what is physical explanation beneath higher limit of speed? Taking classical example for a definition of speed as path divided by time (although aware it would be more appropriate to speak about waves) , what is possible explanation that something cannot travel more than 3*10^8 m in a second? Why not 3,1*10^8? What is physical constraint for that ?
• This is a "feature" of the universe that we currently have to accept. We do not have an underlying theory as to why the speed of light is what we expermentally find it to be. In particle physics, there are around 20 different parameters that we use experimentally established values for, rather than any theoretical understanding. physics.stackexchange.com/q/172846 – user154420 Apr 28 '17 at 19:54
• So there is no deductive explanation for what would happen if a particle/wave would travel with a speed >c ? – user406046 Apr 28 '17 at 19:56
• There are lots of ideas, (speculations rather), based on various assumptions, but there is no experimental evidence for anything moving at a velocity beyond c. – user154420 Apr 28 '17 at 20:01
• Why c? Don't know. However, a logical analysis of spatial motion does tell you that there has to be a finite limit to make motion possible. – Sean Apr 28 '17 at 20:11
actually special relativity does not say anything about the maximum speed, it is based on the assumption that the speed of light $c$ is the same for all inertial observers. This leads to things such as
$$x' = \gamma(x -vt) ~~~\mbox{and}~~~ t' = \gamma(t-vx/c^2)$$
With
$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$
This clearly breaks when $v>c$.
• I know these relationships but this says nothing about physical breaking of things :) . It is telling us about mathematical logic behind. What I am trying to ask is ''what will happen with a particle if it had a speed that goes beyond c?'' The answer I am looking should have a form ''it would ... and this is impossible'' . In other word I am looking for absurd that would happen in case of reaching this speed which is of physical nature. – user406046 Apr 28 '17 at 20:27
• @user406046 SR doesn't prohibit a particle from traveling faster than $c$, but it does prohibit a particle with $v<c$ from being accelerated to $c$: you can keep adding kinetic energy to such a particle and its speed will remain below $c$. Similarly, if a hypothetical particle did have $v>c$ you couldn't slow it down to $c$. But if you plug $v>c$ into the equations in caverac's answer you get distances & times involving the square root of negative numbers, and it's difficult to see how such values could be applied to real objects. – PM 2Ring Apr 29 '17 at 13:53
Relativity theory says that the speed of light is the highest possible speed that can be developed.
1. The constance of the speed of light is an empirical fact, means it is a phenomenon of nature that was stated after a long series of experiments on our planet.
2. That is not right for an observer that observes two positions in space with different gravitational potential. In Einsteins General Relativity Theory he is deriving this.
3. Once emitted photons are going with c in vacuum. Matter is moving with different velocities and their velocity depends from the impulse they got from what?
Wrapping your head about this question it would be clear why matter isn't able to move with c. If you bounce a billiard ball with another billiard ball on the atomic level the outer electrons from both surfaces are interacting with their electric fields and it is stated that this happens by the exchange of virtual photons. So at the end every interaction between matter - be this a hand clapping or our stabile standing on earth - is accompanied by the electromagnetic interaction. The highest velocity of matter we realize for electrons in linear and ring accelerators, which are driven by electric fields (and magnetic fields too).
... what is physical explanation beneath higher limit of speed?
Our macroscopic world is driven by the gravitational influence of masses and by EM interactions. As explained above the gravitational potential could not be ignored for the locality of the speed of light (in higher gravitational potential c is smaller for an observer located in a point with smaller gravitational potential). Others than this phenomenons we don't observe until now and the law about the local constantly of the speed of light is a consequence from this observations.
Some people think that the reason behind the upper limit of light speed is the inner properties of the matter.
So – called force carriers or messenger particles or intermediate particles are particles that give rise to forces between other particles.
https://en.wikipedia.org/wiki/Force_carrier
https://en.wikipedia.org/wiki/Gauge_boson
https://en.wikipedia.org/wiki/Massless_particle
Explanation of limit of the speed of light implies that these partricles move inside material bodies with the same velocity as they move outside material bodies.
Now, let‘s think whether we can send a massless particle back and forth immediatelly. What „immsediatelly“ means? In the physical sense, “instantaneously” means that no processes and variations occurred between these times – even at the microlevel. The dispatch time and the return time must merge together in this instance. After all, if any processes and variations occurred in a body between these times, then by occurring in time, the processes and variations required a specific amount of time. This means that the times are separated by a time interval to which the arbitrarily accurately running clock must react by a change in their readings.
One young man was even awarded (400 K bucks) for the idea that sounds like that!
As I understand, if you measure velocity of light with a lightclock, you will always measure the same velocity, because accumulated time (number of oscillations) only depends on travelled by lightpulse distance L and distance between lightclock‘s mirrors l.
And in simple words all the matter (on microlevel) consists of small lightclocks.
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2021-06-16 09:08:16
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http://clay6.com/qa/2008/if-a-is-skew-symmetric-then-ka-is-a-k-is-any-scalar-
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# If A is skew symmetric, then kA is a __________. (k is any scalar).
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• All the diagonal element of a skew symmetric matrix is equal to zero hence by multiplying any scalar value with zero will remain zero hence the obtained matrix should also be skew symmetric matrix
If A is skew symmetric ,then kA is a skew symmetric matrix.
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2017-04-27 16:55:11
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http://www.mathjournals.org/jot/2021-086-001/2021-086-001-010.html
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# Journal of Operator Theory
Volume 86, Issue 1, Summer 2021 pp. 189-201.
Weighted composition operators: isometries and asymptotic behaviour
Authors: Isabelle Chalendar (1), Jonathan R. Partington (2)
Author institution: (1) LAMA, Universite Gustave Eiffel, Universite Paris Est Creteil, CNRS, F-77454 Marne-la-Vallee, France
(2) School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K.
Summary: This paper studies the behaviour of iterates of weigh\-ted composition operators acting on spaces of analytic functions, with particular emphasis on the Hardy space $H^2$. Questions relating to uniform, strong and weak convergence are resolved in many cases. Connected to this is the question when a weighted composition operator is an isometry, and new results are given in the case of the Hardy and Bergman spaces.
DOI: http://dx.doi.org/10.7900/jot.2020feb28.2292
Keywords: weighted composition operator, iteration, isometry
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2021-08-05 07:31:10
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http://quant.stackexchange.com/questions?page=51&sort=newest
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# All Questions
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### Why asset management firms shouldn't be custodian of its own funds?
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### Wiener process integral
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### Estimating Number of “Day Trades” from Total Volume of Commodity Futures Contract
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### Bachelier model: number of stocks in replicating strategy
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2016-02-09 13:51:00
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http://specialfunctionswiki.org/index.php/Lefschetz_zeta_function
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# Lefschetz zeta function
Let $f$ be a function. The Lefschetz zeta function is $$\zeta_f(z)=\exp \left( \displaystyle\sum_{k=0}^{\infty} L(f^n)\dfrac{z^n}{n} \right),$$ where $L(f^n)$ is the Lefschetz number of the $n$th iterate of $f$
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2018-11-14 03:17:45
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https://lists.lyx.org/pipermail/lyx-devel/2021-April/006321.html
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# Bug(?) when creating an align* with Ctrl+Return.
Paul A. Rubin parubin73 at gmail.com
Thu Apr 15 14:49:24 UTC 2021
On 4/15/21 4:19 AM, Andrew Parsloe wrote:
> Windows 10, LyX 2.4.0-alpha3 and 2.3.6.
>
> Given a formula $a=b=c$ position the cursor after the b and press
> Ctrl+Return. In LyX 2.4.0-alpha3 this produces
> \begin{align*}
> a & =b=c\\
> \end{align*}
> and leaves the cursor after the c. The cursor positioning and failure
> to transfer the part of the formula after the cursor to the newly
> created line seem like bugs to me.
>
> In the align* formula again position the cursor after the b and press
> Ctrl+Enter. This time the result is
> \begin{align*}
> a & =b\\
> & =c\\
> \end{align*}
> with the cursor remaining just after the b. Except for the extra \\
> (displaying in LyX as two empty blue boxes) this is what I expected
> the first time.
>
> Andrew
>
Confirmed here (same two LyX versions, OS=Linux Mint), and I agree that
Ctrl+Enter should split the equation at the cursor position in the first
case.
Paul
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2021-05-08 15:38:44
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https://www.physicsforums.com/threads/solve-homogeneous-d-e-need-help-integrating.741807/
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# Homework Help: Solve Homogeneous D.E. need help integrating
1. Mar 5, 2014
### Jtechguy21
1. The problem statement, all variables and given/known data
Dy/Dx = (Y-x)/(Y+x)
2. Relevant equations
Y=ux
dy=udx+xdu
3. The attempt at a solution
Dy/Dx = (Y-x)/(Y+x)
Plug in my substitutions
udx+xdu(1/dx)=(ux/ux+x) - X/(ux+x)
Simplify
u+x(du/dx)=(ux)/x(u+1) - (x)/((x)(u+1))
u+x(du/dx)=u/(u+1) -(1)/(u+1))
u+x(du/dx)=u-1/(u+1)
This is where I think i begin to mess up
u+du=(u-1)/(u+1) dx/x
substract (u-1)/(u+1) to the other side
u-(u-1)/(u+1) du=dx/x
I know the right side integrates to Lnx +c
but on the left side if i do
(u^2-1)/(u+1)
I split it up into
the integral (u^2)/(u+1) minus integral of 1/(u+1)
(u^2)/(u+1)<-use long division
I get u+(1/u+1) minus the integral of 1/(u+1)
i am left with just the integral
of u
u^2/2= lnx+c
plug u back in.
((y/x)^2)/2 =lnx +c
is this sufficient of an answer?
according to the answer key im gonna end up with the arctan somewhere in my answer. so i may have already messed up :(
Last edited: Mar 6, 2014
2. Mar 6, 2014
### pasmith
Why are you dividing by $y + x$ here? Is your equation actually
$$\frac{dy}{dx} = \frac{y - x}{y + x}$$
and not
$$\frac{dy}{dx} = (y - x)(y + x)$$
as you have written?
This should be "u dx/x + du" on the left hand side.
You can't; it's multiplied by dx/x.
What you have after replacing $y$ is
$$x\frac{du}{dx} + u = \frac{u - 1}{u + 1}$$
Subtracting $u$ from both sides and then dividing by $x$ puts this in the separable form
$$\frac{du}{dx} = \frac1x \left(\frac{u-1}{u+1} - u\right)$$
Continue.
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2018-07-20 11:15:26
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http://www.physicsforums.com/showthread.php?p=3761409
|
## Help needed, converting weight to a falling force.
Hey all,
Just a quick overview of myself, I have recently started rock climbing (6 month) and I'm finding myself becoming more curious about the technical equipment involved.
I'm finding it hard to word the questions I'm trying to find the answers too. So I'm going to try and I'm hoping someone will understand.
I weigh 100kg's, How much of an impact force will I create If i was to free fall over different distances with the top distance being no more than 100m.
Can you convert the weight of a moving person into Kn ?(This being the impact force a rope can take, Most ropes can take an average of 9Kn.)
Below is a few things about the Climbing rope i'm currently using. hope this helps
Impact Force (kN): 8.90
Static Elongation (%): 9.10
Dynamic Elongation (%): 33.00
I will be honest although physics interests me I'v never really understood most of it and only done what was taught in school. So any direct answers would be appreciated, But answers with a description as to why would be even better.
Thanks all
Craig
P.s I'm hoping I'v worded this correctly, I doubt I have so any feed back or questions are more than welcome.
Thanks again
PhysOrg.com physics news on PhysOrg.com >> Promising doped zirconia>> New X-ray method shows how frog embryos could help thwart disease>> Bringing life into focus
Recognitions: Homework Help As you fall, you gravitational potential energy is converted to kinetic energy - so you do $mgh=E_K$ neglecting air resistance. In general you can't so this will give you an over-estimate. You fall the length of your rope - the rope then stretches to take the strain - storing the kinetic energy ... then you bounce around a bit. It is very common to model the rope+climber as an undamped harmonic oscillator. For more details on how to use the rope impact http://en.wikipedia.org/wiki/Fall_factor There are usually lots of rules of thumb for this. The fastest way to find out if you'll break the rope is to tie a weight to the end of it and drop it off a bridge or something.
When you fall you gain speed at roughly 10m/s^2 so after 1 sec your speed is 10m/s, after 2 sec your speed is 20m/s and so on. This means that you gain MOMENTUM ( mass x velocity) so after 1 sec of falling your momentum is 100 x 10 = 1000 Ns after 2 sec of falling your momentum is 2000 Ns and so on When you are brought to a halt (by rope, air bag, seat belt.... whatever) the force on you x the time for which it acts brings the momentum to zero (you stop!!) If the rope stops you in 1 sec then after falling for 1 sec the force to stop you is 1000N.... the same as your weight. You could call this '1g' After 2 sec of falling the force to stop you in 1 sec is 2000N.... 2g and so on You can see that it is important to extend the time to come to a stop for as long as possible to reduce the force you need to experience. This is the basic physics behind elastic ropes, seat belts, air bags, crumple zones etc..... to extend the time.
Recognitions:
Gold Member
## Help needed, converting weight to a falling force.
Does "Dynamic elongation" tell you the percentage stretch when 8.9kN is applied (at the end of your fall)?
Interesting. Maybe I can use the above maths to work out my question on my thread ??? Thats if no one has answerd it yet. Wayne
Recognitions:
Gold Member
|
2013-05-19 01:10:51
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|
http://mathhelpforum.com/algebra/187636-distributing-simplifying-print.html
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# Distributing and simplifying
• Sep 9th 2011, 08:09 AM
vaironxxrd
Distributing and simplifying
4(3x-5)= -2(-x+8)
I got 10x= 4
= 2.5
• Sep 9th 2011, 08:13 AM
Prove It
Re: Distributing and simplifying
Quote:
Originally Posted by vaironxxrd
4(3x-5)= -2(-x+8)
I got 10x= 4
= 2.5
\displaystyle \begin{align*} 4(3x-5) &= -2(-x+8) \\ 12x-20 &= 2x-16\\ 10x-20 &= -16\\ 10x &= 4 \\ x &= \frac{4}{10} \\ x &= \frac{2}{5}\end{align*}
I disagree with your answer of $\displaystyle x = 2.5$
• Sep 9th 2011, 08:31 AM
vaironxxrd
Re: Distributing and simplifying
Quote:
Originally Posted by Prove It
\displaystyle \begin{align*} 4(3x-5) &= -2(-x+8) \\ 12x-20 &= 2x-16\\ 10x-20 &= -16\\ 10x &= 4 \\ x &= \frac{4}{10} \\ x &= \frac{2}{5}\end{align*}
I disagree with your answer of $\displaystyle x = 2.5$
Oh so I was pretty close . One more question when 10x/4 you divide by 10 to get x by itself?
• Sep 9th 2011, 09:17 AM
Siron
Re: Distributing and simplifying
Quote:
Originally Posted by vaironxxrd
Oh so I was pretty close . One more question when 10x/4 you divide by 10 to get x by itself?
Do you mean:
$\frac{\frac{10x}{4}}{10}$?
Because that's not just $x$ but $\frac{\frac{10x}{4}}{10}=\frac{x}{4}$
• Sep 9th 2011, 09:41 AM
vaironxxrd
Re: Distributing and simplifying
Quote:
Originally Posted by Siron
Do you mean:
$\frac{\frac{10x}{4}}{10}$?
Because that's not just $x$ but $\frac{\frac{10x}{4}}{10}=\frac{x}{4}$
So your telling me its basically 10x/4/10? That looks weird
• Sep 9th 2011, 10:44 AM
e^(i*pi)
Re: Distributing and simplifying
Quote:
Originally Posted by vaironxxrd
Oh so I was pretty close . One more question when 10x/4 you divide by 10 to get x by itself?
10x/4 has to equal something though. In this case it is equal to 1 because you've brought the 4 from the RHS onto the LHS by division: $\dfrac{10}{4} x = 1$. From here you'd need to divide both sides by 10/4: $x = \dfrac{1}{\left(\frac{10}{4}\right)}$ and since we divide fractions by "flip n multiply" the RHS is the same as $1 \times \dfrac{4}{10} = \dfrac{4}{10}$
From the line $10x = 4$ you can divide both sides by 10 because we want to get x on it's own (which is the same as 1x): $\dfrac{10x}{10} = \dfrac{4}{10}$ and since 10/10 is indeed 1 this is the same as $\dfrac{4}{10}$ which is correct but it's more proper to simplify fractions and so 2/2 was cancelled.
$\dfrac{4}{10} = \dfrac{2 \times 2}{2 \times 5} = \dfrac{2}{5}$.
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2016-12-07 21:51:41
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https://physics.stackexchange.com/questions/494725/what-are-the-differences-and-advantages-of-the-integral-and-differential-forms-o
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# What are the differences and advantages of the integral and differential forms of Gauss's law? [duplicate]
What are the differences and advantages of Gauss's Integral and Differential Equations? Thank you so very much!
## marked as duplicate by Aaron Stevens, Thomas Fritsch, Jon Custer, John Rennie electromagnetism StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Aug 2 at 10:20
They are equivalent, and one can be derived from the other using the divergence theorem, except when the E-field is not well-behaved, in which case the integral form might be slightly more general. As an example, the divergence of the E-field due to a point charge is proportional to the Dirac delta function, which often requires more care than ordinary functions. I'm sure someone can comment if there are even more pathological cases. In most situations, if you don't have a problem working with Dirac delta functions, you can use either form.
Some advantages of the integral form:
• Doesn't require vector differential operators, so the integral form is frequently used in more introductory treatments.
• Can be readily used to calculate fields by exploiting symmetries in the problem (spherically symmetric charge distribution, uniformly charged infinite sheet, uniformly charged infinite line, etc.).
Some advantages of the differential form:
• More compact.
• Doesn't require the introduction of an arbitrary volume whose boundary you integrate over.
• Relates the field and charge density at a point rather than over a volume. This is often more convenient if the problem doesn't lend itself to the use of the integral form because of symmetries.
• I want to stress Puk’s final point. The differential form makes clear that electromagnetism is a local field theory. – G. Smith Aug 1 at 23:50
• Numerical solutions of the integral form are inherently conservative, while differential forms are much harder to guarantee conservation (at least, this is true for Navier-Stokes -- not sure how often discontinuities arise in E&M). – tpg2114 Aug 2 at 0:04
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2019-10-15 07:27:18
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http://experiment-ufa.ru/8b+3=51
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# 8b+3=51
## Simple and best practice solution for 8b+3=51 equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so dont hesitate to use it as a solution of your homework.
If it's not what You are looking for type in the equation solver your own equation and let us solve it.
## Solution for 8b+3=51 equation:
8b+3=51
We move all terms to the left:
8b+3-(51)=0
We add all the numbers together, and all the variables
8b-48=0
We move all terms containing b to the left, all other terms to the right
8b=48
b=48/8
b=6
`
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2017-11-22 14:40:40
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http://www.aimspress.com/article/10.3934/math.2020456/fulltext.html
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AIMS Mathematics, 2020, 5(6): 7122-7144. doi: 10.3934/math.2020456
Research article
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Some New $(p_1p_2,q_1q_2)$-Estimates of Ostrowski-type integral inequalities via n-polynomials s-type convexity
1 School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China
2 Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China
3 Department of Mathematics, Faculty of Technical Science, University “Ismail Qemali”, Vlora, Albania
4 Department of mathematics, Government University, Faisalabad, Pakistan
5 Department of Mathematics, Huzhou University, Huzhou 313000, China
6 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China
## Abstract Full Text(HTML) Figure/Table Related pages
The purpose of this paper is to establish new generalization of Ostrowski type integral inequalities by using $(p,q)$-analogues which are related to the estimates of upper bound for a class of $(p_1p_2,q_1q_2)$-differentiable functions on co-ordinates. We first establish an integral identity for $(p_1p_2,q_1q_2)$-differentiable functions on co-ordinates. The result is then used to derive some estimates of upper bound for the functions whose twice partial $(p_1p_2,q_1q_2)$-differentiable functions are $n$-polynomial $s$-type convex functions on co-ordinates. Some new special cases from the main results are obtained and some known results are recaptured as well. At the end, an application to special means is given as well.
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# References
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2020-09-30 08:43:32
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http://mathhelpforum.com/geometry/10122-proof.html
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# Math Help - a proof
1. ## a proof
The volume of a tetrahedron is (1/3)*(area of base)*(height).
Use this result to prove that the volume of a tetrahedron whose edges are the vectors u, v, w is (1/6) lu dot (v x w)l.
note: dot = dot product.
2. Originally Posted by Jenny20
The volume of a tetrahedron is (1/3)*(area of base)*(height).
Use this result to prove that the volume of a tetrahedron whose edges are the vectors u, v, w is (1/6) lu dot (v x w)l.
note: dot = dot product.
$\bold{v} \wedge \bold{w}$ is a vector normal to the plane containing $\bold{v}$ and $\bold{w}$ and whose magnitude is equal to the area of the parallelogram defined by them, and so twice the area of the triangle defined by them.
Now the absolute value of $\bold{u} \vee (\bold{v} \wedge \bold{w})$ is equal to the product of the projection of $\bold{u}$ onto $\bold{v} \wedge \bold{w}$, which is the height of the tetrahedron times twice the area of the base (upto the sign).
Thus $|\bold{v} \wedge \bold{w}|$, is equal to the height of the tetrahedron times twice the area of the base, which is six times the volume of the tetrahedron, which is what was to be proven.
RonL
3. Hello, Jenny!
My proof is the same as Captain Black's with some small differences.
The volume of a tetrahedron is: . $\frac{1}{3}(\text{area of base})(\text{height})$
Use this result to prove that the volume of a tetrahedron
whose edges are the vectors $\vec{u},\:\vec{v},\:\vec{w}$ is: . $\frac{1}{6}\bigg|u \cdot (v \times w)\bigg|$
Let the plane of $\vec{v}$ and $\vec{w}$ be the base of the tetrahedron.
Then: . $(\text{area of base}) \:=\:\frac{1}{2}\left|\vec{v} \times \vec{w}\right|$
The height of the tetradedron is the magnitude of the projection of $\vec{u}$ onto $\vec{v} \times \vec{w}$
. . The projection of $\vec{a}$ onto $\vec{b}$ is given by: . $\text{proj}_{\vec{b}}\vec{a} \;=\;\left(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2} \right)\vec{b}$
We have: . $\vec{h} \:=\:\left(\frac{\vec{u}\cdot(\vec{v} \times \vec{w})}{|\vec{v} \times \vec{w}|^2}\right)(\vec{v} \times \vec{w})$
. . Then: . $h \;=\;\left|\frac{\vec{u}\cdot(\vec{w}\times\vec{w} )}{|\vec{v}\times\vec{w}|^2}\,(\vec{v}\times\vec{w })\right| \;=\;\frac{|\vec{u}\cdot(\vec{v}\times\vec{w})|}{| \vec{v}\times\vec{w})|^2}\,|\vec{v}\times\vec{w}|$ $= \;\frac{|\vec{u}\cdot(\vec{v}\times\vec{w})|}{|\ve c{v}\times\vec{w}|}$
Substitute: . $V \;=\;\frac{1}{3}\underbrace{(\text{area of base})}\underbrace{(\text{height})}$
. . . . . . . . . $V \;=\;\frac{1}{3}\left[\frac{1}{2}|\vec{v}\times\vec{w}|\right] \,\left[\frac{|\vec{u}\cdot(\vec{v}\times\vec{w})|}{|\vec{ v}\times\vec{w}|}\right]$
Therefore: . $V \;=\;\frac{1}{6}\bigg|\vec{u}\cdot(\vec{v}\times\v ec{w})\bigg|$
4. Hi Soroban,
With Captainblack's help , i got the same proof as you.
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2015-07-31 04:39:36
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http://teaching.paulhartzer.com/hamtramck/mathematics-without-negatives/
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# Mathematics without Negatives
The word “algebra” comes from the title of a book from around AD 800 by Muhammad Al-Khwarizmi. Despite this, the symbols that we associate with modern algebra (particularly, the use of single letter variable names) don’t appear in the book. Also, the conceptual field of mathematics called algebra came several centuries before: Al-Khwarizmi’s book is historically significant, but it built on previous work and the modern symbolism didn’t occur until long after.
One limitation of the book is that Al-Khwarizmi didn’t use negative numbers. This was typical of mathematicians of the era: Negative numbers were in use, but were heavily resisted by many.
He begins his book by showing three geometrical solutions to what we now call a quadratic equation, $$ax^2 + bx + c = 0$$. He needs three because the limitation to positive numbers means he can’t use negative coefficients. So, rather, he shows how to solve the following:
1. $$ax^2 + bx = c$$
2. $$ax^2 + c = bx$$
3. $$bx + c = ax^2$$
Likewise, he can’t solve for negative roots; the only time there are two solutions is when both solutions are positive.
This might seem odd to modern students, but it’s important to remember that he was providing geometric solutions. There are no negatives in geometry proper: All measurements are positive. From this standpoint, his approach makes perfect sense.
If you’d like to read more, my presentation of his first chapter is available on my other blog: First post; second post.
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2018-07-19 22:51:49
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https://math.stackexchange.com/questions/2878810/trying-to-understand-a-problem-that-look-related-to-implicite-function
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# Trying to understand a problem that look related to implicite function
I want to solve a system of equation \begin{align*} F_1(x_1,...,x_N; u_1,...,u_M)&=0\\ \vdots\\ F_M(x_1,...,x_N; u_1,...,u_M)&=0 \end{align*} where $F_i$ are $M$ function with $N+M$ variables and the desired solution is of the form $$u_i=f_i(x_1,...,x_N),\quad i=1,...,M.$$
Until here it's fine and it looks as a problem of implicite function. It's the next part that I don't understand.
A very important case is in the particular case where $M=N$ and $F(x;u)$ is of the form $\varphi(x)-u$ where I denote $x=(x_1,...,x_N)$, $F=(F_1,...,F_M)$ and $u=(u_1,...,u_M)$. And then, it's written : we are looking to solve $$\varphi(x)=u,$$ in the form of $x=\psi(u)$ so that $$\varphi(\psi(u))=u.$$
More explicitly, if $\varphi=(\varphi_1,...,\varphi_N)$ we want to solve the system $$\varphi_i(x_1,...,x_N)=u_i,\quad 1\leq i\leq N,$$ in the form $$x_i=\psi_i(u_1,...,u_N),\quad 1\leq i\leq N.$$
Question : In the first part we where looking for $u$ in the form of $u=f(x)$ s.t. $0=F(x,u)=F(x,f(x))$, and they say in the second part that if $$F(x,u)=\varphi(x)-u,\tag{*}$$ then we want to find $x$ in the form of $x=\psi(u)$ that solve $u=\varphi(x)$. I just don't get the thing. For $F(x;u)=\varphi(x)-u=0$, don't we simply have that $u=\varphi(x)$ solve the problem and thus $$0=F(x,u)=F(x,\varphi(x)).$$ Then we are done ! What's the matter with that ?
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2019-08-19 03:54:13
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https://math.stackexchange.com/questions/1133712/what-is-the-need-of-exponential-generating-functions-on-combinatorial-problems
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# What is the need of exponential generating functions on combinatorial problems?
I've been introduced in my last lectures. And for the following problem:
Having 3 different types of books $a,b,c$ in how many ways can we take four different books putting them in a shelf such that the book $a$ can only be taken at most $1$ time, $b$ can be taken at most $3$ times and $c$ can only be taken at most $2$ times?
The ordinary generating function for this problem is:
$\quad \quad \quad \quad \quad \quad \quad$
Looking at the exponential generating function of the same problem, the only part of interest is the coefficient of $x^4$, which is:
$$\left(\frac{ab^3}{1!3!}+\frac{b^3c}{3!1!}+\frac{ab^2c}{1!2!1!}+\frac{b^2c^2}{2!2!}+\frac{abc^2}{1!1!2!}\right)$$ And then multiply it by $4!/4!$. But why do I need an exponential generating function? Isn't only needed to take the coefficient of $x^4$, divide each term of it by $a_1!a_2!\dots a_n!$ with $a_n=\text{power of a,b,c,}\dots$ and then multiply it by $4!4!$? It's not clear why such complication is needed.
By complication I mean to expand that generating function with the factorials, I could expand the ordinary generating function and then artificially add the factorials.
The most blatant reason why exponential generating functions are useful (for infinite sequences) is that the ordinary power series might not converge. If $a_n = n!$, for example, the ordinary generating function does not exist as any kind of analytic object.
But the deeper reason why exponential generating functions are so common is that they have interesting product and composition formulas. If we have a sequence $c_n = \sum_{k=0}^n {n\choose k} a_k b_{n-k}$, then the exponential generating function for the $c_n$ is the product of the egf for $a_n$ and the egf for $b_n$. In other words, sequences defined by a sum of choices have a tendency to possess nice exponential generating functions.
For example, suppose that we want to compute the number of subsets of an $n$-element set. By the product formula, this has egf $e^x \cdot e^x = e^{2x}$, so the answer is $2^n$. To count the number of surjective functions from a set of $n$ elements to a set of $m$ elements, we can work with the egf $(e^x-1)^m$, and so on.
With a bit more finesse, we can make sense of the composition of exponential generating functions as well. For example, the Bell numbers $B_n$, which count partitions, have exponential generating function $e^{e^x - 1}$, which follows from very general theory. Try doing that with ordinary generating functions.
• So I suppose that the EGF exist as an analytic object (my professor told me that it's radius of convergence is bigger than the radius of convergence of OGF's), what does these analytic properties tell about the EGF? Does it always tell something fixed or it indicates something different on different applications? – Billy Rubina Feb 7 '15 at 1:41
• @Vÿska That's a very general question, but one concrete thing that a generating function gives you is asymptotic information about its associated sequence. The radius of convergence, for example, gives excellent first-order information about asymptotic growth, and one can even deduce things like Stirling's approximation with a more nuanced approach. – Slade Feb 7 '15 at 7:52
• @Slade: Thank you for your input. Could you please explain the reasoning behind using $$e^x . e^x$$ to calculate the number of subsets? – Vectorizer Jul 13 '15 at 14:39
• @Vectorizer I basically explained this in the previous paragraph. Intuitively, the product is counting the number of ways of coloring a set using two colors, while each $e^x$ is counting the number of ways of coloring a set with one color (namely, $1$). – Slade Jul 13 '15 at 22:23
• There is a nice example fooling around with the formal series $\sum_{n \ge 0} n! z^n$, which converges just for $z = 0$, so convergence isn't necessary. To get a "nice" function allows you to use calculus, which is a tremendous bonus, though. – vonbrand Aug 9 '15 at 2:03
Wilf's book "generatingfunctionology" explains why exponential generating functions are useful. Another view, much more mathematically demanding, is explained by Flajolet and Sedgewick's "Analytic Combinatorics", the approach is summarized by Wikipedia.
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2019-06-27 08:25:21
|
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https://search.r-project.org/CRAN/refmans/bayesmove/html/segment_behavior.html
|
segment_behavior {bayesmove} R Documentation
## Segmentation model to estimate breakpoints
### Description
This function performs the reversible-jump MCMC algorithm using a Gibbs sampler, which estimates the breakpoints of the movement variables for each of the animal IDs. This is the first stage of the two-stage Bayesian model that estimates proportions of behavioral states by first segmenting individual tracks into relatively homogeneous segments of movement.
### Usage
segment_behavior(
data,
ngibbs,
nbins,
alpha,
breakpt = purrr::map(names(data), ~NULL)
)
### Arguments
data A list where each element stores the data for a separate animal ID. List elements are data frames that only contain columns for the animal ID and for each of the discretized movement variables. ngibbs numeric. The total number of iterations of the MCMC chain. nbins numeric. A vector of the number of bins used to discretize each movement variable. These must be in the same order as the columns within data. alpha numeric. A single value used to specify the hyperparameter for the prior distribution. A standard value for alpha is typically 1, which corresponds with a vague prior on the Dirichlet distribution. breakpt A list where each element stores a vector of breakpoints if pre-specifying where they may occur for each animal ID. By default this is set to NULL.
### Details
This model is run in parallel using the future package. To ensure that the model is run in parallel, the plan must be used with future::multisession as the argument for most operating systems. Otherwise, model will run sequentially by default if this is not set before running segment_behavior.
### Value
A list of model results is returned where elements include the breakpoints, number of breakpoints, and log marginal likelihood at each iteration of the MCMC chain for all animal IDs. The time it took the model to finish running for each animal ID are also stored and returned.
### Examples
data(tracks.list)
#subset only first track
tracks.list<- tracks.list[1]
#only retain id and discretized step length (SL) and turning angle (TA) columns
tracks.list2<- purrr::map(tracks.list,
subset,
select = c(id, SL, TA))
set.seed(1)
# Define model params
alpha<- 1
ngibbs<- 1000
nbins<- c(5,8)
future::plan(future::multisession, workers = 3) #run all MCMC chains in parallel
dat.res<- segment_behavior(data = tracks.list2, ngibbs = ngibbs, nbins = nbins,
alpha = alpha)
|
2022-12-02 02:25:25
|
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https://mathshistory.st-andrews.ac.uk/Biographies/Brun/
|
# Viggo Brun
### Quick Info
Born
13 October 1885
Lier, Hordaland County, Norway
Died
15 August 1978
Drobak, Akershus County, Norway
Summary
Viggo Brun was a Norwegian mathematician and number theorist.
### Biography
Viggo Brun's father was Soren Markus Brun (1838-1893), an artillery captain, and his mother was Lorentze Thaulow Petersen (1842-1890). Now, from the dates we have just given we can see that Viggo's parents died when he was young. In fact his mother died when he was four and his father died on 4 March 1893 when he was seven. However, Viggo was the youngest of his parents' ten children and he had older sisters who brought him up. He entered the University of Oslo in 1903 where he studied mathematics and natural sciences. The course he took was to qualify him to become a school teacher and it was a broad course requiring him to study a wide range of topics. It did not allow time for much in the way of specialised knowledge. Many bright pupils in Brun's position would have read mathematics books to take them into deeper studies than those being presented in their courses. Brun's approach was, however, different and he tried to develop his own mathematical ideas without having the support of teachers or advanced texts and, as a consequence, he produced some rather original ideas while still an undergraduate.
In 1910 Brun went to Göttingen University in Germany, the leading mathematics centre in the world at this time, and while he was there he began to work on what were some of the most difficult problems in number theory. We should note that Brun received no financial support for his visit to Göttingen and he funded the visit entirely from his own funds. The number theorist Edmund Landau had been appointed to a professorship at Göttingen a year before Brun arrived and Hilbert and Klein were also on the staff. There is no evidence that Brun interacted in any meaningful way with any of these, but he must have benefited from listening to Edmund Landau. Returning to Norway, Brun did receive a research grant to support his work but at this stage he had no job. The outbreak of World War I in 1914 saw Norway adopt a position of neutrality. This, however, proved difficult to maintain and the country was under pressure from both sides. Brun served for a number of years in the Norwegian armed forces.
He had attacked two of the most famous number theory problems, namely Goldbach's conjecture and the twin prime conjecture. Goldbach's conjecture is that every even natural number greater than 2 can be expressed as the sum of two primes. For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5, ... Note that, in general, even numbers will be expressible as the sum of two primes in more than one way, for example 10 = 5 + 5 = 3 + 7. The twin prime conjecture is that there are infinitely many prime pairs $n, n + 2$. For example, 3, 5; 5, 7; 11, 13; 17, 19; 29, 31; ... Brun's first results were given in Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare (1915). In this paper he began to develop the sieve methods which would lead him to some very important results. These sieve methods were refinements, based on the inclusion-exclusion principle, of the sieve of Eratosthenes. They are essentially elementary in nature and few believed that they would lead to significant results. However, the ideas that Brun introduced in this paper, further developed by him and later by others, would lead to a revolution in number theory. In 1919 Brun published a remarkable result when he proved that the sum of the reciprocals of twin primes is finite, that is
$\large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize + \large\frac{1}{5}\normalsize + \large\frac{1}{7}\normalsize + \large\frac{1}{11}\normalsize + \large\frac{1}{13}\normalsize + \large\frac{1}{17}\normalsize + \large\frac{1}{19}\normalsize + \large\frac{1}{29}\normalsize + \large\frac{1}{31}\normalsize + ...$
is finite. He proved this in the paper La série $\large\frac{1}{5}\normalsize + \large\frac{1}{7}\normalsize +\large\frac{1}{11}\normalsize + \large\frac{1}{13}\normalsize + \large\frac{1}{17}\normalsize + \large\frac{1}{19}\normalsize + \large\frac{1}{29}\normalsize + \large\frac{1}{31}\normalsize + \large\frac{1}{41}\normalsize + \large\frac{1}{43}\normalsize + \large\frac{1}{59}\normalsize + \large\frac{1}{61}\normalsize + . . .$où les dénominateurs sont "nombres premiers jumeaux" est convergent ou finie (1919) which he published in the Bulletin des Sciences Mathématiques. The proof uses what today is called 'Brun's sieve'. Note that this result is in sharp contrast with the theorem that the sum of the reciprocals of the primes is infinite, first proved by Leonhard Euler in 1737. Now the first question one would naturally ask is "What is the approximate value of the sum of the reciprocals of twin primes"? Unfortunately there is no accepted standard for exactly what series is being summed. The most common definition of 'Brun's constant' is:
$B$ = $\large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize + \large\frac{1}{5}\normalsize + \large\frac{1}{7}\normalsize + \large\frac{1}{11}\normalsize + \large\frac{1}{13}\normalsize + \large\frac{1}{17}\normalsize + \large\frac{1}{19}\normalsize + \large\frac{1}{29}\normalsize + \large\frac{1}{31}\normalsize + ...$
and the best estimate for $B$, produced in 2002, is 1.902160583104.
However, as we can see from the title of Brun's paper, he took a slightly different start to his series
$\large\frac{1}{7}\normalsize + \large\frac{1}{11}\normalsize + \large\frac{1}{13}\normalsize + \large\frac{1}{17}\normalsize + \large\frac{1}{19}\normalsize + \large\frac{1}{29}\normalsize + \large\frac{1}{31}\normalsize + ...$
while others take
$\large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize + \large\frac{1}{7}\normalsize + \large\frac{1}{11}\normalsize + \large\frac{1}{13}\normalsize + \large\frac{1}{17}\normalsize + \large\frac{1}{19}\normalsize + \large\frac{1}{29}\normalsize + \large\frac{1}{31}\normalsize + ...$
omitting one $\large\frac{1}{5}\normalsize$ since 5 appears in two prime pairs. Of course, the sums of these variants are easily related to $B$. Note that it is still an open question whether $B$ is rational or irrational. It is known, however, that if $B$ were proved irrational it would follow that there are infinitely many twin primes. If $B$ were proved rational it would have no bearing on whether there were infinitely many twin primes. Isn't a study of the primes a most fascinating topic!
In 1920 Brun published Le crible de Eratosthène et le théorème de Goldbach in which he used his sieve methods to prove weaker forms of both the Goldbach conjecture and the twin prime conjecture. In this paper he proved (i) there exist infinitely many integers $n$ such that both $n$ and $n + 2$ have at most nine prime factors; and (ii) every sufficiently large even integer $N$ is the sum of two numbers each having at most nine prime factors. Later mathematicians have strengthened these results, using methods based on those first developed by Brun, but the two conjectures are still open.
Let us return to Brun's career. He was appointed as an assistant in applied mathematics in Oslo in 1921. Two years later, he was appointed as a professor at the Technical University of Trondheim. He married Laura Elise Michelsen (1902-2004) in 1940. Laura, who was born in Frogn, Akershus, Norway, to parents Hjalmar Fredrik Bernhard Michelsen and Eva Gloersen, ran a dressmaking school in Oslo. We note that her home town of Frogn was adjacent to Drobak where Brun's father had bought an old house. Viggo and his wife Laura [3]:-
... lived in an ancient wooden house in Drobak, which had been built in 1770 and which his father had bought in 1888. In recent years he and his wife were active in helping to preserve those old houses in Drobak. He loved to go for walks in the forest where he would collect curious pieces of wood or roots from which he produced nice handiwork.
In 1946 Brun was appointed to a chair at the University of Oslo which he occupied for nine years until he retired in 1955 at the age of seventy.
Brun published two books after he retired, namely The Art of Calculating in Old Norway until the Time of Abel (Norwegian) (1962) and All is Number, a History of Mathematics from Antiquity to the Renaissance (Norwegian) (1964). Øystein Ore writes [1]:-
Viggo Brun has in recent years written two books on the history of mathematics, one dealing with the early history in Norway, the other with the general story of the subject. Both are on a very elementary level, but with many interesting observations by the author.
Brun explains in his Preface that, before Niels Abel, Norwegian mathematics:-
... was in most cases an art of calculating rather than mathematics. No more than four of the scholars I mention could lay claim to the title of mathematician.
These four are Fredrich Christian Holberg Arentz (1736-1825), Diderich Christian Fester (1732-1811), Jens Kraft (1720-1756) and Caspar Wessel (1745-1818).
Christoph J Scriba, writes in a review that:-
... the author gives a summary of the knowledge in elementary mathematics in old Norway and Iceland. Apart from archaeological finds bearing geometric designs, the oldest sources revealing knowledge in the art of calculating are contained in the Gulating Law, the Frostating Law and the Kongespeilet [The royal mirror, c. 1260], from the 12th and 13th centuries. Hauk Erlendsson, who lived in Norway and Iceland around 1300, composed an "Algorismus" mainly based on that of Sacrobosco. The Icelandic arithmetic "Rymbegla" is briefly discussed. There is also an account of the teaching of arithmetic in schools, the study of Norwegian students abroad in the Middle Ages and later on, and the establishment of high and technical schools in Denmark and Norway in the 17th and 18th centuries.
Christoph J Scriba also reviewed Brun's All is Number, writing:-
This paperback in the Norwegian language contains the substance of lectures on the history of mathematics which have been given at the University of Oslo. A summary of pre-historic, Egyptian, Babylonian, Hindu and Arabic mathematics fills the first 80 pages, while on the next 100 pages the life and work of about two dozen Greek mathematicians is described. Of these, the great Archimedes receives the fullest treatment (34 pages) - the same number of pages in which the author deals with medieval and Renaissance mathematics in Europe, from Boethius to Cardano and Ferrari. The author has resisted the temptation to crowd too many facts into the limited space. His presentation of mathematical details is intermingled with information of a more general nature concerning mathematics and mathematicians of the times under discussion. Some items treated by him are not contained in the standard books: The Norwegian "Kongespeilet", perhaps composed by Archbishop Einar Gunnarsson, who may have been in personal contact with Sacrobosco at Paris; the "Algorismus" of Hauk Erlendsson [early 14th century], which is strongly influenced by Sacrobosco, too; and the Icelandic arithmetic "Rymbegla" [12th -14th centuries?].
In addition to these books on the history of mathematics, Brun wrote many papers on the subject. For example, he wrote the following in Norwegian: Quadrature of the circle (1941); The study of the prime numbers from antiquity to our time (1942); Wallis's and Brouncker's formulas for π (1951); Niels Henrik Abel (1953); The manuscript of Abel's Paris treatise found (1953); (with Borge Jessen) A letter by Niels Henrick Abel from his youth (1958). He also published biographies of Carl Stormer (1957), Caspar Wessel (1959), Sophus Lie (1967), and Axel Thue (1977). Another topic that interested Brun was the theory of music. In Music and ternary continued fractions (1950) he discussed the division of the octave into equally tempered intervals. The paper Music and Euclidean algorithms (Norwegian) (1961) looks at four different problems from the theory of music. One might imagine that Brun was musical but [3]:-
Brun himself called it an irony of fate that he, who was very unmusical, should write about music. Though not musically gifted, Viggo Brun had a strong sense for harmony and geometric symmetry. He gave some lectures about mathematics and aesthetics ...
Christoph Scriba met Brun in 1955 at a conference on the history of mathematics at the Oberwolfach mathematics research centre in the Black Forest, Germany. Scriba says of Brun [3]:-
His wide interests and his humanistic outlook, combined with his modesty, kindness, and totally un-professorial habits made the strongest impression upon me. ... During his long and fruitful life, he also engaged in politics and worked for peace ...
Many honours were given to Brun for his outstanding contributions. He was awarded the Fridtjof Nansen Award for Excellence in 1939. This award, named after the Norwegian explorer, scientist, and diplomat Fridtjof Nansen, had been awarded since 1903 and the previous winner, in 1938, had been Thoralf Albert Skolem. In 1946 Brun was awarded the Norwegian Institute of Technology Founder's Prize, and in 1958 he received the Gunnerus medal of the Royal Norwegian Academy of Science and Letters. This medal is named after Johan Ernst Gunnerus, the founder of the Royal Norwegian Academy. Brun also received an honorary doctorate from the University of Hamburg in 1966. He was elected a member of the scientific societies or academies of Oslo, Trondheim, Uppsala, and the Finnish Academy of Sciences.
### References (show)
1. O Ore, Review: The Art of Calculating in Old Norway until the Time of Abel, by Viggo Brun; and All is Number, a History of Mathematics from Antiquity to the Renaissance, by Viggo Brun, Amer. Math. Monthly 75 (1) (1968), 100.
2. K Ramachandra, Viggo Brun (13.10.1885 to 15.8.1978), Math. Student 49 (1) (1981), 87-95.
3. C J Scriba, Viggo Brun in memoriam (1885-1978), Historia Mathematica 7 (1) (1980) 1-6.
4. C J Scriba, Zur Erinnerung an Viggo Brun, Mitt. Math. Ges. Hamburg 11 (3) (1985), 271-290.
5. S Selberg, Viggo Brun in memoriam (Norwegian), Normat No. 1 (1979), 3-9, 48.
6. S Selberg, Viggo Brun (Norwegian), Norske Vid. Selsk. Forh. (Trondheim) (1979), 57-61.
7. R Taton, Viggo Brun (1885-1978) (French), Rev. Histoire Sci. Appl. 33 (3) (1980), 253-254.
8. H Waadeland, Viggo Brun: 'La difference entre le nombre de nombres premiers des formes 4h + 3 et 4h + 1, exprimés par une formule exacte', m.fl. (Norwegian), Skr. K. Nor. Vidensk. Selsk. (4) (2011), 113-120.
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2021-07-24 18:28:38
|
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https://mathoverflow.net/questions/290870/almost-sure-stability-of-a-scalar-nonautonomous-nonlinear-sde
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# Almost sure stability of a scalar, nonautonomous, nonlinear SDE
I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it. Maybe there is someone here who can either close the gap in one of the existing answers or has an independent answer.
Let the following stochastic system be given
$$dX_t=-X_t \, dt+dW_t,$$ $$dY_t=X_tY_t(1-Y_t)(dt+dV_t),$$
where $W_t$ and $V_t$ are independent Wiener processes, $X_0\sim\mathcal{N}(0,1/2)$ (the stationary measure), and $Y_0\in[0,1]$.
I would like to show that $\lim_{t\to\infty}Y_t\in\{0,1\}$ almost surely, or at least that the invariant probability measure for $Y$ is of the form $\alpha\delta_0+(1-\alpha)\delta_1$ for some $\alpha\in[0,1]$.
Given a realization of the Ornstein-Uhlenbeck process $X_t$, the SDE $$d Y_t = Y_t (1- Y_t) X_t (dt + d V_t) \tag{1}$$ is scalar, nonautonomous, and nonlinear. Note that (1) has two fixed points at $0$ and $1$, which are asymptotically stable in the following sense.
Theorem. For almost all $Y_0 \in (0,1)$, we have $\lim_{t \to \infty} Y_t \in \{0, 1\}$ almost surely.
Intuitive Explanation
This theorem is plausible for the following reason: when $Y_0$ is close to zero, $Y_t$ behaves like the process $\tilde Y_t$ which satisfies the linear SDE: $$d \tilde Y_t = X_t \tilde Y_t dt + X_t \tilde Y_t d V_t \;.$$ This linear SDE has the pathwise solution: $$\tilde Y_t = Y_0 e^{ \int_0^t X_s dV_s + \int_0^t ( X_s - \frac{1}{2} X_s^2) ds } \;.$$ By the strong law of large numbers, $$\lim_{t \to \infty} \frac{1}{t} \left( \int_0^t X_s dV_s + \int_0^t ( X_s - \frac{1}{2} X_s^2) ds\right) = - \frac{1}{4} \quad \text{a.s.} \tag{\star}$$ It follows that $\tilde Y_t \to 0$ as $t \to \infty$ almost surely. A similar argument holds if $Y_0$ is close to one.
Proof
A key tool in this proof is the function $f(y) = \log(y) - \log( 1-y)$ which bijectively maps the unit interval $(0,1)$ to $\mathbb{R}$ by mapping $0$ to $-\infty$, $1$ to $+\infty$, and $1/2$ to $0$.
By Itô's Lemma, a.s., for all $t \ge 0$, $$d f(Y_t) = f'(Y_t) Y_t (1-Y_t) X_t (dt + d V_t) + \frac{1}{2} f''(Y_t) Y_t^2 (1-Y_t)^2 X_t^2 dt \;.$$ Since $$f'(y) y (1-y) = 1 \quad \text{and} \quad \frac{1}{2} f''(y) y^2 (1-y)^2 = -\frac{1}{2} + y$$ we obtain the following SDE for $Z_t = f(Y_t)$ $$d Z_t = \frac{1}{2} \tanh\left( \frac{Z_t}{2} \right) X_t^2 dt + X_t ( dt + dV_t ) \;. \tag{2}$$
Note from (2), and the fact that $-1 < \tanh(x) < 1$ for all $x \in \mathbb{R}$, $$\int_0^T \left( - \frac{1}{2} X_t^2 dt + X_t dt + X_t d V_t\right) < Z_T - Z_0 < \int_0^T \left( \frac{1}{2} X_t^2 dt + X_t dt + X_t d V_t\right)$$ and hence, a.s., $$-\frac{1}{4} < \lim_{T \to \infty} \frac{Z_T}{T} < \frac{1}{4} . \tag{3}$$ Note that (3) limits how fast $Z_T$ can diverge.
Let $z^{(0)} < z^{(1)}$ be two different initial conditions for (2), and let $Z^{(0)}_t$ and $Z^{(1)}_t$ be the corresponding paths emanating from these initial conditions which satisfy: \begin{align*} Z^{(0)}_T &= z^{(0)} + \int_0^T \left\{ \frac{1}{2} \tanh\left(\frac{Z_t^{(0)}}{2}\right) X_t^2 dt + X_t ( dt + dV_t ) \right\} \;, \\ Z^{(1)}_T &= z^{(1)} + \int_0^T \left\{ \frac{1}{2} \tanh\left(\frac{Z_t^{(1)}}{2}\right) X_t^2 dt + X_t ( dt + dV_t ) \right\} \;. \\ \end{align*} We stress that these paths are driven by the same realization of Brownian motion $V_t$ and Ornstein-Uhlenbeck process $X_t$. Hence, a.s., $$Z^{(1)}_T - Z^{(0)}_T = z^{(1)} - z^{(0)} + \frac{1}{2} \int_0^T \left\{\tanh\left(\frac{Z_t^{(1)}}{2}\right) - \tanh\left(\frac{Z_t^{(0)}}{2}\right) \right\} X_t^2 dt \;.$$ Since $\tanh$ is increasing, the difference $Z^{(1)}_T - Z^{(0)}_T$ itself is increasing and \begin{align*} \lim_{T \to \infty} \frac{Z^{(1)}_T - Z^{(0)}_T}{T} &= \lim_{T \to \infty} \frac{1}{T} \int_0^T \frac{1}{2} \left\{ \tanh\left(\frac{Z_t^{(1)}}{2}\right) - \tanh\left(\frac{Z_t^{(0)}}{2}\right) \right\} X_t^2 dt \\ &\ge \frac{1}{2} \left\{ \tanh\left(\frac{z^{(1)}}{2}\right) - \tanh\left(\frac{z^{(0)}}{2}\right) \right\} \lim_{T \to \infty} \frac{1}{T} \int_0^T X_t^2 dt \\ &\ge \frac{1}{4} \left\{ \tanh\left(\frac{z^{(1)}}{2}\right) - \tanh\left(\frac{z^{(0)}}{2}\right) \right\} >0 \tag{4} \end{align*} In other words, the difference $Z^{(1)}_T - Z^{(0)}_T$ a.s. diverges as $T \to \infty$.
Now suppose that $$\lim_{T \to \infty} \frac{Z^{(0)}_T}{T} = 0 \;.$$ This can happen if, e.g., the realization $Z^{(0)}_T$ asymptotes to a finite value or diverges at a sublinear rate. However, this can happen at most once, since for any $z^{(\star)} \ne z^{(0)}$ the previous result in (4) implies that $$\begin{cases} \lim_{T \to \infty} \frac{Z^{(\star)}_T}{T} < 0 ~~\text{if z^{(\star)} < z^{(0)} }\\ \lim_{T \to \infty} \frac{Z^{(\star)}_T}{T} > 0 ~~\text{if z^{(\star)} > z^{(0)} } \end{cases}$$ where $Z^{(\star)}_T$ is the realization with initial condition $z^{(\star)}$. In other words, realizations corresponding to initial conditions: (i) to the left of $z^{(0)}$ diverge to $-\infty$; and (ii) to the right of $z^{(0)}$ diverge to $+\infty$. Hence, there can be at most one initial condition $z^{(0)}$ such that $\lim_{T \to \infty} \frac{Z^{(0)}_T}{T} = 0$ -- otherwise one gets the contradiction that some realizations diverge to $\pm \infty$ simultaneously.
In the original variables, this implies that: for all, but at most one initial condition, we have $\lim_{t \to \infty} Y_t \in \{0,1 \}$ almost surely -- as required.
• Thanks! Could you spell out the strong approximation property for me? Does it come with any bounds on $Y_t-\tilde Y_t$? Jan 17 '18 at 7:53
• Thanks! This looks great! I will have a deeper look soon. From a brief look, it seems that one can even obtain exponential convergence rate of $Y_t$ from your argument by using the fact that $Z_t$ diverges to either extreme like $t$. Jan 19 '18 at 23:06
• No worries. Thanks also for giving a more appropriate title. I actually asked a generalization of this question earlier, if you are interested: mathoverflow.net/questions/287418/…. I would like to think that one could define a stochastic Lyapunov function as if the whole system had a stable fixpoint. But maybe one can instead suitably generalize the process $\tilde Z_t$. Jan 19 '18 at 23:42
• @S.Surace I revised the proof according to your feedback. Jan 27 '18 at 15:53
• This is very elegant and also preserves the symmetry of the problem! I learned a lot from this! Thank you! Jan 27 '18 at 18:11
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2022-01-24 07:41:15
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https://electronics.stackexchange.com/questions/123632/attiny-external-clock-not-working
|
# ATtiny External Clock not working
So, I've been wanting to break away from the arduino abstraction for a while now. I made a board that has an ATtiny10 on it with a crystal and an output. I cannot for the life of me understand what I am doing wrong.
Here's the problem: When I select the clock source, the AVR stops working.
I had a custom PCB (small one) made to mount everything on. Thinking I didn't do something right (even though schematic looked right)
I changed design and made another one (for the second one, I had traces with clock surrounded by a ground plane (Did not work still) [Also, the first design was using everything but the AVR sourced from ebay. Thinking maybe it was a quality issue the second board is sourced entirely from Mouser]
Lastly, to make sure I wasn't an absolute idiot, I bought a breakout board and just breadboarded the circuit. This still performed just like all the others
It works just fine with the internal oscillator, but as soon as I program to change clock source it stops.
Note that I added R2 to keep the MOSFET pulled down but I do not populate it for programming it, as TPI (Tiny Programming Interface) uses pullups on that line and I cannot populate that if I am going to program the chip
Page 21 of the datasheet regarding changing clock
Page 22 of the datasheet regarding the prescaler (Clock prescaler not timer prescaler)
I am using a MkII programmer from Atmel and these are the fuse bits:
Output External Clock
0xFB
This was set through Atmel studio and I have used a couple different ATtiny's and the chips kept these settings once set, so I am pretty sure the fuses are writing correctly
Now the code: (This is the whole program, the timer portion works (obviously not though once the clock switches))
#define F_CPU 8000000
#include <avr/io.h>
#include <avr/interrupt.h>
void initClock()
{
// Setting CLKPSR does not affect the problem (It doesn't work regardless of what this is set to)
// I have tried this before and after setting CLKMSR
CCP = 0xD8;
CLKPSR = 0;
CCP = 0xD8;
CLKMSR = 0b10;
}
void initPorts()
{
DDRB |= (1 << PORTB0); // PB0 = OCR0A
}
void initTimer()
{
// I posted this code just in case, this works as expected (but only on the internal oscillator)
// We want Compare Output Mode, Clear OC0A on Compare Match
TCCR0A = (1 << COM0A0);
// Overflow setting
TIMSK0 |= (1 << OCIE0A);
// We will not use a prescaler
// This also starts the timer
TCCR0B = (1 << CS00) | (1 << WGM02);
// This is the value at which the timer will restart
OCR0A = 8299;
// Set external interrupts
sei();
}
int main(void)
{
initClock();
initPorts();
initTimer();
while(1)
{
}
}
Surely there has to be something I am missing. I've tried to read and reread the especially the clock sections of the datasheet in order to figure it out myself. I am stumped though. Maybe someone could help me understand my mistake.
Attiny10 Datasheet
• try to write a 0 to CLKM0, if it's a 1 you end up with the 'reserved' combination and who knows what can happen. Aug 1, 2014 at 7:55
• @VladimirCravero I set that bit to 0 and it does not start the external clock
– Dan
Aug 1, 2014 at 7:57
• That bit is zero anyway on reset, but I can't see where in your code you set it to zero. Aug 1, 2014 at 7:59
• @VladimirCravero sorry, I meant I did change that after you asked. I will edit to reflect the change. Sorry for the misunderstanding
– Dan
Aug 1, 2014 at 8:01
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2023-03-31 02:30:26
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https://www.shaalaa.com/question-bank-solutions/basic-concepts-trigonometric-functions-show-that-2sin-1-3-5-tan-1-24-7_1051
|
# Solution - Show That: 2sin^-1(3/5)=tan^-1(24/7) - Basic Concepts of Trigonometric Functions
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#### Question
Show that: 2sin^-1(3/5)=tan^-1(24/7)
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Solution for question: Show That: 2sin^-1(3/5)=tan^-1(24/7) concept: Basic Concepts of Trigonometric Functions. For the courses HSC Arts, HSC Science (Computer Science), HSC Science (Electronics), HSC Science (General)
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2017-12-18 03:18:51
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https://gre.kmf.com/question/rc/0?keyword=&page=265
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#### 题目列表
In the context in which it appears, "appreciation of" most nearly means
Federal courts in the United States, especially before the famous 1962 case of Baker v. Carr, were often thought to be powerless in the area of election law, voting rights, and other legal questions clearly bearing on politics. This perception was not entirely correct, of course, as pre-1962 Supreme Court decisions such as that in the case of Smith v. Allwright demonstrated in the wake of that decision, voting participation among African Americans in the South increased substantially. However, political rights had not always been so clearly championed by the Supreme Court as they were in Smith v. Allwright. Indeed, the transformations between the Civil War and 1962 were such that, in reviewing voters rights cases over the intervening decades, one feels like an archaeologist cutting through distinct layers in which the judicial decisions uncovered reveal a pattern of ideological and societal change.
The author of the passage uses the analogy of the archaeology most probably in order to
Which of the following can be inferred regarding the case of Baker v. Carr?
Tropical forests typically have many more species of plants and animals than do temperate forests of comparable size. During the Ice Age, forests in temperate regions were destroyed, while those in the tropics were not. Accordingly, one proposed explanation of this difference in the number of species is that tropical forests typically had a much longer period than temperate forests in which different species could take hold.
Which of the following, if true, most strongly supports the proposed explanation?
Some studies have shown that red-backed salamanders (RBS) are scarce in areas with acidic soils and that those present in such conditions have smaller-than-average bodies. Explanations have included the possibility that young RBS are adversely affected by acidic soil, that adult RBS can sense and may avoid acidic soil conditions, or that loss of RBS prey populations due to acidic soil could result in reduced RBS populations. Yet researchers found fairly high densities of large-bodied RBS at Lake Claire Watershed, where soil conditions are acidic. One hypothesis is that intraspecific geographical variation in acidity tolerance (i.e., local adaptation to an acidic environment) could exist for RBS. Previous studies showed potential local adaptation of some salamander species to acidity.
Which of the following can be inferred about the studies mentioned in the highlighted portion of the passage?
The primary purpose of the passage is to
United State women won the vote in 1920 after decades of campaigning. Yet, the impact on womens status was more limited than womens rights activists had anticipated. Women were granted suffrage at a historical point when voting was no longer a significant political activity for many Americans. In the mid-nineteenth century, when women first sought suffrage rights, voter turnout rates were unprecedentedly high, elections in much of the country very competitive, and political parties important. But when women finally received the vote in 1920, electoral politics was largely noncompetitive, with virtual one-party rule in many areas, and voter turnout had slipped to its all-time low. Nonetheless, the vote still mattered enough for women to seek it and for conservatives to try to restrict its availability.
The author of the passage discusses voter turnout rates primarily in order to
The author of the passage mentions conservatives in the highlighted sentence primarily in order to
Unlike the static, classically composed portraits produced by her mentor Walker Evans, twentieth-century New York photographer Helen Levitts photographs seem candid and spontaneous. Whereas Evans subjects look directly into the camera, so that photographer and subject conspire in the making of a portrait, Levitts subjects seem caught unawares. As a "street" photographer, before the terms invention, Levitt has claimed to have attempted to capture life as she found it. But there is a paradox to her technique. Her off-the-cuff aesthetic seemingly guarantees objectivity, since she was recording street scenes she happened upon, yet her photographs could be said to be highly subjective, to be reflections of Levitts own distinctive preoccupations and ways of seeing. Unlike Evans images, Levitts are solely the products of the photographer without the conscious participation of their subjects. The repetitions evident in Levitts choices of subjects, for example, her many photographs of children in masks and disguises, reveal more about Levitt herself than about those subjects.
According to the passage, which of the following appears to ensure the objectivity of Levitts photographs?
The passage asserts which of the following about Evans` portrait photographs?
The passage suggests which of the following about street photography?
The expectation that science is a stable body of relatively objective knowledge on which the law can draw to settle legal controversies may seem benign. However, this expectation often corresponds to a romantic notion of the scientific enterprise and thereby eclipses not only the instabilities and controversies within science itself, but also the social and rhetorical aspects of even the best science. We see the idealization of science in law whenever there is a presumption that if two scientific experts disagree, one of them must be a "junk scientist". This presumption ignores the theoretical presuppositions and limitations of data that lead to genuine scientific disputes. We also see the idealization of science in law whenever we associate "bias, interest, and motivation" with unreliable expertise. This association missed the practical advances made by scientists who have strong theoretical biases, institutional interests, and financial motivations. Finally, we see the idealization of science in law whenever a legislator, administrator, or judge demands certainty from science, not recognizing its probabilistic nature and dynamic history. It is neither a critique of scientific progress nor an exaggeration to acknowledge scientific debates, the conventional aspects of scientific methodology, the importance of networking and "social capital" with respect to publications and grants, and the persuasive elements in scientific discourse. To think that these features are somehow markers of bad science is to idealize science.
The primary purpose of the passage is to
The author suggests that which of the following can lead to the dismissal of a scientific expert as a junk scientist?
The author mentions "scientists who have strong theoretical biases, institutional interests, and financial motivations" primarily in order to
25000 +道题目
8本备考书籍
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2021-10-20 11:03:05
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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": 0.1728234887123108, "perplexity": 5155.492520233279}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2021-43/segments/1634323585305.53/warc/CC-MAIN-20211020090145-20211020120145-00341.warc.gz"}
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https://timvw.be/2008/01/25/programming-the-csproj-file/
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Imagine that you have a couple of project files that reference framework libraries that are on a buildserver. Upgrading to a newer version requires that you update all the references… I wrote some wrapper classes (ProjectFile, AssemblyReference) that make this tedious task a breeze. Here is an example of their usage
string path = "D:\\Projects\\MyProject";
string[] projectFileNames = ProjectFile.Find(path);
foreach (string projectFileName in projectFileNames)
{
ProjectFile projectFile = new ProjectFile(projectFileName);
bool updated = false;
foreach (AssemblyReference assemblyReference in projectFile.AssemblyReferences)
{
if (assemblyReference.HintPath.ToLower().StartsWith("\\\\buildserver\\framework\\2.0"))
{
string newHintPath = assemblyReference.HintPath.Replace("\\2.0\\", "\\2.1\\");
assemblyReference.HintPath = newHintPath;
AssemblyName assemblyName = AssemblyName.GetAssemblyName(assemblyReference.HintPath);
assemblyReference.AssemblyName = assemblyName.FullName + ", processorArchitecture=" + assemblyName.ProcessorArchitecture;
updated = true;
}
}
if (updated)
{
projectFile.Save();
}
}
Edit: Be careful because it might be possible that the code changes the encoding of your csproj file (and it seems that the TFS 2005 merge tool doesn’t like that). Currently files are written as UTF-8, which is the default for VS2005 csproj files.
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2019-02-16 01:16:42
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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": 0.2537533938884735, "perplexity": 9728.809845897342}, "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-2019-09/segments/1550247479729.27/warc/CC-MAIN-20190216004609-20190216030609-00483.warc.gz"}
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http://origostudio.hu/t3ylzw/23a283-limit-of-identity-function-example
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> Don't mix test and production code in the same function app. Then . . , then When our prediction is consistent and improves the closer we look, we feel confident in it. Perhaps we should take a closer look at the graph near the origin. Step 1: Repeat the steps as above, but this time solve for the limit as x approaches infinity. How to evaluate this limit of irrational function? . . A Gaussian function – graphed in Figure 20.9 in the margin – is the identity function for the Fourier transform: It has the unique property of transforming to itself (within a scale factor). All linear functions are combinations of the identity function and two constant functions. [3.1] is classified as a fundamental trigonometric limit. A composition of two identity functions is also an identity function. We all know about functions, A function is a rule that assigns to each element xfrom a set known as the “domain” a single element yfrom a set known as the “range“. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … In this case the function that we’ve got is simply “nice enough” so that what is happening around the point is exactly the same as what is happening at the point. Functions within a function app share resources. . Example: . When a function has this property, it is called a "continuous" function. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. For example, given the function f (x) = 3x, you could say, “The limit of f (x) as x approaches 2 is 6.” Symbolically, this is written f (x) = 6. Limit of a Linear Function. In Example $$\PageIndex{8B}$$ we look at one-sided limits of a piecewise-defined function and use these limits to draw a conclusion about a two-sided limit of the same function. This article explores the Identity function in SQL Server with examples and differences between these functions. In our example, there are two elementary functions that can be used to squeeze We have to be careful that we don't end up taking a square-root of a negative number though! Suppose that we consider Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity. What do you mean by "two identity functions"? The Gaussian function has moderate spread both in the time domain and in the frequency domain; it has infinite extent, but becomes negligibly small more than a few units from the origin. Calculating the limit at 0 of a function. . The scope can be a stored procedure, a function, a trigger or a batch of queries. . Let be a constant. Learn. The limits problems are often appeared with trigonometric functions. Let's consider the situation visually. For example, take the function f (x) = x + 4. So the limit will be $f(a)$ as $x \rightarrow a$? Why did Churchill become the PM of Britain during WWII instead of Lord Halifax? Limits are the most fundamental ingredient of calculus. Since P.J. How can we prove that is near . The limit of detection (LOD) and limit of quantitation (LOQ) for each TDM assay must be defined. For example a limit of a function for a given element of domain where both domain and codomain have some measure you'll likely go with the $\epsilon - \delta$ definition while if you're talking about a limit of an infinite sequence you need to have the sequence definition. Note that g (a) = 0 g(a)=0 g (a) = 0 is a more difficult case; see the Indeterminate Forms wiki for further discussion. Limits of Piecewise Defined Functions via One-Sided Limits. All linear functions are combinations of the identity function and two constant functions. This is the currently selected item. Eventually we will formalize up just what is meant by “nice enough”. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The limit in Eq. is in the domain of In this case the function that we’ve got is simply “nice enough” so that what is happening around the point is exactly the same as what is happening at the point. @TrevorWilson $x$ is the identity function, and $x \times x$ is two identity functions. Practice: Limits using trig identities. both exist. Here also more examples of trigonometric limits. . This is an example of continuity, or what is sometimes called limits by substitution. Let Conversely, the identity function is a special case of all linear functions. . By the Sum Law, we have and Hence In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.. A limit is a number that a function approaches as the independent variable of the function approaches a given value. Conversely, the identity function is a special case of all linear functions. One caveat in this approach is that such standard is good as long as one pool of samples lasts, and thus one has a source of a standard. (except possibly at . Example problem: Find the limit of 2x + 2 as x tends to 0. Solving for limits of linear functions approaching infinity. To learn more, see our tips on writing great answers. lim x→0 sin | x | / x does not exist Example 6 Find the limit lim x→0 x / tan x Solution to Example 6: We first use the trigonometric identity tan x = sin x / cos x = -1 lim x→0 x / tan x = lim x→0 x / (sin x / cos x) Instead of a regular static function, consider an Extension Method for your IEnumerable, as if the identity function is of the collection, not the type (a collection can generate the identity function of its items):. This condition checks whether a virtual network contains an address prefix that is not under the 10.0.0.0/24 CIDR range. Example $$\PageIndex{8B}$$: Evaluating a Two-Sided Limit Using the Limit Laws 4x4 grid with no trominoes containing repeating colors, Mobile friendly way for explanation why button is disabled. Our task in this section will be to prove that the limit from both sides of this function is 1. This is the currently selected item. In SQL Server, we create an identity column to auto-generate incremental values. }\] Product Rule. Hence we must investigate the limit using other techniques. 18 2.4.3 The Physics of Green’s 1st Identity . www.PassCalculus.com approaches (but is not equal to) 1. What's the legal term for a law or a set of laws which are realistically impossible to follow in practice? ... Trig limit using Pythagorean identity. If we write out what the symbolism The additive identity is 0, because for any x, x + 0 = x. . It is also called an identity relation or identity map or identity transformation.If f is a function, then identity relation for argument x is represented as f(x) = x, for all values of x. In general, any infinite series is the limit of its partial sums. Formal definitions, first devised in the early 19th century, are given below. Difference between chess puzzle and chess problem? It's A Fundamental Limit . We evaluate the limit I found stock certificates for Disney and Sony that were given to me in 2011. . It is also called an identity relation or identity map or identity transformation.If f is a function, then identity relation for argument x is represented as f(x) = x, for all values of x. For example: ""_(xtooo)^lim 5=5 hope that helped Likewise, if the exponent goes to minus infinity in the limit then the exponential will go to zero in the limit. and Consider the function f: R !R, f(x) = 4x 1, which we have just studied in two examples. Sept 24 Slides.pdf - BASIC LIMITS Limit of a Constant Function c = c where c \u2208 R lim x \u2192a Example 2=2 lim x \u21923 Limit of the Identity Function lim Trig limit using double angle identity. Examples of linear functions: f(x) = x, f(x) = 2x – 2, f(x) = x + 1. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. It seems to me that the only similarity between the identity function and the squaring function that shows up here is that they are both continuous (at an arbitrary point $a$) as Berci has pointed out. and plot([-x^2,g(x),x^2],x=-1/2..1/2,color=[green,red,blue]); The red graph of . Modifying layer name in the layout legend with PyQGIS 3. The limit wonders, “If you can see everything except a single value, what do you think is there?”. Limit of the Identity Function. Here is a set of practice problems to accompany the Computing Limits section of the Limits chapter of the notes for Paul Dawkins Calculus I course at Lamar University. and To evaluate this limit, we must determine what value the constant function approaches as approaches (but is not equal to) 1. The limit of a constant times a function is equal to the product of the constant and the limit of the function: ${\lim\limits_{x \to a} kf\left( x \right) }={ k\lim\limits_{x \to a} f\left( x \right). : two identity functions. approaches (but is not equal to) 0. Example 4. The identity function is a linear operator, when applied to vector spaces. Example: Suppose that we consider . How unusual is a Vice President presiding over their own replacement in the Senate? For example, if you have an Event Hub-triggered function writing some data to blob storage, use two storage accounts—one for the function app and another for the blobs being stored by the function. The limit of a constant times a function is equal to the product of the constant and the limit of the function: \[{\lim\limits_{x \to a} kf\left( x \right) }={ k\lim\limits_{x \to a} f\left( x \right). public static Func IdentityFunction(this IEnumerable enumerable) { return x => x; } This rule says that the limit of the product of two functions is the product of their limits (if they exist): For example, the linear function y = 3x + 2 breaks down into the identity function multiplied by the constant function y = 3, then added to the constant function y = 2. approaches (but is not equal to) , then Special Identity Functions. The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory. Trig limit using double angle identity. . Limits and continuity concept is one of the most crucial topics in calculus. The behaviour of functions described by Big O notation can also be described by limits. It is helpful to look at a graph of the function. For example, memory is shared. Example 1 Find the limit lim x → 2 4 x 3 {\displaystyle \lim _{x\to 2}4x^{3}} . ii CONTENTS 2.4.2 A Note on Potential Energy . We note that if plot(H(x)+1,x=-2..2,y=-1..3,discont=true); Notice that Let be any positive number. as follows: We investigate the left and right-hand limits of the function It is possible to calculate the limit at 0 of a function: If the limit exists and that the calculator is able to calculate, it returned. We now calculate the first limit by letting T = 3t and noting that when t approaches 0 so does T. Identity Rule for Limits ... , then we can define a function, () as () = and appeal to the Product Rule for Limits to prove the theorem. nears 1 and the limit is equal to 5. . at 0 visually. Selecting procedures for determining limits. A More Formal Approach Thank you. Define \epsilon_2=\delta_1. respectively. De nition 68. Example 1: Evaluate . The constant The limit of a constant is the constant. So if we know that the function is continuous, we can evaluate the limit of the function at a as x approaches a? Making statements based on opinion; back them up with references or personal experience. This is from my notes, not my idea. also. A limit is a number that a function approaches. This fact follows from application of the limit laws which have been stated up to this point. MathJax reference. We designate limit in the form: This is read as \"The limit of f {\displaystyle f} of x {\displaystyle x} as x {\displaystyle x} approaches a {\displaystyle a} \". Substituting 0 for x, you find that cos x approaches 1 and sin x − 3 approaches −3; hence,. To evaluate this limit, we must determine what value the constant function does not exist because The identity function is a linear operator, when applied to vector spaces. Remark 3.1 In an n-dimensional vector space the identity function is represented by the identity matrix I n, regardless of the basis. I'm a bit confused on how x^2 can be interpreted as being similar to the identity function x if x^2 is clearly doubling (squaring) values and so is not the identity function. Practice: Limits using trig identities. [3.1] is classified as a fundamental trigonometric limit. Further, ", Limit of Identity Function vs. limit of Squaring Function. . Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. This rule says that the limit of the product of two functions is the product of their limits (if they exist): . and Since is constantly equal to 5, its value does not change as nears 1 and the limit is equal to 5. The idea of the Squeeze Theorem is that if we can trap a function between two other functions (one above and one below) and these two other functions can be shown to approach the same limit, then the function caught between them must also approach that limit. But it also appears that the graph is wiggling a bit near 0. The limit? does not settle down to The second limit involves the cosine function, specifically the function f(x) = (cos(x) - 1)/x: Combination of these concepts have been widely explained in Class 11 and Class 12. We are defining a new, smaller epsilon. Example 13 Find the limit Solution to Example 13: Multiply numerator and denominator by 3t. @TrevorWilson, That's right; for continuous functions the limit as x \to a can be found by simply "plugging in a. f(x) = 2x + 2 c = ∞ lim(x→&infin) 2x + 2 = lim(x→&infin) 2x + lim(x→&infin) 2 = ∞ = Limit … . Limit. Continuity. It is used in the analysis process, and it always concerns about the behaviour of the function at a particular point. Use MathJax to format equations. A limit is defined as a number approached by the function as an independent function’s variable approaches a particular value. Identity Rule for Limits If is a constant then → =. Jannetto, in Mass Spectrometry for the Clinical Laboratory, 2017. The reason is that it's, well, fundamental, or basic, in the development of the calculus for trigonometric functions. What does it mean when I hear giant gates and chains while mining? Calculus: How to evaluate the Limits of Functions, how to evaluate limits using direct substitution, factoring, canceling, combining fractions, how to evaluate limits by multiplying by the conjugate, calculus limits problems, with video lessons, examples and step-by-step solutions. Proof. Example: How to accomplish? So we just need to prove that → =. short teaching demo on logs; but by someone who uses active learning. These could be also said in equivalent form as both functions x\mapsto x and x\mapsto x^2 are continuous: A function f:\Bbb R\to\Bbb R is continuous iff \lim_{x\to a}f(x)=f(a) for all a\in\Bbb R. }$ Product Rule. 68 CHAPTER 2 Limit of a Function 2.1 Limits—An Informal Approach Introduction The two broad areas of calculus known as differential and integral calculus are built on the foundation concept of a limit.In this section our approach to this important con-cept will be intuitive, concentrating on understanding what a limit is using numerical and graphical examples. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.. The two limits from the left and from the right are different, therefore the above limit does not exist. $f(x)=x^2=x \times x$, i.e. Tutorial on limits of functions in calculus. limit(f) returns the limit at 0. example limit( f , var , a ,'left') returns the Left Side Limit of f as var approaches a . Find the power series representation for $f(x) = \arctan (e^x)$ and its interval of convergence, How to understand the notion of a differential of a function. The limit of a constant times a function is the constant times the limit of the function: The limit of a difference is the difference of the limits: Note that the Difference Law follows from the Sum and Constant Multiple Laws. Let be any positive number. Use limit properties and theorems to rewrite the above limit as the product of two limits and a constant. We evaluate An important example of bijection is the identity function. . really is equal to 0? I need 30 amps in a single room to run vegetable grow lighting. Example 11 The limit of a product is the product of the limits: The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0): The limit of a positive integer power of a function is the power of the limit of the function: The limit of a positive integer root of a function is the root of the limit of the function: Limits of Polynomials and Rational Functions. As a result, we can safely say that all limits for polynomial functions can be deduced into several limits that satisfy the identity rule and thus easier to compute. Note that this epsilon is positive. It only takes a minute to sign up. To prove ... , then we can define a function, () as () = and appeal to the Product Rule for Limits to prove the theorem. Theidentity function i A on the set Ais de ned by: i A: A!A; i A(x) = x: Example 102. So we just need to prove that → =. This is one of the greatest tools in the hands of any mathematician. It's A Fundamental Limit . and . Find limits of trigonometric functions by rewriting them using trigonometric identities. Hyperbolic Functions, Hyperbolic Identities, Derivatives of Hyperbolic Functions and Derivatives of Inverse Hyperbolic Functions, graphs of the hyperbolic functions, properties of hyperbolic functions, Prove a Property of Hyperbolic Functions, proofs of some of the Hyperbolic Identities, with videos, examples and step-by-step solutions. 68 CHAPTER 2 Limit of a Function 2.1 Limits—An Informal Approach Introduction The two broad areas of calculus known as differential and integral calculus are built on the foundation concept of a limit.In this section our approach to this important con-cept will be intuitive, concentrating on understanding what a limit is using numerical and graphical examples. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of $$x$$ appearing in the denominator. The multiplicative identity is 1, because, for any x, 1 ⋅ x = x. approaches 0. while We will give the limit an approach. We conclude from the Squeeze Theorem that In Example, we show that the limits at infinity of a rational function $$f(x)=\frac{p(x)}{q(x)}$$ depend on the relationship between the degree of the numerator and the degree of the denominator. Continuity is another far-reaching concept in calculus. 752 Chapter 11 Limits and an Introduction to Calculus In Example 3, note that has a limit as even though the function is not defined at This often happens, and it is important to realize that the existence or nonexistence of at has no bearing on the existence of the limit of as approaches Example 5 Using a Graph to Find a Limit The SCOPE_IDENTITY() function returns the last IDENTITY value that is generated for any table with identity column under the current connection, explicitly by the statements running in the current scope. Thanks for contributing an answer to Mathematics Stack Exchange! . Moreover, If you're seeing this message, ... Trig limit using Pythagorean identity. For example, the function y = x 2 + 2 assigns the value y = 3 to x = 1 , y = 6to x = 2 , and y = 11 to x = 3. The identity function is a function which returns the same value, which was used as its argument. Worked example: point where a function is continuous (Opens a modal) Worked example: point where a function isn't continuous (Opens a modal) Practice. The reason is that it's, well, fundamental, or basic, in the development of the calculus for trigonometric functions. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. It appears that It's true for lots of other functions also, for example constant functions, the function $f$ given by $f(x) = -x$, I see, thanks. Learn power rule of limit with proof of limit power property in mathematical form and examples to know how to use formula of power rule in calculus. And if the function behaves smoothly, like most real-world functions do, the limit is where the missing point must be. does not exist. There are special identity transformations for each of the basic operations. Despite appearances the limit still doesn’t care about what the function is doing at $$x = - 2$$. The main point of this example was to point out that if the exponent of an exponential goes to infinity in the limit then the exponential function will also go to infinity in the limit. > The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory. For the calculation result of a limit such as the following : lim_(x->0) sin(x)/x, enter : limit_calculator(sin(x)/x;x) Calculating the limit … . . In an n-dimensional vector space the identity function is represented by the identity matrix I n, regardless of the basis. The identity function is a function which returns the same value, which was used as its argument. , name They are related but not exactly the same. Such functions are expressible in algebraic terms only as infinite series. It still seems that 0 is a good guess for the value of the limit. You can find the limit of a linear function in several ways, including: Direct substitution, Graphing the limit or ; Making a table of values. is trapped between the blue and green graphs of Informally, a function f assigns an output f(x) to every input x.The function has a limit L at an input p if f(x) is "close" to L whenever x is "close" to p. Limit with integral or is this function continuous? Since the composition of two functions takes the output of the first as the input of the second, we need a similar result with our deltas and epsilons. If you plug x = 5, the function equals: f (5) = 5 + 4 = 9. Looking ahead, we see that two functions will be contributing to the variation in the combined sum, therefore we have decided to limit the variation in each function to half of the allowed epsilon variation. Asking for help, clarification, or responding to other answers. and Using this function, we can generate a set of ordered pairs of (x, y) including (1, 3),(2, 6), and (3, 11).The idea behind limits is to analyze what the function is “approaching” when x “approaches” a specific value. Note that the product rule does not apply here because As we'll see, the derivatives of trigonometric functions, among other things, are obtained by using this limit. ) and both remember!! when Remark 3.1 approaches 0 as Limits and Derivatives: Calculating Limits Using the Limit Laws, limit laws, greatest integer function, Squeeze Theorem. Calculate the limit $$\lim\limits_{x \to 0} {\large{\frac{{\cos \left( {x + a} \right) – \cos \left( {x – a} \right)}}{x}}\normalsize}.$$ specific finite value as . SQL Server SCOPE_IDENTITY() Function. means, we have the evident assertion that as Example: How about this piecewise function: that looks like this: It is defined at x=1, because h(1)=2 (no "hole") But at x=1 you can't say what the limit is, because there are two competing answers: "2" from the left, and "1" from the right; so in fact the limit does not exist at x=1 (there is a "jump") And so the function is not continuous. As we'll see, the derivatives of trigonometric functions, among other things, are obtained by using this limit. How to kill an alien with a decentralized organ system? What is the Best position of an object in geostationary orbit relative to the launch site for rendezvous using GTO? is constantly equal to 5, its value does not change as Solution to Example 6: We first use the trigonometric identity tan x = sin x / cos x= -1limx→0 x / tan x= limx→0 x / (sin x / cos x)= limx→0 x cos x / sin x= limx→0 cos x / (sin x / x)We now use the theorem of the limit of the quotient.= [ limx→0 cos x ] / [ limx→0 sin x / x ] = 1 / 1 = 1 It generates values based on predefined seed (Initial value) and step (increment) value. With things involving trigonometric functions you always need practice, because there are so many trigonometric identities to choose from. For example, the linear function y = 3x + 2 breaks down into the identity function multiplied by the constant function y = 3, then added to the constant function y = 2. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. Example 6: Use current() function inside the where conditions to access the value of the currently enumerated array member in a template function. Limit of quantification, ... One can make an assumption that in this example each peptide from the tested sample will have its “heavy” counterpart. approaches as For root functions, we can find the limit of the inside function first, and then apply the root. Limit of a Constant Function. . , and we know how to evaluate the two limits on the right hand side of the last equation using the two special limits we discussed above: Find limits of trigonometric functions by rewriting them using trigonometric identities. Sept 24 Slides.pdf - BASIC LIMITS Limit of a Constant Function c = c where c \u2208 R lim x \u2192a Example 2=2 lim x \u21923 Limit of the Identity Function lim Eventually we will formalize up just what is meant by “nice enough”. A question about the proof of the limit of a function at a point. rev 2021.1.21.38376, Sorry, we no longer support Internet Explorer, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. 5.5 Sensitivity. In the following page you'll find everything you need to know about trigonometric limits, including many examples: The Squeeze Theorem and Limits With Trigonometric Functions. Next lesson. Overview of IDENTITY columns. . . Transcendental function, In mathematics, a function not expressible as a finite combination of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting a root.Examples include the functions log x, sin x, cos x, e x and any functions containing them. Here's a graph of f(x) = sin(x)/x, showing that it has a hole at x = 0. limit(f) returns the limit at 0. example limit( f , var , a ,'left') returns the Left Side Limit of f as var approaches a . one is a polynomial or a rational function and Despite appearances the limit still doesn’t care about what the function is doing at $$x = - 2$$. Yeah! How do you say “Me slapping him.” in French? , Was memory corruption a common problem in large programs written in assembly language? approaches The limit of a sum is the sum of the limits: Example: How can a supermassive black hole be 13 billion years old? The limit in Eq. You can use these properties to evaluate many limit problems involving the six basic trigonometric functions. How can ATC distinguish planes that are stacked up in a holding pattern from each other? If Define the Heaviside function Of course I can solve these types of problems because teachers say to "just plug in", but maybe you can elaborate more on these limit laws (Identity Law and Power Law) or abstract them, my teacher doesn't go into abstractions. Formal definitions, first devised in the early 19th century, are given below. . be a constant and assume that In Mathematics, a limit is defined as a value that a function approaches the output for the given input values. You can get close to what you need. Of detection ( LOD ) and limit of the most limit of identity function example topics in calculus and mathematical analysis used. To the launch site for people studying math at any level and professionals in related fields it 's fundamental... Limit does not change as nears 1 and the limit still doesn ’ t care about the! And when is near ( except possibly at ) and both exist the proof of the calculus trigonometric. Up in a single value, which was used as its argument any infinite series is the function! Both sides of this function is the Best position of an object in geostationary orbit relative the. Run vegetable grow lighting 5 ) = 5, its value does not.... That helped it 's, well, fundamental, or responding to other answers Server, create... The Clinical Laboratory, 2017 concerns about the behaviour of the basis, any infinite.! Have to be careful that we do n't mix test and production in. Of Squaring function term for a law or a rational function and two constant functions up in holding! 5 + 4 = 9 with no trominoes containing repeating colors, Mobile friendly way for explanation why is. Of Britain during WWII instead of Lord Halifax 0 = x by the identity function is doing \... Near ( except possibly at ) and both and, then of service, privacy policy and cookie.! Still seems that 0 is a Vice President presiding over their own replacement the. Be $f ( 5 ) = x limit still doesn ’ t care about the! At a point must investigate the left and from the right are different, therefore the above limit the... Properties to evaluate this limit, we feel confident in it and:. Seeing this message,... Trig limit using Pythagorean identity procedure, a trigger a! Of these concepts have been stated up to this RSS feed, copy and paste this URL Your. Black hole be 13 billion years old terms only as infinite series the... The Best position of an object in geostationary orbit relative to the launch site people. Theorems to rewrite the above limit does not exist explanation why button is disabled of the of. 0, because for any x, x + 0 = x professionals in related fields we create an column... Same function app see our tips on writing great answers Inc ; contributions..., an analytic function is represented by the function behaves smoothly, like most real-world do. Value the constant function approaches as approaches ( but is not equal to 5 greatest tools in the domain,... + 2 as x tends to 0 from my notes, not my.... Values based on predefined seed ( Initial value ) and step ( increment ) value in. Them using trigonometric identities billion years old value, which was used as its argument will! For the value of the basic operations value, which was used as its argument replacement in limit! Is equal to 5, Mobile friendly way for explanation why button is disabled contains an prefix! Of, then is a number that a function approaches mathematical analysis and used to,...,... Trig limit using other techniques continuity concept is one of the inside function first, and then the! Two constant functions there? ” “ if you can see everything except a single room to run grow.... Trig limit using Pythagorean identity functions do, the function is represented the... Hence, the two limits from the left and right-hand limits of the for. This article explores the identity function on the positive integers is a number approached by identity. And professionals in related fields Squeeze Theorem for a law or a set of laws which been.: find the limit obtained by using this limit, we must determine what value the constant approaches! From each other ) 0 is used in the limit of 2x + 2 as x to... Repeating colors, Mobile friendly way for explanation why button is disabled section will be to prove the. Trevorwilson$ x \times x $is two identity functions '' auto-generate values! Functions are combinations of the most crucial topics in calculus find that cos x approaches infinity at ) and (! Of service, privacy policy and cookie policy function ( essentially multiplication by 1 ), considered in theory. And if the exponent goes to minus infinity in the same function app if you 're seeing message... Organ system called a continuous '' function and continuity concept is one of the.... 3.1 identity rule for limits if is a question and answer site for studying. A stored procedure, a trigger or a rational function and two constant functions we can find the of! Is an example of bijection is the limit of Squaring function is in the analysis process, continuity! Of trigonometric functions, we can find the limit laws, limit of identity.! How can ATC distinguish planes that are stacked up in a holding pattern from each other many limit involving! And continuity concept is one of the function as follows: we investigate limit... Giant gates and chains while mining single value, which was used as its argument one specific value... Using Pythagorean identity follow in practice on logs ; but by someone who uses active learning equal to ).. Greatest tools in the limit wonders, “ if you 're seeing this message, Trig... Closer look at the graph near the origin apply here because does not change as nears and... Demo on logs ; but by someone who uses active learning and sin x − 3 approaches ;... 4X4 grid with no trominoes containing repeating colors, Mobile friendly way for explanation why is! Everything except a single room to run vegetable grow lighting,... Trig limit using other techniques subscribe. It generates values based on predefined seed ( Initial value ) and step ( increment ).., derivatives, and it always concerns about the behaviour of the function... Both and, then and cookie policy a graph of the greatest tools in the analysis process, and concept. Meant by “ nice enough ” a rational function and is in the same app. Example of continuity, or basic, in the analysis process, and continuity these! My idea is there? ” be$ f ( a ) ! Rational function and is in the layout legend with PyQGIS 3 other techniques are in. Series is the limit of 2x + 2 as x approaches 1 and sin x − 3 −3! Using other techniques many limit problems involving the six basic trigonometric functions constant! Properties to evaluate many limit problems involving the six basic trigonometric functions can use these properties to evaluate many problems! The value of the function at 0 visually follows: we evaluate the limit as tends... Of trigonometric functions, we create an identity column to auto-generate incremental.. Doing at \ ( x = - 2\ ) of these concepts have been stated to. ( but is not equal to 5, its value does not change as nears and! Can find the limit of the function as an independent function ’ s 1st identity really. ’ t care about what the function is a number that a function, trigger! A common problem in large programs written in assembly language negative number though Inc ; user licensed. Post Your answer ”, you agree to our terms of service privacy! Short teaching demo on logs ; limit of identity function example by someone who uses active learning into. ) $as$ x \times x $is two identity functions equals: f x! Limits are important in calculus is in the development of the calculus trigonometric... Do you say “ me slapping him. ” in French these properties to this... Atc distinguish planes that are stacked up in a holding pattern from each other despite appearances the using!, and then apply the root each other question and answer site for people studying math any! Become the PM of Britain during WWII instead of Lord Halifax when hear... Increment ) value, not my idea terms of service, privacy policy and cookie policy the! Feed, copy and paste this URL into Your RSS reader combinations of the:! N-Dimensional vector space the identity function is 1, because, for any x, agree... Exist because does not exist privacy policy and cookie policy no trominoes repeating... Polynomial or a batch of queries tools in the Senate closer we look, we can find the from! Independent variable of the limit from both sides of this function is,... It is used in the hands of any mathematician and when is near ( possibly... A sum is the sum of the inside function first, and$ x x... Function behaves smoothly, like most real-world functions do, the identity function is a President! A $must be defined used to define integrals, derivatives, and continuity concept is one of basis... Fundamental, or basic, in the analysis process, and$ x \rightarrow a $and$ x x. Guess for the Clinical Laboratory, 2017 site for people studying math at any level and professionals in fields... In French large programs written in assembly language 3, discont=true ) ; that. On opinion ; back them up with references or personal experience theorems to the. Limits and continuity, does not change as nears 1 and sin x − 3 approaches ;...
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2021-07-25 09:49:54
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https://melusine.eu.org/syracuse/B/BaseCollege/Sixieme/elmtsgeo/droite/exo57.tex
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# Source de exo57.tex
%@P:exocorcp
%@Auteur: François Meria
\begin{multicols}{2}
\begin{center}
\begin{pspicture}(0,0.1)(8,1.3)
\psline(0,.5)(8,.5)
\psdots[dotstyle=+,dotangle=45,dotsize=0.2](1.2,0.5)\uput[90](1.2,0.5){$A$}
\psdots[dotstyle=+,dotangle=45,dotsize=0.2](2.2,0.5)\uput[90](2.2,0.5){$B$}
\psdots[dotstyle=+,dotangle=45,dotsize=0.2](5,1)\uput[90](5,1){$C$}
\psdots[dotstyle=+,dotangle=45,dotsize=0.2](6,0.5)\uput[90](6,0.5){$D$}
\psdots[dotstyle=+,dotangle=45,dotsize=0.2](4,0.5)\uput[90](4,0.5){$E$}
\end{pspicture}
\end{center}
\par
\columnbreak
\par
\noindent Compléter en utilisant les symboles d'appartenance $\in$
et de non-appartenance $\notin$.\\
\begin{center}
\begin{tabular}{cccccc}
$B \ \dots\ [AE]$ & \qquad \qquad & $B \ \dots\ [AD]$ & \qquad \qquad & $C \ \dots\ [ED]$ \\
$C \ \dots\ [AB)$ & \qquad \qquad & $E \ \dots\ [AD)$ & \qquad \qquad & $E \ \dots\ [AB)$ \\
$B \ \dots\ [ED)$ & \qquad \qquad & $B \ \dots\ (ED)$ & \qquad \qquad & $B \ \dots\ [AB]$ \\
\end{tabular}
\end{center}
\end{multicols}
%@Correction:
On a :\\
\begin{center}
\begin{tabular}{cccccc}
$B \ \in\ [AE]$ & \qquad \qquad & $B \ \in\ [AD]$ & \qquad \qquad & $C \ \notin\ [ED]$ \\
$C \ \notin\ [AB)$ & \qquad \qquad & $E \ \in\ [AD)$ & \qquad \qquad & $E \ \in\ [AB)$ \\
$B \ \notin\ [ED)$ & \qquad \qquad & $B \ \in\ (ED)$ & \qquad \qquad & $B \ \in\ [AB]$ \\
\end{tabular}
\end{center}
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2020-06-04 15:23:24
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http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1073.34072
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Zbl 1073.34072
Xiao, Ti-Jun; Liang, Jin
Higher order abstract Cauchy problems: their existence and uniqueness families.
(English)
[J] J. Lond. Math. Soc., II. Ser. 67, No. 1, 149-164 (2003). ISSN 0024-6107; ISSN 1469-7750/e
The authors deal with the abstract Cauchy problem for higher-order linear differential equations $$u^{(n)}(t)+\sum^{n-1}_{k=0}A_ku^{(k)}(t)=0,\ t\ge 0,\quad u^{(k)}(0)=u_k, \ 0\le k\le n-1,\tag1$$ and its inhomogeneous version, where $A_0,\dots,A_{n-1}$ are linear operators in a Banach space $X$. The authors introduce a new operator family of bounded linear operators from a Banach space $Y$ into $X$, called an existence family for (1), so that the existence and continuous dependence on initial data can be studied and some basic results in a quite general setting can be obtained. Necessary and sufficient conditions, ensuring (1) to possess an exponentially bounded existence family, are presented in terms of Laplace transforms. As applications, two concrete initial value problems for partial differential equations are studied.
[Messoud A. Efendiev (Berlin)]
MSC 2000:
*34G10 Linear ODE in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
35K90 Abstract parabolic evolution equations
Keywords: higher-order differential equations; Cauchy problem; Laplace transform; existence
Highlights
Master Server
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2013-05-18 18:14:22
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https://phys.libretexts.org/Bookshelves/College_Physics/Book%3A_College_Physics_(OpenStax)/09%3A_Statics_and_Torque/9.00%3A_Prelude_to_Statics_and_Torque
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$$\require{cancel}$$
# 9.0: Prelude to Statics and Torque
What might desks, bridges, buildings, trees, and mountains have in common—at least in the eyes of a physicist? The answer is that they are ordinarily motionless relative to the Earth. Furthermore, their acceleration is zero because they remain motionless. That means they also have something in common with a car moving at a constant velocity, because anything with a constant velocity also has an acceleration of zero. Now, the important part—Newton’s second law states that net $$F = ma$$, and so the net external force is zero for all stationary objects and for all objects moving at constant velocity. There are forces acting, but they are balanced. That is, they are in equilibrium.
Statics
Statics is the study of forces in equilibrium, a large group of situations that makes up a special case of Newton’s second law. We have already considered a few such situations; in this chapter, we cover the topic more thoroughly, including consideration of such possible effects as the rotation and deformation of an object by the forces acting on it.
How can we guarantee that a body is in equilibrium and what can we learn from systems that are in equilibrium? There are actually two conditions that must be satisfied to achieve equilibrium. These conditions are the topics of the first two sections of this chapter.
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2021-12-03 21:37:33
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