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https://en.wikipedia.org/wiki/Tail%20value%20at%20risk | In financial mathematics, tail value at risk (TVaR), also known as tail conditional expectation (TCE) or conditional tail expectation (CTE), is a risk measure associated with the more general value at risk. It quantifies the expected value of the loss given that an event outside a given probability level has occurred.
Background
There are a number of related, but subtly different, formulations for TVaR in the literature. A common case in literature is to define TVaR and average value at risk as the same measure. Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at , the value at risk of level . Under some other settings, TVaR is the conditional expectation of loss above a given value, whereas the expected shortfall is the product of this value with the probability of it occurring. The former definition may not be a coherent risk measure in general, however it is coherent if the underlying distribution is continuous. The latter definition is a coherent risk measure. TVaR accounts for the severity of the failure, not only the chance of failure. The TVaR is a measure of the expectation only in the tail of the distribution.
Mathematical definition
The canonical tail value at risk is the left-tail (large negative values) in some disciplines and the right-tail (large positive values) in other, such as actuarial science. This is usually due to the differing conventions of treating losses as large negative or positive values. Using the negative value convention, Artzner and others define the tail value at risk as:
Given a random variable which is the payoff of a portfolio at some future time and given a parameter then the tail value at risk is defined by
where is the upper -quantile given by . Typically the payoff random variable is in some Lp-space where to guarantee the existence of the expectation. The typical values for are 5% and 1%.
Formulas for continuous probability distributions
Closed-form formulas exist for calculating TVaR when the payoff of a portfolio or a corresponding loss follows a specific continuous distribution. If follows some probability distribution with the probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) , the left-tail TVaR can be represented as
For engineering or actuarial applications it is more common to consider the distribution of losses , in this case the right-tail TVaR is considered (typically for 95% or 99%):
Since some formulas below were derived for the left-tail case and some for the right-tail case, the following reconciliations can be useful:
and
Normal distribution
If the payoff of a portfolio follows normal (Gaussian) distribution with the p.d.f. then the left-tail TVaR is equal to where is the standard normal p.d.f., is the standard normal c.d.f., so is the standard normal quantile.
If the loss of a portfolio follows normal distribution, the right-tail TVaR is equal |
https://en.wikipedia.org/wiki/Josue%20%28footballer%2C%20born%201987%29 | Josue Souza Santos, or simply Josue (born July 10, 1987), is a Brazilian striker. Since June 2012 he has played for A.D. San Carlos.
Club statistics
References
External links
1987 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Anagennisi Karditsa F.C. players
Expatriate men's footballers in Japan
J2 League players
Sagan Tosu players
FC Machida Zelvia players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kim%20Sang-woo%20%28footballer%2C%20born%201987%29 | Kim-Sang Woo (born May 18, 1987 in Jinju) is a South Korean midfielder. He currently plays for Gimhae City FC.
Club statistics
References
External links
1987 births
Living people
South Korean men's footballers
South Korean expatriate men's footballers
J2 League players
Tokushima Vortis players
Korea National League players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan
Men's association football midfielders
People from Jinju
Footballers from South Gyeongsang Province |
https://en.wikipedia.org/wiki/Boris%20Kushner%20%28mathematician%29 | Boris Abramovich Kushner (; December 10, 1941May 7, 2019) was a mathematician, poet and essayist. His primary contribution in mathematics was in the field of Constructive Mathematical Analysis and the Theory of Constructive Numbers and Functions. He has published several books of poetry (in Russian) and a number of music, literary, and political essays (Russian and English). Dr. Kushner taught at the University of Pittsburgh at Johnstown, Pennsylvania.
References
B.A. Kushner, E. Mendelson (Translator). "Lectures on constructive mathematical analysis". American Mathematical Society, 1984.
Boris Kushner, Yulia Kushner (Illustrator). "Prichina Pechali - the Reason of Sadness: Selected Poems 1993-1996". VIA Press, Baltimore, 1999.
External links
Russian mathematicians
Russian male poets
1941 births
2019 deaths |
https://en.wikipedia.org/wiki/Wally%20Smith%20%28mathematician%29 | Walter Laws Smith (November 12, 1926 – March 6, 2023) was a British-born American mathematician, known for his contributions to applied probability theory.
Biography
Smith was born in London on November 12, 1926.
Smith received a B.A. in mathematics (1947) from Cambridge University, having gained First Class in the Mathematical Tripos Part 1 and Part 2. He then received an M.A. (1951) and Ph.D (1953) from Cambridge. His dissertation was entitled Stochastic Sequences of Events advised by Henry Daniels and D. R. Cox, with whom he published the book Queues (1961) and also published with in his early years. He became a professor of statistics at The University of North Carolina Chapel Hill (1954–56 and 1958–), and he was an emeritus professor in the Department of Statistics and Operations Research.
Smith was a fellow of the Institute of Mathematical Statistics, a fellow of the American Statistical Association (1966), a winner of the Adams Prize at the University of Cambridge (1960), a Sir Winston Churchill overseas fellow and a recipient of a Guggenheim Fellowship (see List of Guggenheim Fellowships awarded in 1974)
Smith died in Chapel Hill, North Carolina, on March 6, 2023, at the age of 96.
Publications
The superimposition of several strictly periodic sequences of events, in Biometrika, 40(?), 1953. With Cox.
A direct proof of a fundamental theorem of renewal theory, in Skandinavisk Aktuartidsskrift, 36(?), 1953
On the superposition of renewal processes, in Biometrika, 41(1–2):91–99, 1954. With Cox.
A note on truncation and sufficient statistics in The Annals of Mathematical Statistics, 28(1):247–252, 1957
On the distribution of Tribolium confusum in a container, in Biometrika, 44(?), 1957. With Cox.
Renewal theory and its ramifications, in Journal of the Royal Statistical Society, 20(2):243–302, 1958
On the elementary renewal theorem for non-identically distributed variables, in Pacific Journal of Mathematics, 14(2):673–699, 1964
Congestion Theory, Proceedings of the Symposium on Congestion Theory, The University of North Carolina Monograph Series in Probability and Statistics., 1965. With William E. Wilkinson (editors).
Necessary conditions for almost sure extinction of a branching process with random environment, Annals of Mathematical Statistics,. 39(?):2136–2140, 1968
Branching processes in Markovian environments in Duke Mathematical Journal 38(4):749–763, 1971. With William E. Wilkinson
Harold Hotelling 1895–1973 in The Annals of Statistics, 6(6):1173–1183, 1978
On transient regenerative processes in Journal of Applied Probability, 23(?):52–70, 1986. With E. Murphree.
References
1926 births
2023 deaths
American statisticians
University of North Carolina at Chapel Hill alumni
University of North Carolina at Chapel Hill faculty
Fellows of the Institute of Mathematical Statistics
Fellows of the American Statistical Association
Alumni of the University of Cambridge
British emigrants to the United States
People from London
Probability the |
https://en.wikipedia.org/wiki/John%20Monroe%20Van%20Vleck | John Monroe Van Vleck (March 4, 1833 – November 4, 1912) was an American mathematician and astronomer.
He taught astronomy and mathematics at Wesleyan University in Middletown, Connecticut for more than 50 years (1853-1912), and served as acting university president twice. The Van Vleck Observatory (at Wesleyan University) and the crater Van Vleck on the Moon are named after him.
Early life
John Monroe Van Vleck was born on March 4, 1833, in Stone Ridge, New York; he was the son of Peter Van Vleck (1806-1872) and Ann Hasbrouck (1803-1854). His maternal grandparents were Joseph Hasbrouck (1754-1831) and Margaret Hoornbeck Hasbrouck (1768-1831). He graduated from Wesleyan University in 1850, and began teaching at Greenwich Academy. The degree of LL.D. was conferred on him by Northwestern University in 1876. From 1851 to 1853 he had been an assistant at the Nautical Almanac Office.
Career
He taught astronomy and mathematics at Wesleyan University in Middletown, Connecticut for more than 50 years, serving as adjunct professor of Mathematics 1853–57, professor of Mathematics and Astronomy 1858–1904, and professor emeritus 1904–12. He served as the acting president for the university on two occasions, 1872–73 and 1887–89, the vice president 1890–93.
In 1904 he was vice-president of the American Mathematical Society.
He was a member of the Connecticut Academy of Arts and Sciences.
In 1869 he was a member of the Solar Eclipse Expedition to Mount Pleasant, Iowa. He was a fellow of the A.A.A.S. His publications include "Tables giving the Positions of the Moon for 1855-'6" and for 1878–91, and similar "Tables giving the Positions of Saturn for 1857 to 1877" contributed to the "American Nautical Almanac".
Honors
The Van Vleck Observatory at Wesleyan University was named after him, as was the crater Van Vleck on the Moon.
Personal life
He was married to Ellen Maria Burr on May 2, 1854. His wife died December 26, 1899, but he lived an additional 12 years. J. M. van Vleck was survived by a son and three daughters:
Anna Van Vleck
Clara Van Vleck
Edward Burr Van Vleck (1863-1943), a leading mathematician in the United States.
Edward taught at the University of Wisconsin–Madison, where he became professor emeritus in 1926.
Jane Van Vleck
See also
John Hasbrouck van Vleck - grandson of John Monroe Van Vleck
Notes
References
"THE SCIENCE FACULTY 1831-1861: Department of Astronomy", Wesleyan University, 2007-12-03, webpage: WU-Fac: states "Prof., 1885-1904" but should be "1858" not 1885.
"John Monroe Van Vleck", Virtualology (from Appletons Encyclopedia), 2001, webpage: www.famousamericans.net/johnmonroevanvleck.
1833 births
1912 deaths
People from Ulster County, New York
Wesleyan University faculty
American mathematicians
American astronomers
Wesleyan University alumni
American people of Dutch descent
American people of German descent
Mathematicians from New York (state) |
https://en.wikipedia.org/wiki/Chong%20Yong-de | Chong Yong-De (Korean: 정용대, Hanja: 鄭容臺, born 4 February 1978), is a former Japanese-born South Korean midfielder.
Club statistics
References
External links
J. League #29
1978 births
Living people
Association football people from Aichi Prefecture
South Korean men's footballers
K League 1 players
J1 League players
J2 League players
Pohang Steelers players
Nagoya Grampus players
Cerezo Osaka players
Kawasaki Frontale players
Yokohama FC players
Hokkaido Consadole Sapporo players
South Korean expatriate men's footballers
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan
Zainichi Korean men's footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Cylindrical%20harmonics | In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, , expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function Vn(k) is the product of three terms, each depending on one coordinate alone. The ρ-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics).
Definition
Each function of this basis consists of the product of three functions:
where are the cylindrical coordinates, and n and k constants that differentiate the members of the set. As a result of the superposition principle applied to Laplace's equation, very general solutions to Laplace's equation can be obtained by linear combinations of these functions.
Since all surfaces with constant ρ, φ and z are conicoid, Laplace's equation is separable in cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation can be expressed as:
and Laplace's equation, divided by V, is written:
The Z part of the equation is a function of z alone, and must therefore be equal to a constant:
where k is, in general, a complex number. For a particular k, the Z(z) function has two linearly independent solutions. If k is real they are:
or by their behavior at infinity:
If k is imaginary:
or:
It can be seen that the Z(k,z) functions are the kernels of the Fourier transform or Laplace transform of the Z(z) function and so k may be a discrete variable for periodic boundary conditions, or it may be a continuous variable for non-periodic boundary conditions.
Substituting for , Laplace's equation may now be written:
Multiplying by , we may now separate the P and Φ functions and introduce another constant (n) to obtain:
Since is periodic, we may take n to be a non-negative integer and accordingly, the the constants are subscripted. Real solutions for are
or, equivalently:
The differential equation for is a form of Bessel's equation.
If k is zero, but n is not, the solutions are:
If both k and n are zero, the solutions are:
If k is a real number we may write a real solution as:
where and are ordinary Bessel functions.
If k is an imaginary number, we may write a real solution as:
where and are modified Bessel functions.
The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions:
where the are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary conditions of the problem. Note that the integral may be replaced by a sum for appropriate boundary conditions. The orthogonality of the is often very useful when finding a solution to a particular problem. The and functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functio |
https://en.wikipedia.org/wiki/Pussyole%20%28Old%20Skool%29 | "Pussyole (Old Skool)" also known as just "Pussyole" and cleanly as just "Old Skool", is the second single taken from British rapper Dizzee Rascal's third studio album Maths + English, and eighth overall. It reached #22 on the UK Singles Chart and topped the UK Indie Singles Chart for a week. The word "pussyole" is a slang term for someone who is weak and unwilling to back up their friends during confrontation.
Wiley
The song is rumoured to be a diss to former friend Wiley of the Roll Deep crew, with whom he had a conflict, which made Dizzee (who was once a member) leave the crew. Wiley responded to the song in a video circulating on YouTube, in which he also takes jabs at rappers Kano and Lethal Bizzle, and then later with the track "Letter 2 Dizzee", from his album Playtime Is Over, to try to end the rift between them.
In the clean version, the chorus is removed, because of the repetition of the phrase "Pussyole".
Samples
The song samples the "Yeah! Woo!" break from Lyn Collins' "Think (About It)". Because of sample clearance issues, the song was removed from the Definitive Jux copies of the album in the US.
Track listing
CD:
"Pussyole (Old Skool)" (explicit version)
"Old Skool (Pussyole)" (clean version)
"My Life" (featuring Newham Generals)
"My Life" (instrumental)
"Pussyole (Old Skool)" (instrumental)
"Pussyole (Old Skool)" (a cappella)
7" Vinyl:
"Pussyole (Old Skool)" (explicit version)
"Old Skool (Pussyole)" (clean version)
Charts
References
2007 singles
Dizzee Rascal songs
XL Recordings singles
Songs written by Dizzee Rascal
2007 songs |
https://en.wikipedia.org/wiki/Ed%C3%ADlson%20%28footballer%2C%20born%201978%29 | Edílson José da Silva (born 8 December 1978) is a Brazilian former footballer who played as a striker.
Career statistics
Club
Honours
Individual
Lebanese Premier League Best Player: 2003–04
Lebanese Premier League Team of the Season: 2003–04
References
External links
Edílson at playmakerstats.com (English version of ogol.com.br)
1978 births
Living people
Olympic Beirut players
Expatriate men's footballers in Lebanon
Brazilian men's footballers
Brazilian expatriate men's footballers
Al Ansar FC players
Lebanese Premier League players
Avispa Fukuoka players
Paraná Clube players
Clube Atlético Sorocaba players
Rio Branco Esporte Clube players
J2 League players
Brazilian expatriate sportspeople in Lebanon
Expatriate men's footballers in Japan
Men's association football forwards |
https://en.wikipedia.org/wiki/Flex%20%28Dizzee%20Rascal%20song%29 | "Flex" is the third single from British rapper Dizzee Rascal's third studio album Maths + English, and ninth overall. The song reached number 23 on the UK Singles Chart, while his previous single placed 1 higher at number 22, but topped the UK Indie Singles Chart for 2 weeks, while his previous single topped it only for a week.
Music video
The video is set in a dream of Dizzee Rascal's after he falls asleep while watching TV, bored of the programs on it. The video parodies The X Factor as "Flex Factor" (though the video censors the "X" logo of The X Factor, which was on the TV Dizzee was watching). It features Reggie Yates as one of the three judges along with a woman and Dizzee Rascal. Rapper Mike Skinner of The Streets also features in the video as one of the performers, along with Peterborough United footballer Gabriel Zakuani and magician Dynamo. Instead of singing as on The X-Factor contestants, including Dizzee, dance on the Flex Factor - but all are considered bad by the judges apart from two girls dressed in black, who end up winning. Each girl kisses Dizzee on both cheeks at the end of the video, which then Dizzee stands up from his seat, pulls his pants up, and goes backstage with them, which Reggie later follows him. The video was directed by co-creator/director of Fonejacker and Facejacker, Ed Tracy.
Track listings
CD
"Flex" (Original)
"Flex" (Dave Spoon Reflex)
"Flex" (Dave Spoon Redub)
"Flex" (Micky Slim Remix)
"Flex" (Micky Slim Dub)
"Flex" (D 70 Remix)
7" Vinyl
"Flex" (Original)
"Pussy'ole (Old Skool)" (Family of Five Remix)
Charts
References
2007 singles
2007 songs
Dizzee Rascal songs
XL Recordings singles
Songs written by Dizzee Rascal |
https://en.wikipedia.org/wiki/Idoneal%20number | In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.
Definition
A positive integer n is idoneal if and only if it cannot be written as ab + bc + ac for distinct positive integers a, b, and c.
It is sufficient to consider the set ; if all these numbers are of the form , , or 2s for some integer s, where is a prime, then is idoneal.
Conjecturally complete listing
The 65 idoneal numbers found by Leonhard Euler and Carl Friedrich Gauss and conjectured to be the only such numbers are
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848 .
Results of Peter J. Weinberger from 1973 imply that at most two other idoneal numbers exist, and that the list above is complete if the generalized Riemann hypothesis holds (some sources incorrectly claim that Weinberger's results imply that there is at most one other idoneal number).
See also
List of unsolved problems in mathematics
Notes
References
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425–430.
L. Euler, "An illustration of a paradox about the idoneal, or suitable, numbers", 1806
G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 55–58 and 64.
O-H. Keller, Ueber die "Numeri idonei" von Euler, Beitraege Algebra Geom., 16 (1983), 79–91. [Math. Rev. 85m:11019]
G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.
P. Ribenboim, "Galimatias Arithmeticae", in Mathematics Magazine 71(5) 339 1998 MAA or, 'My Numbers, My Friends', Chap.11 Springer-Verlag 2000 NY
J. Steinig, On Euler's ideoneal numbers, Elemente Math., 21 (1966), 73–88.
A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 188.
P. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith., 22 (1973), 117–124.
Ernst Kani, Idoneal Numbers And Some Generalizations, Ann. Sci. Math. Québec 35, No 2, (2011), 197-227.
External links
K. S. Brown, Mathpages, Numeri Idonei
M. Waldschmidt, Open Diophantine problems
Integer sequences
Unsolved problems in number theory
Leonhard Euler |
https://en.wikipedia.org/wiki/W.%20R.%20Ritchie | William Riley Ritchie Sr. (January 25, 1876 – January 18, 1970) was an American college football player and coach, mathematics professor, and civil engineer. He was the second head football coach at Baylor University, serving for on year, in 1901 and compiling a record of 5–3. He was also the chairman of Baylor's mathematics department. Ritchie graduated in 1900 from the University of Georgia, where he played football.
In 1906, Ritchie was residing at Campbell, Texas, working as a mathematics professor at Henry College. He later worked a civil engineer for the Missouri–Kansas–Texas Railroad subsequently went into various businesses in banking, livestock, logging, and oil. Ritchie retired in 1954 and moved to Carmichael, California in 1960. He died on January 18, 1970, at a hospital in Sacramento, California.
Head coaching record
References
1876 births
1970 deaths
19th-century players of American football
American football tackles
American railway civil engineers
Baylor Bears football coaches
Georgia Bulldogs football players
People from Carmichael, California
People from Rabun County, Georgia
Coaches of American football from Georgia (U.S. state)
Players of American football from Georgia (U.S. state) |
https://en.wikipedia.org/wiki/Herman%20Otto%20Hartley | Herman Otto Hartley (born Hermann Otto Hirschfeld in Berlin, Germany; 1912–1980) was a German American statistician. He made significant contributions in many areas of statistics, mathematical programming, and optimization. He also founded Texas A&M University's Department of Statistics.
Hartley's earliest papers appeared under the name H.O. Hirschfeld. His father having been born in England, Hartley had dual nationality. He cleverly translated his German last name Hirschfeld (Hirsch = Hart, Feld = field = lea = ley) into English.
Career
In 1934, at the age of 22, Hartley earned a Ph.D. in mathematics from the University of Berlin, followed by a Ph.D. in mathematical statistics from the University of Cambridge in 1940 and a Doctorate of Science in mathematical statistics from University College London in 1954. He began his independent academic career at UCL, where he met Egon Pearson, with whom he collaborated to produce the classic two-volume Biometrika Tables for Statisticians, and also developed Hartley's F-max test for equality of variances.
A one-year Visiting Research Professor in Statistics position at then-Iowa State College brought Hartley to the United States in 1953 and to the forefront of a major statistics program. The position was extended after that initial year to include nine more years, during which he became deeply involved in research and teaching. His early computational talent enabled him to play a prominent part in instituting computing both for scientific and administrative purposes at Iowa State, which for the first time had university-wide service in data processing and numerical analysis. He also was a remarkably active consultant on statistics to a wide variety of scientists on campus.
After a decade at Iowa State, Hartley came to Texas A&M University, where he was appointed in 1963 as a distinguished professor and founding director of the Institute of Statistics. He was tasked with leading the Graduate Institute of Statistics, which had been formed a year earlier with only a handful of faculty, two graduate students, and the lofty mandate of providing statistical research, consulting, and instruction for all of Texas A&M University. In the ensuing decade and a half, he built his initial faculty of four into a group of 16, directed more than 30 doctoral students, and attracted significant research funding.
Hartley was short in stature, and in giving lectures, he would often begin with an audience icebreaker, asking, "Can you hear me? Can you see me?" He brought many distinguished statisticians to Texas A&M during his tenure, including Pearson, a slight man of "considerable height" who towered above Hartley when standing side by side, earning him the classic introduction from Hartley: "Never were there two more appropriate statisticians to work on the concept of range statistics."
Hartley remained active at Texas A&M until 1979, when he accepted a full-time visiting professor position at Duke University while als |
https://en.wikipedia.org/wiki/Nikodym%20set | In mathematics, a Nikodym set is a subset of the unit square in with complement of Lebesgue measure zero (i.e. with an area of 1), such that, given any point in the set, there is a straight line that only intersects the set at that point. The existence of a Nikodym set was first proved by Otto Nikodym in 1927. Subsequently, constructions were found of Nikodym sets having continuum many exceptional lines for each point, and Kenneth Falconer found analogues in higher dimensions.
Nikodym sets are closely related to Kakeya sets (also known as Besicovitch sets).
The existence of Nikodym sets is sometimes compared with the Banach–Tarski paradox. There is, however, an important difference between the two: the Banach–Tarski paradox relies on non-measurable sets.
Mathematicians have also researched Nikodym sets over finite fields (as opposed to ).
References
Measure theory
Paradoxes |
https://en.wikipedia.org/wiki/Journal%20de%20Math%C3%A9matiques%20Pures%20et%20Appliqu%C3%A9es | The Journal de Mathématiques Pures et Appliquées () is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville (editor: 1836–1874). The journal was originally published by Charles Louis Étienne Bachelier. After Bachelier's death in 1853, publishing passed to his son-in-law, Louis Alexandre Joseph Mallet, and the journal was marked Mallet-Bachelier. The publisher was sold to Gauthier-Villars (:fr:Gauthier-Villars) in 1863, where it remained for many decades. The journal is currently published by Elsevier. According to the 2018 Journal Citation Reports, its impact factor is 2.464. Articles are written in English or French.
References
External links
Online access
http://sites.mathdoc.fr/JMPA/
Index of freely available volumes
Up to 1945, volumes of Journal de Mathématiques Pures et Appliquées are available online free in their entirety from Internet Archive or Bibliothèque nationale de France. Recent volumes (from 1997 onward) are made freely available on the journal's website after 48 months.
"J. Math. Pures Appl. Ser. 9 Vol 76-103 (1997 - Jun 2015)" ScienceDirect
Mathematics journals
Publications established in 1836
Elsevier academic journals
Monthly journals
Multilingual journals |
https://en.wikipedia.org/wiki/Beit%20Ummar | Beit Ummar () is a Palestinian town located eleven kilometers northwest of Hebron in the Hebron Governorate of the State of Palestine. According to the Palestinian Central Bureau of Statistics, in 2017, the town had a population of 16,977 inhabitants. Over 4,800 residents of the town are under the age of 18. Since the Second Intifada, unemployment ranges between 60 and 80 percent due mostly to the inability of residents to work in Israel and a depression in the Palestinian economy. A part of the city straddles Road 60 and due to this, several propositions of house demolition have occurred.
Beit Ummar is mostly agricultural and is noted for its many grape vines. This has a major aspect on their culinary tradition of stuffed grape leaves known as waraq al-'inib and a grape syrup called dibs. Beit Ummar also has cherry, plum, apple and olive orchards.
History
Beit Ummar is believed to be the site of Biblical village of Maarath.
A church, tentatively dating to the 5th century CE, (but with changes probably done in the 8th century) was excavated in the 1930s at Khirbat Asida, to the east of the centre of Beit Ummar.
According to some traditions, the town was named after the Islamic Caliph Umar ibn al-Khattab because he supposedly frequented the town. Many of the town's predominantly Muslim residents are descendants of Arab Christian families who converted during the 7th century Muslim conquest. Christian ruins in the old city are a testament to this conversion over 1,000 years ago.
The main mosque in Beit Ummar houses the tomb of Nabi Matta. Matta refers to Amittai, the father of Jonah. Mujir ad-Din writes that Matta was "a holy man from the people of the house of the prophecy." Nearby Halhul houses the purported tomb of Jonah with the inscription reading "Yunus ibn Matta" or "Jonah son of Amittai", confirming that Matta is indeed the Arabic name for Amittai (some suggested it referred to the apostle Matthew); the Beit Ummar tomb is dedicated to Amittai.
In 1226, the Ayyubid sultan al-Mu'azzam built a mosque with a minaret under the supervision of Jerusalem governor Rashid ad-Din al-Mu'azzami. The Mamluks constructed some additions to the mosque and engraved several inscriptions on its surface.
Ottoman era
In 1838, Edward Robinson noted the village from Al-Dawayima.
Victor Guérin visited the village in 1863, and found it to have about 450 inhabitants. Swiss orientalist Albert Socin, noting an official Ottoman village list circa 1870, wrote that Beit Ummar had a total of 44 houses and a population of 133, though the population count included only men. Hartmann found that Bet Ummar had 60 houses.
In 1883, the PEF's Survey of Western Palestine (SWP) described Beit Ummar as a "small but conspicuous village standing on the watershed, and visible from some distance on the north. An ancient road passes through it. Half a mile north-east is a good spring, ‘Ain Kufin. The mosque has a small tower to it. The surrounding neighbourhood is covered with br |
https://en.wikipedia.org/wiki/Ordinal%20definable%20set | In mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first-order formula. Ordinal definable sets were introduced by .
A drawback to this informal definition is that it requires quantification over all first-order formulas, which cannot be formalized in the language of set theory. However there is a different way of stating the definition that can be so formalized. In this approach, a set S is formally defined to be ordinal definable if there is some collection of ordinals α1, ..., αn such that and can be defined as an element of by a first-order formula φ taking α2, ..., αn as parameters. Here denotes the set indexed by the ordinal α1 in the von Neumann hierarchy. In other words, S is the unique object such that φ(S, α2...αn) holds with its quantifiers ranging over .
The class of all ordinal definable sets is denoted OD; it is not necessarily transitive, and need not be a model of ZFC because it might not satisfy the axiom of extensionality. A set is hereditarily ordinal definable if it is ordinal definable and all elements of its transitive closure are ordinal definable. The class of hereditarily ordinal definable sets is denoted by HOD, and is a transitive model of ZFC, with a definable well ordering. It is consistent with the axioms of set theory that all sets are ordinal definable, and so hereditarily ordinal definable. The assertion that this situation holds is referred to as V = OD or V = HOD. It follows from V = L, and is equivalent to the existence of a (definable) well-ordering of the universe. Note however that the formula expressing V = HOD need not hold true within HOD, as it is not absolute for models of set theory: within HOD, the interpretation of the formula for HOD may yield an even smaller inner model.
HOD has been found to be useful in that it is an inner model that can accommodate essentially all known large cardinals. This is in contrast with the situation for core models, as core models have not yet been constructed that can accommodate supercompact cardinals, for example.
References
Set theory |
https://en.wikipedia.org/wiki/Countably%20generated | In mathematics, the term countably generated can have several meanings:
An algebraic structure (group, module, algebra) having countably many generators, see generating set
Countably generated space, a topological space in which the topology is determined by its countable subsets
Countably generated module. (Kaplansky's theorem says that a projective module is a direct sum of countably generated modules.) |
https://en.wikipedia.org/wiki/Castro%20Hlongwane%2C%20Caravans%2C%20Cats%2C%20Geese%2C%20Foot%20%26%20Mouth%20and%20Statistics | Castro Hlongwane, Caravans, Cats, Geese, Foot & Mouth and Statistics: HIV/Aids and the Struggle for the Humanisation of the African is an anonymously-authored document that was distributed to party members during the 51st National Conference of the African National Congress. The 114-page document alleges that presidential spokesperson Parks Mankahlana and AIDS/HIV icon Nkosi Johnson had died because of consumption of antiretrovirals. It was alleged that Peter Mokaba, a noted pro-AIDS reappraisal politician, had co-authored the document, even though it was written in a manner that was typical of a high-ranking leader within the party. The paper was derided by ANC member Dr. Saadiq Kariem as "ludicrous", and international criticism of the stance eventually forced Mbeki to back off from his public stance on AIDS reappraisal.
In 2007, a biography on Thabo Mbeki mentioned the author's secret contact with the president during the writing of the biography. Mbeki did not directly claim authorship of the document, but said it reflected his views.
External links
ANC archive of the 51st conference website
Story in the Guardian
History of the African National Congress
2002 documents |
https://en.wikipedia.org/wiki/Monge%27s%20theorem | In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear.
For any two circles in a plane, an external tangent is a line that is tangent to both circles but does not pass between them. There are two such external tangent lines for any two circles. Each such pair has a unique intersection point in the extended Euclidean plane. Monge's theorem states that the three such points given by the three pairs of circles always lie in a straight line. In the case of two of the circles being of equal size, the two external tangent lines are parallel. In this case Monge's theorem asserts that the other two intersection points must lie on a line parallel to those two external tangents. In other words, if the two external tangents are considered to intersect at the point at infinity, then the other two intersection points must be on a line passing through the same point at infinity, so the line between them takes the same angle as the external tangent.
Proofs
The simplest proof employs a three-dimensional analogy. Let the three circles correspond to three spheres of different radii; the circles correspond to the equators that result from a plane passing through the centers of the spheres. The three spheres can be sandwiched uniquely between two planes. Each pair of spheres defines a cone that is externally tangent to both spheres, and the apex of this cone corresponds to the intersection point of the two external tangents, i.e., the external homothetic center. Since one line of the cone lies in each plane, the apex of each cone must lie in both planes, and hence somewhere on the line of intersection of the two planes. Therefore, the three external homothetic centers are collinear.
Monge's theorem can also be proved by using Desargues' theorem.
Another easy proof uses Menelaus' theorem, since the ratios can be calculated with the diameters of each circle, which will be eliminated by cyclic forms when using Menelaus' theorem.
Desargues' theorem also asserts that 3 points lie on a line, and has a similar proof using the same idea of considering it in 3 rather than 2 dimensions and writing the line as an intersection of 2 planes.
See also
Homothetic centers of circles
Problem of Apollonius, constructs a circle (not necessarily unique) given three other circles
References
Bibliography
External links
Monge's Circle Theorem at MathWorld
Monge's theorem at cut-the-knot
Three Circles and Common Tangents at cut-the-knot
Euclidean plane geometry
Articles containing proofs
Theorems about circles |
https://en.wikipedia.org/wiki/Algebraic-group%20factorisation%20algorithm | Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the unknown prime factors p1, p2, ... By the Chinese remainder theorem, arithmetic modulo N corresponds to arithmetic in all the reduced groups simultaneously.
The aim is to find an element which is not the identity of the group modulo N, but is the identity modulo one of the factors, so a method for recognising such one-sided identities is required. In general, one finds them by performing operations that move elements around and leave the identities in the reduced groups unchanged. Once the algorithm finds a one-sided identity all future terms will also be one-sided identities, so checking periodically suffices.
Computation proceeds by picking an arbitrary element x of the group modulo N and computing a large and smooth multiple Ax of it; if the order of at least one but not all of the reduced groups is a divisor of A, this yields a factorisation. It need not be a prime factorisation, as the element might be an identity in more than one of the reduced groups.
Generally, A is taken as a product of the primes below some limit K, and Ax is computed by successive multiplication of x by these primes; after each multiplication, or every few multiplications, the check is made for a one-sided identity.
The two-step procedure
It is often possible to multiply a group element by several small integers more quickly than by their product, generally by difference-based methods; one calculates differences between consecutive primes and adds consecutively by the . This means that a two-step procedure becomes sensible, first computing Ax by multiplying x by all the primes below a limit B1, and then examining p Ax for all the primes between B1 and a larger limit B2.
Methods corresponding to particular algebraic groups
If the algebraic group is the multiplicative group mod N, the one-sided identities are recognised by computing greatest common divisors with N, and the result is the p − 1 method.
If the algebraic group is the multiplicative group of a quadratic extension of N, the result is the p + 1 method; the calculation involves pairs of numbers modulo N. It is not possible to tell whether is actually a quadratic extension of without knowing the factorisation of N. This requires knowing whether t is a quadratic residue modulo N, and there are no known methods for doing this without knowledge of the factorisation. However, provided N does not have a very large number of factors, in which case another method should be used first, picking random t (or rather picking A with t = A2 − 4) will accidentally hit a quadratic non-residue fairly quickly. If t is a quadratic residue, the p+1 method degenerates to a slower form of the p − 1 method.
If the algebraic group is an elliptic curve, t |
https://en.wikipedia.org/wiki/Gury%20Marchuk | Gury Ivanovich Marchuk (; 8 June 1925 – 24 March 2013) was a Soviet and Russian scientist in the fields of computational mathematics, and physics of atmosphere. Academician (since 1968); the President of the USSR Academy of Sciences in 1986–1991. Among his notable prizes are the USSR State Prize (1979), Demidov Prize (2004), Lomonosov Gold Medal (2004).
Marchuk was born in Orenburg Oblast, Russia. A member of the Communist Party of the Soviet Union since 1947, Academician Marchuk was elected to the Central Committee of the Party as a candidate member in 1976 and as a full member in 1981. He was elected as deputy of the Supreme Soviet of the Union of Soviet Socialist Republics in 1979. He was appointed to succeed Vladimir Kirillin as chairman of the State Committee for Science and Technology (GKNT) in 1980.
Marchuk was a proponent of the Integrated Long-Term Programme (ILTP) of Cooperation in Science & Technology that was established in 1987 as a scientific cooperative venture between India and the Soviet Union. The programme allowed the scientists of the countries to collaboratively undertake research in areas as diverse as healthcare and lasers. Marchuk co-chaired the programme's Joint Council with Prof. C.N.R. Rao for 25 years and was made an honorary member of India's National Academy of Sciences. In 2002, the Government of India conferred the Padma Bhushan on him.
Honours and awards
Hero of Socialist Labour (1975)
Honorary Citizen of Obninsk (1985)
Four Orders of Lenin (1967, 1971, 1975, 1985)
Keldysh Gold Medal — for his work "The development and creation of new methods of mathematical modeling" (1981)
Karpinski International Prize (1988)
Chebyshev Gold Medal — for outstanding performance in mathematics (1996)
Lomonosov Gold Medal (Moscow State University, 2004) - for his outstanding contribution to the creation of new models and methods for solving problems in the physics of nuclear reactors, the physics of the atmosphere and ocean, and immunology
Cavalier silver sign "Property of Siberia"
Lenin Prize in Science (1961)
Friedman Prize (1975)
USSR State Prize (1979)
State Prize of the Russian Federation (2000)
Demidov Prize (2004)
Honorary Doctorates of the University of Toulouse (1973), Charles University (Prague, 1978), Dresden University of Technology (1978), Technical University of Budapest (1978)
Foreign Member of the Bulgarian Academy of Sciences (1977), German Academy of Sciences at Berlin (1977), Czechoslovak Academy of Sciences (1977), Polish Academy of Sciences (1988)
Order of Merit for the Fatherland, 2nd and 4th classes
Jubilee Medal "300 Years of the Russian Navy"
Medal "For the Victory over Germany in the Great Patriotic War 1941–1945"
Medal "For Valiant Labour in the Great Patriotic War 1941-1945"
Commander of the Legion of Honour
Order of Georgi Dimitrov
Padma Bhushan (2002)
Vilhelm Bjerknes Medal (2008)
References
External links
Gury Marchuk — scientific works on the website Math-Net.Ru
Scientif |
https://en.wikipedia.org/wiki/Josep%20Dom%C3%A8nech%20i%20Estap%C3%A0 | Josep Domènech i Estapà (; Tarragona, 1858 – Cabrera de Mar, 1917) was a Catalan architect.
He graduated in 1881, and became professor of geodesy (1888) and descriptive geometry (1895) at the University of Barcelona, and member of the Acadèmia de Ciències i Arts (1883), of which he subsequently became president(1914).
His works in Barcelona include the church of Sant Andreu del Palomar (1881, with Pere Falqués), Teatre Poliorama and Reial Acadèmia de les Ciències (1883), Palau de la Justícia - Palace of Justice courthouse (1887-1908, with Enric Sagnier i Villavecchia), Palau Montaner, now the Delegación del Gobierno Español (Delegation of the Spanish Government) in Barcelona (1889-1896, with Lluís Domènech i Montaner), the University of Barcelona's Faculty of Medicine (1904), Modelo prison (1904, with Salvador Vinyals i Sabaté), the Amparo de Santa Lucía / Empar de Santa Llúcia home for the blind, which eventually became the Museu de la Ciència de Barcelona, now known as CosmoCaixa Barcelona (1904-1909), the Fabra Observatory (1906), Catalana de Gas i electricitat building and water tower (1908), the Church of Our Lady of Carmen (Església de la Mare de Déu del Carme) and Carmelite convent (1910-1921, finished by his son Josep Domènech i Mansana) and Magoria station (1912). He also headed the construction of the Hospital Clínic (1895-1906), based on a design by Ignasi C. Bartrolí (1881). In the town of Viladrau, he built the Hotel Bofill (1898).
He created his own style through the modification of Classical motifs, distinct both from Eclecticism and Modernisme, but was accepted by the establishment. He wrote several books, including Tratado de geometría descriptiva and El modernismo arquitectónico (1911).
Gallery
External links
Sergio Fuentes Milà, "L’estació de La Magòria. Un cas paradigmàtic de l’arquitectura industrial de Domènech i Estapà (1911-1912)", IX Jornades d'Arqueologia Industrial de Catalunya, desembre 2013 (En premsa)
Architects from Catalonia
Modernisme architects
Art Nouveau architects
1917 deaths
1858 births |
https://en.wikipedia.org/wiki/Principal%20root%20of%20unity | In mathematics, a principal n-th root of unity (where n is a positive integer) of a ring is an element satisfying the equations
In an integral domain, every primitive n-th root of unity is also a principal -th root of unity. In any ring, if n is a power of 2, then any n/2-th root of −1 is a principal n-th root of unity.
A non-example is in the ring of integers modulo ; while and thus is a cube root of unity, meaning that it is not a principal cube root of unity.
The significance of a root of unity being principal is that it is a necessary condition for the theory of the discrete Fourier transform to work out correctly.
References
Algebraic numbers
Cyclotomic fields
Polynomials
1 (number)
Complex numbers |
https://en.wikipedia.org/wiki/De%20Longchamps%20point | In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle about the circumcenter.
Definition
Let the given triangle have vertices , , and , opposite the respective sides , , and , as is the standard notation in triangle geometry. In the 1886 paper in which he introduced this point, de Longchamps initially defined it as the center of a circle orthogonal to the three circles , , and , where is centered at with radius and the other two circles are defined symmetrically. De Longchamps then also showed that the same point, now known as the de Longchamps point, may be equivalently defined as the orthocenter of the anticomplementary triangle of , and that it is the reflection of the orthocenter of around the circumcenter.
The Steiner circle of a triangle is concentric with the nine-point circle and has radius 3/2 the circumradius of the triangle; the de Longchamps point is the homothetic center of the Steiner circle and the circumcircle.
Additional properties
As the reflection of the orthocenter around the circumcenter, the de Longchamps point belongs to the line through both of these points, which is the Euler line of the given triangle. Thus, it is collinear with all the other triangle centers on the Euler line, which along with the orthocenter and circumcenter include the centroid and the center of the nine-point circle.
The de Longchamp point is also collinear, along a different line, with the incenter and the Gergonne point of its triangle. The three circles centered at , , and , with radii , , and respectively (where is the semiperimeter) are mutually tangent, and there are two more circles tangent to all three of them, the inner and outer Soddy circles; the centers of these two circles also lie on the same line with the de Longchamp point and the incenter. The de Longchamp point is the point of concurrence of this line with the Euler line, and with three other lines defined in a similar way as the line through the incenter but using instead the three excenters of the triangle.
The Darboux cubic may be defined from the de Longchamps point, as the locus of points such that , the isogonal conjugate of , and the de Longchamps point are collinear. It is the only cubic curve invariant of a triangle that is both isogonally self-conjugate and centrally symmetric; its center of symmetry is the circumcenter of the triangle. The de Longchamps point itself lies on this curve, as does its reflection the orthocenter.
References
External links
Triangle centers |
https://en.wikipedia.org/wiki/Ordered%20geometry | Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry).
History
Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). Euclid anticipated Pasch's approach in definition 4 of The Elements: "a straight line is a line which lies evenly with the points on itself".
Primitive concepts
The only primitive notions in ordered geometry are points A, B, C, ... and the ternary relation of intermediacy [ABC] which can be read as "B is between A and C".
Definitions
The segment AB is the set of points P such that [APB].
The interval AB is the segment AB and its end points A and B.
The ray A/B (read as "the ray from A away from B") is the set of points P such that [PAB].
The line AB is the interval AB and the two rays A/B and B/A. Points on the line AB are said to be collinear.
An angle consists of a point O (the vertex) and two non-collinear rays out from O (the sides).
A triangle is given by three non-collinear points (called vertices) and their three segments AB, BC, and CA.
If three points A, B, and C are non-collinear, then a plane ABC is the set of all points collinear with pairs of points on one or two of the sides of triangle ABC.
If four points A, B, C, and D are non-coplanar, then a space (3-space) ABCD is the set of all points collinear with pairs of points selected from any of the four faces (planar regions) of the tetrahedron ABCD.
Axioms of ordered geometry
There exist at least two points.
If A and B are distinct points, there exists a C such that [ABC].
If [ABC], then A and C are distinct (A ≠ C).
If [ABC], then [CBA] but not [CAB].
If C and D are distinct points on the line AB, then A is on the line CD.
If AB is a line, there is a point C not on the line AB.
(Axiom of Pasch) If ABC is a triangle and [BCD] and [CEA], then there exists a point F on the line DE for which [AFB].
Axiom of dimensionality:
For planar ordered geometry, all points are in one plane. Or
If ABC is a plane, then there exists a point D not in the plane ABC.
All points are in the same plane, space, etc. (depending on the dimension one chooses to work within).
(Dedekind's Axiom) For every partition of all the points on a line into two nonempty sets such that no point of either lies between two points of the other, there is a point of one set which lies between every other point of that set and every point of the other set.
These axioms are closely related to Hilbert's axioms of order. For a comprehensive survey of axiomatizations of ordered geometry see Victor (2011).
Results
Sylvester's problem of collinear points
The Sylvester–Gallai theorem can be proven within ordere |
https://en.wikipedia.org/wiki/Pythagorean%20addition | In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two sides. According to the Pythagorean theorem, for a triangle with sides and , this length can be calculated as
where denotes the Pythagorean addition operation.
This operation can be used in the conversion of Cartesian coordinates to polar coordinates. It also provides a simple notation and terminology for some formulas when its summands are complicated; for example, the energy-momentum relation in physics becomes
It is implemented in many programming libraries as the hypot function, in a way designed to avoid errors arising due to limited-precision calculations performed on computers. In its applications to signal processing and propagation of measurement uncertainty, the same operation is also called addition in quadrature; it is related to the quadratic mean or "root mean square".
Applications
Pythagorean addition (and its implementation as the hypot function) is often used together with the atan2 function to convert from Cartesian coordinates to polar coordinates :
If measurements have independent errors respectively, the quadrature method gives the overall error,
whereas the upper limit of the overall error is
if the errors were not independent.
This is equivalent of finding the magnitude of the resultant of adding orthogonal vectors, each with magnitude equal to the uncertainty, using the Pythagorean theorem.
In signal processing, addition in quadrature is used to find the overall noise from independent sources of noise. For example, if an image sensor gives six digital numbers of shot noise, three of dark current noise and two of Johnson–Nyquist noise under a specific condition, the overall noise is
digital numbers, showing the dominance of larger sources of noise.
The root mean square of a finite set of numbers is just their Pythagorean sum, normalized to form a generalized mean by dividing by .
Properties
The operation is associative and commutative, and
This means that the real numbers under form a commutative semigroup.
The real numbers under are not a group, because can never produce a negative number as its result, whereas each element of a group must be the result of applying the group operation to itself and the identity element. On the non-negative numbers, it is still not a group, because Pythagorean addition of one number by a second positive number can only increase the first number, so no positive number can have an inverse element. Instead, it forms a commutative monoid on the non-negative numbers, with zero as its identity.
Implementation
Hypot is a mathematical function defined to calculate the length of the hypotenuse of a right-angle triangle. It was designed to avoid errors arising due to limited-precision calculations performed on computers. Calculating the length of the hypotenuse of a triangle is possible using the square root function on |
https://en.wikipedia.org/wiki/Mathematics%20of%20Operations%20Research | Mathematics of Operations Research is a quarterly peer-reviewed scientific journal established in February 1976. It focuses on areas of mathematics relevant to the field of operations research such as continuous optimization, discrete optimization, game theory, machine learning, simulation methodology, and stochastic models. The journal is published by INFORMS (Institute for Operations Research and the Management Sciences). the journal has a 2017 impact factor of 1.078.
History
The journal was established in 1976. The founding editor-in-chief was Arthur F. Veinott Jr. (Stanford University). He served until 1980, when the position was taken over by Stephen M. Robinson, who held the position until 1986. Erhan Cinlar served from 1987 to 1992, and was followed by Jan Karel Lenstra (1993-1998). Next was Gérard Cornuéjols (1999-2003) and Nimrod Megiddo (2004-2009). Finally came Uri Rothblum (2009-2012), Jim Dai (2012-2018), and the current editor-in-chief Katya Scheinberg (2019–present).
The journal's three initial sections were game theory, stochastic systems, and mathematical programming. Currently, the journal has four sections: continuous optimization, discrete optimization, stochastic models, and game theory.
Notable papers
The following papers have been cited most frequently:
Roger B. Myerson, "Optimal Auction Design", vol 6:1, 58-73
A. Ben-Tal and Arkadi Nemirovski, "Robust Convex Optimization", vol 23:4, 769-805
M. R. Garey, D. S. Johnson, and Ravi Sethi, "The Complexity of Flowshop and Jobshop Scheduling", vol 1:2, 117-129
References
External links
Academic journals established in 1976
Media related to game theory
Operations research
Systems journals
Mathematics journals
INFORMS academic journals |
https://en.wikipedia.org/wiki/List%20of%20Seattle%20Seahawks%20records | This article details statistics relating to the Seattle Seahawks NFL football team, including career, single season and game records.
Offense
Passing
Most pass attempts, career: Russell Wilson, 4,735
Most pass attempts, season: Geno Smith, 572 (2022)
Most pass attempts, rookie season: Rick Mirer, 486 (1993)
Most pass attempts, game: Matt Hasselbeck, 55 (2002)
Most pass completions, career: Russell Wilson, 3,079
Most pass completions, season: Geno Smith, 399 (2022)
Most pass completions, rookie season: Rick Mirer, 274 (1993)
Most pass completions, game: Matt Hasselbeck, 39 (2009)
Highest completion percentage, career (min. 500 attempts): Geno Smith, 69.6
Highest completion percentage, season (min. 200 attempts): Geno Smith, 69.8 (2022)
Highest completion percentage, rookie season (min. 200 attempts): Russell Wilson, 64.1 (2012)
Highest completion percentage, game (min. 15 attempts): Russell Wilson, 88.6 (2020)
Most passing yards, career: Russell Wilson, 37,059
Most passing yards, season: Geno Smith, 4,282 (2022)
Most passing yards, rookie season: Russell Wilson, 3,118 (2012)
Most passing yards, game: Russell Wilson, 452 (2017)
Highest yards per attempt, career (min. 500 attempts): Russell Wilson, 7.8
Highest yards per attempt, season (min. 200 attempts): Dave Krieg, 8.8 (1983)
Highest yards per attempt, rookie season (min. 200 attempts): Russell Wilson, 7.9 (2012)
Highest yards per attempt, game (min. 15 attempts): Russell Wilson, 14.6 (2018)
Most passing touchdowns, career: Russell Wilson, 267
Most passing touchdowns, season: Russell Wilson, 40 (2020)
Most passing touchdowns, rookie season: Russell Wilson, 26 (2012) (tied NFL record)
Most passing touchdowns, game: 5 (four players), most recently Russell Wilson (2020)
Most passes intercepted, career: Dave Krieg, 148
Most passes intercepted, season: Jim Zorn, 27 (1976)
Most passes intercepted, rookie season: Jim Zorn, 27 (1976)
Most passes intercepted, game: Jim Zorn, 6 (1976)
Lowest percentage passes had intercepted, career (min. 500 attempts): Russell Wilson and Geno Smith, 1.8
Lowest percentage passes had intercepted, season (min. 200 attempts): Seneca Wallace, 1.2 (2008)
Lowest percentage passes had intercepted, rookie season (min. 200 attempts): Russell Wilson, 2.5 (2012)
Highest passer rating, career (min. 500 attempts): Russell Wilson, 101.8
Highest passer rating, season (min. 200 attempts): Russell Wilson, 110.9 (2018)
Highest passer rating, rookie season (min. 200 attempts): Russell Wilson, 100.0 (2012)
Highest passer rating, game (min. 10 attempts): Russell Wilson, 158.3 (2018)
Most games, 300+ passing yards, career: Russell Wilson, 21
Most games, 400+ passing yards, career: Dave Krieg, 4
Most games, 300+ passing yards, season: Russell Wilson, 5 (2020)
Most games, 400+ passing yards, season: Matt Hasselbeck, 2 (2002)
Most games, 1+ passing TD's, career: Russell Wilson, 137
Most games, 2+ passing TD's, career: Russell Wilson, 92
Most games, 3+ passing TD's, career: Russell Wil |
https://en.wikipedia.org/wiki/Toeplitz%20algebra | In operator algebras, the Toeplitz algebra is the C*-algebra generated by the unilateral shift on the Hilbert space l2(N). Taking l2(N) to be the Hardy space H2, the Toeplitz algebra consists of elements of the form
where Tf is a Toeplitz operator with continuous symbol and K is a compact operator.
Toeplitz operators with continuous symbols commute modulo the compact operators. So the Toeplitz algebra can be viewed as the C*-algebra extension of continuous functions on the circle by the compact operators. This extension is called the Toeplitz extension.
By Atkinson's theorem, an element of the Toeplitz algebra Tf + K is a Fredholm operator if and only if the symbol f of Tf is invertible. In that case, the Fredholm index of Tf + K is precisely the winding number of f, the equivalence class of f in the fundamental group of the circle. This is a special case of the Atiyah-Singer index theorem.
Wold decomposition characterizes proper isometries acting on a Hilbert space. From this, together with properties of Toeplitz operators, one can conclude that the Toeplitz algebra is the universal C*-algebra generated by a proper isometry; this is Coburn's theorem.
References
C*-algebras |
https://en.wikipedia.org/wiki/Os%C3%A9as | Oséas Reis dos Santos (born May 14, 1971 in Salvador, Bahia, Brazil), known as Oséas, is a retired Brazilian football player.
Club statistics
National team statistics
Honors
Team
Copa Libertadores Winner: 1999
Intercontinental Cup Runners-up: 1999
Individual
Brazilian 2nd Division League Top Scorer: 1995
External links
1971 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Club Athletico Paranaense players
Sociedade Esportiva Palmeiras players
Cruzeiro Esporte Clube players
Santos FC players
Sport Club Internacional players
Brasiliense FC players
Expatriate men's footballers in Japan
J1 League players
Vissel Kobe players
Albirex Niigata players
Copa Libertadores-winning players
Brazil men's international footballers
Campeonato Brasileiro Série A players
Men's association football forwards
Footballers from Salvador, Bahia |
https://en.wikipedia.org/wiki/Casey%27s%20theorem | In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey.
Formulation of the theorem
Let be a circle of radius . Let be (in that order) four non-intersecting circles that lie inside and tangent to it. Denote by the length of the exterior common bitangent of the circles . Then:
Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem.
Proof
The following proof is attributable to Zacharias. Denote the radius of circle by and its tangency point with the circle by . We will use the notation for the centers of the circles.
Note that from Pythagorean theorem,
We will try to express this length in terms of the points . By the law of cosines in triangle ,
Since the circles tangent to each other:
Let be a point on the circle . According to the law of sines in triangle :
Therefore,
and substituting these in the formula above:
And finally, the length we seek is
We can now evaluate the left hand side, with the help of the original Ptolemy's theorem applied to the inscribed quadrilateral :
Further generalizations
It can be seen that the four circles need not lie inside the big circle. In fact, they may be tangent to it from the outside as well. In that case, the following change should be made:
If are both tangent from the same side of (both in or both out), is the length of the exterior common tangent.
If are tangent from different sides of (one in and one out), is the length of the interior common tangent.
The converse of Casey's theorem is also true. That is, if equality holds, the circles are tangent to a common circle.
Applications
Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry. For example, the shortest known proof of Feuerbach's theorem uses the converse theorem.
References
External links
Shailesh Shirali: "'On a generalized Ptolemy Theorem'". In: Crux Mathematicorum, Vol. 22, No. 2, pp. 49-53
Theorems about circles
Euclidean geometry
Articles containing proofs |
https://en.wikipedia.org/wiki/Quantaloid | In mathematics, a quantaloid is a category enriched over the category Sup of suplattices. In other words, for any objects a and b the morphism object between them is not just a set but a complete lattice, in such a way that composition of morphisms preserves all joins:
The endomorphism lattice of any object in a quantaloid is a quantale, whence the name.
References
Category theory |
https://en.wikipedia.org/wiki/U.%20Narayan%20Bhat | U. Narayan Bhat (born 1934) is an Indian-born Mathematician, known for his contributions to queueing theory and reliability theory.
Academic career
He received a B.A. in mathematics (1953) and B.T. in education (1954) from the University of Madras, an M.A. in statistics (1958) from Karnatak University in Dharwar and Ph.D. in Mathematical statistics from the University of Western Australia on the dissertation Some Simple and Bulk Queueing Systems: A Study of Their Transient Behavior (1965). He worked at Michigan State University (1965–66), Case Western Reserve University (1966–69), and Southern Methodist University (1969–2005). Bhat is a fellow of the American Statistical Association and the Institute for Operations Research and the Management Sciences and an elected
member of the International Statistical Institute.
U. Narayan Bhat was a dean of research and graduate studies at Southern Methodist University and then was named interim dean for the university's Dedman College .
Books
A Study of the Queueing Systems M/G/1 and GI/M/1, (Springer Verlag, 1968)
Elements of Applied Stochastic Processes (Wiley, 1972)
Introduction to Operations Research Models (W. B. Saunders & Co., 1977). With L. Cooper and L. J. LeBlanc
Queueing and Related Models (Oxford University Press, 1992). Editor with I. V. Basawa
Elements of Applied Stochastic Processes (Wiley, 2002). With Gregory K. Miller
Introduction to queueing theory (Birkhauser, 2008)
Publications
Further Results for the Queue with Poisson Arrivals, Operations Research, Vol. 11(3), (1963), 380-386 (with Narahari Umanath Prabhu).
Imbedded Markov Chain Analysis of Single-Server Bulk Queues, Journal of the Australian Math, Soc., Vol. 4(2), (1964), 244-263.
On Single-Server Bulk Queueing Processes with Binomial Input, Operations Research, Vol. 12(4), (1964), 527-533.
On a Stochastic Process Occurring in Queueing Systems, Journal of Applied Probability, Vol. 2(2), (1965), 467-469.
Statistical Analysis of Queueing Systems in Frontiers in Queuing by Dshalalow etc. (1997). (with G.K. Miller and S. Subba Rao).
Estimation of Renewal Processes with Unobservable Gamma or Erlang Interarrival Times, J. Stat. Plan. and Inf., 61 (1997), 355-372 (with G. K. Miller).
Maximum Likelihood Estimation for Single Server Queues from Waiting Time Data, Queueing Systems (journal), 24, (1997), 155-167 (with I. V. Basawa and R. Lund).
Estimation of the Coefficient of Variation for Unobservable Service Times in the M/G/1 Queue, Journal of Mathematical Sciences, Vol. 1, 2002 (with G. K. Miller).
References
On Google scholar
External links
Biography of U. Narayan Bhat from the Institute for Operations Research and the Management Sciences (INFORMS)
University of Western Australia alumni
Michigan State University faculty
Case Western Reserve University faculty
Southern Methodist University faculty
Queueing theorists
Indian emigrants to the United States
Elected Members of the International Statistical Institute
1934 births
L |
https://en.wikipedia.org/wiki/Coherent%20topology | In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps.
Definition
Let be a topological space and let be a family of subsets of each having the subspace topology. (Typically will be a cover of .) Then is said to be coherent with (or determined by ) if the topology of is recovered as the one coming from the final topology coinduced by the inclusion maps
By definition, this is the finest topology on (the underlying set of) for which the inclusion maps are continuous.
is coherent with if either of the following two equivalent conditions holds:
A subset is open in if and only if is open in for each
A subset is closed in if and only if is closed in for each
Given a topological space and any family of subspaces there is a unique topology on (the underlying set of) that is coherent with This topology will, in general, be finer than the given topology on
Examples
A topological space is coherent with every open cover of More generally, is coherent with any family of subsets whose interiors cover As examples of this, a weakly locally compact space is coherent with the family of its compact subspaces. And a locally connected space is coherent with the family of its connected subsets.
A topological space is coherent with every locally finite closed cover of
A discrete space is coherent with every family of subspaces (including the empty family).
A topological space is coherent with a partition of if and only is homeomorphic to the disjoint union of the elements of the partition.
Finitely generated spaces are those determined by the family of all finite subspaces.
Compactly generated spaces (in the sense of Definition 1 in that article) are those determined by the family of all compact subspaces.
A CW complex is coherent with its family of -skeletons
Topological union
Let be a family of (not necessarily disjoint) topological spaces such that the induced topologies agree on each intersection
Assume further that is closed in for each Then the topological union is the set-theoretic union
endowed with the final topology coinduced by the inclusion maps . The inclusion maps will then be topological embeddings and will be coherent with the subspaces
Conversely, if is a topological space and is coherent with a family of subspaces that cover then is homeomorphic to the topological union of the family
One can form the topological union of an arbitrary family of topological spaces as above, but if the topologies do not agree on the intersections then the inclusions will not necessarily be embeddings.
One can also describe the topological union by m |
https://en.wikipedia.org/wiki/Nicolaus%20Mulerius | Nicolaus Mulerius (25 December 1564, Bruges – 5 September 1630, Groningen) was a professor of medicine and mathematics at the University of Groningen.
Education and career
Mulerius was born Nicolaas Des Muliers, son of Pierre Des Muliers and Claudia Le Vettre. He grew up in Bruges, where he was taught by Jacobus Cruquius, among others. Mulerius first studied Philology, Philosophy and Theology and from 1582 he also studied Medicine and Mathematics at the University of Leiden, where Lipsius, Vulcanius, Snellius and Heurnius were teachers. In 1589 he married Christina Six and set up practice for 13 years in Harlingen. In 1603 he became leading physician in Groningen, in 1608 he took the position as school master of the Leeuwarden gymnasium. From 1614 he was professor in medicine and mathematics at the Groningen University. From 1619 – 1621 and 1626 – 1630, he was in charge of the library of the University of Groningen.
Publications
In 1616, Nicolaus Mulerius published a textbook on astronomy reminiscent of the Sphere by Johannes de Sacrobosco.
Also in 1616, he published the third, updated and annotated edition of Nicolaus Copernicus' De revolutionibus orbium coelestium.
An account of the life of Ubbo Emmius, written by Nicolaus Mulerius, was published, with the lives of other professors of Groningen, at Groningen in 1638, eight years after Mulerius' death.
Personal life
Mulerius married Christina Maria Six (1566-1645) in 1589 in Amsterdam. Among their children were (1599-1647), who would become professor at Groningen in physics and botany from 1628, and Carolus Mulerius (1601-1638) who wrote the first grammar of Spanish in Dutch.
Works
Naturae tabulae Frisicae lunae-solares quadruplices, quibus accessere solis ..., 1611
Institutionum astronomicarum libri duo, 1616
Iudæorum annus lunæ-solaris: et Turc-Arabum annus merê lunaris,1630
Naturae tabulae Frisicae lunae-solares quadruplices, quibus accessere solis ..., 1611
Nicolai Mulerii ... Exercitationes in Apocalypsin s. Johannis apostoli, 1691
References
Sources
Lynn Thorndike : History of Magic and Experimental Science.
Tabitta van Nouhuys: The Age of Two-Faced Janus: The Comets of 1577 and 1618 and the Decline of ... 1998
Klaas van Berkel, Albert Van Helden, L. C. Palm: A History of Science in the Netherlands: Survey, Themes and Reference
A.J. van der Aa, Biographisch Woordenboek der Nederlanden, 13th edition, 1876 (Dutch)
1564 births
1630 deaths
17th-century writers in Latin
17th-century Dutch physicians
Physicians of the Spanish Netherlands
17th-century Dutch astronomers
Academic staff of the University of Groningen
Physicians from Bruges
Leiden University alumni
Scientists from Bruges
16th-century Dutch astronomers |
https://en.wikipedia.org/wiki/Alexandrov%20space | In geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are locally compact complete length spaces where the lower curvature bound is defined via comparison of geodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces.
One can show that the Hausdorff dimension of an Alexandrov space with curvature ≥ k is either a non-negative integer or infinite. One can define a notion of "angle" and "tangent cone" in these spaces.
Alexandrov spaces with curvature ≥ k are important as they form the limits (in the Gromov-Hausdorff metric) of sequences of Riemannian manifolds with sectional curvature ≥ k, as described by Gromov's compactness theorem.
Alexandrov spaces with curvature ≥ k were introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov in 1948 and should not be confused with Alexandrov-discrete spaces named after the Russian topologist Pavel Alexandrov. They were studied in detail by Burago, Gromov and Perelman in 1992 and were later used in Perelman's proof of the Poincaré conjecture.
References
Metric geometry
Differential geometry
Riemannian manifolds |
https://en.wikipedia.org/wiki/Berlin%20Mathematical%20School | The Berlin Mathematical School (BMS) is a joint graduate school of the three renowned mathematics departments of the public research universities in Berlin: Freie Universität Berlin, Humboldt-Universität zu Berlin, and Technische Universität Berlin.
In October 2006, the BMS was awarded one of the 18 prestigious graduate school awards by the Excellence Initiative of the German Federal Government for its innovative concept, its strong cross-disciplinary focus, and its outstanding teaching schedule tailored to the needs of students in an international environment. This was reconfirmed in June 2012 when the German Research Foundation announced that the BMS would also receive funding for a second period until 2017. Since 2019, the BMS is the graduate school in the Cluster of Excellence MATH+, which is funded by the Excellence Strategy.
The BMS Chair is Jürg Kramer (HU), and the deputy Chairs are John M. Sullivan (TU) and Holger Reich (FU).
Cooperation
BMS students enjoy access to exclusive seminars, workshops and lectures in English not only at the participating universities, but also at their academic partners:
the Research Training Groups (RTG)
the International Max Planck Research Schools (IMPRS)
the Zuse Institute Berlin (ZIB)
the Weierstrass Institute for Applied Analysis and Stochastics (WIAS)
or the DFG Collaborative Research Centers:
Discretization in Geometry (SFB 109)
Scaling Cascades in Complex Systems (SFB 1114).
PhD in Mathematics at the BMS
The BMS PhD study program guides a student with a bachelor's degree through a structured course program, an oral qualifying exam, then directly to a doctoral degree in four to five years.
Phase I is the first part of the program and includes a lecture program created specifically for the BMS and coordinated among the three universities. Applicants who hold a bachelor's degree, Vordiplom, or equivalent can apply for Phase I of the BMS. Each semester, seven to ten Basic Courses are offered in English. During Phase I, every BMS student should complete at least five Basic Courses, plus two Advanced Courses (including one seminar) within three to four semesters. At the end of Phase I the Qualifying Exam takes place: an oral exam which is mandatory for admission to the research phase. Phase I students have a BMS faculty member as a mentor, who is assigned by the Admissions Committee. The Phase I mentor guides the student for the entire duration of Phase I, through the BMS Qualifying Exam until the start of Phase II.
Phase II is the research phase of the BMS PhD program and, to apply, students are expected to have a master's degree and meet the regular admission requirements of the Berlin universities' PhD programs. Each student is registered as a PhD student at one of the three universities and are expected to finish Phase II within four to six semesters. Phase II students have a thesis supervisor who provides support in all aspects of the PhD thesis, and gives advice on choosing the right con |
https://en.wikipedia.org/wiki/Sylvia%20Richardson | Sylvia Therese Richardson is a French/British Bayesian statistician and is currently Professor of Biostatistics and Director of the MRC Biostatistics Unit at the University of Cambridge. In 2021 she became the president of the Royal Statistical Society for the 2021–22 year.
Education
Richardson completed her PhD at the University of Nottingham in 1978 with a thesis entitled "Ergodic properties of stopping time transformations”. She then went to study at Université Paris-Sud supervised by Jean Bretagnolle was awarded a Doctorat d'État for a thesis entitled "Processus spatialement dépendants: convergence vers la normalité, tests d'association et applications" in 1989.
Career
Richardson has been the MRC Research Professor of Biostatistics at the University of Cambridge, bye-fellow of Emmanuel College and Director of the Medical Research Council Biostatistics Unit since 2012. Previously, she was chair in Biostatistics at Imperial College London from 2000 and before that she was Directeur de Recherches at INSERM and held lectureships at Warwick University and the University of Paris V.
She is co-editor of the volume Markov Chain Monte Carlo in Practice with Wally Gilks and David Spiegelhalter.
Research
Richardson has made significant contributions to Bayesian statistical methodology and the application of Markov chain Monte Carlo. Her expertise is in spatial statistics with applications to geographic epidemiology and in biostatistics with applications in biochemical modeling, in particular modeling of gene expression data.
Recognition
Richardson was awarded the Guy Medal of The Royal Statistical Society in Silver in 2009.
She is also a Fellow of the Institute of Mathematical Statistics, of the International Society for Bayesian Analysis and of the Academy of Medical Sciences.
She was appointed Commander of the Order of the British Empire (CBE) in the 2019 Birthday Honours for services to medical statistics.
References
Year of birth missing (living people)
French statisticians
Bayesian statisticians
Women statisticians
Academics of Imperial College London
Living people
Fellows of the Institute of Mathematical Statistics
Fellows of the Academy of Medical Sciences (United Kingdom)
Commanders of the Order of the British Empire
French emigrants to England
Naturalised citizens of the United Kingdom
British statisticians
Mathematical statisticians |
https://en.wikipedia.org/wiki/Beurling%20algebra | In mathematics, the term Beurling algebra is used for different algebras introduced by , usually it is an algebra of periodic functions with Fourier series
Example
We may consider the algebra of those functions f where the majorants
of the Fourier coefficients an are summable. In other words
Example
We may consider a weight function w on such that
in which case
is a unitary commutative Banach algebra.
These algebras are closely related to the Wiener algebra.
References
Fourier series
Algebras |
https://en.wikipedia.org/wiki/Geometric%20flow | In the mathematical field of differential geometry, a geometric flow, also called a geometric evolution equation, is a type of partial differential equation for a geometric object such as a Riemannian metric or an embedding. It is not a term with a formal meaning, but is typically understood to refer to parabolic partial differential equations.
Certain geometric flows arise as the gradient flow associated to a functional on a manifold which has a geometric interpretation, usually associated with some extrinsic or intrinsic curvature. Such flows are fundamentally related to the calculus of variations, and include mean curvature flow and Yamabe flow.
Examples
Extrinsic
Extrinsic geometric flows are flows on embedded submanifolds, or more generally
immersed submanifolds. In general they change both the Riemannian metric and the immersion.
Mean curvature flow, as in soap films; critical points are minimal surfaces
Curve-shortening flow, the one-dimensional case of the mean curvature flow
Willmore flow, as in minimax eversions of spheres
Inverse mean curvature flow
Intrinsic
Intrinsic geometric flows are flows on the Riemannian metric, independent of any embedding or immersion.
Ricci flow, as in the solution of the Poincaré conjecture, and Richard S. Hamilton's proof of the uniformization theorem
Calabi flow, a flow for Kähler metrics
Yamabe flow
Classes of flows
Important classes of flows are curvature flows, variational flows (which extremize some functional), and flows arising as solutions to parabolic partial differential equations. A given flow frequently admits all of these interpretations, as follows.
Given an elliptic operator the parabolic PDE yields a flow, and stationary states for the flow are solutions to the elliptic partial differential equation
If the equation is the Euler–Lagrange equation for some functional then the flow has a variational interpretation as the gradient flow of and stationary states of the flow correspond to critical points of the functional.
In the context of geometric flows, the functional is often the norm of some curvature.
Thus, given a curvature one can define the functional which has Euler–Lagrange equation for some elliptic operator and associated parabolic PDE
The Ricci flow, Calabi flow, and Yamabe flow arise in this way (in some cases with normalizations).
Curvature flows may or may not preserve volume (the Calabi flow does, while the Ricci flow does not), and if not, the flow may simply shrink or grow the manifold, rather than regularizing the metric. Thus one often normalizes the flow, for instance, by fixing the volume.
See also
Harmonic map heat flow
References |
https://en.wikipedia.org/wiki/Pugh%27s%20closing%20lemma | In mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows:
Let be a diffeomorphism of a compact smooth manifold . Given a nonwandering point of , there exists a diffeomorphism arbitrarily close to in the topology of such that is a periodic point of .
Interpretation
Pugh's closing lemma means, for example, that any chaotic set in a bounded continuous dynamical system corresponds to a periodic orbit in a different but closely related dynamical system. As such, an open set of conditions on a bounded continuous dynamical system that rules out periodic behaviour also implies that the system cannot behave chaotically; this is the basis of some autonomous convergence theorems.
See also
Smale's problems
References
Further reading
Dynamical systems
Lemmas in analysis
Limit sets |
https://en.wikipedia.org/wiki/Hadi%20Jafari | Hadi Jafari (, born 4 August 1982) is an Iranian football midfielder who currently plays for Sepahan F.C. in the Iran Pro League.
Club career
Club career statistics
Assist Goals
Honours
Club
Iran's Premier Football League
Winner: 1
2009/10 with Sepahan
References
1982 births
Living people
Iranian men's footballers
Men's association football midfielders
Sepahan S.C. footballers
Foolad Natanz F.C. players
Gostaresh Foulad F.C. players
Sportspeople from Isfahan province |
https://en.wikipedia.org/wiki/Farshad%20Bahadorani | Farshad Bahadorani (, born August 28, 1982) is an Iranian football midfielder who currently plays for Zob Ahan in the Iran Pro League.
Club career
Club career statistics
Last Update 24 August 2012
Assist Goals
References
1982 births
Living people
Iranian men's footballers
Men's association football midfielders
Sepahan S.C. footballers
Zob Ahan Esfahan F.C. players
F.C. Aboomoslem players
Footballers from Isfahan |
https://en.wikipedia.org/wiki/Weierstrass%20ring | In mathematics, a Weierstrass ring, named by Nagata after Karl Weierstrass, is a commutative local ring that is Henselian, pseudo-geometric, and such that any quotient ring by a prime ideal is a finite extension of a regular local ring.
Examples
The Weierstrass preparation theorem can be used to show that the ring of convergent power series over the complex numbers in a finite number of variables is a Wierestrass ring. The same is true if the complex numbers are replaced by a perfect field with a valuation.
Every ring that is a finitely-generated module over a Weierstrass ring is also a Weierstrass ring.
References
Bibliography
Commutative algebra |
https://en.wikipedia.org/wiki/Henselian%20ring | In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative.
Some standard references for Hensel rings are , , and .
Definitions
In this article rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings.
A local ring R with maximal ideal m is called Henselian if Hensel's lemma holds. This means that if P is a monic polynomial in R[x], then any factorization of its image P in (R/m)[x] into a product of coprime monic polynomials can be lifted to a factorization in R[x].
A local ring is Henselian if and only if every finite ring extension is a product of local rings.
A Henselian local ring is called strictly Henselian if its residue field is separably closed.
By abuse of terminology, a field with valuation is said to be Henselian if its valuation ring is Henselian. That is the case if and only if extends uniquely to every finite extension of (resp. to every finite separable extension of , resp. to , resp. to ).
A ring is called Henselian if it is a direct product of a finite number of Henselian local rings.
Properties
Assume that is an Henselian field. Then every algebraic extension of is henselian (by the fourth definition above).
If is a Henselian field and is algebraic over , then for every conjugate of over , . This follows from the fourth definition, and from the fact that for every K-automorphism of , is an extension of . The converse of this assertion also holds, because for a normal field extension , the extensions of to are known to be conjugated.
Henselian rings in algebraic geometry
Henselian rings are the local rings with respect to the Nisnevich topology in the sense that if is a Henselian local ring, and is a Nisnevich covering of , then one of the is an isomorphism. This should be compared to the fact that for any Zariski open covering of the spectrum of a local ring , one of the is an isomorphism. In fact, this property characterises Henselian rings, resp. local rings.
Likewise strict Henselian rings are the local rings of geometric points in the étale topology.
Henselization
For any local ring A there is a universal Henselian ring B generated by A, called the Henselization of A, introduced by , such that any local homomorphism from A to a Henselian ring can be extended uniquely to B. The Henselization of A is unique up to unique isomorphism. The Henselization of A is an algebraic substitute for the completion of A. The Henselization of A has the same completion and residue field as A and is a flat module over A. If A is Noetherian, reduced, normal, regular, or excellent then so is its Henselization. For example, the Henselization of the ring of polynomials k[x,y,...] localized at the point (0,0,...) is the ring of algebraic formal po |
https://en.wikipedia.org/wiki/Polynomial%20Diophantine%20equation | In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation for which one seeks solutions restricted to be polynomials in the indeterminate. A Diophantine equation, in general, is one where the solutions are restricted to some algebraic system, typically integers. (In another usage ) Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made initial studies of integer Diophantine equations.
An important type of polynomial Diophantine equations takes the form:
where a, b, and c are known polynomials, and we wish to solve for s and t.
A simple example (and a solution) is:
A necessary and sufficient condition for a polynomial Diophantine equation to have a solution is for c to be a multiple of the GCD of a and b. In the example above, the GCD of a and b was 1, so solutions would exist for any value of c.
Solutions to polynomial Diophantine equations are not unique. Any multiple of (say ) can be used to transform and into another solution :
Some polynomial Diophantine equations can be solved using the extended Euclidean algorithm, which works as well with polynomials as it does with integers.
References
Algebra |
https://en.wikipedia.org/wiki/Nisnevich%20topology | In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles.
Definition
A morphism of schemes is called a Nisnevich morphism if it is an étale morphism such that for every (possibly non-closed) point x ∈ X, there exists a point y ∈ Y in the fiber such that the induced map of residue fields k(x) → k(y) is an isomorphism. Equivalently, f must be flat, unramified, locally of finite presentation, and for every point x ∈ X, there must exist a point y in the fiber such that k(x) → k(y) is an isomorphism.
A family of morphisms {uα : Xα → X} is a Nisnevich cover if each morphism in the family is étale and for every (possibly non-closed) point x ∈ X, there exists α and a point y ∈ Xα s.t. uα(y) = x and the induced map of residue fields k(x) → k(y) is an isomorphism. If the family is finite, this is equivalent to the morphism from to X being a Nisnevich morphism. The Nisnevich covers are the covering families of a pretopology on the category of schemes and morphisms of schemes. This generates a topology called the Nisnevich topology. The category of schemes with the Nisnevich topology is notated Nis.
The small Nisnevich site of X has as underlying category the same as the small étale site, that is to say, objects are schemes U with a fixed étale morphism U → X and the morphisms are morphisms of schemes compatible with the fixed maps to X. Admissible coverings are Nisnevich morphisms.
The big Nisnevich site of X has as underlying category schemes with a fixed map to X and morphisms the morphisms of X-schemes. The topology is the one given by Nisnevich morphisms.
The Nisnevich topology has several variants which are adapted to studying singular varieties. Covers in these topologies include resolutions of singularities or weaker forms of resolution.
The cdh topology allows proper birational morphisms as coverings.
The h topology allows De Jong's alterations as coverings.
The l′ topology allows morphisms as in the conclusion of Gabber's local uniformization theorem.
The cdh and l′ topologies are incomparable with the étale topology, and the h topology is finer than the étale topology.
Equivalent conditions for a Nisnevich cover
Assume the category consists of smooth schemes over a qcqs (quasi-compact and quasi-separated) scheme, then the original definition due to NisnevichRemark 3.39, which is equivalent to the definition above, for a family of morphisms of schemes to be a Nisnevich covering is if
Every is étale; and
For all field , on the level of -points, the (set-theoretic) coproduct of all covering morphisms is surjective.
The following yet another equivalent condition for Nisnevich covers is due to Lurie: The Nisnevich topology is generated by all f |
https://en.wikipedia.org/wiki/Yamabe%20flow | In differential geometry, the Yamabe flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold. First introduced by Richard S. Hamilton,
Yamabe flow is for noncompact manifolds, and is the
negative L2-gradient flow of the (normalized) total scalar curvature, restricted to a given conformal class: it can be interpreted as deforming a Riemannian metric to a conformal metric of constant scalar curvature, when this flow converges.
The Yamabe flow was introduced in response to Richard S. Hamilton's own work on the Ricci flow and Rick Schoen's solution of the Yamabe problem on manifolds of positive conformal Yamabe invariant.
Main results
The fixed points of the Yamabe flow are metrics of constant scalar curvature in the given conformal class. The flow was first studied in the 1980s in unpublished notes of Richard Hamilton. Hamilton conjectured that, for every initial metric, the flow converges to a conformal metric of constant scalar curvature. This was verified by Rugang Ye in the locally conformally flat case. Later, Simon Brendle proved convergence of the flow for all conformal classes and arbitrary initial metrics. The limiting constant-scalar-curvature metic is typically no longer a Yamabe minimizer in this context. While the compact case is settled, the flow on complete, non-compact manifolds is not completely understood, and remains a topic of current research.
Notes
Geometric flow |
https://en.wikipedia.org/wiki/Hojat%20Zadmahmoud | Hojatolah Zadmahmoud (, born September 13, 1983) is an Iranian football midfielder who currently plays for Gostaresh Foolad F.C. in the Azadegan League.
Club career
Club career statistics
Last update 4 May 2011
Assist Goals
References
1983 births
Living people
Iranian men's footballers
Men's association football midfielders
Persian Gulf Pro League players
Azadegan League players
Esteghlal Ahvaz F.C. players
Sepahan S.C. footballers
Foolad F.C. players
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Mohsen%20Hamidi | Mohsen Hamidi (, born September 30, 1985) is an Iranian football midfielder who most recently played for Sanat Naft in the Persian Gulf Pro League.
Club career
Club career statistics
Assist Goals
References
External links
1985 births
Living people
Iranian men's footballers
Men's association football midfielders
Foolad F.C. players
Shahin Bushehr F.C. players
Sepahan S.C. footballers
Iranian Arab sportspeople
Aluminium Hormozgan F.C. players
Sanat Naft Abadan F.C. players
PAS Hamedan F.C. players
Esteghlal Ahvaz F.C. players
Gostaresh Foulad F.C. players
Footballers from Ahvaz |
https://en.wikipedia.org/wiki/Quarters%20of%20Paris | Each of the 20 arrondissements of Paris is officially divided into 4 quartiers. Outside administrative use (census statistics and the localisation of post offices and other government services), they are very rarely referenced by Parisians themselves, and have no specific administration or political representation attached to them.
References
Bibliography
Districts of Paris
History of Paris
Arrondissements of Paris |
https://en.wikipedia.org/wiki/Burqin%2C%20Palestine | Burqin () is a Palestinian town in the northern West Bank located 5 km west of Jenin. According to the Palestinian Central Bureau of Statistics (PCBS) census, its population was 5,685 in 2007 and 7,126 in 2017. The majority of Burqin's residents are Muslims, and 20 Christian families live in the town. The Byzantine-era Burqin Church or St. George's Church is one of the oldest churches in the world.
History
Burqin is an ancient site, situated on a slope, with old stones reused in the town houses.
It was mentioned under the name Burqana, in the 14th century BCE Amarna letters, as one of several cities conquered by the Canaanite warlord Lab'ayu in the Dothan Valley and southern Jezreel Valley.
Pottery sherds from the Early Bronze I, Early Bronze IIB, Late Bronze III, Iron Age I, Iron Age II, late Roman, Byzantine, Umayyad/Abbasid, Medieval and early Ottoman era have been found.
Ottoman era
In 1517, the village was included in the Ottoman Empire with the rest of Palestine, and in the 1596 tax-records it appeared as Bruqin, located in the Nahiya of Jabal Sami of the Liwa of Nablus. The population was 23 households and 4 bachelors, all Muslim. They paid a tax rate of 33.3% on agricultural products, which included wheat, barley, summer crops, olive trees, occasional revenues, goats and beehives; a total of 7980 akçe.
In 1799, Pierre Jacotin placed the village, named Berkin, nearly straight west of Jenin on his map. In 1838 Edward Robinson placed Burqin as being in the District of Jenin, also called "Haritheh esh-Shemaliyeh".
In 1863, when Victor Guérin visited, he found the village to have about 1,000 inhabitants, all Muslim with the exception of 90 Greek Orthodox Christians. He further noted that "Some 30 excavated cisterns are evidence that this village sits upon an ancient settlement."
In 1870/1871 (1288 AH), an Ottoman census listed the village in the nahiya (sub-district) of al-Sha'rawiyya al-Sharqiyya.
In 1872, Claude Reignier Conder visited Burqin during his surveying work in Palestine. He was met by the local curé and shown the church. It had a stone screen on the east, shutting off three apses.
In 1882, the PEF's Survey of Western Palestine described Burkin as "A village of Greek Christians, with a small modern church for the Greek rite. It stands on the side of a white hill, with a good well below on the north, and olives near it."
British Mandate era
In the 1922 census of Palestine, conducted by the British Mandate authorities, Burqin had a population of 883; 871 Muslims and 12 Christians males, where all the Christians were Orthodox. This had increased in the 1931 census to a population of 1,086; 1,010 Muslim and 76 Christians, in a total of 227 inhabited houses.
In the 1945 statistics the population were 1,540; 1,430 Muslims and 110 Christians, with a total of 19,447 dunams of land, according to an official land and population survey. Of this, 3,902 dunams were used for plantations and irrigable land, 11,219 dunams for |
https://en.wikipedia.org/wiki/Levi%20Risamasu | Levi Risamasu (born 23 November 1982) is a Dutch former footballer who played as a midfielder.
Career statistics
Source:
References
Living people
1982 births
People from Nieuwerkerk aan den IJssel
Men's association football midfielders
Dutch men's footballers
Dutch people of Indonesian descent
Dutch people of Moluccan descent
NAC Breda players
AGOVV players
Excelsior Rotterdam players
Eredivisie players
Eerste Divisie players
Footballers from South Holland |
https://en.wikipedia.org/wiki/National%20Office%20of%20Statistics | The National Office of Statistics (NOS, , ONS, ) is the Algerian ministry charged with the collection and publication of statistics related to the economy, population, and society of Algeria at national and local levels. Its head office is in Algiers.
History
It was established after the independence of Algeria in 1964, and originally named National Commission for the Census of the Population (CNRP, ).
In 1966, the office carried out the first census of the Algerian population after the independence of the country. Its missions, as well as its name, have evolved in parallel with the demographic, economic and social evolution of Algeria for which the office collects, processes and publishes statistics in these fields.
References
External links
Publications in English
National Office of Statistics
Algeria
Government agencies of Algeria |
https://en.wikipedia.org/wiki/J.%20Peter%20May | Jon Peter May (born September 16, 1939 in New York) is an American mathematician working in the fields of algebraic topology, category theory, homotopy theory, and the foundational aspects of spectra. He is known, in particular, for the May spectral sequence and for coining the term operad. The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer).
Education and career
May received a Bachelor of Arts degree from Swarthmore College in 1960 and a Doctor of Philosophy degree from Princeton University in 1964. His thesis, written under the direction of John Moore, was titled The cohomology of restricted Lie algebras and of Hopf algebras: Application to the Steenrod algebra.
From 1964 to 1967, May taught at Yale University. He has been a faculty member at the University of Chicago since 1967, and a professor since 1970.
Awards
In 2012 he became a fellow of the American Mathematical Society. He has advised over 60 doctoral students, including Mark Behrens, Andrew Blumberg, Frederick Cohen, Ib Madsen, Emily Riehl, Michael Shulman, and Zhouli Xu.
References
External links
May's homepage at the University of Chicago
Jon Peter May at the Mathematics Genealogy Project
20th-century American mathematicians
21st-century American mathematicians
Topologists
University of Chicago faculty
Yale University faculty
Princeton University alumni
Swarthmore College alumni
Fellows of the American Mathematical Society
1939 births
Living people
Mathematicians from New York (state) |
https://en.wikipedia.org/wiki/Pfaffian%20function | In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally introduced by Askold Khovanskii in the 1970s, but are named after German mathematician Johann Pfaff.
Basic definition
Some functions, when differentiated, give a result which can be written in terms of the original function. Perhaps the simplest example is the exponential function, f(x) = ex. If we differentiate this function we get ex again, that is
Another example of a function like this is the reciprocal function, g(x) = 1/x. If we differentiate this function we will see that
Other functions may not have the above property, but their derivative may be written in terms of functions like those above. For example, if we take the function h(x) = ex log(x) then we see
Functions like these form the links in a so-called Pfaffian chain. Such a chain is a sequence of functions, say f1, f2, f3, etc., with the property that if we differentiate any of the functions in this chain then the result can be written in terms of the function itself and all the functions preceding it in the chain (specifically as a polynomial in those functions and the variables involved). So with the functions above we have that f, g, h is a Pfaffian chain.
A Pfaffian function is then just a polynomial in the functions appearing in a Pfaffian chain and the function argument. So with the Pfaffian chain just mentioned, functions such as F(x) = x3f(x)2 − 2g(x)h(x) are Pfaffian.
Rigorous definition
Let U be an open domain in Rn. A Pfaffian chain of order r ≥ 0 and degree α ≥ 1 in U is a sequence of real analytic functions f1,..., fr in U satisfying differential equations
for i = 1, ..., r where Pi, j ∈ R[x1, ..., xn, y1, ..., yi] are polynomials of degree ≤ α. A function f on U is called a Pfaffian function of order r and degree (α, β) if
where P ∈ R[x1, ..., xn, y1, ..., yr] is a polynomial of degree at most β ≥ 1. The numbers r, α, and β are collectively known as the format of the Pfaffian function, and give a useful measure of its complexity.
Examples
The most trivial examples of Pfaffian functions are the polynomials in R[X]. Such a function will be a polynomial in a Pfaffian chain of order r = 0, that is the chain with no functions. Such a function will have α = 0 and β equal to the degree of the polynomial.
Perhaps the simplest nontrivial Pfaffian function is f(x) = ex. This is Pfaffian with order r = 1 and α = β = 1 due to the equation f = f.
Inductively, one may define f1(x) = exp(x) and fm+1(x) = exp(fm(x)) for 1 ≤ m < r. Then fm′ = f1f2···fm. So this is a Pfaffian chain of order r and degree α = r.
All of the algebraic functions are Pfaffian on suitable domains, as are the hyperbolic functions. The trigonometric functions on bounded intervals are Pfaffian, but they must be formed indirectly. For example, the function cos(x) is a polynomial in the Pfaffian chain tan(x/2), cos2(x/2) on the |
https://en.wikipedia.org/wiki/Piskacek%27s%20sign | In medicine, Piskacek's sign is an indication of pregnancy. This sign, however, may or may not be a concrete probability of pregnancy along with other signs of early pregnancy. Other signs of early pregnancy include Goodell, Hegar, von Braun Fernwald, Hartman sign and Chadwick.
Implantation of Zygote is eccentric so that growth of uterus is unequal in early pregnancy known as Piskacek's sign.
Specifically, Piskacek's sign consists noting a palpable lateral bulge or soft prominence one of the locations where the fallopian tube meets the uterus. Piskacek's sign can be noted in the seventh to eight week of gestation. Non pregnant uterus is pyriform in shape. By 12 weeks of gestation it becomes globular. In lateral implantation, there is asymmetrical enlargement of the uterus. One half of the uterus where the implantation occurred is firm while the other half is soft. This is known as Piskacek's sign.
The sign is named after Ludwig Piskaçek.
References
Medical signs
Obstetrics
Midwifery |
https://en.wikipedia.org/wiki/Khaled%20Saad | Khaled Saad Salem Al-Malta'ah () is a Jordanian former footballer.
Career statistics
International
Scores and results list Oman's goal tally first.
References
External links
1981 births
Living people
Jordanian men's footballers
Jordan men's international footballers
Jordanian expatriate men's footballers
Men's association football defenders
2004 AFC Asian Cup players
Nejmeh SC players
Expatriate men's footballers in Lebanon
Jordanian expatriate sportspeople in Lebanon
Zamalek SC players
Expatriate men's footballers in Egypt
Jordanian expatriate sportspeople in Egypt
Salalah SC players
Fanja SC players
Expatriate men's footballers in Oman
Jordanian expatriate sportspeople in Oman
Al-Faisaly SC players
Egyptian Premier League players
Lebanese Premier League players |
https://en.wikipedia.org/wiki/Geometallurgy | Geometallurgy relates to the practice of combining geology or geostatistics with metallurgy, or, more specifically, extractive metallurgy, to create a spatially or geologically based predictive model for mineral processing plants. It is used in the hard rock mining industry for risk management and mitigation during mineral processing plant design. It is also used, to a lesser extent, for production planning in more variable ore deposits.
There are four important components or steps to developing a geometallurgical program,:
the geologically informed selection of a number of ore samples
laboratory-scale test work to determine the ore's response to mineral processing unit operations
the distribution of these parameters throughout the orebody using an accepted geostatistical technique
the application of a mining sequence plan and mineral processing models to generate a prediction of the process plant behavior
Sample selection
The sample mass and size distribution requirements are dictated by the kind of mathematical model that will be used to simulate the process plant, and the test work required to provide the appropriate model parameters. Flotation testing usually requires several kg of sample and grinding/hardness testing can required between 2 and 300 kg.
The sample selection procedure is performed to optimize granularity, sample support, and cost. Samples are usually core samples composited over the height of the mining bench. For hardness parameters, the variogram often increases rapidly near the origin and can reach the sill at distances significantly smaller than the typical drill hole collar spacing. For this reason the incremental model precision due to additional test work is often simply a consequence of the central limit theorem, and secondary correlations are sought to increase the precision without incurring additional sampling and testing costs. These secondary correlations can involve multi-variable regression analysis with other, non-metallurgical, ore parameters and/or domaining by rock type, lithology, alteration, mineralogy, or structural domains.
Test work
The following tests are commonly used for geometallurgical modeling:
Bond ball mill work index test
Modified or comparative Bond ball mill index
Bond rod mill work index and Bond low energy impact crushing work index
SAGDesign test
SMC test
JK drop-weight test
Point load index test
Sag Power Index test (SPI(R))
MFT test
FKT, SKT, and SKT-WS tests
Geostatistics
Block kriging is the most common geostatistical method used for interpolating metallurgical index parameters and it is often applied on a domain basis. Classical geostatistics require that the estimation variable be additive, and there is currently some debate on the additive nature of the metallurgical index parameters measured by the above tests. The Bond ball mill work index test is thought to be additive because of its units of energy; nevertheless, experimental blending results sh |
https://en.wikipedia.org/wiki/Tarry%20point | In geometry, the Tarry point for a triangle is a point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangle's first Brocard triangle . The Tarry point lies on the other endpoint of the diameter of the circumcircle drawn through the Steiner point. The point is named for Gaston Tarry.
See also
Concurrent lines
Notes
Triangle centers |
https://en.wikipedia.org/wiki/Brocard%20circle | In geometry, the Brocard circle (or seven-point circle) is a circle derived from a given triangle. It passes through the circumcenter and symmedian of the triangle, and is centered at the midpoint of the line segment joining them (so that this segment is a diameter).
Equation
In terms of the side lengths , , and of the given triangle, and the areal coordinates for points inside the triangle (where the -coordinate of a point is the area of the triangle made by that point with the side of length , etc), the Brocard circle consists of the points satisfying the equation
Related points
The two Brocard points lie on this circle, as do the vertices of the Brocard triangle.
These five points, together with the other two points on the circle (the circumcenter and symmedian), justify the name "seven-point circle".
The Brocard circle is concentric with the first Lemoine circle.
Special cases
If the triangle is equilateral, the circumcenter and symmedian coincide and therefore the Brocard circle reduces to a single point.
History
The Brocard circle is named for Henri Brocard, who presented a paper on it to the French Association for the Advancement of Science in Algiers in 1881.
References
External links
See also
Nine-point circle
Circles defined for a triangle |
https://en.wikipedia.org/wiki/Alternating-direction%20implicit%20method | In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve parabolic and elliptic partial differential equations, and is a classic method used for modeling heat conduction and solving the diffusion equation in two or more dimensions. It is an example of an operator splitting method.
ADI for matrix equations
The method
The ADI method is a two step iteration process that alternately updates the column and row spaces of an approximate solution to . One ADI iteration consists of the following steps:1. Solve for , where 2. Solve for , where .
The numbers are called shift parameters, and convergence depends strongly on the choice of these parameters. To perform iterations of ADI, an initial guess is required, as well as shift parameters, .
When to use ADI
If and , then can be solved directly in using the Bartels-Stewart method. It is therefore only beneficial to use ADI when matrix-vector multiplication and linear solves involving and can be applied cheaply.
The equation has a unique solution if and only if , where is the spectrum of . However, the ADI method performs especially well when and are well-separated, and and are normal matrices. These assumptions are met, for example, by the Lyapunov equation when is positive definite. Under these assumptions, near-optimal shift parameters are known for several choices of and . Additionally, a priori error bounds can be computed, thereby eliminating the need to monitor the residual error in implementation.
The ADI method can still be applied when the above assumptions are not met. The use of suboptimal shift parameters may adversely affect convergence, and convergence is also affected by the non-normality of or (sometimes advantageously). Krylov subspace methods, such as the Rational Krylov Subspace Method, are observed to typically converge more rapidly than ADI in this setting, and this has led to the development of hybrid ADI-projection methods.
Shift-parameter selection and the ADI error equation
The problem of finding good shift parameters is nontrivial. This problem can be understood by examining the ADI error equation. After iterations, the error is given by
Choosing results in the following bound on the relative error:
where is the operator norm. The ideal set of shift parameters defines a rational function that minimizes the quantity . If and are normal matrices and have eigendecompositions and , then
.
Near-optimal shift parameters
Near-optimal shift parameters are known in certain cases, such as when and , where and are disjoint intervals on the real line. The Lyapunov equation , for example, satisfies these assumptions when is positive defini |
https://en.wikipedia.org/wiki/Topological%20indistinguishability | In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are points in X, and Nx is the set of all neighborhoods that contain x, and Ny is the set of all neighborhoods that contain y, then x and y are "topologically indistinguishable" if and only if Nx = Ny.
(See Hausdorff's axiomatic neighborhood systems.)
Intuitively, two points are topologically indistinguishable if the topology of X is unable to discern between the points.
Two points of X are topologically distinguishable if they are not topologically indistinguishable. This means there is an open set containing precisely one of the two points (equivalently, there is a closed set containing precisely one of the two points). This open set can then be used to distinguish between the two points. A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. This is the weakest of the separation axioms.
Topological indistinguishability defines an equivalence relation on any topological space X. If x and y are points of X we write x ≡ y for "x and y are topologically indistinguishable". The equivalence class of x will be denoted by [x].
Examples
For T0 spaces (in particular, for Hausdorff spaces) the notion of topological indistinguishability is trivial, so one must look to non-T0 spaces to find interesting examples. On the other hand, regularity and normality do not imply T0, so we can find examples with these properties. In fact, almost all of the examples given below are completely regular.
In an indiscrete space, any two points are topologically indistinguishable.
In a pseudometric space, two points are topologically indistinguishable if and only if the distance between them is zero.
In a seminormed vector space, x ≡ y if and only if ‖x − y‖ = 0.
For example, let L2(R) be the space of all measurable functions from R to R which are square integrable (see Lp space). Then two functions f and g in L2(R) are topologically indistinguishable if and only if they are equal almost everywhere.
In a topological group, x ≡ y if and only if x−1y ∈ cl{e} where cl{e} is the closure of the trivial subgroup. The equivalence classes are just the cosets of cl{e} (which is always a normal subgroup).
Uniform spaces generalize both pseudometric spaces and topological groups. In a uniform space, x ≡ y if and only if the pair (x, y) belongs to every entourage. The intersection of all the entourages is an equivalence relation on X which is just that of topological indistinguishability.
Let X have the initial topology with respect to a family of functions . Then two points x and y in X will be topologically indistinguishable if the family does not separate them (i.e. for all ).
Given any equivalence relation on a set X there is a topology on X for which the notion of topological indistinguishability agrees with the given equivalence relation. One can simply take the equivalence |
https://en.wikipedia.org/wiki/Trace%20diagram | In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the trace and determinant of a matrix. Several results in linear algebra, such as Cramer's Rule and the Cayley–Hamilton theorem, have simple diagrammatic proofs. They are closely related to Penrose's graphical notation.
Formal definition
Let V be a vector space of dimension n over a field F (with n≥2), and let Hom(V,V) denote the linear transformations on V. An n-trace diagram is a graph , where the sets Vi (i = 1, 2, n) are composed of vertices of degree i, together with the following additional structures:
a ciliation at each vertex in the graph, which is an explicit ordering of the adjacent edges at that vertex;
a labeling V2 → Hom(V,V) associating each degree-2 vertex to a linear transformation.
Note that V2 and Vn should be considered as distinct sets in the case n = 2. A framed trace diagram is a trace diagram together with a partition of the degree-1 vertices V1 into two disjoint ordered collections called the inputs and the outputs.
The "graph" underlying a trace diagram may have the following special features, which are not always included in the standard definition of a graph:
Loops are permitted (a loop is an edge that connects a vertex to itself).
Edges that have no vertices are permitted, and are represented by small circles.
Multiple edges between the same two vertices are permitted.
Drawing conventions
When trace diagrams are drawn, the ciliation on an n-vertex is commonly represented by a small mark between two of the incident edges (in the figure above, a small red dot); the specific ordering of edges follows by proceeding counter-clockwise from this mark.
The ciliation and labeling at a degree-2 vertex are combined into a single directed node that allows one to differentiate the first edge (the incoming edge) from the second edge (the outgoing edge).
Framed diagrams are drawn with inputs at the bottom of the diagram and outputs at the top of the diagram. In both cases, the ordering corresponds to reading from left to right.
Correspondence with multilinear functions
Every framed trace diagram corresponds to a multilinear function between tensor powers of the vector space V. The degree-1 vertices correspond to the inputs and outputs of the function, while the degree-n vertices correspond to the generalized Levi-Civita symbol (which is an anti-symmetric tensor related to the determinant). If a diagram has no output strands, its function maps tensor products to a scalar. If there are no degree-1 vertices, the diagram is said to be closed and its corresponding function may be identified with a scalar.
By definition, a trace diagram's function is computed using signed graph coloring. For each edge coloring of the graph's edges by n labels, so that no two edges adjacen |
https://en.wikipedia.org/wiki/Birkhoff%E2%80%93Grothendieck%20theorem | In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over is a direct sum of holomorphic line bundles. The theorem was proved by , and is more or less equivalent to Birkhoff factorization introduced by .
Statement
More precisely, the statement of the theorem is as the following.
Every holomorphic vector bundle on is holomorphically isomorphic to a direct sum of line bundles:
The notation implies each summand is a Serre twist some number of times of the trivial bundle. The representation is unique up to permuting factors.
Generalization
The same result holds in algebraic geometry for algebraic vector bundle over for any field .
It also holds for with one or two orbifold points, and for chains of projective lines meeting along nodes.
Applications
One application of this theorem is it gives a classification of all coherent sheaves on . We have two cases, vector bundles and coherent sheaves supported along a subvariety, so where n is the degree of the fat point at . Since the only subvarieties are points, we have a complete classification of coherent sheaves.
See also
Algebraic geometry of projective spaces
Euler sequence
Splitting principle
K-theory
Jumping line
References
Further reading
External links
Roman Bezrukavnikov. 18.725 Algebraic Geometry (LEC # 24 Birkhoff–Grothendieck, Riemann-Roch, Serre Duality) Fall 2015. Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons BY-NC-SA.
Vector bundles
Theorems in projective geometry
Theorems in algebraic geometry
Theorems in complex geometry |
https://en.wikipedia.org/wiki/Zeroth-order%20logic | Zeroth-order logic is a branch of logic without variables or quantifiers. Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus, but an alternative definition extends propositional logic by adding constants, operations, and relations on non-Boolean values. Every zeroth-order language in this broader sense is complete and compact.
References
Propositional calculus
Systems of formal logic |
https://en.wikipedia.org/wiki/Mass%20point | Mass point may refer to:
Mass point geometry
Point mass in physics
The values of a probability mass function in probability and statistics |
https://en.wikipedia.org/wiki/Finite%20topological%20space | In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements.
Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can
lend good insight to a variety of questions".
Topologies on a finite set
Let be a finite set. A topology on is a subset of (the power set of ) such that
and .
if then .
if then .
In other words, a subset of is a topology if contains both and and is closed under arbitrary unions and intersections. Elements of are called open sets. The general description of topological spaces requires that a topology be closed under arbitrary (finite or infinite) unions of open sets, but only under intersections of finitely many open sets. Here, that distinction is unnecessary. Since the power set of a finite set is finite there can be only finitely many open sets (and only finitely many closed sets).
A topology on a finite set can also be thought of as a sublattice of which includes both the bottom element and the top element .
Examples
0 or 1 points
There is a unique topology on the empty set ∅. The only open set is the empty one. Indeed, this is the only subset of ∅.
Likewise, there is a unique topology on a singleton set {a}. Here the open sets are ∅ and {a}. This topology is both discrete and trivial, although in some ways it is better to think of it as a discrete space since it shares more properties with the family of finite discrete spaces.
For any topological space X there is a unique continuous function from ∅ to X, namely the empty function. There is also a unique continuous function from X to the singleton space {a}, namely the constant function to a. In the language of category theory the empty space serves as an initial object in the category of topological spaces while the singleton space serves as a terminal object.
2 points
Let X = {a,b} be a set with 2 elements. There are four distinct topologies on X:
{∅, {a,b}} (the trivial topology)
{∅, {a}, {a,b}}
{∅, {b}, {a,b}}
{∅, {a}, {b}, {a,b}} (the discrete topology)
The second and third topologies above are easily seen to be homeomorphic. The function from X to itself which swaps a and b is a homeomorphism. A topological space homeomorphic to one of these is called a Sierpiński space. So, in fact, there are only three inequivalent topologies on a two-point set: the trivial one, the discrete one, and the Sierpiński topology.
The specialization preorder on the Sierpiński space {a,b} with {b} open is given by: a ≤ a, b ≤ b, and a ≤ b.
3 points
Let X = {a,b,c} be a set with 3 elements. There are 29 distinct topologies on X but only 9 inequivalent topologies:
{∅, {a,b,c}}
{∅, {c}, {a,b,c}}
{∅, {a,b}, {a,b,c}}
{∅, {c}, {a,b}, {a,b,c}}
{∅, {c}, {b,c}, {a,b, |
https://en.wikipedia.org/wiki/Perfectly%20normal | Perfectly normal may refer to:
Perfectly Normal, a 1990 Canadian comedy film directed by Yves Simoneau
Perfectly normal space, a type of topology |
https://en.wikipedia.org/wiki/Euclid%27s%20orchard | In mathematics, informally speaking, Euclid's orchard is an array of one-dimensional "trees" of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclid's orchard is the set of line segments from to , where and are positive integers.
The trees visible from the origin are those at lattice points , where and are coprime, i.e., where the fraction is in reduced form. The name Euclid's orchard is derived from the Euclidean algorithm.
If the orchard is projected relative to the origin onto the plane (or, equivalently, drawn in perspective from a viewpoint at the origin) the tops of the trees form a graph of Thomae's function. The point projects to
The solution to the Basel problem can be used to show that the proportion of points in the grid that have trees on them is approximately and that the error of this approximation goes to zero in the limit as goes to infinity.
See also
Opaque forest problem
References
External links
Euclid's Orchard, Grade 9-11 activities and problem sheet, Texas Instruments Inc.
Project Euler related problem
Greek_mathematics
Lattice points |
https://en.wikipedia.org/wiki/Character%20variety | In the mathematics of moduli theory, given an algebraic, reductive, Lie group and a finitely generated group , the -character variety of is a space of equivalence classes of group homomorphisms from to :
More precisely, acts on by conjugation, and two homomorphisms are defined to be equivalent (denoted ) if and only if their orbit closures intersect. This is the weakest equivalence relation on the set of conjugation orbits, , that yields a Hausdorff space.
Formulation
Formally, and when the reductive group is defined over the complex numbers , the -character variety is the spectrum of prime ideals of the ring of invariants (i.e., the affine GIT quotient).
Here more generally one can consider algebraically closed fields of prime characteristic. In this generality, character varieties are only algebraic sets and are not actual varieties. To avoid technical issues, one often considers the associated reduced space by dividing by the radical of 0 (eliminating nilpotents). However, this does not necessarily yield an irreducible space either. Moreover, if we replace the complex group by a real group we may not even get an algebraic set. In particular, a maximal compact subgroup generally gives a semi-algebraic set. On the other hand, whenever is free we always get an honest variety; it is singular however.
Examples
An interesting class of examples arise from Riemann surfaces: if is a Riemann surface then the -character variety of , or Betti moduli space, is the character variety of the surface group
.
For example, if and is the Riemann sphere punctured three times, so is free of rank two, then Henri G. Vogt, Robert Fricke, and Felix Klein
proved that the character variety is ; its coordinate ring is isomorphic to the complex polynomial ring in 3 variables, . Restricting to gives a closed real three-dimensional ball (semi-algebraic, but not algebraic).
Another example, also studied by Vogt and Fricke–Klein is the case with and is the Riemann sphere punctured four times, so is free of rank three. Then the character variety is isomorphic to the hypersurface in given by the equation
This character variety appears in the theory of the sixth Painleve equation, and
has a natural Poisson structure such that are Casimir functions, so the symplectic leaves are affine cubic surfaces of the form
Variants
This construction of the character variety is not necessarily the same as that of Marc Culler and Peter Shalen (generated by evaluations of traces), although when they do agree, since Claudio Procesi has shown that in this case the ring of invariants is in fact generated by only traces. Since trace functions are invariant by all inner automorphisms, the Culler–Shalen construction essentially assumes that we are acting by on even if
.
For instance, when is a free group of rank 2 and , the conjugation action is trivial and the -character variety is the torus
But the trace algebra is a strictly small subalgebra (there are |
https://en.wikipedia.org/wiki/Quantile%20regression | Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median (or other quantiles) of the response variable. Quantile regression is an extension of linear regression used when the conditions of linear regression are not met.
Advantages and applications
One advantage of quantile regression relative to ordinary least squares regression is that the quantile regression estimates are more robust against outliers in the response measurements. However, the main attraction of quantile regression goes beyond this and is advantageous when conditional quantile functions are of interest. Different measures of central tendency and statistical dispersion can be used to more comprehensively analyze the relationship between variables.
In ecology, quantile regression has been proposed and used as a way to discover more useful predictive relationships between variables in cases where there is no relationship or only a weak relationship between the means of such variables. The need for and success of quantile regression in ecology has been attributed to the complexity of interactions between different factors leading to data with unequal variation of one variable for different ranges of another variable.
Another application of quantile regression is in the areas of growth charts, where percentile curves are commonly used to screen for abnormal growth.
History
The idea of estimating a median regression slope, a major theorem about minimizing sum of the absolute deviances and a geometrical algorithm for constructing median regression was proposed in 1760 by Ruđer Josip Bošković, a Jesuit Catholic priest from Dubrovnik. He was interested in the ellipticity of the earth, building on Isaac Newton's suggestion that its rotation could cause it to bulge at the equator with a corresponding flattening at the poles. He finally produced the first geometric procedure for determining the equator of a rotating planet from three observations of a surface feature. More importantly for quantile regression, he was able to develop the first evidence of the least absolute criterion and preceded the least squares introduced by Legendre in 1805 by fifty years.
Other thinkers began building upon Bošković's idea such as Pierre-Simon Laplace, who developed the so-called "methode de situation." This led to Francis Edgeworth's plural median - a geometric approach to median regression - and is recognized as the precursor of the simplex method. The works of Bošković, Laplace, and Edgeworth were recognized as a prelude to Roger Koenker's contributions to quantile regression.
Median regression computations for larger data sets are quite tedious compared to the least squares method, for which reason it has historically generated a lack of popularity among statisticians, until |
https://en.wikipedia.org/wiki/Argument%20%28complex%20analysis%29 | In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in Figure 1.
It is a multivalued function operating on the nonzero complex numbers.
To define a single-valued function, the principal value of the argument (sometimes denoted Arg z) is used. It is often chosen to be the unique value of the argument that lies within the interval .
Definition
An argument of the complex number , denoted , is defined in two equivalent ways:
Geometrically, in the complex plane, as the 2D polar angle from the positive real axis to the vector representing . The numeric value is given by the angle in radians, and is positive if measured counterclockwise.
Algebraically, as any real quantity such that for some positive real (see Euler's formula). The quantity is the modulus (or absolute value) of , denoted ||:
The names magnitude, for the modulus, and phase, for the argument, are sometimes used equivalently.
Under both definitions, it can be seen that the argument of any non-zero complex number has many possible values: firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of radians (a complete circle) are the same, as reflected by figure 2 on the right. Similarly, from the periodicity of and , the second definition also has this property. The argument of zero is usually left undefined.
Alternative definition
The complex argument can also be defined algebraically in terms of complex roots as:
This definition removes reliance on other difficult-to-compute functions such as arctangent as well as eliminating the need for the piecewise definition. Because it's defined in terms of roots, it also inherits the principal branch of square root as its own principal branch. The normalization of by dividing by isn't necessary for convergence to the correct value, but it does speed up convergence and ensures that is left undefined.
Principal value
Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for by circling the origin any number of times. This is shown in figure 2, a representation of the multi-valued (set-valued) function , where a vertical line (not shown in the figure) cuts the surface at heights representing all the possible choices of angle for that point.
When a well-defined function is required, then the usual choice, known as the principal value, is the value in the open-closed interval , that is from to radians, excluding rad itself (equiv., from −180 to +180 degrees, excluding −180° itself). This represents an angle of up to half a complete circle from the positive real axis in either direction.
Some authors define the range of the principal value as being in the closed-open interval .
Notation
Th |
https://en.wikipedia.org/wiki/Government%20Degree%20College%20Shakargarh | Government Degree College Shakargarh is located in Shakargarh, Punjab, Pakistan. It was established in 1964 and offers courses in mathematics, sciences, language, computer science, languages, history, religion and philosophy. The college was nationalized during the government of Pakistan Prime Minister Zulfikar Ali Bhutto.
References
Public universities and colleges in Punjab, Pakistan |
https://en.wikipedia.org/wiki/OxMetrics | OxMetrics is an econometric software including the Ox programming language for econometrics and statistics, developed by Jurgen Doornik and David Hendry. OxMetrics originates from PcGive, one of the first econometric software for personal computers, initiated by David Hendry in the 1980s at the London School of Economics.
OxMetrics builds on the Ox programming language of Jurgen Doornik developed at University of Oxford. describes the history of econometric software packages.
OxMetrics is a family of software packages for the econometric and financial analysis of time series, forecasting, econometric model selection and for the statistical analysis of cross-sectional data and panel data.
The main modules apart from PcGive for dynamic econometric models (ARDL, VAR, GARCH, Switching, Autometrics), panel data models (DPD), limited dependent models, are STAMP for structural time series modelling, "SsfPack" for State space methods and "G@RCH" for financial volatility modelling. present many empirical examples in PcGive for OxMetrics in their econometrics textbook. give modern examples in their time series analysis textbook.
See also
Econometric software
Comparison of statistical packages
References
External links
OxMetrics Homepage
PcGive
STAMP software
G@RCH software
Comparison of mathematical programs for data analysis ScientificWeb
Support
Ox mailing list
Econometrics software
Statistical programming languages
Proprietary commercial software for Linux |
https://en.wikipedia.org/wiki/Aubin%E2%80%93Lions%20lemma | In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of nonlinear evolutionary partial differential equations. Typically, to prove the existence of solutions one first constructs approximate solutions (for example, by a Galerkin method or by mollification of the equation), then uses the compactness lemma to show that there is a convergent subsequence of approximate solutions whose limit is a solution.
The result is named after the French mathematicians Jean-Pierre Aubin and Jacques-Louis Lions. In the original proof by Aubin, the spaces X0 and X1 in the statement of the lemma were assumed to be reflexive, but this assumption was removed by Simon, so the result is also referred to as the Aubin–Lions–Simon lemma.
Statement of the lemma
Let X0, X and X1 be three Banach spaces with X0 ⊆ X ⊆ X1. Suppose that X0 is compactly embedded in X and that X is continuously embedded in X1. For , let
(i) If then the embedding of into is compact.
(ii) If and then the embedding of into is compact.
See also
Lions–Magenes lemma
Notes
References
(Theorem II.5.16)
(Sect.7.3)
(Proposition III.1.3)
Banach spaces
Theorems in functional analysis
Lemmas in analysis
Measure theory |
https://en.wikipedia.org/wiki/Type%20inhabitation | In type theory, a branch of mathematical logic, in a given typed calculus, the type inhabitation problem for this calculus is the following problem: given a type and a typing environment , does there exist a -term M such that ? With an empty type environment, such an M is said to be an inhabitant of .
Relationship to logic
In the case of simply typed lambda calculus, a type has an inhabitant if and only if its corresponding proposition is a tautology of minimal implicative logic. Similarly, a System F type has an inhabitant if and only if its corresponding proposition is a tautology of intuitionistic second-order logic.
Girard's paradox shows that type inhabitation is strongly related to the consistency of a type system with Curry–Howard correspondence. To be sound, such a system must have uninhabited types.
Formal properties
For most typed calculi, the type inhabitation problem is very hard. Richard Statman proved that for simply typed lambda calculus the type inhabitation problem is PSPACE-complete. For other calculi, like System F, the problem is even undecidable.
See also
Curry–Howard isomorphism
References
Lambda calculus
Type theory |
https://en.wikipedia.org/wiki/List%20of%20FC%20Barcelona%20records%20and%20statistics | Futbol Club Barcelona is a professional association football club based in Barcelona, Catalonia, Spain. Founded by a group of Swiss, German, English and Catalan footballers led by Joan Gamper, the club has become a symbol of Catalan culture and Catalanism, hence the motto "Més que un club" (More than a club). The official Barça anthem is the "Cant del Barça", written by Jaume Picas and Josep Maria Espinàs. Unlike many other football clubs, the socis, who are the members and supporters of the club, own and operate Barcelona. It is the world's fourth richest football club in terms of revenue, with an annual turnover of €582.1 million in the 2020–21 season.
Barcelona played its first friendly match on 8 December 1899 against the English colony in Barcelona in the old velodrome in Bonanova. Initially, Barcelona played against other local clubs in various Catalan tournaments. In 1929, the club became one of the founding members of La Liga, Spain's first national league, and has since achieved the distinction of being one of only three clubs to have never been relegated, along with Real Madrid and Athletic Bilbao. Barcelona is also the only European club to have played continental football every season since 1955. They hold a long-standing rivalry with Real Madrid, with matches between the two teams referred to as "El Clásico" (El Clàssic in Catalan). Matches against city rivals Espanyol are known as the "Derbi barceloní".
Barcelona has amassed various records in regional, domestic and continental tournaments since its founding. During the time the club played in regional competitions until the end of the Catalan championship in 1940, it won a record 23 titles from a possible 38. In 2009, Barcelona achieved an unprecedented sextuple by winning La Liga, the Copa del Rey, the UEFA Champions League, the Supercopa de España, the UEFA Super Cup and the FIFA Club World Cup in one calendar year. Additionally, Barça has won the coveted continental treble, consisting of La Liga, the Copa del Rey and the UEFA Champions League in the aforementioned 2009 and again 2015, becoming the first European club to have won the treble twice.
Barcelona has signed several high-profile players, setting the world record in transfer fees on three occasions with the purchase of Johan Cruyff from Ajax in 1973, Diego Maradona from Boca Juniors in 1982 and Ronaldo from PSV Eindhoven in 1996. The club's players have received seven FIFA World Player of the Year awards, twelve Ballon d'Or awards, three UEFA Men's Player of the Year awards and eight European Golden Shoe awards.
Honours
FC Barcelona won their first trophy in 1902 when they lifted the Copa Macaya, which was the predecessor to the Catalan Championship. The club won the Catalan Championship a record 23 times during the 40-year span of the tournament.
When the national league was established in 1929, the importance of the regional league declined, and it was abandoned in 1940. From then on, Barcelona did not participat |
https://en.wikipedia.org/wiki/C.%20P.%20Ramanujam | Chakravarthi Padmanabhan Ramanujam (9 January 1938 – 27 October 1974) was an Indian mathematician who worked in the fields of number theory and algebraic geometry. He was elected a fellow of the Indian Academy of Sciences in 1973.
Like his namesake Srinivasa Ramanujan, Ramanujam also had a very short life.
As David Mumford put it, Ramanujam felt that the spirit of mathematics demanded of him not merely routine developments but the right theorem on any given topic. "He wanted mathematics to be beautiful and to be clear and simple. He was sometimes tormented by the difficulty of these high standards, but in retrospect, it is clear to us how often he succeeded in adding to our knowledge, results both new, beautiful and with a genuinely original stamp".
Early life and education
Ramanujam was born to a Tamil family on 9 January 1938 in Madras (now Chennai), India, as the eldest of seven, to Chakravarthi Srinivasa Padmanabhan. He finished his schooling in Town Higher Secondary School, Kumbakonam and joined Loyola College in Madras in 1952. He wanted to specialise in mathematics and he set out to master it with vigour and passion. He also enjoyed music and his favourite musician was M. D. Ramanathan, a maverick concert musician. His teacher and friend at this time was Father (Fr.) Charles Racine (1897-1976), Loyola College a missionary who had obtained his doctorate under the supervision of Élie Cartan. He had been taught mathematics by Father Charles Racine in his final honours years at Loyola College and he encouraged Ramanujam to apply for entry to the School of Mathematics at the Tata Institute in Bombay. In his letter of recommendation Father Charles Racine wrote:-"He has certainly originality of mind and the type of curiosity which is likely to suggest that he will develop into a good research worker if given sufficient opportunity."With Father Charles Racine's encouragement and recommendation, Ramanujam applied and was admitted to the graduate school at the Tata Institute of Fundamental Research in Bombay. His father had wanted him to join the Indian Statistical Institute in Calcutta as he had passed the entrance exam meritoriously.
Career
Ramanujam set out for Mumbai at the age of eighteen to pursue his interest in mathematics. He and his friend and schoolmate Raghavan Narasimhan, and S. Ramanan joined TIFR together in 1957. At the Tata Institute there was a stream of first-rate visiting mathematicians from all over the world. It was a tradition for some graduate student to write up the notes of each course of lectures. Accordingly, Ramanujam wrote up in his first year, the notes of Max Deuring's lectures on Algebraic functions of one variable. It was a nontrivial effort and the notes were written clearly and were well received. The analytical mind was much in evidence in this effort as he could simplify and extend the notes within a short time period. "He could reduce difficult solutions to be simple and elegant due to his deep knowledge |
https://en.wikipedia.org/wiki/10-cube | In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.
It can be named by its Schläfli symbol {4,38}, being composed of 3 9-cubes around each 8-face. It is sometimes called a dekeract, a portmanteau of tesseract (the 4-cube) and deka- for ten (dimensions) in Greek, It can also be called an icosaronnon or icosa-10-tope as a 10 dimensional polytope, constructed from 20 regular facets.
It is a part of an infinite family of polytopes, called hypercubes. The dual of a dekeract can be called a 10-orthoplex or decacross, and is a part of the infinite family of cross-polytopes.
Cartesian coordinates
Cartesian coordinates for the vertices of a dekeract centered at the origin and edge length 2 are
(±1,±1,±1,±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9) with −1 < xi < 1.
Other images
Derived polytopes
Applying an alternation operation, deleting alternating vertices of the dekeract, creates another uniform polytope, called a 10-demicube, (part of an infinite family called demihypercubes), which has 20 demienneractic and 512 enneazettonic facets.
References
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
External links
Multi-dimensional Glossary: hypercube Garrett Jones
10-polytopes |
https://en.wikipedia.org/wiki/TEAMS | Teams is the plural form of team.
TEAMS or teams may also refer to:
Tests of Engineering Aptitude, Mathematics, and Science, a competition sponsored by Junior Engineering Technical Society
TEAMS (cable system), a Kenyan fibre optic cable system
TEAMS, "The Consortium for the Teaching of the Middle Ages", originally a committee of the Medieval Academy of America
Texas Educational Assessment of Minimum Skills, a standardized test used in Texas prior to 1990
Microsoft Teams, a computing platform for businesses and education
See also
Team (disambiguation) (including senses of TEAM) |
https://en.wikipedia.org/wiki/%C3%82nderson%20Lima%20%28footballer%2C%20born%201973%29 | Ânderson Lima Veiga (born March 18, 1973), or simply Ânderson Lima, is a Brazilian football midfielder. He is well known as being a free-kick specialist in Brazil.
Club statistics
Honours
Brazil
South American Under-17 Championship: 1988
South American Under-20 Championship: 1991
Santos
Torneio Rio-São Paulo: 1997
Copa Conmebol: 1998
Grêmio
Copa do Brasil: 2001
Campeonato Gaúcho: 2001
São Caetano
Campeonato Paulista: 2004
Coritiba
Campeonato Brasileiro Série B: 2007
References
External links
1973 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
São Paulo state football team players
Clube Atlético Juventus players
Guarani FC players
Santos FC players
São Paulo FC players
Grêmio Foot-Ball Porto Alegrense players
Associação Desportiva São Caetano players
Albirex Niigata players
Coritiba Foot Ball Club players
Operário Futebol Clube (MS) players
Clube Atlético Bragantino players
Associação Chapecoense de Futebol players
Brazil men's youth international footballers
Men's association football midfielders
Footballers from São Paulo |
https://en.wikipedia.org/wiki/Rick%20Durrett | Richard Timothy Durrett is an American mathematician known for his research and
books on mathematical probability theory, stochastic processes and their
application to mathematical ecology and population genetics.
Education and career
He received his BS and MS at Emory University in 1972 and 1973 and his Ph.D. at Stanford University in 1976 under advisor Donald
Iglehart. From 1976 to 1985 he taught at UCLA. From 1985 until 2010 was on the faculty at Cornell University, where his students included Claudia Neuhauser. Since 2010, Durrett has been a professor at Duke University.
He was elected to the United States National Academy of Sciences in 2007. In 2012 he became a fellow of the American Mathematical Society.
Durrett is the founder of the Cornell Probability Summer Schools.
Selected publications
Books
Durrett, R. Probability. Theory and examples. Wadsworth & Brooks/Cole, Pacific Grove, CA (1991). 453 pp. ; 4th edition, 2010
Durrett, R. Probability models for DNA sequence evolution. Springer-Verlag, New York (2002). 240 pp. ; 2nd edition, 2008
Durrett, R. Stochastic Calculus: A Practical Introduction. CRC Press (1996). 341 pp.
Durrett, R. Random Graph Dynamics. Cambridge University Press (2006). 222 pp.
Papers
(This article has over 1100 citations.)
References
External links
Personal Home Page at Duke University.
Cornell Probability Summer Schools.
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Mathematical Society
Members of the United States National Academy of Sciences
Cornell University faculty
Probability theorists
Living people
Emory University alumni
Year of birth missing (living people)
Stanford University alumni
University of California, Los Angeles faculty
Duke University faculty
American mathematicians |
https://en.wikipedia.org/wiki/Thomas%20Archer%20Hirst | Thomas Archer Hirst FRS (22 April 1830 – 16 February 1892) was a 19th-century English mathematician, specialising in geometry. He was awarded the Royal Society's Royal Medal in 1883.
Life
Thomas Hirst was born in Heckmondwike, Yorkshire, England, where both his parents came from families in the wool trade. He was the youngest of four sons. The family moved to Wakefield so that the boys could attend a better school. Thomas attended Wakefield Proprietary School for four years from 1841. Of these days, he said "... I could obtain the most rudimentary and necessary instruction. I remember, however, that here mathematics was my favourite study ..." He left the school at fifteen to work as an apprentice engineer in Halifax, surveying for proposed railway lines. It was there that he met John Tyndall, ten years older than Hirst and working as an engineer in the same firm.
In his late teens, at the instigation of Tyndall, Hirst decided to go to Germany for education, initially in chemistry. He eventually received a doctorate in mathematics from the University of Marburg in 1852 (tutor: Friedrich Ludwig Stegmann). In 1853, he attended geometry lectures by Jakob Steiner at University of Berlin. Hirst married Anna Martin in 1854, and spent much of the decade of the 1850s on the European continent, where he socialised with many mathematicians, and used his inherited wealth to support himself.
From 1860 to 1864, Hirst taught at University College School, but resigned because he wanted more time for his mathematical research. He was appointed Professor of Physics at University College London in 1865, and he succeeded Augustus De Morgan to the Chair of Mathematics at UCL in 1867. In 1873 he was appointed as the first Director of Studies at the new Royal Naval College, Greenwich. He retired from that post in 1882, to be succeeded by William Davidson Niven.
From the 1860s onwards, Hirst also allocated much of his time in England to the administrative committees of British science. He was an active member of the governing councils of the Royal Society, the British Association for the Advancement of Science, and the London Mathematical Society. He was the founding president of an association to reform school mathematics curricula and also worked to promote the education of women. Alongside his old friend Tyndall, Hirst was a member of T. H. Huxley's London X-Club. He died in London in 1892, four weeks after he had made the last entry in his journal, and was buried on the eastern side of Highgate Cemetery.
Vestiges
In his early days, Hirst wrote extensively in his notebooks (sometimes called the Journal), recording everything he read and much of what he was thinking about. This extraordinary record of about fifty years is preserved in the library of the Royal Institution. As a result, we know much about the development of his mind before he became a professional mathematician. We know, for example, what the effect was of his reading the Vestiges of the Natura |
https://en.wikipedia.org/wiki/Boron%20triiodide | Boron triiodide is a chemical compound of boron and iodine with chemical formula BI3. It has a trigonal planar molecular geometry.
Preparation
Boron triiodide can be prepared by the reaction of boron with iodine at 209.5 °C or 409.1 °F.
It can also be prepared by reacting hydroiodic acid with boron trichloride:
(reaction requires high temperature)
Another method is by reacting lithium borohydride with iodine. As well as boron triiodide, this reaction also produces lithium iodide, hydrogen and hydrogen iodide:
Properties
In its pure state, boron triiodide forms colorless, otherwise reddish, shiny, air and hydrolysis-sensitive crystals, which have a hexagonal crystal structure (a = 699.09 ± 0.02 pm, c = 736.42 ± 0.03 pm, space group P63/m (space group no. 176)). Boron triiodide is a strong Lewis acid and soluble in carbon disulfide.
Boron triiodide reacts with water and decomposes to boric acid and hydriodic acid:
Its dielectric constant is 5.38 and its heat of vaporization is 40.5 kJ/mol. At extremely high pressures, BI3 becomes metallic at ~23 GPa and is a superconductor above ~27 GPa.
Applications
Boron triiodide can be used to produce other chemical compounds and as a catalyst (for example in coal liquefaction).
References
External links
MSDS (link is broken)
Boron compounds
Iodides
Boron halides |
https://en.wikipedia.org/wiki/Coupon%20collector%27s%20problem | In probability theory, the coupon collector's problem describes "collect all coupons and win" contests. It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? An alternative statement is: Given n coupons, how many coupons do you expect you need to draw with replacement before having drawn each coupon at least once? The mathematical analysis of the problem reveals that the expected number of trials needed grows as . For example, when n = 50 it takes about 225 trials on average to collect all 50 coupons.
Solution
Calculating the expectation
Let time T be the number of draws needed to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. Then . Think of T and ti as random variables. Observe that the probability of collecting a coupon is . Therefore, has geometric distribution with expectation . By the linearity of expectations we have:
Here Hn is the n-th harmonic number. Using the asymptotics of the harmonic numbers, we obtain:
where is the Euler–Mascheroni constant.
Using the Markov inequality to bound the desired probability:
The above can be modified slightly to handle the case when we've already collected some of the coupons. Let k be the number of coupons already collected, then:
And when then we get the original result.
Calculating the variance
Using the independence of random variables ti, we obtain:
since (see Basel problem).
Bound the desired probability using the Chebyshev inequality:
Tail estimates
A stronger tail estimate for the upper tail be obtained as follows. Let denote the event that the -th coupon was not picked in the first trials. Then
Thus, for , we have . Via a union bound over the coupons, we obtain
Extensions and generalizations
Pierre-Simon Laplace, but also Paul Erdős and Alfréd Rényi, proved the limit theorem for the distribution of T. This result is a further extension of previous bounds. A proof is found in.
Donald J. Newman and Lawrence Shepp gave a generalization of the coupon collector's problem when m copies of each coupon need to be collected. Let Tm be the first time m copies of each coupon are collected. They showed that the expectation in this case satisfies:
Here m is fixed. When m = 1 we get the earlier formula for the expectation.
Common generalization, also due to Erdős and Rényi:
In the general case of a nonuniform probability distribution, according to Philippe Flajolet et al.
This is equal to
where m denotes the number of coupons to be collected and PJ denotes the probability of getting any coupon in the set of coupons J.
See also
McDonald's Monopoly – an example of the coupon collector's problem that further increases the challenge by making some coupons of the set rarer
Watterson estimator
Birthday problem
Notes
References
.
.
.
.
.
.
.
Ex |
https://en.wikipedia.org/wiki/Field%20arithmetic | In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a and its absolute Galois group.
It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups.
Fields with finite absolute Galois groups
Let K be a field and let G = Gal(K) be its absolute Galois group. If K is algebraically closed, then G = 1. If K = R is the real numbers, then
Here C is the field of complex numbers and Z is the ring of integer numbers.
A theorem of Artin and Schreier asserts that (essentially) these are all the possibilities for finite absolute Galois groups.
Artin–Schreier theorem. Let K be a field whose absolute Galois group G is finite. Then either K is separably closed and G is trivial or K is real closed and G = Z/2Z.
Fields that are defined by their absolute Galois groups
Some profinite groups occur as the absolute Galois group of non-isomorphic fields. A first example for this is
This group is isomorphic to the absolute Galois group of an arbitrary finite field. Also the absolute Galois group of the field of formal Laurent series C((t)) over the complex numbers is isomorphic to that group.
To get another example, we bring below two non-isomorphic fields whose absolute Galois groups are free (that is free profinite group).
Let C be an algebraically closed field and x a variable. Then Gal(C(x)) is free of rank equal to the cardinality of C. (This result is due to Adrien Douady for 0 characteristic and has its origins in Riemann's existence theorem. For a field of arbitrary characteristic it is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden.)
The absolute Galois group Gal(Q) (where Q are the rational numbers) is compact, and hence equipped with a normalized Haar measure. For a Galois automorphism s (that is an element in Gal(Q)) let Ns be the maximal Galois extension of Q that s fixes. Then with probability 1 the absolute Galois group Gal(Ns) is free of countable rank. (This result is due to Moshe Jarden.)
In contrast to the above examples, if the fields in question are finitely generated over Q, Florian Pop proves that an isomorphism of the absolute Galois groups yields an isomorphism of the fields:
Theorem. Let K, L be finitely generated fields over Q and let a: Gal(K) → Gal(L) be an isomorphism. Then there exists a unique isomorphism of the algebraic closures, b: Kalg → Lalg, that induces a.
This generalizes an earlier work of Jürgen Neukirch and Koji Uchida on number fields.
Pseudo algebraically closed fields
A pseudo algebraically closed field (in short PAC) K is a field satisfying the following geometric property. Each absolutely irreducible algebraic variety V defined over K has a K-rational point.
Over PAC fields there is a firm link between arithmetic properties of the field and group theoretic properties of its |
https://en.wikipedia.org/wiki/Israel%20Lyons | Israel Lyons the Younger (1739–1775), mathematician and botanist, was born at Cambridge, the son of Israel Lyons the elder (died 1770). He was regarded as a prodigy, especially in mathematics, and Robert Smith, master of Trinity College, took him under his wing and paid for his attendance.
Biography
Due to his Ashkenazi Jewish origins, Lyons was not permitted to become an official member of the University of Cambridge. Nevertheless, his brilliance resulted in his publication Treatise on Fluxions at the age of 19, and his enthusiasm for botany resulted in a published survey of Cambridge flora a few years later. An Oxford undergraduate, Joseph Banks, paid Lyons to deliver a series of botany lectures at the University of Oxford. Lyons was selected by the Astronomer Royal to compute astronomical tables for the Nautical Almanac. Later, Banks secured Lyons a position as the astronomer for the 1773 North Pole voyage led by Constantine Phipps, 2nd Baron Mulgrave.
Lyons married, in March 1774, Phoebe Pearson, daughter of Newman Pearson of Over, Cambridgeshire, and settled in Rathbone Place, London. There he died of measles on 1 May 1775, at the age of only 36, while preparing a complete edition of Edmond Halley's works sponsored by the Royal Society.
See also
European and American voyages of scientific exploration
References
Lynn B. Glyn, "Israel Lyons: A Short but Starry Career. The Life of an Eighteenth-Century Jewish Botanist and Astronomer," Notes and Records of the Royal Society of London, Vol. 56, No. 3, 2002, pp. 275–305.
People associated with the University of Cambridge
18th-century English mathematicians
1739 births
1775 deaths
English botanical writers
Deaths from measles
18th-century British botanists
Jewish British scientists
Jewish biologists
English Ashkenazi Jews |
https://en.wikipedia.org/wiki/Bol%C3%ADvar%20Municipality%2C%20T%C3%A1chira | Bolívar Municipality is one of the 29 municipalities that makes up the western Venezuelan state of Táchira and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 60,149. The town of San Antonio del Táchira is the shiretown of the Bolívar Municipality.
Name
The municipality is one of several in Venezuela named "Bolívar Municipality" in honour of Venezuelan independence hero Simón Bolívar.
Demographics
The Bolívar Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 60,149 (up from 50,209 in 2000). This amounts to 5.1% of the state's population. The municipality's population density is .
Government
The mayor of the Bolívar Municipality is Juan Vicente Cañas Alviarez, elected on October 31, 2004, with 48% of the vote. He replaced Ramon Vivas shortly after the elections. The municipality is divided into four parishes; Bolívar, Palotal, Juan Vicente Gómez, and Isaías Medina Angarita (parishes Juan Vicente Gómez and Isaías Medina Angarita were officially separated from the Bolívar parish on January 25, 1995).
References
External links
bolivar-tachira.gob.ve
Information on the Bolívar Municipality
More information on the Bolívar Municipality
Municipalities of Táchira |
https://en.wikipedia.org/wiki/Paul%20Zimmermann%20%28mathematician%29 | Paul Zimmermann (born 13 November 1964) is a French computational mathematician, working at INRIA.
Zimmermann co-authored the book Computational Mathematics with SageMath used by Mathematical students worldwide.
His interests include asymptotically fast arithmetic—he wrote a book on algorithms for computer arithmetic with Richard Brent. He has developed some of the fastest available code for manipulating polynomials over GF(2), and for calculating hypergeometric constants to billions of decimal places. He is associated with the CARAMEL project to develop efficient arithmetic, in a general context and in particular in the context of algebraic curves of small genus; arithmetic on polynomials of very large degree turns out to be useful in algorithms for point-counting on such curves. He is also interested in computational number theory. In particular, he has contributed to some of the record computations in integer factorisation and discrete logarithm.
He has been an active developer of the GMP-ECM implementation of the elliptic curve method for integer factorisation and of MPFR, an arbitrary precision floating point library with correct rounding. He is also a coauthor of the CADO-NFS software tool, which was used to factor RSA-240 in record time.
In a 2014 blog post, Zimmermann said that he would refuse invitations to review papers submitted to gold (author-pays) open access and hybrid open access journals, because he disagrees with the publication mechanism.
References
External links
http://www.loria.fr/~zimmerma/
Living people
1964 births
French mathematicians
Free software people
Free software programmers
GNU people |
https://en.wikipedia.org/wiki/Instituto%20de%20Investigaciones%20en%20Matem%C3%A1ticas%20Aplicadas%20y%20Sistemas | The Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, or IIMAS ("Applied Mathematics and Systems Research Institute") is the research institute of the UNAM in Mexico City which focuses on computer science, applied mathematics, and robotics and control engineering.
History
The IIMAS was founded as the Centro de Cálculo Electrónico (Electronic Calculation Center) in 1958 under the auspices of UNAM rector Nabor Carrillo Flores with the purpose of housing and operating the first mainframe acquired by the university. In 1973 the institute acquired its current name and refocused its primary activities from computing services to applied mathematics and computer science research. In the early nineties the IIMAS moved to its current home.
Staff
The IIMAS has an average of 60 researchers aided by 40 technical staff. It is currently headed by Héctor Benítez Pérez, Ph.D.
Location and facilities
The IIMAS is located in Ciudad Universitaria in Mexico City, nearby the Engineering School and the Science School.
It consists in two buildings, the second one built later on to host the graduate programs and the new library.
Graduate studies
The institute currently offers graduate programs in four areas conjointly with the schools of science and engineering, as well as with the Earth Sciences Institute. Graduates students may focus on engineering, computer science, applied mathematics, or Earth sciences.
Current research
Research is divided among six departments:
Mathematics and mechanics
Mathematical physics
Mathematical modeling of social systems
Probability and statistics
Computer science
Computer systems and automation engineering
References
External links
IIMAS homepage
National Autonomous University of Mexico |
https://en.wikipedia.org/wiki/Delannoy%20number | In mathematics, a Delannoy number describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east. The Delannoy numbers are named after French army officer and amateur mathematician Henri Delannoy.
The Delannoy number also counts the number of global alignments of two sequences of lengths and , the number of points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and, in cellular automata, the number of cells in an m-dimensional von Neumann neighborhood of radius n while the number of cells on a surface of an m-dimensional von Neumann neighborhood of radius n is given with .
Example
The Delannoy number D(3,3) equals 63. The following figure illustrates the 63 Delannoy paths from (0, 0) to (3, 3):
The subset of paths that do not rise above the SW–NE diagonal are counted by a related family of numbers, the Schröder numbers.
Delannoy array
The Delannoy array is an infinite matrix of the Delannoy numbers:
{| class="wikitable" style="text-align:right;"
|-
!
! width="50" | 0
! width="50" | 1
! width="50" | 2
! width="50" | 3
! width="50" | 4
! width="50" | 5
! width="50" | 6
! width="50" | 7
! width="50" | 8
|-
! 0
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|-
! 1
| 1 || 3 || 5 || 7 || 9 || 11 || 13 || 15 || 17
|-
! 2
| 1 || 5 || 13 || 25 || 41 || 61 || 85 || 113 || 145
|-
! 3
| 1 || 7 || 25 || 63 || 129 || 231 || 377 || 575 || 833
|-
! 4
| 1 || 9 || 41 || 129 || 321 || 681 || 1289 || 2241 || 3649
|-
! 5
| 1 || 11 || 61 || 231 || 681 || 1683 || 3653 || 7183 || 13073
|-
! 6
| 1 || 13 || 85 || 377 || 1289 || 3653 || 8989 || 19825 || 40081
|-
! 7
| 1 || 15 || 113 || 575 || 2241 || 7183 || 19825 || 48639 || 108545
|-
! 8
| 1 || 17 || 145 || 833 || 3649 || 13073 || 40081 || 108545 || 265729
|-
! 9
| 1 || 19 || 181 || 1159 || 5641 || 22363 || 75517 || 224143 || 598417
|}
In this array, the numbers in the first row are all one, the numbers in the second row are the odd numbers, the numbers in the third row are the centered square numbers, and the numbers in the fourth row are the centered octahedral numbers. Alternatively, the same numbers can be arranged in a triangular array resembling Pascal's triangle, also called the tribonacci triangle, in which each number is the sum of the three numbers above it:
1
1 1
1 3 1
1 5 5 1
1 7 13 7 1
1 9 25 25 9 1
1 11 41 63 41 11 1
Central Delannoy numbers
The central Delannoy numbers D(n) = D(n,n) are the numbers for a square n × n grid. The first few central Delannoy numbers (starting with n=0) are:
1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, ... .
Computation
Delannoy numbers
For diagonal (i.e. northeast) steps, there must be steps in the direction and steps in the direction in order to reach the point ; as these steps can be performed in any order, the number of such |
https://en.wikipedia.org/wiki/Extraneous%20and%20missing%20solutions | In mathematics, an extraneous solution (or spurious solution) is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem. A missing solution is a solution that is a valid solution to the problem, but disappeared during the process of solving the problem. Both are frequently the consequence of performing operations that are not invertible for some or all values of the variables, which prevents the chain of logical implications in the proof from being bidirectional.
Extraneous solutions: multiplication
One of the basic principles of algebra is that one can multiply both sides of an equation by the same expression without changing the equation's solutions. However, strictly speaking, this is not true, in that multiplication by certain expressions may introduce new solutions that were not present before. For example, consider the following equation:
If we multiply both sides by zero, we get,
This is true for all values of x, so the solution set is all real numbers. But clearly not all real numbers are solutions to the original equation. The problem is that multiplication by zero is not invertible: if we multiply by any nonzero value, we can reverse the step by dividing by the same value, but division by zero is not defined, so multiplication by zero cannot be reversed.
More subtly, suppose we take the same equation and multiply both sides by x. We get
This quadratic equation has two solutions, − 2 and 0. But if zero is substituted for x into the original equation, the result is the invalid equation 2 = 0. This counterintuitive result occurs because in the case where x=0, multiplying both sides by x multiplies both sides by zero, and so necessarily produces a true equation just as in the first example.
In general, whenever we multiply both sides of an equation by an expression involving variables, we introduce extraneous solutions wherever that expression is equal to zero. But it is not sufficient to exclude these values, because they may have been legitimate solutions to the original equation. For example, suppose we multiply both sides of our original equation x + 2 = 0 by x + 2. We get
which has only one real solution: x = −2, and this is a solution to the original equation, so it cannot be excluded, even though x + 2 is zero for this value of x.
Extraneous solutions: rational
Extraneous solutions can arise naturally in problems involving fractions with variables in the denominator. For example, consider this equation:
To begin solving, we multiply each side of the equation by the least common denominator of all the fractions contained in the equation. In this case, the least common denominator is . After performing these operations, the fractions are eliminated, and the equation becomes:
Solving this yields the single solution x = −2. However, when we substitute the solution back into the original equation, we obtain:
The equation then becomes:
This eq |
https://en.wikipedia.org/wiki/Mathematics%20%28UIL%29 | Mathematics (sometimes referred to as General Math, to distinguish it from other mathematics-related events) is one of several academic events sanctioned by the University Interscholastic League. It is also a competition held by the Texas Math and Science Coaches Association, using the same rules as the UIL.
Mathematics is designed to test students' understanding of advanced mathematics. The UIL contest began in 1943, and is among the oldest of all UIL academic contests.
Eligibility
Students in Grade 6 through Grade 12 are eligible to enter this event. For competition purposes, separate divisions are held for Grades 6-8 and Grades 9-12, with separate subjects covered on each test as follows:
The test for Grades 6-8 covers numeration systems, arithmetic operations involving whole numbers, integers, fractions, decimals, exponents, order of operations, probability, statistics, number theory, simple interest, measurements and conversions, plus possibly geometry and algebra problems (as appropriate for the grade level).
The test for Grades 9-12 covers algebra I and II, geometry, trigonometry, math analysis, analytic geometry, pre-calculus, and elementary calculus.
For Grades 6-8 each school may send up to three students per division. In order for a school to participate in team competition in a division, the school must send three students in that division.
For Grades 9-12 each school may send up to four students; however, in districts with more than eight schools the district executive committee can limit participation to three students per school. In order for a school to participate in team competition, the school must send at least three students.
Rules and Scoring
At the junior high level, the test consists of 50 questions and is limited to only 30 minutes. At the high school level, the test consists of 60 questions and is limited to only 40 minutes. Both tests are multiple choice.
There is no intermediate time signal given; at the end of the allotted time the students must immediately stop writing (they are not allowed to finish incomplete answers started before the stop signal). If contestants are in the process of writing down an answer, they
may finish; they may not do additional work on a test question.
The questions can be answered in any order; a skipped question is not scored.
Calculators are permitted provided they are (or were) commercially available models, run quietly, and do not require auxiliary power. One calculator plus one spare is permitted.
Five points are awarded for each correct answer at the junior high level while six points are awarded at the high school level. Two points are deducted for each wrong answer. Skipped or unanswered questions are not scored.
Determining the Winner
Elementary and Junior High
Scoring is posted for only the top six individual places and the top three teams.
There are no tiebreakers for either individual or team competition.
High School Level
The top three individuals and |
https://en.wikipedia.org/wiki/Anzo%C3%A1tegui%20Municipality | The Anzoátegui Municipality is one of the nine municipalities (municipios) that makes up the Venezuelan state of Cojedes and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 17,030. The town of Cojedes is the shire town of the Anzoátegui Municipality.
Demographics
The Anzoátegui Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 16,333 (up from 14,354 in 2000). This amounts to 5.4% of the state's population. The municipality's population density is .
Government
The mayor of the Anzoátegui Municipality is Luis Linares, re-elected on October 31, 2004, with 44% of the vote. The municipality is divided into two parishes; Cojedes and Juan de Mata Suárez.
References
External links
anzoategui-cojedes.gob.ve
Municipalities of Cojedes (state) |
https://en.wikipedia.org/wiki/Hardy%E2%80%93Ramanujan%20theorem | In mathematics, the Hardy–Ramanujan theorem, proved by Ramanujan and checked by Hardy states that the normal order of the number ω(n) of distinct prime factors of a number n is log(log(n)).
Roughly speaking, this means that most numbers have about this number of distinct prime factors.
Precise statement
A more precise version states that for every real-valued function ψ(n) that tends to infinity as n tends to infinity
or more traditionally
for almost all (all but an infinitesimal proportion of) integers. That is, let g(x) be the number of positive integers n less than x for which the above inequality fails: then g(x)/x converges to zero as x goes to infinity.
History
A simple proof to the result was given by Pál Turán, who used the Turán sieve to prove that
Generalizations
The same results are true of Ω(n), the number of prime factors of n counted with multiplicity.
This theorem is generalized by the Erdős–Kac theorem, which shows that ω(n) is essentially normally distributed.
References
Theorems in analytic number theory
Theorems about prime numbers |
https://en.wikipedia.org/wiki/Glyn%20Harman | Glyn Harman (born 2 November 1956) is a British mathematician working in analytic number theory. One of his major interests is prime number theory. He is best known for results on gaps between primes and the greatest prime factor of p + a, as well as his lower bound for the number of Carmichael numbers up to X. His monograph Prime-detecting Sieves (2007) was published by Princeton University Press. He has also written a book Metric Number Theory (1998). As well, he has contributed to the field of Diophantine approximation.
Harman also proved that there are infinitely many primes (additive primes) whose sum of digits is prime. (the sequence A046704 in the OEIS).
Harman retired at the end of 2013 from being a professor at Royal Holloway, University of London. Previously he was a professor at Cardiff University.
Harman is married, and has three sons, and used to live in Wokingham, Berkshire before moving to Harrow, Middlesex/Greater London and then Uxbridge.
References
External links
Home page of Glyn Harman
Academics of Royal Holloway, University of London
Academics of Cardiff University
Living people
20th-century British mathematicians
21st-century British mathematicians
Number theorists
1956 births
People educated at Ashford County Grammar School |
https://en.wikipedia.org/wiki/Ayacucho%20Municipality%2C%20T%C3%A1chira | The Ayacucho Municipality is one of the 29 municipalities that makes up the western Venezuelan state of Táchira and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 60,454. The town of Colón is the municipal seat of the Ayacucho Municipality.
Demographics
The Ayacucho Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 60,454 (up from 50,992 in 2000). This amounts to 5.1% of the state's population. The municipality's population density is .
Government
The mayor of the Ayacucho Municipality is Gabino Paz Guerrero, re-elected on October 31, 2004, with 47% of the vote. The municipality is divided into three parishes; Ayacucho, Rivas Berti, San Pedro del Río.
References
External links
ayacucho-tachira.gob.ve
Municipalities of Táchira |
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