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Geodetic datum
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Geodesy
Fundamentals
Geodesy
Geodynamics
Geomatics
Cartography
History
Concepts
Geographical distance
Geoid
Figure of the Earth
Geodetic datum
Geodesic
Geographic coordinate system
Horizontal position representation
Latitude / Longitude
Map projection
Reference ellipsoid
Satellite geodesy
Spatial reference system
Technologies
Global Navigation Satellite System (GNSS)
Global Positioning System (GPS)
GLONASS (Russian)
BeiDou (BDS) (Chinese)
Galileo (European)
Indian Regional Navigation
Satellite System (IRNSS) (India)
Quasi-Zenith Satellite System (QZSS) (Japan)
Legenda (satellite system)
Standards (History)
NGVD 29 Sea Level Datum 1929
OSGB36 Ordnance Survey Great Britain 1936
SK-42 Systema Koordinat 1942 goda
ED50 European Datum 1950
SAD69 South American Datum 1969
GRS 80 Geodetic Reference System 1980
NAD 83 North American Datum 1983
WGS 84 World Geodetic System 1984
NAVD 88 N. American Vertical Datum 1988
ETRS89 European Terrestrial Reference
System 1989
GCJ-02 Chinese encrypted datum 2002
International Terrestrial Reference System
Spatial Reference System Identifier (SRID)
Universal Transverse Mercator (UTM)
v
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A geodetic datum or geodetic system is a coordinate system, and a set of reference points, used to locate places on the Earth (or similar objects). An approximate definition of sea level is the datum WGS 84, an ellipsoid, whereas a more accurate definition is Earth Gravitational Model 2008 (EGM2008), using at least 2,159 spherical harmonics. Other datums are defined for other areas or at other times; ED50 was defined in 1950 over Europe and differs from WGS 84 by a few hundred meters depending on where in Europe you look. Mars has no oceans and so no sea level, but at least two martian datums[which?] have been used to locate places there.[citation needed]
Datums are used in geodesy, navigation, and surveying by cartographers and satellite navigation systems to translate positions indicated on maps (paper or digital) to their real position on Earth. Each starts with an ellipsoid (stretched sphere), and then defines latitude, longitude and altitude coordinates. One or more locations on the Earth's surface are chosen as anchor "base-points".
The difference in co-ordinates between datums is commonly referred to as datum shift. The datum shift between two particular datums can vary from one place to another within one country or region, and can be anything from zero to hundreds of meters (or several kilometers for some remote islands). The North Pole, South Pole and Equator will be in different positions on different datums, so True North will be slightly different. Different datums use different interpolations for the precise shape and size of the Earth (reference ellipsoids).
Because the Earth is an imperfect ellipsoid, localised datums can give a more accurate representation of the area of coverage than WGS 84. OSGB36, for example, is a better approximation to the geoid covering the British Isles than the global WGS 84 ellipsoid.[1] However, as the benefits of a global system outweigh the greater accuracy, the global WGS 84 datum is becoming increasingly adopted.[2]
Horizontal datums are used for describing a point on the Earth's surface, in latitude and longitude or another coordinate system. Vertical datums measure elevations or depths.
City of Chicago Datum Benchmark
Contents
[hide]
1 Definition
1.1 Horizontal datum
1.2 Vertical datum
2 Geodetic coordinates
2.1 Geodetic versus geocentric latitude
3 Earth reference ellipsoid
3.1 Defining and derived parameters
3.2 Parameters for some geodetic systems
4 Conversion calculations
5 Reference datums
6 Examples
7 See also
8 Footnotes
9 References
10 Further reading
11 External links
Definition[edit]
In surveying and geodesy, a datum is a reference system or an approximation of the Earth's surface against which positional measurements are made for computing locations. Horizontal datums are used for describing a point on the Earth's surface, in latitude and longitude or another coordinate system. Vertical datums are used to measure elevations or underwater depths.
Horizontal datum[edit]
The horizontal datum is the model used to measure positions on the Earth. A specific point on the Earth can have substantially different coordinates, depending on the datum used to make the measurement. There are hundreds of local horizontal datums around the world, usually referenced to some convenient local reference point. Contemporary datums, based on increasingly accurate measurements of the shape of the Earth, are intended to cover larger areas. The WGS 84 datum, which is almost identical to the NAD83 datum used in North America and the ETRS89 datum used in Europe, is a common standard datum.[citation needed]
For example, in Sydney there is a 200 metres (700 feet) difference between GPS coordinates configured in GDA (based on global standard WGS 84) and AGD (used for most local maps), which is an unacceptably large error for some applications, such as surveying or site location for scuba diving.[3]
Vertical datum[edit]
Further information: Chart datum
Vertical datums in Europe
A vertical datum is used as a reference point for elevations of surfaces and features on the Earth including terrain, bathymetry, water levels, and man-made structures. Vertical datums are either: tidal, based on sea levels; gravimetric, based on a geoid; or geodetic, based on the same ellipsoid models of the Earth used for computing horizontal datums.
In common usage, elevations are often cited in height above sea level, although what “sea level” actually means is a more complex issue than might at first be thought: the height of the sea surface at any one place and time is a result of numerous effects, including waves, wind and currents, atmospheric pressure, tides, topography, and even differences in the strength of gravity due to the presence of mountains etc.
For the purpose of measuring the height of objects on land, the usual datum used is mean sea level (MSL). This is a tidal datum which is described as the arithmetic mean of the hourly water elevation taken over a specific 19 years cycle. This definition averages out tidal highs and lows (caused by the gravitational effects of the sun and the moon) and short term variations. It will not remove the effects of local gravity strength, and so the height of MSL, relative to a geodetic datum, will vary around the world, and even around one country. Countries tend to choose the mean sea level at one specific point to be used as the standard “sea level” for all mapping and surveying in that country. (For example, in Great Britain, the national vertical datum, Ordnance Datum Newlyn, is based on what was mean sea level at Newlyn in Cornwall between 1915 and 1921).[citation needed] However, zero elevation as defined by one country is not the same as zero elevation defined by another (because MSL is not the same everywhere), which is why locally defined vertical datums differ from one another.
A different principle is used when choosing a datum for nautical charts. For safety reasons, a mariner must be able to know the minimum depth of water that could occur at any point. For this reason, depths and tides on a nautical chart are measured relative to chart datum, which is defined to be a level below which tide rarely falls. Exactly how this is chosen depends on the tidal regime in the area being charted and on the policy of the hydrographic office producing the chart in question; a typical definition is Lowest Astronomical Tide (the lowest tide predictable from the effects of gravity), or Mean Lower Low Water (the average lowest tide of each day), although MSL is sometimes used in waters with very low tidal ranges.
Conversely, if a ship is to safely pass under a low bridge or overhead power cable, the mariner must know the minimum clearance between the masthead and the obstruction, which will occur at high tide. Consequently, bridge clearances etc. are given relative to a datum based on high tide, such as Highest Astronomical Tide or Mean High Water Springs.
Sea level does not remain constant throughout geological time, and so tidal datums are less useful when studying very long-term processes. In some situations sea level does not apply at all — for instance for mapping Mars' surface — forcing the use of a different "zero elevation", such as mean radius.
A geodetic vertical datum takes some specific zero point, and computes elevations based on the geodetic model being used, without further reference to sea levels. Usually, the starting reference point is a tide gauge, so at that point the geodetic and tidal datums might match, but due to sea level variations, the two scales may not match elsewhere. An example of a gravity-based geodetic datum is NAVD88, used in North America, which is referenced to a point in Quebec, Canada. Ellipsoid-based datums such as WGS 84, GRS80 or NAD83 use a theoretical surface that may differ significantly from the geoid.
Geodetic coordinates[edit]
Main article: Geodetic coordinates
The same position on a spheroid has a different angle for latitude depending on whether the angle is measured from the normal line segment IP of the ellipsoid (angle α) or the line segment AP from the center (angle β). Note that the "flatness" of the spheroid (orange) in the image is greater than that of the Earth; as a result, the corresponding difference between the "geodetic" and "geocentric" latitudes is also exaggerated.
In geodetic coordinates, the Earth's surface is approximated by an ellipsoid, and locations near the surface are described in terms of latitude ( ϕ {\displaystyle \ \phi } ), longitude ( λ {\displaystyle \ \lambda } ) and height ( h {\displaystyle h} ).[footnotes 1]
Geodetic versus geocentric latitude[edit]
Main article: latitude
It is important to note that geodetic latitude ( ϕ {\displaystyle \ \phi } ) (resp. altitude) is different from geocentric latitude ( ϕ ′ {\displaystyle \ \phi ^{\prime }} ) (resp. altitude). Geodetic latitude is determined by the angle between the equatorial plane and normal to the ellipsoid, whereas geocentric latitude is determined by the angle between the equatorial plane and line joining the point to the centre of the ellipsoid (see figure). Unless otherwise specified latitude is geodetic latitude.
Earth reference ellipsoid[edit]
Defining and derived parameters[edit]
Main article: Reference ellipsoid § Ellipsoid parameters
The ellipsoid is completely parameterised by the semi-major axis a {\displaystyle a} and the flattening f {\displaystyle f} .
Parameter
Symbol
Semi-major axis a
Reciprocal of flattening 1/f
From a and f it is possible to derive the semi-minor axis b, first eccentricity e and second eccentricity e′ of the ellipsoid
Parameter
Value
semi-minor axis b = a(1 − f)
First eccentricity squared e2 = 1 − b2/a2 = 2f − f2
Second eccentricity squared e′2 = a2/b2 − 1 = f(2 − f)/(1 − f)2
Parameters for some geodetic systems[edit]
Main article: Earth ellipsoid § Historical Earth ellipsoids
Australian Geodetic Datum 1966 [AGD66] and Australian Geodetic Datum 1984 (AGD84)
AGD66 and AGD84 both use the parameters defined by Australian National Spheroid (see below)
Australian National Spheroid (ANS)
ANS Defining Parameters
Parameter
Notation
Value
semi-major axis a 6 378 160.000 m
Reciprocal of Flattening 1/f 298.25
Geocentric Datum of Australia 1994 (GDA94)
GDA94 uses the parameters defined by GRS80 (see below)
Geodetic Reference System 1980 (GRS80)
GRS80 Parameters
Parameter
Notation
Value
semi-major axis a 6 378 137 m
Reciprocal of flattening 1/f 298.257 222 101
see GDA Technical Manual document for more details; the value given above for the flattening is not exact.
World Geodetic System 1984 (WGS 84)
The Global Positioning System (GPS) uses the World Geodetic System 1984 (WGS 84) to determine the location of a point near the surface of the Earth.
WGS 84 Defining Parameters
Parameter
Notation
Value
semi-major axis a 6 378 137.0 m
Reciprocal of flattening 1/f 298.257 223 563
WGS 84 derived geometric constants
Constant
Notation
Value
Semi-minor axis b 6 356 752.3142 m
First eccentricity squared e2 6.694 379 990 14x10−3
Second eccentricity squared e′2 6.739 496 742 28x10−3
see The official World Geodetic System 1984 document for more details.
A more comprehensive list of geodetic systems can be found here
Conversion calculations[edit]
Further information: Geographic coordinate conversion
Datum conversion is the process of converting the coordinates of a point from one datum system to another. Datum conversion may frequently be accompanied by a change of grid projection.
Reference datums[edit]
See also: Reference ellipsoid
A reference datum is a known and constant surface which is used to describe the location of unknown points on the Earth. Since reference datums can have different radii and different center points, a specific point on the Earth can have substantially different coordinates depending on the datum used to make the measurement. There are hundreds of locally developed reference datums around the world, usually referenced to some convenient local reference point. Contemporary datums, based on increasingly accurate measurements of the shape of the Earth, are intended to cover larger areas. The most common reference Datums in use in North America are NAD27, NAD83, and WGS 84.
The North American Datum of 1927 (NAD 27) is "the horizontal control datum for the United States that was defined by a location and azimuth on the Clarke spheroid of 1866, with origin at (the survey station) Meades Ranch (Kansas)." ... The geoidal height at Meades Ranch was assumed to be zero, as sufficient gravity data was not available, and this was needed to relate surface measurements to the datum. "Geodetic positions on the North American Datum of 1927 were derived from the (coordinates of and an azimuth at Meades Ranch) through a readjustment of the triangulation of the entire network in which Laplace azimuths were introduced, and the Bowie method was used." (http://www.ngs.noaa.gov/faq.shtml#WhatDatum ) NAD27 is a local referencing system covering North America.
The North American Datum of 1983 (NAD 83) is "The horizontal control datum for the United States, Canada, Mexico, and Central America, based on a geocentric origin and the Geodetic Reference System 1980 (GRS80). "This datum, designated as NAD 83 ...is based on the adjustment of 250,000 points including 600 satellite Doppler stations which constrain the system to a geocentric origin." NAD83 may be considered a local referencing system.
WGS 84 is the World Geodetic System of 1984. It is the reference frame used by the U.S. Department of Defense (DoD) and is defined by the National Geospatial-Intelligence Agency (NGA) (formerly the Defense Mapping Agency, then the National Imagery and Mapping Agency). WGS 84 is used by DoD for all its mapping, charting, surveying, and navigation needs, including its GPS "broadcast" and "precise" orbits. WGS 84 was defined in January 1987 using Doppler satellite surveying techniques. It was used as the reference frame for broadcast GPS Ephemerides (orbits) beginning January 23, 1987. At 0000 GMT January 2, 1994, WGS 84 was upgraded in accuracy using GPS measurements. The formal name then became WGS 84 (G730), since the upgrade date coincided with the start of GPS Week 730. It became the reference frame for broadcast orbits on June 28, 1994. At 0000 GMT September 30, 1996 (the start of GPS Week 873), WGS 84 was redefined again and was more closely aligned with International Earth Rotation Service (IERS) frame ITRF 94. It was then formally called WGS 84 (G873). WGS 84 (G873) was adopted as the reference frame for broadcast orbits on January 29, 1997.[4] Another update brought it to WGS84(G1674).
The WGS 84 datum, within two meters of the NAD83 datum used in North America, is the only world referencing system in place today. WGS 84 is the default standard datum for coordinates stored in recreational and commercial GPS units.
Users of GPS are cautioned that they must always check the datum of the maps they are using. To correctly enter, display, and to store map related map coordinates, the datum of the map must be entered into the GPS map datum field.
Examples[edit]
Examples of map datums are:
WGS 84, 72, 64 and 60 of the World Geodetic System
NAD83, the North American Datum which is very similar to WGS 84
NAD27, the older North American Datum, of which NAD83 was basically a readjustment [1]
OSGB36 of the Ordnance Survey of Great Britain
ED50, the European Datum
GDA94, the Australian Datum[5]
JGD2011, the Japanese Datum, adjusted for changes caused by 2011 Tōhoku earthquake and tsunami[6]
Tokyo97, the older Japanese Datum[7]
KGD2002, the Korean Datum[8]
TWD67 and TWD97, different datum currently used in Taiwan.[9]
BJS54 and XAS80, old geodetic datum used in China[10]
GCJ-02 and BD-09, Chinese encrypted geodetic datum.
PZ-90.11, the current geodetic reference used by GLONASS[11]
GTRF, the geodetic reference used by Galileo[12]
CGCS2000, or CGS-2000, the geodetic reference used by BeiDou Navigation Satellite System[12][13][14]
International Terrestrial Reference Frames (ITRF88, 89, 90, 91, 92, 93, 94, 96, 97, 2000, 2005, 2008, 2014), different realizations of the ITRS.[15][16]
Hong Kong Principal Datum, a vertical datum used in Hong Kong.[17][18]
See also[edit]
Axes conventions
Engineering datum
Geographic coordinate conversion
Figure of the Earth
Ordnance Datum
World Geodetic System
Footnotes[edit]
Jump up ^ About the right/left-handed order of the coordinates, i.e., ( λ , ϕ ) {\displaystyle \ (\lambda ,\phi )} or ( ϕ , λ ) {\displaystyle \ (\phi ,\lambda )} , see Spherical coordinate system#Conventions.
References[edit]
Jump up ^ "Geoid—Help | ArcGIS for Desktop". desktop.arcgis.com. Retrieved 2017-01-23.
Jump up ^ "Datums—Help | ArcGIS for Desktop". desktop.arcgis.com. Retrieved 2017-01-23.
Jump up ^ McFadyen, GPS and Diving
Jump up ^ WGS 84 on the site of National Geodetic Survey
Jump up ^ http://www.geoproject.com.au/gda.faq.html
Jump up ^ http://club.informatix.co.jp/?p=998
Jump up ^ http://vldb.gsi.go.jp/sokuchi/software/
Jump up ^ Yang, H.; Lee, Y.; Choi, Y.; Kwon, J.; Lee, H.; Jeong, K. (2007). "The Korean Datum Change to a World Geodetic System". AGU Spring Meeting Abstracts. Bibcode:2007AGUSM.G33B..03Y. CS1 maint: Uses authors parameter (link)
Jump up ^ http://www.sunriver.com.tw/grid_tm2.htm
Jump up ^ Analysis of Conversion Method and Map Merging from BJS54 XA80 Surveying and Mapping Results to CGCS2000
Jump up ^ THE TRANSITION TO USING THE TERRESTRIAL GEOCENTRIC COORDINATE SYSTEM “PARAMETRY ZEMLI 1990” (PZ-90.11) IN OPERATING THE GLOBAL NAVIGATION SATELLITE SYSTEM (GLONASS) HAS BEEN IMPLEMENTED
^ Jump up to: a b Use of international references for GNSS operations and applications
Jump up ^ Handbook of Satellite Orbits: From Kepler to GPS, Table 14.2
Jump up ^ BeiDou Navigation Satellite System Signal In Space Interface Control Document, Open Service Signal (Version 2.0) section 3.2
Jump up ^ http://www.quickclose.com.au/stanaway07pres.pdf
Jump up ^ http://itrf.ensg.ign.fr/general.php
Jump up ^ "Vertical Datum used in China – Hong Kong – onshore".
Jump up ^ Explanatory Notes on Geodetic Datums in Hong Kong
Further reading[edit]
List of geodetic parameters for many systems from University of Colorado
Gaposchkin, E. M. and Kołaczek, Barbara (1981) Reference Coordinate Systems for Earth Dynamics Taylor & Francis ISBN 9789027712608
Kaplan, Understanding GPS: principles and applications, 1 ed. Norwood, MA 02062, USA: Artech House, Inc, 1996.
GPS Notes
Introduction to GPS Applications
P. Misra and P. Enge, Global Positioning System Signals, Measurements, and Performance. Lincoln, Massachusetts: Ganga-Jamuna Press, 2001.
Peter H. Dana: Geodetic Datum Overview – Large amount of technical information and discussion.
UK Ordnance Survey
US National Geodetic Survey
External links[edit]
Look up datum in Wiktionary, the free dictionary.
GeographicLib includes a utility CartConvert which converts between geodetic and geocentric (ECEF) or local Cartesian (ENU) coordinates. This provides accurate results for all inputs including points close to the center of the Earth.
A collection of geodetic functions that solve a variety of problems in geodesy in Matlab.
NGS FAQ – What is a geodetic datum?
Convert coordinates between different datums
About the surface of the Earth on kartoweb.itc.nl
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Bridget Jones
From Wikipedia, the free encyclopedia
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This article is about the fictional character. For the actor and writer, see Bridget Jones Nelson. For the English musician Bridgette Jones, see Fluffy (band).
Bridget Jones
First appearance
Bridget Jones's Diary
Last appearance
Bridget Jones's Baby
Created by
Helen Fielding
Portrayed by
Renée Zellweger
Information
Full name
Bridget Rose Jones
Gender
Female
Occupation
Reporter
Family
Colin Jones (father)
Pamela Jones (mother)
Spouse(s)
Mark Darcy
Children
William "Billy" Darcy
Mabel Darcy
Nationality
British English
Residence
London
Bridget Jones is a franchise based on a fictional character of the same name. British writer Helen Fielding started her Bridget Jones's Diary column in The Independent in 1995, while chronicling the life of Jones as a thirtysomething single woman in London as she tries to make sense of life and love with the help of a surrogate "urban family" of friends in the 1990s. The column lampooned the obsession of women with women's magazines such as Cosmopolitan and wider social trends in Britain at the time. Fielding published the novelisation of the column in 1996, followed by a sequel in 1999 called Bridget Jones: The Edge of Reason.
Both novels were adapted for the big screen in 2001 and 2004, starring Renée Zellweger as Bridget Jones, and Hugh Grant and Colin Firth as the men in her life: Daniel Cleaver and Mark Darcy, respectively. After Fielding had ceased to work for The Daily Telegraph in late 1998, the feature began again in The Independent on 4 August 2005 and finished in June 2006. Helen Fielding released a third novel in 2013, Bridget Jones: Mad About the Boy, which is set 14 years after the events of the second novel.
The fictional character "Bridget Jones" was named on the 2016 Woman's Hour Power List as one of seven women judged to have had the biggest impact on women's lives over the past 70 years, alongside Margaret Thatcher, Barbara Castle, Helen Brook, Germaine Greer, Jayaben Desai and Beyoncé.[1]
Contents
[hide]
1 Plot summary
1.1 Original column and novelizations
1.2 Return to the Independent
2 History
2.1 Original column
2.1.1 Novelizations
2.2 Return to The Independent
2.3 Connections to Pride and Prejudice
3 Adaptations
3.1 Screen
3.2 Musical
4 References
5 External links
Plot summary[edit]
Original column and novelizations[edit]
Bridget Jones graduated from Bangor University. She is a 32-year-old single woman whose life is a satirized version of the stereotypical single London 30-something in the 1990s trying to go out there and look for love. She has some bad habits—smoking and drinking too much—but she annually writes her New Year's resolutions in her diary, determined to stop smoking, drink no more than 14 alcohol units a week, and eat more "pulses" and try her best to lose weight. In the two novels and screen adaptations, Bridget's mother is bored with her life as a housewife in the country and leaves Bridget's father. Bridget repeatedly flirts with her boss, Daniel Cleaver. A successful barrister named Mark Darcy also keeps popping into Bridget's life, being extremely awkward, and sometimes coming off a bit rude. After Bridget and Mark reach an understanding of each other and find a sort of happiness together, she gains some self-esteem and cuts down on her cigarette consumption. However, Bridget's obsession with self-help books plus several misunderstandings cannot keep the couple together forever.
Return to the Independent[edit]
The new Independent column was set in the then-present day of 2005 and 2006, with references being made to events such as River Thames whale,[2] and has dropped some of the motifs of the original diary, particularly the alcohol unit and calorie counts. Despite the time advance, Cleaver and Darcy were still the two men in Jones' life[3] ("I'm not sleeping with them both at once," she explains later to her friend Shaz. "I accidentally slept with each of them separately"),[4] and the plot line launched into a pregnancy. As Fielding said, "she's heading in a different direction."[5] The column is continued into 2006. In the last entry, Bridget Jones gave birth to a baby boy, fathered by Daniel Cleaver, and moved in with him.[6] However, Mark was not entirely out of the picture, as he previously suggested that he would like to adopt the child.[7] The column finished with the note, "Bridget is giving every attention to the care of her newborn son – and is too busy to keep up her Diary for the time being."[6]
History[edit]
Original column[edit]
In the mid-'90s, Charles Leadbeater, at the time the features editor of the English newspaper The Independent, offered Helen Fielding, then a journalist on The Independent on Sunday, a weekly column about urban life in London designed to appeal to young professional women. Fielding accepted and Bridget Jones was born on 28 February 1995.[8] The instantaneous popularity of the columns led to publication of the first book, Bridget Jones's Diary, in 1996.
The column appeared regularly every Wednesday on the pages of The Independent for almost three years: the last one was published on 10 September 1997. A couple of months later, on 15 November 1997, Helen Fielding resumed her weekly diary in The Daily Telegraph.[9] Fielding ceased to work for The Daily Telegraph on 19 December 1998.
Novelizations[edit]
Main articles: Bridget Jones's Diary and Bridget Jones: The Edge of Reason
The column was made into a novel in 1996, Bridget Jones's Diary. The plot is loosely based on Pride and Prejudice by Jane Austen. Critics assert that Fielding's book arguably began the popular fiction movement known as chick lit. The second book, Bridget Jones: The Edge of Reason, was published on 1998, and was based on the plot of another of Austen's novels, Persuasion.
Fielding named the character of Mark Darcy after the Pride and Prejudice character Fitzwilliam Darcy and described him exactly like Colin Firth, who played Mr Darcy in the 1995 BBC adaptation.[10] Mark Darcy was also partly modelled on a friend of Fielding's called Mark Muller, a barrister with chambers in Gray's Inn Square; and the Kurdish revolutionary leader whom Darcy defends in the movie was inspired by a real case of Muller's.[11] The character of Shazzer was reportedly based on Sharon Maguire, who is a friend of Fielding and would become director of the film.[12]
A third book was published in October 2013, titled Bridget Jones: Mad About the Boy. The novel is set in present-day London; Bridget is 51, still keeping a diary, but is also immersed in texting and experimenting with social media.[13] It is revealed in the book that Mark Darcy had died five years earlier and that they have two children, Billy and Mabel, aged seven and five respectively.
Return to The Independent[edit]
The feature began again in The Independent on 4 August 2005 with a "Sunday 31 July" entry. A book containing all of the original columns was given away with the paper the following Saturday. This relaunch of the column is also printed in the Irish Independent. The International Herald Tribune reviewed the new column rather favourably, commenting that Fielding's satire was in good form.[3] The Daily Mail reviewed it less favourably, saying it lacked the genuine quality of the first two novels.[14]
Connections to Pride and Prejudice[edit]
Helen Fielding (as Bridget Jones) wrote of her love of the 1995 BBC adaptation of Pride and Prejudice in her Bridget Jones's Diary column during the original British broadcast,[15] mentioning her "simple human need for Darcy to get off with Elizabeth" and regarding the couple as her "chosen representatives in the field of shagging, or rather courtship".[16] Fielding loosely reworked the plot of Pride and Prejudice in her 1996 novelization of the column, naming Bridget's uptight love interest "Mark Darcy" and describing him exactly like Colin Firth.[10][15] Following a first meeting with Firth during his filming of Fever Pitch in 1996, Fielding asked him to collaborate in what would become an eight-page interview between Bridget Jones and Firth in her 1999 sequel novel, Bridget Jones: The Edge of Reason. Conducting the real interview with Firth in Rome, Fielding lapsed into Bridget Jones mode and obsessed over Darcy in his wet shirt. Firth participated in the following editing process of what critics would consider "one of the funniest sequences in the diary's sequel".[17][18] Both novels make various other references to the BBC serial.[19]
Pride and Prejudice screenwriter Andrew Davies collaborated on the screenplays for the 2001 and 2004 Bridget Jones films, which would show Crispin Bonham-Carter (Mr Bingley in Pride and Prejudice) and Lucy Robinson (Mrs Hurst) in minor roles. The self-referential in-joke between the projects intrigued Colin Firth to accept the role of Mark Darcy,[15] as it gave him an opportunity to ridicule and liberate himself from his Pride and Prejudice character.[20] Film critic James Berardinelli would later state that Firth "plays this part [of Mark Darcy] exactly as he played the earlier role, making it evident that the two Darcys are essentially the same".[21] The producers never found a solution to incorporate the Jones Firth interview in the second film but shot a spoof interview with Firth as himself and Renée Zellweger staying in-character as Bridget Jones after a day's wrap. The scene is available as a deleted scene on DVD.[22]
Adaptations[edit]
Screen[edit]
Main article: Bridget Jones (film series)
The first novel was turned into a movie of the same name in 2001, directed by Sharon Maguire. The movie starred Renée Zellweger as Bridget, Hugh Grant as Daniel Cleaver and Colin Firth as Mark Darcy. Before the film was released, a considerable amount of controversy surrounded the casting of the American Zellweger as what some saw as a quintessentially British heroine: however, her performance is widely considered to be of a high standard, including a perfect English accent, and garnered Zellweger an Academy Award nomination. A second film was released in 2004, directed by Beeban Kidron. There are many differences between the books and the films. A third movie, Bridget Jones's Baby, was released on 16 September 2016.[23] Patrick Dempsey was cast as Jack Qwant. [24]
Musical[edit]
Main article: Bridget Jones' Diary (musical)
A musical adaptation of the franchise is currently in Workshop productions in London, featuring Sheridan Smith in the title role of Bridget. The show is due to open in London's West End, although no date has been officially confirmed.
References[edit]
Jump up ^ "Margaret Thatcher tops Woman's Hour Power List", BBC News (Arts & Entertainment), 14 December 2016.
Jump up ^ Jones, Bridget (Fielding, Helen). "Bridget Jones's Diary – Monday 23 January". The Independent. CS1 maint: Uses authors parameter (link)
^ Jump up to: a b James, Caryn (20 August 2005). "Bridget suffers through a new serial". International Herald Tribune. Retrieved 2008-06-22.
Jump up ^ Jones, Bridget (Fielding, Helen). "Bridget Jones's Diary – Wednesday 17 August". The Independent. CS1 maint: Uses authors parameter (link)
Jump up ^ "Bridget Jones back in new diary". BBC News. 4 August 2005. Retrieved 2008-06-22.
^ Jump up to: a b Jones, Bridget (Fielding, Helen). "Bridget Jones's Diary – Saturday 10 June". The Independent. CS1 maint: Uses authors parameter (link)
Jump up ^ Jones, Bridget (Fielding, Helen). "Bridget Jones's Diary – Tuesday 21 March". The Independent. [telephone call] 'Bridget, it's Mark. You know what I said the other night - about adopting the baby? I did mean it you know. It's only the wretched Cleaver element which gets in the way. Maybe we should talk.' CS1 maint: Uses authors parameter (link)
Jump up ^ Independent Columns 1995
Jump up ^ Telegraph Columns 1997
^ Jump up to: a b Pearlman, Cindy (4 May 2001). "Brit Colin Firth is newest hottie". Chicago Sun-Times. |access-date= requires |url= (help)
Jump up ^ Cartner-Morley, Jess (11 April 2001). "The real Mr Darcy". The Guardian. Retrieved 2008-06-19.
Jump up ^ Director Sharon Maguire in the audio commentary for Bridget Jones's Diary.
Jump up ^ Capon, Felicity (28 May 2013). "Bridget Jones returns in new title, Mad About the Boy". The Telegraph. Retrieved 2013-07-30.
Jump up ^ Boshoff, Alison (1 June 2006). "Is Helen Fielding about to bump off Bridget Jones?". Daily Mail. Retrieved 2008-06-22.
^ Jump up to: a b c Steiner, Susie (31 March 2001). "Twice Shy". The Guardian. Retrieved 2008-05-20.
Jump up ^ Jones, Bridget (Fielding, Helen) (25 October 1995). "Bridget Jones's Diary". The Independent: 5. CS1 maint: Uses authors parameter (link)
Jump up ^ Ryan, Tom (6 March 2004). "Renaissance man". The Age. Retrieved 2008-05-25.
Jump up ^ Grice, Elizabeth (3 April 2001). "He's back - without the breeches". The Daily Telegraph. Retrieved 2008-06-03.
Jump up ^ Salber, Cecilia (Winter 2001). "Bridget Jones and Mark Darcy: Art Imitating Art ... Imitating Art". Jane Austen Society of North America. Retrieved 2008-06-19.
Jump up ^ Faillaci, Sara (16 October 2003). "Me Sexy?". Vanity Fair (Italy).
Jump up ^ Berardinelli, James (2001). "Bridget Jones's Diary". ReelViews. Retrieved 2008-05-21.
Jump up ^ Kidron, Beeban (2004). Bridget Jones: The Edge of Reason – Deleted scenes (DVD). United International Pictures.
Jump up ^ "Third Bridget Jones in pipeline". BBC News. 15 July 2009. Retrieved 2009-07-15.
Jump up ^ "Patrick Dempsey Set to Star in Bridget Jones' Baby". US Magazine. 9 September 2015.
External links[edit]
Bridget Jones Online Archive
[hide]
v
t
e
Bridget Jones by Helen Fielding
Novels
Bridget Jones's Diary (1996)
Bridget Jones: The Edge of Reason (1999)
Bridget Jones: Mad About the Boy (2013)
Bridget Jones's Baby: The Diaries (2016)
Films
Bridget Jones's Diary (2001)
Bridget Jones: The Edge of Reason (2004)
Bridget Jones's Baby (2016)
Other
Bridget Jones' Diary (musical)
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Inequality (mathematics)
From Wikipedia, the free encyclopedia
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Not to be confused with Inequation.
"Less than" and "Greater than" redirect here. For the use of the "<" and ">" signs as punctuation, see Bracket. For the Nine Inch Nails, see Less Than (song).
For the UK insurance brand "More Th>n", see More Than (company).
This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (May 2017) (Learn how and when to remove this template message)
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The feasible regions of linear programming are defined by a set of inequalities.
In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality).
The notation a ≠ b means that a is not equal to b.
It does not say that one is greater than the other, or even that they can be compared in size.
If the values in question are elements of an ordered set, such as the integers or the real numbers, they can be compared in size.
The notation a < b means that a is less than b.
The notation a > b means that a is greater than b.
In either case, a is not equal to b. These relations are known as strict inequalities. The notation a < b may also be read as "a is strictly less than b".
In contrast to strict inequalities, there are two types of inequality relations that are not strict:
The notation a ≤ b means that a is less than or equal to b (or, equivalently, not greater than b, or at most b); "not greater than" can also be represented by the symbol for "greater than" bisected by a vertical line, "not." (The unicode for ≤ is "U+ 2264".)
The notation a ≥ b means that a is greater than or equal to b (or, equivalently, not less than b, or at least b),; "not less than" can also be represented by the symbol for "less than" bisected by a vertical line, "not." (The unicode for ≥ is "U+ 2265".)
In engineering sciences, a less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude.
The notation a ≪ b means that a is much less than b. (In measure theory, however, this notation is used for absolute continuity, an unrelated concept.)
The notation a ≫ b means that a is much greater than b.
Contents
[hide]
1 Properties
1.1 Transitivity
1.2 Converse
1.3 Addition and subtraction
1.4 Multiplication and division
1.5 Additive inverse
1.6 Multiplicative inverse
1.7 Applying a function to both sides
2 Ordered fields
3 Chained notation
4 Sharp inequalities
5 Inequalities between means
6 Power inequalities
6.1 Examples
7 Well-known inequalities
8 Complex numbers and inequalities
9 Vector inequalities
10 General existence theorems
11 See also
12 Notes
13 References
14 External links
Properties[edit]
Inequalities are governed by the following properties. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and (in the case of applying a function) monotonic functions are limited to strictly monotonic functions.
Transitivity[edit]
The transitive property of inequality states:
For any real numbers a, b, c:
If a ≥ b and b ≥ c, then a ≥ c.
If a ≤ b and b ≤ c, then a ≤ c.
If either of the premises is a strict inequality, then the conclusion is a strict inequality:
If a ≥ b and b > c, then a > c
If a > b and b ≥ c, then a > c
Since a = b implies a ≥ b these imply:
If a = b and b > c, then a > c
If a > b and b = c, then a > c
Converse[edit]
The relations ≤ and ≥ are each other's converse:
For any real numbers a and b:
If a ≤ b, then b ≥ a.
If a ≥ b, then b ≤ a.
Addition and subtraction[edit]
If x < y, then x + a < y + a.
A common constant c may be added to or subtracted from both sides of an inequality:
For any real numbers a, b, c
If a ≤ b, then a + c ≤ b + c and a − c ≤ b − c.
If a ≥ b, then a + c ≥ b + c and a − c ≥ b − c.
i.e., the real numbers are an ordered group under addition.
Multiplication and division[edit]
If x < y and a > 0, then ax < ay.
If x < y and a < 0, then ax > ay.
The properties that deal with multiplication and division state:
For any real numbers, a, b and non-zero c:
If c is positive, then multiplying or dividing by c does not change the inequality:
If a ≥ b and c > 0, then ac ≥ bc and a/c ≥ b/c.
If a ≤ b and c > 0, then ac ≤ bc and a/c ≤ b/c.
If c is negative, then multiplying or dividing by c inverts the inequality:
If a ≥ b and c < 0, then ac ≤ bc and a/c ≤ b/c.
If a ≤ b and c < 0, then ac ≥ bc and a/c ≥ b/c.
More generally, this applies for an ordered field; see #Ordered fields.
Additive inverse[edit]
The properties for the additive inverse state:
For any real numbers a and b, negation inverts the inequality:
If a ≤ b, then −a ≥ −b.
If a ≥ b, then −a ≤ −b.
Multiplicative inverse[edit]
The properties for the multiplicative inverse state:
For any non-zero real numbers a and b that are both positive or both negative:
If a ≤ b, then 1/a ≥ 1/b.
If a ≥ b, then 1/a ≤ 1/b.
If one of a and b is positive and the other is negative, then:
If a < b, then 1/a < 1/b.
If a > b, then 1/a > 1/b.
These can also be written in chained notation as:
For any non-zero real numbers a and b:
If 0 < a ≤ b, then 1/a ≥ 1/b > 0.
If a ≤ b < 0, then 0 > 1/a ≥ 1/b.
If a < 0 < b, then 1/a < 0 < 1/b.
If 0 > a ≥ b, then 1/a ≤ 1/b < 0.
If a ≥ b > 0, then 0 < 1/a ≤ 1/b.
If a > 0 > b, then 1/a > 0 > 1/b.
Applying a function to both sides[edit]
The graph of y = ln x
Any monotonically increasing function may be applied to both sides of an inequality (provided they are in the domain of that function) and it will still hold. Applying a monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.
If the inequality is strict (a < b, a > b) and the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. The rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.
A few examples of this rule are:
Exponentiating both sides of an inequality by n > 0 when a and b are positive real numbers:
a ≤ b ⇔ an ≤ bn.
a ≤ b ⇔ a−n ≥ b−n.
Taking the natural logarithm to both sides of an inequality when a and b are positive real numbers:
a ≤ b ⇔ ln(a) ≤ ln(b).
a < b ⇔ ln(a) < ln(b).
This is true because the natural logarithm is a strictly increasing function.
Ordered fields[edit]
If (F, +, ×) is a field and ≤ is a total order on F, then (F, +, ×, ≤) is called an ordered field if and only if:
a ≤ b implies a + c ≤ b + c;
0 ≤ a and 0 ≤ b implies 0 ≤ a × b.
Note that both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field, because −1 is the square of i and would therefore be positive.
The non-strict inequalities ≤ and ≥ on real numbers are total orders. That is, given arbitrary a,b∈R, at least one of a≤b and b≤a holds; at the same time, at least one of a≥b and b≥a holds. The strict inequalities < and > on real numbers are strict total orders. That is, < on R has trichotomy property: given arbitrary a,b∈R, exactly one of a<b, b<a and a=b is true; likewise, > on R has the trichotomy property.
Chained notation[edit]
The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, a < b + e < c is equivalent to a − e < b < c − e.
This notation can be generalized to any number of terms: for instance, a1 ≤ a2 ≤ ... ≤ an means that ai ≤ ai+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to ai ≤ aj for any 1 ≤ i ≤ j ≤ n.
When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1/2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1/2.
Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For instance, a < b = c ≤ d means that a < b, b = c, and c ≤ d. This notation exists in a few programming languages such as Python.
Sharp inequalities[edit]
An inequality is said to be sharp, if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ, if ψ⇒φ holds, then ψ⇔φ also holds. For instance, the inequality ∀a∈ℝ. a2 ≥ 0 is sharp, whereas the inequality ∀a∈ℝ. a2 ≥ −1 is not sharp.[citation needed]
Inequalities between means[edit]
See also: Inequality of arithmetic and geometric means
There are many inequalities between means. For example, for any positive numbers a1, a2, …, an we have H ≤ G ≤ A ≤ Q, where
H = n 1 / a 1 + 1 / a 2 + ⋯ + 1 / a n {\displaystyle H={\frac {n}{1/a_{1}+1/a_{2}+\cdots +1/a_{n}}}} (harmonic mean),
G = a 1 ⋅ a 2 ⋯ a n n {\displaystyle G={\sqrt[{n}]{a_{1}\cdot a_{2}\cdots a_{n}}}} (geometric mean),
A = a 1 + a 2 + ⋯ + a n n {\displaystyle A={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}} (arithmetic mean),
Q = a 1 2 + a 2 2 + ⋯ + a n 2 n {\displaystyle Q={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\cdots +a_{n}^{2}}{n}}}} (quadratic mean).
Power inequalities[edit]
A "power inequality" is an inequality containing terms of the form ab, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.
Examples[edit]
For any real x,
e x ≥ 1 + x . {\displaystyle e^{x}\geq 1+x.}
If x > 0 and p > 0, then
( x p − 1 ) / p ≥ ln  ( x ) ≥ ( 1 − 1 / x p ) / p . {\displaystyle (x^{p}-1)/p\geq \ln(x)\geq (1-{1}/{x^{p}})/p.}
In the limit of p → 0, the upper and lower bounds converge to ln(x).
If x > 0, then
x x ≥ ( 1 e ) 1 / e . {\displaystyle x^{x}\geq \left({\frac {1}{e}}\right)^{1/e}.}
If x ≥ 1, then
x x x ≥ x . {\displaystyle x^{x^{x}}\geq x.}
If x, y, z > 0, then
( x + y ) z + ( x + z ) y + ( y + z ) x > 2. {\displaystyle (x+y)^{z}+(x+z)^{y}+(y+z)^{x}>2.}
For any real distinct numbers a and b,
e b − e a b − a > e ( a + b ) / 2 . {\displaystyle {\frac {e^{b}-e^{a}}{b-a}}>e^{(a+b)/2}.}
If x, y > 0 and 0 < p < 1, then
x p + y p > ( x + y ) p . {\displaystyle x^{p}+y^{p}>(x+y)^{p}.}
If x, y, z > 0, then
x x y y z z ≥ ( x y z ) ( x + y + z ) / 3 . {\displaystyle x^{x}y^{y}z^{z}\geq (xyz)^{(x+y+z)/3}.}
If a, b > 0, then
a a + b b ≥ a b + b a . {\displaystyle a^{a}+b^{b}\geq a^{b}+b^{a}.}
This inequality was solved by I.Ilani in JSTOR,AMM,Vol.97,No.1,1990.
If a, b > 0, then
a e a + b e b ≥ a e b + b e a . {\displaystyle a^{ea}+b^{eb}\geq a^{eb}+b^{ea}.}
This inequality was solved by S.Manyama in AJMAA,Vol.7,Issue 2,No.1,2010 and by V.Cirtoaje in JNSA, Vol.4, Issue 2, 130–137, 2011.
If a, b, c > 0, then
a 2 a + b 2 b + c 2 c ≥ a 2 b + b 2 c + c 2 a . {\displaystyle a^{2a}+b^{2b}+c^{2c}\geq a^{2b}+b^{2c}+c^{2a}.}
If a, b > 0, then
a b + b a > 1. {\displaystyle a^{b}+b^{a}>1.}
This result was generalized by R. Ozols in 2002 who proved that if a1, ..., an > 0, then
a 1 a 2 + a 2 a 3 + ⋯ + a n a 1 > 1 {\displaystyle a_{1}^{a_{2}}+a_{2}^{a_{3}}+\cdots +a_{n}^{a_{1}}>1}
(result is published in Latvian popular-scientific quarterly The Starry Sky, see references).
Well-known inequalities[edit]
See also: List of inequalities
Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:
Azuma's inequality
Bernoulli's inequality
Bell's inequality
Boole's inequality
Cauchy–Schwarz inequality
Chebyshev's inequality
Chernoff's inequality
Cramér–Rao inequality
Hoeffding's inequality
Hölder's inequality
Inequality of arithmetic and geometric means
Jensen's inequality
Kolmogorov's inequality
Markov's inequality
Minkowski inequality
Nesbitt's inequality
Pedoe's inequality
Poincaré inequality
Samuelson's inequality
Triangle inequality
Complex numbers and inequalities[edit]
The set of complex numbers C {\displaystyle \mathbb {C} } with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that ( C , + , × , ≤ ) {\displaystyle (\mathbb {C} ,+,\times ,\leq )} becomes an ordered field. To make ( C , + , × , ≤ ) {\displaystyle (\mathbb {C} ,+,\times ,\leq )} an ordered field, it would have to satisfy the following two properties:
if a ≤ b then a + c ≤ b + c
if 0 ≤ a and 0 ≤ b then 0 ≤ a b
Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ −a). In either case 0 ≤ a2; this means that i 2 > 0 {\displaystyle i^{2}>0} and 1 2 > 0 {\displaystyle 1^{2}>0} ; so − 1 > 0 {\displaystyle -1>0} and 1 > 0 {\displaystyle 1>0} , which means ( − 1 + 1 ) > 0 {\displaystyle (-1+1)>0} ; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:
a ≤ b {\displaystyle a\leq b} if R e ( a ) < R e ( b ) {\displaystyle \mathrm {Re} (a)<\mathrm {Re} (b)} or ( R e ( a ) = R e ( b ) {\displaystyle \left(\mathrm {Re} (a)=\mathrm {Re} (b)\right.} and I m ( a ) ≤ I m ( b ) ) {\displaystyle \left.\mathrm {Im} (a)\leq \mathrm {Im} (b)\right)}
It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.
Vector inequalities[edit]
Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors x , y ∈ R n {\displaystyle x,y\in \mathbb {R} ^{n}} (meaning that x = ( x 1 , x 2 , … , x n ) T {\displaystyle x=\left(x_{1},x_{2},\ldots ,x_{n}\right)^{\mathsf {T}}} and y = ( y 1 , y 2 , … , y n ) T {\displaystyle y=\left(y_{1},y_{2},\ldots ,y_{n}\right)^{\mathsf {T}}} where x i {\displaystyle x_{i}} and y i {\displaystyle y_{i}} are real numbers for i = 1 , … , n {\displaystyle i=1,\ldots ,n} ), we can define the following relationships.
x = y {\displaystyle x=y} if x i = y i {\displaystyle x_{i}=y_{i}} for i = 1 , … , n {\displaystyle i=1,\ldots ,n}
x < y {\displaystyle x<y} if x i < y i {\displaystyle x_{i}<y_{i}} for i = 1 , … , n {\displaystyle i=1,\ldots ,n}
x ≤ y {\displaystyle x\leq y} if x i ≤ y i {\displaystyle x_{i}\leq y_{i}} for i = 1 , … , n {\displaystyle i=1,\ldots ,n} and x ≠ y {\displaystyle x\neq y}
x ≦ y {\displaystyle x\leqq y} if x i ≤ y i {\displaystyle x_{i}\leq y_{i}} for i = 1 , … , n {\displaystyle i=1,\ldots ,n}
Similarly, we can define relationships for x > y {\displaystyle x>y} , x ≥ y {\displaystyle x\geq y} , and x ≧ y {\displaystyle x\geqq y} . We note that this notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).
The trichotomy property (as stated above) is not valid for vector relationships. For example, when x = [ 2 , 5 ] T {\displaystyle x=\left[2,5\right]^{\mathsf {T}}} and y = [ 3 , 4 ] T {\displaystyle y=\left[3,4\right]^{\mathsf {T}}} , there exists no valid inequality relationship between these two vectors. Also, a multiplicative inverse would need to be defined on a vector before this property could be considered. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.
General existence theorems[edit]
For a general system of polynomial inequalities, one can find a condition for a solution to exist. Firstly, any system of polynomial inequalities can be reduced to a system of quadratic inequalities by increasing the number of variables and equations (for example by setting a square of a variable equal to a new variable). A single quadratic polynomial inequality in n − 1 variables can be written as:
X T A X ≥ 0 {\displaystyle X^{T}AX\geq 0}
where X is a vector of the variables X = ( x , y , z , … , 1 ) T {\displaystyle X=(x,y,z,\ldots ,1)^{T}} and A is a matrix. This has a solution, for example, when there is at least one positive element on the main diagonal of A.
Systems of inequalities can be written in terms of matrices A, B, C, etc. and the conditions for existence of solutions can be written as complicated expressions in terms of these matrices. The solution for two polynomial inequalities in two variables tells us whether two conic section regions overlap or are inside each other. The general solution is not known but such a solution could be theoretically used to solve such unsolved problems as the kissing number problem. However, the conditions would be so complicated as to require a great deal of computing time or clever algorithms.
See also[edit]
Binary relation
Bracket (mathematics), for the use of similar ‹ and › signs as brackets
Fourier–Motzkin elimination
Inclusion (set theory)
Inequation
Interval (mathematics)
List of inequalities
List of triangle inequalities
Partially ordered set
Relational operators, used in programming languages to denote inequality
Notes[edit]
References[edit]
Hardy, G., Littlewood J.E., Pólya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0-521-05206-8. CS1 maint: Multiple names: authors list (link)
Beckenbach, E.F., Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0-394-01559-2. CS1 maint: Multiple names: authors list (link)
Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. Springer-Verlag. ISBN 0-387-98404-6. CS1 maint: Multiple names: authors list (link)
Grinshpan, A. Z. (2005), "General inequalities, consequences, and applications", Advances in Applied Mathematics, 34 (1): 71–100, doi:10.1016/j.aam.2004.05.001
Murray S. Klamkin. "'Quickie' inequalities" (PDF). Math Strategies.
Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
Harold Shapiro (2005, 1972–1985). "Mathematical Problem Solving". The Old Problem Seminar. Kungliga Tekniska högskolan. Check date values in: |date= (help)
"3rd USAMO". Archived from the original on 2008-02-03.
Pachpatte, B.G. (2005). Mathematical Inequalities. North-Holland Mathematical Library. 67 (first ed.). Amsterdam, The Netherlands: Elsevier. ISBN 0-444-51795-2. ISSN 0924-6509. MR 2147066. Zbl 1091.26008.
Ehrgott, Matthias (2005). Multicriteria Optimization. Springer-Berlin. ISBN 3-540-21398-8.
Steele, J. Michael (2004). The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press. ISBN 978-0-521-54677-5.
External links[edit]
Wikimedia Commons has media related to Inequalities (mathematics).
Hazewinkel, Michiel, ed. (2001) [1994], "Inequality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Graph of Inequalities by Ed Pegg, Jr., Wolfram Demonstrations Project.
AoPS Wiki entry about Inequalities
Authority control
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