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https://en.wikipedia.org/wiki/Coordinate-induced%20basis
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In mathematics, a coordinate-induced basis is a basis for the tangent space or cotangent space of a manifold that is induced by a certain coordinate system. Given the coordinate system , the coordinate-induced basis of the tangent space is given by
and the dual basis of the cotangent space is
References
D.J. Hurley, M.A. Vandyck Topics in Differential Geometry: a New Approach Using D-Differentiation (2002 Springer) p. 5
Differential geometry
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https://en.wikipedia.org/wiki/Nicholas%20Shepherd-Barron
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Nicholas Ian Shepherd-Barron, FRS (born 17 March 1955), is a British mathematician working in algebraic geometry. He is a professor of mathematics at King's College London.
Education and career
Shepherd-Barron was a scholar of Winchester College. He obtained his B.A. at Jesus College, Cambridge in 1976, and received his Ph.D. at the University of Warwick under the supervision of Miles Reid in 1981.
In 2013, he moved from the University of Cambridge to King's College London.
Research
Shepherd-Barron works in various aspects of algebraic geometry, such as: singularities in the minimal model program; compactification of moduli spaces; the rationality of orbit spaces, including the moduli spaces of curves of genus 4 and 6; the geography of algebraic surfaces in positive characteristic, including a proof of Raynaud's conjecture; canonical models of moduli spaces of abelian varieties; the Schottky problem at the boundary; the relation between algebraic groups and del Pezzo surfaces; the period map for elliptic surfaces.
In 2008, with the number theorists Michael Harris and Richard Taylor, he proved the original version of the Sato–Tate conjecture and its generalization to totally real fields, under mild assumptions.
Awards and honors
Shepherd-Barron was elected Fellow of the Royal Society in 2006.
Personal life
He is the son of John Shepherd-Barron, a Scottish inventor, who was responsible for inventing the first cash machine in 1967.
Notes
References
1955 births
Living people
20th-century British mathematicians
21st-century British mathematicians
Algebraic geometers
People educated at Winchester College
Alumni of Jesus College, Cambridge
Fellows of Trinity College, Cambridge
Fellows of the Royal Society
Academics of King's College London
Professors of the University of Cambridge
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https://en.wikipedia.org/wiki/Emma%20Lehmer
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Emma Markovna Lehmer (née Trotskaia) (November 6, 1906 – May 7, 2007) was a mathematician known for her work on reciprocity laws in algebraic number theory. She preferred to deal with complex number fields and integers, rather than the more abstract aspects of the theory.
Biography
She was born in Samara, Russian Empire, but her father's job as a representative with a Russian sugar company moved the family to Harbin, China in 1910. Emma was tutored at home until the age of 14, when a school was opened locally. She managed to make her way to the US for her higher education.
At UC Berkeley, she started out in engineering in 1924, but found her niche in mathematics. One of her professors was Derrick N. Lehmer, the number theorist well known for his work on prime number tables and factorizations. While working for him at Berkeley finding pseudosquares, she met his son, her future husband Derrick H. Lehmer. Upon her graduation summa cum laude with a B.A. in Mathematics (1928), Emma married the younger Lehmer. They moved to Brown University, where Emma received her M.Sc., and Derrick his Ph.D., both in 1930. Emma did not obtain a Ph.D. herself; she claimed there were many advantages to not holding a doctorate.
The Lehmers had two children, Laura (1932) and Donald (1934).
Contributions
Lehmer did independent mathematical work, including a translation from Russian to English of Pontryagin's book Topological Groups. She worked closely with her husband on many projects; 21 of her 56 publications were joint work with him. Her publications were mainly in number theory and computation, with emphasis on reciprocity laws, special primes, and congruences.
She proved that there were infinitely many Fibonacci pseudoprimes.
Paul Halmos, in his book I want to be a mathematician: An automathography, wrote about Lehmer's translation of Pontryagin's Topological Groups: "I read the English translation by Mrs. Lehmer (usually referred to as Emma Lemma)...". Several later publications repeated Halmos' reference to reinforce the significance of Lehmer's translation.
During World War II, she authored the paper "Simplified Rule for Determining Spacing in Train Bombing on Stationary Targets" and co-authored three others for the Statistical Laboratory at the University of California.
With her husband, she co-founded the West Coast Number Theory conference.
Emma and Derrick Lehmer both have Erdős number two. They published a joint paper with John Brillhart in 1964 on bounds on consecutive power residues. Brillhart published a paper on the Rudin-Shapiro sequence with Erdős and Morton in 1983.
Notes
Notable Women in Mathematics, a Biographical Dictionary, edited by Charlene Morrow and Teri Perl, Greenwood Press, 1998. pp 123–128
External links
Biographies of Women Mathematicians
The Princeton Mathematics Community in the 1930s
1906 births
2007 deaths
Russian centenarians
Emigrants from the Russian Empire to the United States
American centenarians
American women
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https://en.wikipedia.org/wiki/Vala%C5%A1kovce
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Valaškovce is a municipality and former village and in Humenné District in the Prešov Region of north-east Slovakia.
History
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Humenné District
Former villages in Slovakia
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https://en.wikipedia.org/wiki/Chabauty%20topology
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In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty, on the set of all closed subgroups of a locally compact group G.
The intuitive idea may be seen in the case of the set of all lattices in a Euclidean space E. There these are only certain of the closed subgroups: others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology.
This topology can be derived from the Vietoris topology construction, a topological structure on all non-empty subsets of a space. More precisely, it is an adaptation of the Fell topology construction, which itself derives from the Vietoris topology concept.
References
Claude Chabauty, Limite d'ensembles et géométrie des nombres. Bulletin de la Société Mathématique de France, 78 (1950), p. 143-151
Topological groups
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https://en.wikipedia.org/wiki/Donaldson%20theory
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In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold. Important consequences of this theorem include the existence of an Exotic R4 and the failure of the smooth h-cobordism theorem in 4 dimensions. The results of Donaldson theory depend therefore on the manifold having a differential structure, and are largely false for topological 4-manifolds.
Many of the theorems in Donaldson theory can now be proved more easily using Seiberg–Witten theory, though there are a number of open problems remaining in Donaldson theory, such as the Witten conjecture and the Atiyah–Floer conjecture.
See also
Kronheimer–Mrowka basic class
Instanton
Floer homology
Yang–Mills equations
References
.
.
.
.
Geometric topology
4-manifolds
Differential topology
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https://en.wikipedia.org/wiki/CLHEP
|
CLHEP (short for A Class Library for High Energy Physics) is a C++ library that provides utility classes for general numerical programming, vector arithmetic, geometry, pseudorandom number generation, and linear algebra, specifically targeted for high energy physics simulation and analysis software.
The project is hosted by CERN and currently managed by a collaboration of researchers from CERN and other physics research laboratories and academic institutions. According to the project's website, CLHEP is in maintenance mode (accepting bug fixes but no further development is expected).
CLHEP was proposed by Swedish physicist Leif Lönnblad in 1992 at a Conference on Computing in High-Energy Physics. Lönnblad is still involved in maintaining CLHEP.
The project has more recently accepted contributions from other projects built on top of CLHEP, including the physics packages Geant4 and ZOOM, and the BaBar experiment at SLAC.
See also
Geant4, a software using CLHEP
FreeHEP, a similar library to CLHEP
COLT, a Java package for High Performance Scientific and Technical Computing, provided by CERN.
References
External links
Project CLHEP website
CLHEP User Guide
CLHEP at CERN
Physics software
CERN software
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https://en.wikipedia.org/wiki/Statistics%20%28disambiguation%29
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Statistics is a mathematical science pertaining to the collection, analysis, interpretation, and presentation of data.
Statistic may also refer to:
Statistic, the result of applying a statistical algorithm to a set of data
Statistic (role-playing games), a piece of data which represents a particular aspect of a fictional character
Statistics (band), an American rock band
"Statistics" (song), by Lyfe Jennings, 2010
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https://en.wikipedia.org/wiki/Nen%C3%AA%20%28footballer%2C%20born%201975%29
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Fábio Camilo de Brito, nicknamed "Nenê", (born 6 June 1975 in São Paulo, Brazil), is a Brazilian former professional footballer who played as a central defender.
Club statistics
Honours
Hertha BSC
DFB-Ligapokal: 2003
Vitória
Campeonato Baiano: 2004, 2005
Urawa Reds
J. League: 2006
Emperor's Cup: 2006
AFC Champions League: 2007
References
External links
1975 births
Living people
Brazilian men's footballers
Men's association football defenders
Urawa Red Diamonds players
Clube Atlético Juventus players
Guarani FC players
Sporting CP footballers
Grêmio Foot-Ball Porto Alegrense players
Esporte Clube Vitória players
Esporte Clube Bahia players
Coritiba Foot Ball Club players
Hertha BSC players
Footballers at the 1995 Pan American Games
Campeonato Brasileiro Série A players
Primeira Liga players
Bundesliga players
J1 League players
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in Japan
Expatriate men's footballers in Japan
Brazilian expatriate sportspeople in Germany
Expatriate men's footballers in Germany
Brazilian expatriate sportspeople in Portugal
Expatriate men's footballers in Portugal
Footballers from São Paulo
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https://en.wikipedia.org/wiki/Matthias%20Mann
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Matthias Mann (born 10 October 1959) is a scientist in the area of mass spectrometry and proteomics.
Early life and education
Born in Germany he studied mathematics and physics at the University of Göttingen. He received his Ph.D. in 1988 at Yale University where he worked in the group of John Fenn, who was later awarded the Nobel Prize in Chemistry.
Career
After a postdoctoral fellowship at the University of Southern Denmark in Odense Mann became group leader at the European Molecular Biology Laboratory (EMBL) in Heidelberg. Later he went back to Odense as a professor of bioinformatics. Since 2005 he has been a director at the Max Planck Institute of Biochemistry in Munich. In addition, he became a principal investigator at the newly founded "Novo Nordisk Foundation Center for Protein Research" in Copenhagen.
From his research group in Munich originated in 2016 PreOmics – a company commercializing sample prep sets, and EVOSEP – a company commercializing protein analysis equipment.
His work has impact in various fields of mass spectrometry-based proteomics:
The peptide sequence tag approach developed at the EMBL was one of the first methods for the identification of peptides based on mass spectra and genome data.
Nano-electrospray (an electrospray technique with very low flow rates) was the first method that allowed femtomole sequencing of proteins from polyacrylamide gels.
A recently developed metabolic labeling technique called SILAC (stable isotope labeling with amino acids in cell culture) is widely used in quantitative proteomics.
Other activities
PharmaFluidics, Member of the Advisory Board (since 2019)
Awards and honors
1991: Malcom Award by the journal Organic Mass Spectrometry
1996: Mattauch Herzog Prize in Mass Spectrometry
1997: Hewlett-Packard Prize for Strategic Research in Automation of Sample Preparation
1998: Edman Prize by the "Methods in Protein Structure Analysis" Society
1999: Bieman Medal for Outstanding Achievement in Mass Spectrometry (American Society for Mass Spectrometry)
1999: Named second most cited scientist in chemistry in the years 1994 to 1996 by the Institute of Scientific Information
1999: Elected visiting professor Harvard Medical School
1999: Elected to the European Molecular Biology Organization (EMBO)
2000: Meyenburg Prize
2001: Bernhard and Matha Rasmussens Memorial award in Cancer Research
2001: Meyenburg Cancer Research Award given by the German Cancer Research Center
2001: Fresenius Prize and Medal for Analytical Chemistry given by the German Chemical Society
2004: Honorary Doctorate, University of Utrecht, Netherlands
2004: Lundbeck Prize
2004: Novo-Nordisk Prize
2005: Anfinsen Award of the Protein Society
2006: "Biochemical Analysis" prize by the German Society for Clinical Chemistry and Laboratory Medicine
2008: HUPO Distinguished Achievement Award in Proteomic Science
2008: Bijvoet Medal of the Bijvoet Center for Biomolecular Research of Utrecht University
2010: Friedrich Wilhelm Jos
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https://en.wikipedia.org/wiki/Simeon%20ben%20Zemah%20Duran
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Simeon ben Zemah Duran, also Tzemach Duran (1361–1444; ), known as Rashbatz () or Tashbatz was a Rabbinical authority, student of philosophy, astronomy, mathematics, and especially of medicine, which he practised for a number of years at Palma de Mallorca. A major 15th century posek, his published decisions in matters of halakha have been widely quoted in halakhic literature for hundreds of years.
Biography
Simeon ben Tzemach was born in the Hebrew month of Adar, 1361. Various accounts put his birthplace as either Barcelona, or the island of Majorca. He was a near relation but not a grandson of Levi ben Gershon. He was a student of Ephraim Vidal, and of Jonah de Maestre, rabbi in Zaragoza or in Calatayud, whose daughter Bongoda he married.
After the 1391 massacre in the Balearic Islands, he fled Spain with his father and sister for Algiers, where, in addition to practicing medicine, he continued his studies during the earlier part of his stay. In 1394 he and the Algerian rabbi Isaac ben Sheshet ("the Rivash") drafted statutes for the Jewish community of Algiers. After the Rivash's retirement, Duran became rabbi of Algiers in 1407. Unlike his predecessor, he refused on principle to accept any confirmation of his appointment by the regent. As Duran had lost all his property during the massacre at Palma, he was forced against his will to accept a salary from the community, not having other means of subsistence. He held this office until his death. His epitaph, written by himself, has been reprinted for the first time, from a manuscript, in Orient, Lit. v. 452. According to Joseph ben Isaac Sambari, Simon was much respected in court circles. He was the father of the Solomon ben Simon Duran.
Duran's Magen Avot was a polemic against Christians and Muslims, of which the fourth chapter of the second part was published separately as Keshet u-Magen ("The Arrow and the Shield").
Works
Simon was prolific. He wrote commentaries on several tractates of the Mishnah and the Talmud and on Alfasi (Nos. 4, 5, 7, 11, 12, and 16 in the list of his works given below); he treated various religious dogmas as well as the synagogal rite of Algiers (Nos. 5, 8, 10, 16), while in his responsa he showed a profound acquaintance with the entire halakic literature. His theologico-philosophical scholarship, as well as his secular learning, is conspicuous in his elaborate work, Magen Abot, in which he also appears as a clever controversialist (No. 7). The same ability is evidenced in his writings against Hasdai Crescas, which afford him an opportunity to defend Maimonides (No. 2), in his commentary on the Pentateuch (No. 6), where he takes occasion to enter into polemics with Levi ben Gershon, and in that on the Book of Job (No. 1), especially the introduction. In his commentary on the Pirke Avot he shows a broad historical sense (No. 7, part iv.) and it is not improbable that the tradition which ascribes to him the historico-didactic poem Seder ha-Mishneh leha-Rambam (No
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https://en.wikipedia.org/wiki/2006%20Wigan%20Warriors%20season
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The Wigan Warriors played in the Super League and Challenge Cup in the 2006 season.
Statistics
Tries
Goals
Points
Appearances
2006 squad
Key players
Key players for Wigan Warriors in 2006 were:
Chris Ashton was brought in to the Wigan squad at the start of 2006 as replacement for the injured Kris Radlinski. Ashton had impressed on his debut for Wigan in 2005, scoring two tries against Huddersfield Giants but in 2006 he showed good skill, pace and talent and impressed many Wigan fans and people within rugby league. He finished the 2006 season as the leading try scorer at Wigan and his support play throughout the season was excellent. Although some criticism was made about his defensive abilities, he did earn a call into the England squad at the end of the year. Ashton was one of the most consistent players for Wigan through the 2006 and was a contender for the Young Player of the Year award.
Michael Dobson was signed by Wigan after his loan finished with Catalans Dragons as a replacement for Denis Moran, who had been released by Wigan a week earlier. Dobson had impressed for Catalans Dragons, but when he signed for Wigan, they were at the bottom of the league and facing relegation at the end of the season. His talents, organisation skills and goal kicking were a big factor in Wigan surviving relegation. In 2006 Michael Dobson was the most consistent goal kicker in Rugby League.
Stuart Fielden signed for Wigan from Bradford Bulls on 22 June 2006 for a Super League record transfer fee of £450,000. Fielden was regarded as one of the best props in the world by many, and his influence in the Wigan team could clearly be seen. He was strong in both attack and defence and offered leadership and inspiration to the rest of the Wigan players. Fielden was one of the main reasons that Wigan avoided relegation in 2006; not only did he provide a physical presence on the pitch, he offered new hope to Wigan fans and players.
Transfers
2006 transfer (in)
2006 transfer (out)
2006 loans (in)
2006 loans (out)
Staff
Chief executive – Carol Banks
Chief administrator – Mary Sharkey
Head coach – Brian Noble
Head coach – Ian Millward (sacked April 2006)
Chairman – Maurice P. Lindsay
References
External links
Wigan RL 2006 Season on the Wigan RL Fansite.
Wigan RL 2006 Senior Academy Team on the Wigan RL Fansite.
Wigan RL 2006 Junior Academy Team on the Wigan RL Fansite.
Official site
Wigan-Warriors fan site
Wigan Warriors seasons
Wigan Warriors season
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https://en.wikipedia.org/wiki/Vladimir%20Abramovich%20Rokhlin
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Vladimir Abramovich Rokhlin (Russian: Влади́мир Абра́мович Ро́хлин) (23 August 1919 – 3 December 1984) was a Soviet mathematician, who made numerous contributions in algebraic topology, geometry, measure theory, probability theory, ergodic theory and entropy theory.
Life
Vladimir Abramovich Rokhlin was born in Baku, Azerbaijan, to a wealthy Jewish family. His mother, Henrietta Emmanuilovna Levenson, had studied medicine in France (she died in Baku in 1923, believed to have been killed during civil unrest provoked by an epidemic). His maternal grandmother, Clara Levenson, had been one of the first female doctors in Russia. His maternal grandfather Emmanuil Levenson was a wealthy businessman (he was also the illegitimate father of Korney Chukovsky, who was thus Henrietta's half-brother). Vladimir Rokhlin's father Abram Veniaminovich Rokhlin was a well-known social democrat (he was imprisoned during Stalin's Great Purge, and executed in 1941).
Vladimir Rokhlin entered Moscow State University in 1935. His advisor was Abraham Plessner. He volunteered for the army in 1941, leading to four years as a prisoner of a German war camp. During this time he was able to hide his Jewish origins from the Nazis. Rokhlin was liberated by the Soviet military in January 1945. He then served as a German language translator for the 5th Army of the Belorussian front. In May 1945 he was sent to a Soviet 'verification camp' for former prisoners of war. In January 1946 he was transferred to another camp to determine if he was an "enemy of the Soviet." Rokhlin was cleared in June 1946 but was forced to remain in the camp as a guard. Due to intercession by mathematicians Andrey Kolmogorov and Lev Pontryagin, he was released in December 1946 and allowed to return to Moscow, after which he returned to mathematics.
In 1959 Rokhlin joined Leningrad State University as a faculty member. He died in 1984 in Leningrad. His students include Viatcheslav Kharlamov, Yakov Eliashberg, Mikhail Gromov, Nikolai V. Ivanov, Anatoly Vershik and Oleg Viro.
Work
Rokhlin's contributions to topology include Rokhlin's theorem, a result of 1952 on the signature of 4-manifolds. He also worked in the theory of characteristic classes, homotopy theory, cobordism theory, and in the topology of real algebraic varieties.
In measure theory, Rokhlin introduced what are now called Rokhlin partitions. He introduced the notion of standard probability space, and characterised such spaces up to isomorphism mod 0. He also proved the famous Rokhlin lemma.
Family
His son Vladimir Rokhlin, Jr. is a well-known mathematician and computer scientist at Yale University.
Rokhlin's uncle was Korney Chukovsky, a well-known Russian poet, most famous for his popular children's books.
See also
Gudkov's conjecture
Notes
References
External links
1919 births
1984 deaths
20th-century Russian mathematicians
Soviet mathematicians
Jewish scientists
Russian Jews
Soviet prisoners of war
Jewish concentration camp survivors
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https://en.wikipedia.org/wiki/Dennis%20Barden
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Dennis Barden is a mathematician at the University of Cambridge working in the fields of geometry and topology. He is known for his classification of the simply connected compact 5-manifolds and, together with Barry Mazur and John R. Stallings, for having proved the s-cobordism theorem. Barden received his Ph.D. from Cambridge in 1964 under the supervision of C. T. C. Wall.
Academic Positions
Barden is a Life Fellow of Girton College, Cambridge and emeritus fellow of Pembroke College. In 1991, he became Director of Studies for mathematics at Pembroke College, succeeding Raymond Lickorish. He held the position until Michaelmas 2003, and in his time saw a great increase in the number of applicants for mathematics, with consistently high performances in Tripos exams. He remains an active supervisor at Pembroke and Girton College.
Selected publications
References
1936 births
Living people
21st-century British mathematicians
20th-century British mathematicians
Topologists
Academics of the University of Cambridge
Alumni of the University of Cambridge
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https://en.wikipedia.org/wiki/Muted%20group%20theory
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Muted group theory (MGT), created by Edwin Ardener and Shirley Ardener in 1975, is a communication theory that focuses on how marginalized groups are muted and excluded via the use of language. The main idea of MGT is that "Language serves its creators better than those in other groups who have to learn to use the language as best they can."
The term mutedness refers to a group's inability to express themselves due to this inequity. The theory describes the relationship between a dominant group and its subordinate group(s) as being as follows: 1) the dominant group contributes mostly to the formulation of the language system, including the social norms and vocabulary, and 2) members from the subordinate group(s) have to learn and use the dominant language to express themselves. However, this translation process may result in the loss and distortion of information as the people from subordinate groups cannot articulate their ideas clearly. The dominant group may also ignore the voice of the marginalized group. All these may eventually lead to the mutedness of the subordinate group. Although this theory was initially developed to study the different situations faced by female and male, it can also be applied to any marginalized group that is muted by the inadequacies of their languages.
Overview
Origin
Background
in 1968, Edwin Ardener pointed out a phenomenon where anthropologists had difficulties reproducing models of the society from women's perspectives, if they (the models) do not conform to those generated by men.
In 1971, Shirley Ardener cited instances from feminism movements to articulate how women as a muted group used body symbolism to justify their actions and arguments in her article "Sexual Insult and Female Militancy".
In 1975, Shirley Ardener reprinted Edwin's paper "The Interpretation of Ritual,1972" and included her sexual insult text in the book "Perceiving Women", for which she wrote an intro.
Genesis
Muted Group Theory was firstly developed in the field of cultural anthropology by the British anthropologist, Edwin Ardener. The first formulation of MGT emerges from one of Edwin Ardener's short essays, entitled "Belief and the Problem of Women," in which Ardener explored the "problem" of women. In social anthropology, the problem of women is divided in two parts: technical and analytical. The technical problem is that although half of the population and society is technically made up of women, ethnographers have often ignored this half of the population. Ardener writes that "those trained in ethnography evidently have a bias towards the kinds of model that men are ready to provide (or to concur in) rather than towards any that women might provide." He also suggests that the reason behind this is that men tend to give a "bounded model of society" akin to the ones that ethnographers are attracted to. Therefore, men are those who produce and control symbolic production in a society. This leads to the analytical part of th
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https://en.wikipedia.org/wiki/Richard%20D.%20Gill
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Richard David Gill (born 1951) is a British-Dutch mathematician. He has held academic positions in the Netherlands. As a probability theorist and statistician, Gill has researched counting processes. He is also known for his consulting and advocacy on behalf of alleged victims of statistical misrepresentation, including the reversal of the murder conviction of a Dutch nurse who had been jailed for six years.
Education
Gill studied mathematics at the University of Cambridge (1970–1973), and subsequently followed the Diploma of Statistics course there (1973–1974). He obtained a Ph.D. in mathematics in 1979, with the thesis Censoring and Stochastic Integrals, which was supervised by Jacobus Oosterhoff of the Vrije Universiteit, which awarded the doctorate.
Gill has said that he was "not much of an activist" as a student, but now feels guilty about not speaking up more at the time about perceived injustices, saying that this is partly because of an incident when working as a statistician in the 1970s when he helped on an experiment that severed the front legs of rats to investigate whether it would lead to the reshaping of their skulls. Gill said that this incident has stayed with him, as "what upset me most is that I didn’t have the strength of character to refuse to do that job".
Career
In 1974 Gill was appointed at the Mathematical Centre (later renamed Centrum Wiskunde & Informatica, or CWI) of Amsterdam. After receiving his Ph.D., he continued to collaborate with Danish and Norwegian statisticians for ten years, co-authoring Statistical models based on counting processes, by Andersen, Borgan, Gill, and Keiding.
Gill became head of the Department of Mathematical Statistics at CWI in 1983. In 1988, Gill moved to the Department of Mathematics of Utrecht University, where he held the chair in mathematical stochastics. His PhD students include Sara van de Geer, and Mark van der Laan (co-advised by Peter Bickel). In 2006, Gill moved to the Department of Mathematics at Leiden University, where he held the chair of mathematical statistics. He retired from Leiden in 2017.
Gill became a citizen of the Netherlands in 1996.
Advocacy
Gill has lobbied for retrials for nurses whose criminal convictions were based in part on statistical evidence, including Lucia de Berk and Benjamin Geen. Gill also said in a 2021 lecture that he suspects Beverley Allitt is innocent, and in a 2020 paper said the case "deserves fresh study". Gill states that his original involvement in campaigning for nurses stemmed from his wife encouraging him to get involved in the de Berk case, recounting her saying "They’re using statistics; you should get involved, do something useful".
De Berk was sentenced in the Netherlands to life imprisonment in 2003, after a legal psychologist gave testimony that there was great likelihood that de Berk committed a string of murders. Gill and other professional statisticians showed this statistical testimony to be fallacious. Continued scrutin
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https://en.wikipedia.org/wiki/Math%20wars
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Math wars is the debate over modern mathematics education, textbooks and curricula in the United States that was triggered by the publication in 1989 of the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics (NCTM) and subsequent development and widespread adoption of a new generation of mathematics curricula inspired by these standards.
While the discussion about math skills has persisted for many decades, the term "math wars" was coined by commentators such as John A. Van de Walle and David Klein. The debate is over traditional mathematics and reform mathematics philosophy and curricula, which differ significantly in approach and content.
Advocates of reform
The largest supporter of reform in the US has been the National Council of Teachers of Mathematics.
One aspect of the debate is over how explicitly children must be taught skills based on formulas or algorithms (fixed, step-by-step procedures for solving math problems) versus a more inquiry-based approach in which students are exposed to real-world problems that help them develop fluency in number sense, reasoning, and problem-solving skills. In this latter approach, conceptual understanding is a primary goal and algorithmic fluency is expected to follow secondarily. Some parents and other stakeholders blame educators saying that failures occur not because the method is at fault, but because these educational methods require a great deal of expertise and have not always been implemented well in actual classrooms.
A backlash, which advocates call "poorly understood reform efforts" and critics call "a complete abandonment of instruction in basic mathematics," resulted in "math wars" between reform and traditional methods of mathematics education.
Critics of reform
Those who disagree with the inquiry-based philosophy maintain that students must first
develop computational skills before they can understand concepts of mathematics. These
skills should be memorized and practiced, using time-tested traditional methods until they become automatic. Time is better spent practicing skills rather than in investigations inventing alternatives, or justifying more than one correct answer or method. In this view, estimating answers is insufficient and, in fact, is considered to be dependent on strong foundational skills. Learning abstract concepts of mathematics is perceived to depend on a solid base of knowledge of the tools of the subject.
Supporters of traditional mathematics teaching oppose excessive dependence on innovations such as calculators or new technology, such as the Logo language. Student innovation is acceptable, even welcome, as long as it is mathematically valid. Calculator use can be appropriate after number sense has developed and basic skills have been mastered. Constructivist methods which are unfamiliar to many adults, and books which lack explanations of methods or solved examples make it difficult to help with homew
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https://en.wikipedia.org/wiki/Fran%C3%A7ois-Romain%20Lh%C3%A9risson
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François-Romain Lhérisson (1798–1859) was a Haitian poet and educator. He was born and later taught in Aquin in southwestern Haiti. He taught a broad range of subjects, including Latin, algebra, geometry, and law. As a poet, Lhérisson is remembered for his poetic songs, such as La Bergère Somnambule.
References
External links
1798 births
1859 deaths
Haitian educators
Haitian male poets
19th-century Haitian poets
19th-century male writers
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https://en.wikipedia.org/wiki/List%20of%20Thor%20and%20Delta%20launches%20%281990%E2%80%931999%29
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Between 1990 and 1999, there were 89 Thor-based rockets launched, of which 85 were successful, giving a 95.5% success rate.
Notable missions
Mars Pathfinder
Mars Climate Orbiter
Launch statistics
Rocket configurations
Launch sites
Launch outcomes
Launch history
1990
There were 13 Thor missiles launched in 1990. All 13 launches were successful.
1991
There were 6 Thor missiles launched in 1991. All 6 launches were successful.
1992
There were 12 Thor missiles launched in 1992. All 12 launches were successful.
1993
There were 7 Thor missiles launched in 1993. All 7 launches were successful.
1994
There were 3 Thor missiles launched in 1994. All 3 launches were successful.
1995
There were 3 Thor missiles launched in 1995. 2 of the 3 launches were successful, giving a 66.7% success rate.
1996
There were 10 Thor missiles launched in 1996. All 10 launches were successful.
1997
There were 11 Thor missiles launched in 1997. 10 of the 11 launches were successful, giving a 90.9% success rate.
1998
There were 13 Thor missiles launched in 1998. 12 of the 13 launches were successful, giving a 92.3% success rate.
1999
There were 11 Thor missiles launched in 1999. 10 of the 11 launches were successful, giving a 90.9% success rate.
References
Lists of Thor and Delta launches
Lists of Delta launches
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https://en.wikipedia.org/wiki/L%C3%A9l%C3%A9%2C%20Cameroon
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Lélé is a town in southern Cameroon, near the junction of the borders of Cameroon, Gabon and Congo-Brazzaville.
Statistics
Population = 794
See also
Lélé River
References
External links
Populated places in South Region (Cameroon)
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https://en.wikipedia.org/wiki/Robert%20Abelson
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Robert Paul Abelson (September 12, 1928 – July 13, 2005) was a Yale University psychologist and political scientist with special interests in statistics and logic.
Biography
He was born in New York City and attended the Bronx High School of Science. He did his undergraduate work at MIT and his Ph.D. in psychology at Princeton University's Department of Psychology under John Tukey and Silvan Tomkins.
From Princeton, Abelson went to Yale, where he stayed for the subsequent five decades of his career. Arriving during the Yale Communication Project, Abelson contributed to the foundation of attitudes studies as co-author of Attitude Organization and Change: An Analysis of Consistency Among Attitude Component, (1960, with Rosenberg, Hovland, McGuire, & Brehm). While at Yale, Abelson was briefly a bass in the Yale Russian Chorus. Abelson also played an instrumental role in the founding of computer science at Yale, chairing a 1967 University Committee that recommended establishing a computer science department.
With Milton J. Rosenberg, he developed the notion of “symbolic psycho-logic," an early attempt, using an idiosyncratic kind of adjacency matrix of a signed graph, at a descriptive (rather than prescriptive) psychological organization of attitudes and attitude consistency, which was key to the development of the field of social cognition.
The notion that beliefs, attitudes, and ideology were deeply connected knowledge structures was contained in Scripts, Plans, Goals, and Understanding (1977, with Roger Schank), a work that has collected several thousand citations, and led to the first interdisciplinary graduate program in cognitive science at Yale. His work on voting behavior in the 1960 and 1964 elections, and the creation of a computer program modeling ideology (the “Goldwater machine”) helped define and build the field of political psychology.
He was the author of Statistics As Principled Argument which includes prescriptions for how statistical analyses should proceed, as well as a description of what statistical analysis is, why we should do it, and how to differentiate good from bad statistical arguments. He was a co-author of several other books in psychology, statistics, and political science.
In 1959, Abelson published a paper to elucidate different ways in which an individual tends to resolve his "belief dilemmas" (Abelson «Modes of Resolution of Belief Dilemmas» Journal of conflict Resolution 1959).
Abelson received the Distinguished Scientific Contribution Award from APA, the Distinguished Scientist Award from SESP, and the Distinguished Scientist Award from the International Society of Political Psychology. He was elected a Fellow of the American Academy of Arts and Sciences in 1978.
He died of complications of Parkinson's disease.
Books
See also
The MAGIC criteria
Abelson's paradox
Bibliography
Ira J. Roseman, Stephen J. Read, "Psychologist at Play: Robert P. Abelson's Life and Contributions to Psychological Scienc
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https://en.wikipedia.org/wiki/Emil%20Viklick%C3%BD
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Emil Viklický (born 23 November 1948) is a Czech jazz pianist and composer.
Career
Viklický was born in Olomouc. He graduated from Palacký University in 1971 with a degree in mathematics. As a student, he devoted a lot of time to playing jazz piano, and in 1974, he was awarded the prize for best soloist at the Czechoslovak Amateur Jazz Festival. The same year, he joined Karel Velebný's SHQ ensemble. In 1976, he was a prizewinner at a jazz improvisation competition in Lyon, and his composition "Green Satin" () won first prize in a music conservatory competition in Monaco. In 1985, his composition "Cacharel" won second prize in the same competition.
In 1977, he was awarded a year's scholarship to study composition and arrangement with Herb Pomeroy at Berklee College of Music in Boston. He then continued his composition studies with Jarmo Sermila, George Crumb, and Václav Kučera. Since returning to Prague, he has led his own ensembles (primarily quartets and quintets), composed and arranged, and—after the death of Karel Velebný—worked as director of the Summer Jazz Workshops in Frýdlant. He has lectured at a similar workshop event in Glamorgan, Wales. Between 1991 and 1995, Viklický was President of the Czech Jazz Society, and since 1994, he has worked with the Ad lib Moravia ensemble, whose performances combine elements of Moravian folk music, modern jazz, and contemporary serious music. In 1996, the ensemble went on a concert tour of Mexico and the United States.
As a pianist
As a pianist, Viklický has performed in numerous international ensembles alongside musicians from the U.S. and other European countries, including the Lou Blackburn International Quartet, the Benny Bailey Quintet, and multi-instrumentalist Scott Robinson. He has made frequent appearances in Finland (with the Finnczech Quartet and in particular with Jarmo Sermila) and Norway (with the Czech-Norwegian Big Band and ), and has performed in the USA, Japan, Mexico, Israel, Germany, Luxembourg, the Netherlands (at the North Sea Festival), and elsewhere. He has also worked with fellow Czech, saxophonist Jaroslav Jakubovič, and often accompanied Czech jazz singer Eva Olmerová, during the last years of her career.
As a composer
As a composer, Viklický has attracted attention abroad primarily for having created a synthesis of the expressive elements of modern jazz with the melodicism and tonalities of Moravian folk song that is distinctly individual in contemporary jazz. Besides this, however, he also composes "straight-ahead" modern jazz as well as chamber and orchestral works that utilize certain elements of New Music, and at times his music requires a combination of classical and jazz performers. He also composes incidental and film music and has produced scores for several full-length feature films, such as the German horror comedy Killer Condom, and television series. Throughout the 1990s, he devoted an increasing amount of time to the composition of contemporary classical mu
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https://en.wikipedia.org/wiki/Siegel%20modular%20form
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In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.
Siegel modular forms are holomorphic functions on the set of symmetric n × n matrices with positive definite imaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables.
Siegel modular forms were first investigated by for the purpose of studying quadratic forms analytically. These primarily arise in various branches of number theory, such as arithmetic geometry and elliptic cohomology. Siegel modular forms have also been used in some areas of physics, such as conformal field theory and black hole thermodynamics in string theory.
Definition
Preliminaries
Let and define
the Siegel upper half-space. Define the symplectic group of level , denoted by as
where is the identity matrix. Finally, let
be a rational representation, where is a finite-dimensional complex vector space.
Siegel modular form
Given
and
define the notation
Then a holomorphic function
is a Siegel modular form of degree (sometimes called the genus), weight , and level if
for all .
In the case that , we further require that be holomorphic 'at infinity'. This assumption is not necessary for due to the Koecher principle, explained below. Denote the space of weight , degree , and level Siegel modular forms by
Examples
Some methods for constructing Siegel modular forms include:
Eisenstein series
Theta functions of lattices (possibly with a pluri-harmonic polynomial)
Saito–Kurokawa lift for degree 2
Ikeda lift
Miyawaki lift
Products of Siegel modular forms.
Level 1, small degree
For degree 1, the level 1 Siegel modular forms are the same as level 1 modular forms. The ring of such forms is a polynomial ring C[E4,E6] in the (degree 1) Eisenstein series E4 and E6.
For degree 2, showed that the ring of level 1 Siegel modular forms is generated by the (degree 2) Eisenstein series E4 and E6 and 3 more forms of weights 10, 12, and 35. The ideal of relations between them is generated by the square of the weight 35 form minus a certain polynomial in the others.
For degree 3, described the ring of level 1 Siegel modular forms, giving a set of 34 generators.
For degree 4, the level 1 Siegel modular forms of small weights have been found. There are no cusp forms of weights 2, 4, or 6. The space of cusp forms of weight 8 is 1-dimensional, spanned by the Schottky form. The space of cusp forms of weight 10 has dimension 1, the
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https://en.wikipedia.org/wiki/Paul%20Wolfskehl
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Paul Friedrich Wolfskehl (30 June 1856 in Darmstadt – 13 September 1906 in Darmstadt), was a physician with an interest in mathematics. He bequeathed 100,000 marks (equivalent to 1,000,000 pounds in 1997 money) to the first person to prove Fermat's Last Theorem.
He was the younger of two sons of a banker, Joseph Carl Theodor Wolfskehl. His elder brother, the jurist Wilhelm Otto Wolfskehl, took over the family bank after the death of his father. From 1875 to 1880 Paul Wolfskehl studied medicine at the Universities of Leipzig, Tübingen and Heidelberg. In 1880 he received his doctorate from the Heidelberg University. At about this time, he began to suffer from multiple sclerosis, which eventually forced him to pursue another career. From 1880 to 1883 he studied mathematics at the universities of Bonn and Bern. In 1887 he habilitated at the Technische Hochschule Darmstadt and became a Privatdozent for mathematics at the university.
There are a number of theories concerning the prize's origin. The most romantic is that he was spurned by a young lady and decided to commit suicide, but was distracted by what he thought was an error in a paper by Ernst Kummer, who had detected a flaw in Augustin Cauchy's attempted proof of Fermat's famous problem. This rekindled his will to live and, in gratitude, he established the prize. This story was traced by Philip Davis and William Chinn in their 1969 book 3.1416 and All That to renowned mathematician Alexander Ostrowski, who supposedly heard it from another, unidentified source. Another more prosaic story has it that Wolfskehl wanted to leave as little as possible to his shrewish wife. Yet another story, told in The Man Who Loved Only Numbers by Paul Hoffman, tells that Wolfskehl actually missed his supposed suicide time because he was in the library studying the Theorem. Upon realizing that, he concluded that the contemplation of mathematics was more rewarding than a beautiful woman so he decided not to kill himself. He bankrolled the Theorem because it "saved his life". On June 27, 1997, the prize was finally won by Andrew Wiles. By then, due in part to the hyperinflation Germany suffered after the end of World War I, the award had dwindled to £30,000.
The play From Abstraction by Robert Thorogood is based on the life of Wolfskehl. It was broadcast on BBC Radio 4 on 1 November 2006 and 29 August 2008.
See also
Andrew Beal, a Dallas banker who has offered $1,000,000 for a proof or disproof of Beal's conjecture
Wiles' proof of Fermat's Last Theorem
Millennium Prize Problems
Notes
References
Ball, W. W. R. and Coxeter, H. S. M., Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 69–73, 1987.
Barner, K. "Paul Wolfskehl and the Wolfskehl Prize". Not. Amer. Math. Soc. 44, 1294–1303, 1997.
Hoffman, P., The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth, New York: Hyperion, pp. 193–199, 1998.
External links
Details about Wolfskehl from Simon Singh
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https://en.wikipedia.org/wiki/Markus%E2%80%93Yamabe%20conjecture
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In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. If the Jacobian matrix of a dynamical system at a fixed point is Hurwitz, then the fixed point is asymptotically stable. Markus-Yamabe conjecture asks if a similar result holds globally. Precisely, the conjecture states that if a continuously differentiable map on an -dimensional real vector space has a fixed point, and its Jacobian matrix is everywhere Hurwitz, then the fixed point is globally stable.
The conjecture is true for the two-dimensional case. However, counterexamples have been constructed in higher dimensions. Hence, in the two-dimensional case only, it can also be referred to as the Markus–Yamabe theorem.
Related mathematical results concerning global asymptotic stability, which are applicable in dimensions higher than two, include various autonomous convergence theorems. Analog of the conjecture for nonlinear control system with scalar nonlinearity is known as Kalman's conjecture.
Mathematical statement of conjecture
Let be a map with and Jacobian which is Hurwitz stable for every .
Then is a global attractor of the dynamical system .
The conjecture is true for and false in general for .
References
Disproved conjectures
Stability theory
Fixed points (mathematics)
Theorems in dynamical systems
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https://en.wikipedia.org/wiki/Tristram%20Jones-Parry
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Tristram Jones-Parry (born 23 July 1947) is a British teacher of mathematics. He was headmaster of Emanuel School (1994–1998) and Westminster School (1998–2005), two independent schools in the UK, and is currently a governor at Hampton Court House School. He was educated at Westminster School and Christ Church, Oxford.
Biography
Upon his departure from Westminster, Jones-Parry received a great amount of attachment and approval from staff and pupils. He was depicted as "an excellent leader and educator, who devoted his time entirely for the sake of education rather than to building excessive relationship with external high-society figures, who bears a warm heart under a stern appearance, who will always be remembered for picking up rubbish from the Little Dean's Yard ground." There was rumour that he was in argument with the school council over issues concerning the allocation of funds gathered from benefactors. The council thought the fund should be used for improving school facilities while he was of the opinion that the money should be provided to smart students unable to afford fees.
He became involved in media and political controversy in 2004, when on retiring from independent schools at the age of 58, after 30 years' teaching experience, applied to teach in a state school in order to "give a bit back", but was rejected by the General Teaching Council on the basis that he had not completed the PGCE teacher training course which is obligatory for teachers in the state sector but not in the independent schools, insisting also that they were obliged to apply this rule rigidly.
Critics of the decision labelled it “totally absurd”, given Jones-Parry's long teaching experience in leading independent schools, and as Headmaster of Westminster - frequently cited as the most academically successful school in the country. The growing media controversy surrounding the issue, further spotlighted what many saw as a particularly onerous example of bureaucratic pedantry, leading to pressure on the then Education Secretary Charles Clarke from his own advisers to change the rules. Following a decision that the rules were necessary in order to preserve standards, the National Union of Teachers and NASUWT endorsed it, and the Teacher Training Agency stated that a fast-track route for qualified teachers already exists.
Jones-Parry was head of the innovative independent Sixth Form and is now a governor at Hampton Court House School. In 2015 Tristram Jones-Parry was interviewed on BBC Breakfast where he highlighted the benefits of later start times for teenagers.
References
External links
2005 farewell interview and article in The Elizabethan, Westminster School's magazine.
1947 births
Living people
People educated at Westminster School, London
Alumni of Christ Church, Oxford
Heads of schools in England
Head Masters of Westminster School
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https://en.wikipedia.org/wiki/Bohemian%20Society%20of%20Sciences
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Bohemian Society of Sciences was the first official scientific organization within Bohemia.
History
The Bohemian Society of Sciences was created from the Private Society for Mathematics, Patriotic History and Natural History, the first scientific society within the frontiers of the later Czechoslovakia. This organization was founded in 1772 and published six volumes of its proceedings before becoming the Bohemian Society of Sciences, and then later becoming the Royal Bohemian Scientific Society in 1784. Its members included Masons and Illuminatis, and the Royal Bohemian Scientific Society it later established some ties with the Private Scientific and Patriotic Society of Moravia.
In the early 18th century, the institution began to become, partially due to its usage of both Czech and German languages, which caused it to lose some of the more radical Czech scientists while the creation of the Vienna Academy caused the loss of some of the German-speaking scientists. By 1847, members of the Royal Bohemian Society of Sciences moved to the Vienna Academy, however, some of the members moved to other academies. Members moving to academies other than the Vienna Academy included: Palacký, Šafařík, Zippe, Presl and Purkyně.
After 1847 the sciences have continued to play a role in the Czech state, continuing through the creation of the Czech Academy of Sciences and Arts, which was created in 1890 through a decree issued by Emperor Franz Joseph, which existed among many other institutions. During World War II, most scientific research was halted due to the Nazi occupation, but was restarted in 1952 with the creation of the Czechoslovak Academy of Sciences, which continues to operate today.
References
Scientific societies based in the Czech Republic
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https://en.wikipedia.org/wiki/E-function
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In mathematics, E-functions are a type of power series that satisfy particular arithmetic conditions on the coefficients. They are of interest in transcendental number theory, and are more special than G-functions.
Definition
A function is called of type , or an -function, if the power series
satisfies the following three conditions:
All the coefficients belong to the same algebraic number field, , which has finite degree over the rational numbers;
For all ,
,
where the left hand side represents the maximum of the absolute values of all the algebraic conjugates of ;
For all there is a sequence of natural numbers such that is an algebraic integer in for , and and for which
.
The second condition implies that is an entire function of .
Uses
-functions were first studied by Siegel in 1929. He found a method to show that the values taken by certain -functions were algebraically independent. This was a result which established the algebraic independence of classes of numbers rather than just linear independence. Since then these functions have proved somewhat useful in number theory and in particular they have application in transcendence proofs and differential equations.
The Siegel–Shidlovsky theorem
Perhaps the main result connected to -functions is the Siegel–Shidlovsky theorem (also known as the Siegel and Shidlovsky theorem), named after Carl Ludwig Siegel and Andrei Borisovich Shidlovsky.
Suppose that we are given -functions, , that satisfy a system of homogeneous linear differential equations
where the are rational functions of , and the coefficients of each and are elements of an algebraic number field . Then the theorem states that if are algebraically independent over , then for any non-zero algebraic number that is not a pole of any of the the numbers are algebraically independent.
Examples
Any polynomial with algebraic coefficients is a simple example of an -function.
The exponential function is an -function, in its case for all of the .
If is an algebraic number then the Bessel function is an -function.
The sum or product of two -functions is an -function. In particular -functions form a ring.
If is an algebraic number and is an -function then will be an -function.
If is an -function then the derivative and integral of are also -functions.
References
Number theory
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https://en.wikipedia.org/wiki/Fernando%20Tarrida%20del%20M%C3%A1rmol
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Fernando Tarrida del Mármol (1861 – 1915) was a mathematics professor born in Cuba and raised in Catalonia best known for proposing "anarchism without adjectives", the idea that anarchists should set aside their debates over the most preferable economic systems and acknowledge their commonality in ultimate aims.
Early life and career
Fernando Tarrida del Mármol was born in 1861 in Cuba, son to Juan Tarrida, a merchant from Sitges, and Margarita Mármol, sister to the future Cuban insurgent leader Donato Mármol. His father became a prominent businessman in Santiago de Cuba, being the founder of the Spanish Circle in that city in January 1869. Following the passing away of Margarita, Juan Tarrida moved back to Spain in 1873, establishing shoe and boot manufacturing plant in the Catalan town of Sitges. Tarrida received a degree in mathematics from the Pau lycée, in southern France. His classmate and later French prime minister Louis Barthou converted him to republicanism. Tarrida moved to the University of Barcelona for a degree in civil engineering, and became a professor of mathematics at Barcelona's Polytechnic.
Despite his family's wealth, he identified more closely with Barcelona's working class and visited their clubs to discuss politics and quality of life. The workers appreciated his charisma and sincerity. By the mid 1880s—Tarrida's twenties—he was a collectivist anarchist who identified with the federalism of Pierre-Joseph Proudhon and Francesc Pi i Margall. Tarrida viewed anarchism beyond political philosophy as an all-encompassing philosophy, or the process by which humanity integrates and develops. He often referred to anarchism in mathematical formula as both the language to clarify his thoughts and to scientifically prove the philosophy's tenets. Tarrida gave public lectures and wrote about anarchism for libertarian journals, and developed a friendship with the Spanish anarchist Anselmo Lorenzo. Barcelona workers chose Tarrida as their delegate to the International Socialist Congress in Paris, 1889.
Tarrida first proposed the idea of "anarchism without adjectives" during a public speech in November 1889. Anarchists often debated their ideal economic conditions, and "anarchism without adjectives" appealed anarchists to abandon these divisions, accommodate other factions, follow the basic principles of anarchism, and instead work together towards their unified cause. He argued that anarchists share opposition to dogma and should therefore let each other freely choose their choice of economic system. Put another way, anarchism was "the axiom" and their economic model was "secondary". Tarrida gave this speech at the Bellas Artes palace as a representative of an affinity group in commemoration of the Chicago Haymarket affair two years prior. Tarrida, himself, did not publicly engage in the factionism between collectivism and communism, though his earlier works adopted a collectivist position. In 1890, the French anarcho-communist journ
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https://en.wikipedia.org/wiki/Conditional%20event%20algebra
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A standard, Boolean algebra of events is a set of events related to one another by the familiar operations and, or, and not. A conditional event algebra (CEA) contains not just ordinary events but also conditional events, which have the form "if A, then B". The usual purpose of a CEA is to enable the defining of a probability function, P, that satisfies the equation P(if A then B) = P(A and B) / P(A).
Motivation
In standard probability theory, an event is a set of outcomes, any one of which would be an occurrence of the event. P(A), the probability of event A, is the sum of the probabilities of all A-outcomes, P(B) is the sum of the probabilities of all B-outcomes, and P(A and B) is the sum of the probabilities of all outcomes that are both A-outcomes and B-outcomes. In other words, and, customarily represented by the logical symbol ∧, is interpreted as set intersection: P(A ∧ B) = P(A ∩ B). In the same vein, or, ∨, becomes set union, ∪, and not, ¬, becomes set complementation, ′. Any combination of events using the operations and, or, and not is also an event, and assigning probabilities to all outcomes generates a probability for every event. In technical terms, this means that the set of events and the three operations together constitute a Boolean algebra of sets, with an associated probability function.
P(if A, then B) is normally interpreted not as an ordinary probability—not, specifically, as P(A′ ∨ B)—but as the conditional probability of B given A, P(B | A) = P(A ∧ B) / P(A). This raises a question: what about a probability like P(if A, then B, and if C, then D)? For this, there is no standard answer. What would be needed, for consistency, is a treatment of if-then as a binary operation, →, such that for conditional events A → B and C → D, P(A → B) = P(B | A), P(C → D) = P(D | C), and P((A → B) ∧ (C → D)) is well-defined and reasonable. This treatment is what conditional event algebras try to provide.
Types of conditional event algebra
Ideally, a conditional event algebra, or CEA, would support a probability function that meets three conditions:
1. The probability function validates the usual axioms.
2. For any two ordinary events A and B, if P(A) > 0, then P(A → B) = P(B | A) = P(A ∧ B) / P(A).
3. For ordinary event A and acceptable probability function P, if P(A) > 0, then PA = P ( ⋅ | A), the function produced by conditioning on A, is also an acceptable probability function.
However, David Lewis showed (1976) that those conditions can only be met when there are just two possible outcomes—as with, say, a single coin flip. With three or more possible outcomes, constructing a probability function requires choosing which of the above three conditions to violate. Interpreting A → B as A′ ∪ B produces an ordinary Boolean algebra that violates 2. With CEAs, the choice is between 1 and 3.
Tri-event CEAs
Tri-event CEAs take their inspiration from three-valued logic, where the identification of logical conjunction, disjunction,
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https://en.wikipedia.org/wiki/Goldie%27s%20theorem
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In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (="finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R.
Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem.
In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all right ideals. This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain.
A consequence of Goldie's theorem, again due to Goldie, is that every semiprime principal right ideal ring is isomorphic to a finite direct sum of prime principal right ideal rings. Every prime principal right ideal ring is isomorphic to a matrix ring over a right Ore domain.
Sketch of the proof
This is a sketch of the characterization mentioned in the introduction. It may be found in .
If R be a semiprime right Goldie ring, then it is a right order in a semisimple ring:
Essential right ideals of R are exactly those containing a regular element.
There are no non-zero nil ideals in R.
R is a right nonsingular ring.
From the previous observations, R is a right Ore ring, and so its right classical ring of quotients Qr exists. Also from the previous observations, Qr is a semisimple ring. Thus R is a right order in Qr.
If R is a right order in a semisimple ring Q, then it is semiprime right Goldie:
Any right order in a Noetherian ring (such as Q) is right Goldie.
Any right order in a Noetherian semiprime ring (such as Q) is itself semiprime.
Thus, R is semiprime right Goldie.
References
External links
PlanetMath page on Goldie's theorem
PlanetMath page on Goldie ring
Theorems in ring theory
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https://en.wikipedia.org/wiki/Semiprime%20ring
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In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals and semiprime rings are the same as reduced rings.
For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form where n is a square-free integer. So, is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but is not (because 12 = 22 × 3, with a repeated prime factor).
The class of semiprime rings includes semiprimitive rings, prime rings and reduced rings.
Most definitions and assertions in this article appear in and .
Definitions
For a commutative ring R, a proper ideal A is a semiprime ideal if A satisfies either of the following equivalent conditions:
If xk is in A for some positive integer k and element x of R, then x is in A.
If y is in R but not in A, all positive integer powers of y are not in A.
The latter condition that the complement is "closed under powers" is analogous to the fact that complements of prime ideals are closed under multiplication.
As with prime ideals, this is extended to noncommutative rings "ideal-wise". The following conditions are equivalent definitions for a semiprime ideal A in a ring R:
For any ideal J of R, if Jk⊆A for a positive natural number k, then J⊆A.
For any right ideal J of R, if Jk⊆A for a positive natural number k, then J⊆A.
For any left ideal J of R, if Jk⊆A for a positive natural number k, then J⊆A.
For any x in R, if xRx⊆A, then x is in A.
Here again, there is a noncommutative analogue of prime ideals as complements of m-systems. A nonempty subset S of a ring R is called an n-system if for any s in S, there exists an r in R such that srs is in S. With this notion, an additional equivalent point may be added to the above list:
R\A is an n-system.
The ring R is called a semiprime ring if the zero ideal is a semiprime ideal. In the commutative case, this is equivalent to R being a reduced ring, since R has no nonzero nilpotent elements. In the noncommutative case, the ring merely has no nonzero nilpotent right ideals. So while a reduced ring is always semiprime, the converse is not true.
General properties of semiprime ideals
To begin with, it is clear that prime ideals are semiprime, and that for commutative rings, a semiprime primary ideal is prime.
While the intersection of prime ideals is not usually prime, it is a semiprime ideal. Shortly it will be shown that the converse is also true, that every semiprime ideal is the intersection of a family of prime ideals.
For any ideal B in a ring R, we can form the following sets:
The set is the definition of the radical of B and is clearly a semiprime ideal containing B, and in fact is the smallest semiprime ideal containing B. The inclusion above is sometimes proper in the general case, but for commutative rings it becomes an equality.
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https://en.wikipedia.org/wiki/Parry%E2%80%93Sullivan%20invariant
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In mathematics, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of incidence matrices in graph theory, and of certain one-dimensional dynamical systems. It provides a partial classification of non-trivial irreducible incidence matrices.
It is named after the English mathematician Bill Parry and the American mathematician Dennis Sullivan, who introduced the invariant in a joint paper published in the journal Topology in 1975.
Definition
Let A be an n × n incidence matrix. Then the Parry–Sullivan number of A is defined to be
where I denotes the n × n identity matrix.
Properties
It can be shown that, for nontrivial irreducible incidence matrices, flow equivalence is completely determined by the Parry–Sullivan number and the Bowen–Franks group.
References
Dynamical systems
Matrices
Algebraic graph theory
Graph invariants
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https://en.wikipedia.org/wiki/Automatic%20sequence
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In mathematics and theoretical computer science, an automatic sequence (also called a k-automatic sequence or a k-recognizable sequence when one wants to indicate that the base of the numerals used is k) is an infinite sequence of terms characterized by a finite automaton. The n-th term of an automatic sequence a(n) is a mapping of the final state reached in a finite automaton accepting the digits of the number n in some fixed base k.
An automatic set is a set of non-negative integers S for which the sequence of values of its characteristic function χS is an automatic sequence; that is, S is k-automatic if χS(n) is k-automatic, where χS(n) = 1 if n S and 0 otherwise.
Definition
Automatic sequences may be defined in a number of ways, all of which are equivalent. Four common definitions are as follows.
Automata-theoretic
Let k be a positive integer, and let D = (Q, Σk, δ, q0, Δ, τ) be a deterministic finite automaton with output, where
Q is the finite set of states;
the input alphabet Σk consists of the set {0,1,...,k-1} of possible digits in base-k notation;
δ : Q × Σk → Q is the transition function;
q0 ∈ Q is the initial state;
the output alphabet Δ is a finite set; and
τ : Q → Δ is the output function mapping from the set of internal states to the output alphabet.
Extend the transition function δ from acting on single digits to acting on strings of digits by defining the action of δ on a string s consisting of digits s1s2...st as:
δ(q,s) = δ(δ(q, s1s2...st-1), st).
Define a function a from the set of positive integers to the output alphabet Δ as follows:
a(n) = τ(δ(q0,s(n))),
where s(n) is n written in base k. Then the sequence a = a(1)a(2)a(3)... is a k-automatic sequence.
An automaton reading the base k digits of s(n) starting with the most significant digit is said to be direct reading, while an automaton starting with the least significant digit is reverse reading. The above definition holds whether s(n) is direct or reverse reading.
Substitution
Let be a k-uniform morphism of a free monoid and let be a coding (that is, a -uniform morphism), as in the automata-theoretic case. If is a fixed point of —that is, if —then is a k-automatic sequence. Conversely, every k-automatic sequence is obtainable in this way. This result is due to Cobham, and it is referred to in the literature as Cobham's little theorem.
k-kernel
Let k ≥ 2. The k-kernel of the sequence s(n) is the set of subsequences
In most cases, the k-kernel of a sequence is infinite. However, if the k-kernel is finite, then the sequence s(n) is k-automatic, and the converse is also true. This is due to Eilenberg.
It follows that a k-automatic sequence is necessarily a sequence on a finite alphabet.
Formal power series
Let u(n) be a sequence over an alphabet Σ and suppose that there is an injective function β from Σ to the finite field Fq, where q = pn for some prime p. The associated formal power series is
Then the sequence u is q-automatic if and only if this f
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https://en.wikipedia.org/wiki/Commutative%20magma
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In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras.
A magma which is both commutative and associative is a commutative semigroup.
A commutative non-associative magma derived from the rock, paper, scissors game
Let , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the binary operation derived from the rules of the game as follows:
For all :
If and beats in the game, then
I.e. every is idempotent.
So that for example:
"paper beats rock";
"scissors tie with scissors".
This results in the Cayley table:
By definition, the magma is commutative, but it is also non-associative, as shown by:
but
i.e.
Other examples
The "mean" operation on the rational numbers (or any commutative number system closed under division) is also commutative but not in general associative, e.g.
but
Generally, the mean operations studied in topology need not be associative.
The construction applied in the previous section to rock-paper-scissors applies readily to variants of the game with other numbers of gestures, as described in the section Variations, as long as there are two players and the conditions are symmetric between them; more abstractly, it may be applied to any trichotomous binary relation (like "beats" in the game). The resulting magma will be associative if the relation is transitive and hence is a (strict) total order;
otherwise, if finite, it contains directed cycles (like rock-paper-scissors-rock) and the magma is non-associative. To see the latter, consider combining all the elements in a cycle in reverse order, i.e. so that each element combined beats the previous one;
the result is the last element combined, while associativity and commutativity would mean that the result only depended on the set of elements in the cycle.
The bottom row in the Karnaugh diagram above gives more example operations, defined on the integers (or any commutative ring).
Derived commutative non-associative algebras
Using the rock-paper-scissors example, one can construct a commutative non-associative algebra over a field : take to be the three-dimensional vector space over whose elements are written in the form
for . Vector addition and scalar multiplication are defined component-wise, and vectors are multiplied using the above rules for multiplying the elements .
The set
i.e.
forms a basis for the algebra . As before, vector multiplication in is commutative, but not associative.
The same procedure may be used to derive from any commutative magma a commutative algebra over on , which will be non-associative if is.
Non-associative algebra
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https://en.wikipedia.org/wiki/Large
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Large means of great size.
Large may also refer to:
Mathematics
Arbitrarily large, a phrase in mathematics
Large cardinal, a property of certain transfinite numbers
Large category, a category with a proper class of objects and morphisms (or both)
Large diffeomorphism, a diffeomorphism that cannot be continuously connected to the identity diffeomorphism in mathematics and physics
Large numbers, numbers significantly larger than those ordinarily used in everyday life
Large ordinal, a type of number in set theory
Large sieve, a method of analytic number theory
Larger sieve, a heightening of the large sieve
Law of large numbers, a result in probability theory
Sufficiently large, a phrase in mathematics
Other uses
Large (film), a 2001 comedy film
Large (surname), an English surname
LARGE, an enzyme
Large, a British English name for the maxima (music), a note length in mensural notation
Large, or G's, or grand, slang for $1,000 US dollars
Large, a community in Jefferson Hills, Pennsylvania
See also
Big (disambiguation)
Giant (disambiguation)
Huge (disambiguation)
Humongous (disambiguation)
Macro (disambiguation)
Size (disambiguation)
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https://en.wikipedia.org/wiki/Huge
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Huge may refer to:
Huge cardinal, a number in mathematics
Huge (Caroline's Spine album), 1996
Huge (Hugh Hopper and Kramer album), 1997
Huge (TV series), a television series on ABC Family
Huge (digital agency)
Huge (magazine), a style magazine published by Kodansha in Japan
Human Genome Equivalent, a genomic sequence as long as the human genome, which can be used as a unit
Huge (film), a 2010 film directed by Ben Miller
The Huge Crew, trio of female bullies from Ned's Declassified School Survival Guide
King Huge, a character in the Adventure Time
See also
Hu Ge (disambiguation)
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https://en.wikipedia.org/wiki/Dawsonville%2C%20Kenya
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Dawsonville is a railway town and junction in Kenya, lying on the main line to Uganda and the branches to Kisumu and Solai.
See also
Railway stations in Kenya
Statistics
Elevation = 1984m
Population = 49,675
References
Populated places in Nakuru County
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https://en.wikipedia.org/wiki/Andrews%E2%80%93Curtis%20conjecture
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In mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators, named after James J. Andrews and Morton L. Curtis who proposed it in 1965. It is difficult to verify whether the conjecture holds for a given balanced presentation or not.
It is widely believed that the Andrews–Curtis conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the Zeeman conjecture on collapsibility implies the Andrews–Curtis conjecture.
References
Combinatorial group theory
Conjectures
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/Non-abelian
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Non-abelian or nonabelian may refer to:
Non-abelian group, in mathematics, a group that is not abelian (commutative)
Non-abelian gauge theory, in physics, a gauge group that is non-abelian
See also
Non-abelian gauge transformation, a gauge transformation
Non-abelian class field theory, in class field theory
Nonabelian cohomology, a cohomology
Abelian (disambiguation)
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https://en.wikipedia.org/wiki/Odd%20Aalen
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Odd Olai Aalen (born 6 May 1947, in Oslo) is a Norwegian statistician and a professor at the Department of Biostatistics at the Institute of Basic Medical Sciences at the University of Oslo.
Life
Aalen completed his examen artium in 1966 at Oslo Cathedral School before studying first mathematics and physics and then statistics in which he graduated at the University of Oslo in 1972.
Work
His research work is geared towards applications in biosciences. Aalen's early work on counting processes and martingales, starting with his 1976 Ph.D. thesis at the University of California, Berkeley, has had profound influence in biostatistics. Inferences for fundamental quantities associated with cumulative hazard rates, in survival analysis and models for analysis of event histories, are typically based on the Nelson–Aalen estimator or appropriate related statistics. The Nelson–Aalen estimator is related to the Kaplan-Meier estimator and generalisations thereof.
Aalen is currently professor emeritus at the Oslo Centre for Biostatistics and Epidemiology at the Faculty of Medicine at the University of Oslo.
Honors and awards
He is an elected member of the Norwegian Academy of Science and Letters.
References
External links
The BMMS Centre
Aalen's home page
1947 births
Living people
Members of the Norwegian Academy of Science and Letters
Norwegian statisticians
Academic staff of the University of Oslo
University of California, Berkeley alumni
University of Oslo alumni
People educated at Oslo Cathedral School
Norwegian expatriates in the United States
Fellows of the American Statistical Association
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https://en.wikipedia.org/wiki/Carleson%20measure
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In mathematics, a Carleson measure is a type of measure on subsets of n-dimensional Euclidean space Rn. Roughly speaking, a Carleson measure on a domain Ω is a measure that does not vanish at the boundary of Ω when compared to the surface measure on the boundary of Ω.
Carleson measures have many applications in harmonic analysis and the theory of partial differential equations, for instance in the solution of Dirichlet problems with "rough" boundary. The Carleson condition is closely related to the boundedness of the Poisson operator. Carleson measures are named after the Swedish mathematician Lennart Carleson.
Definition
Let n ∈ N and let Ω ⊂ Rn be an open (and hence measurable) set with non-empty boundary ∂Ω. Let μ be a Borel measure on Ω, and let σ denote the surface measure on ∂Ω. The measure μ is said to be a Carleson measure if there exists a constant C > 0 such that, for every point p ∈ ∂Ω and every radius r > 0,
where
denotes the open ball of radius r about p.
Carleson's theorem on the Poisson operator
Let D denote the unit disc in the complex plane C, equipped with some Borel measure μ. For 1 ≤ p < +∞, let Hp(∂D) denote the Hardy space on the boundary of D and let Lp(D, μ) denote the Lp space on D with respect to the measure μ. Define the Poisson operator
by
Then P is a bounded linear operator if and only if the measure μ is Carleson.
Other related concepts
The infimum of the set of constants C > 0 for which the Carleson condition
holds is known as the Carleson norm of the measure μ.
If C(R) is defined to be the infimum of the set of all constants C > 0 for which the restricted Carleson condition
holds, then the measure μ is said to satisfy the vanishing Carleson condition if C(R) → 0 as R → 0.
References
External links
Measures (measure theory)
Norms (mathematics)
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https://en.wikipedia.org/wiki/Arnoldo%20Frigessi
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Arnoldo Frigessi di Rattalma (born 1959) is an Italian statistician based in Norway, where he is a professor at the Department of Biostatistics (now called Oslo Centre for Biostatistics and Epidemiology) with the Institute of Basic Medical Research at the University of Oslo. He has also a position at the Oslo University Hospital and is affiliated with the Norwegian Computing Centre. He led the centre Statistics for Innovation, which was created in 2007 as one of 14 designated national centres for research-based innovation, funded by the Norwegian Research Council, until 2014. Frigessi succeeded in obtaining funding for a second centre of the same type, BigInsight, which started in 2014 and will operate for 8 years, again under his leadership. Frigessi develops new methods in statistics and machine learning and stochastic models to study principles, dynamics and patterns of complex dependence. His approach is often Bayesian and computationally intensive. He has developed theory for Markov Chain Monte Carlo methods, inferential methods for pair copula constructions, methods for the analysis of multiple genomic data types, the first digital twin of a breast tumor useful for personalised treatment. His work has been central to the national response to the COVID-19 pandemics in Norway, as a key member of the modelling group at the National Intritute of Public Health of Norway.
Frigessi is a fellow of the Norwegian Academy of Technological Sciences. and was elected member of the Norwegian Academy of Science and Letters in 2008. In 2021 he received the Sverdrup Prize from the Norwegian Statistical Association, elected Fellow of the Institute of Mathematical Statistics and is knighted Cavaliere Ordine al Merito della Repubblica Italiana. He is married to Ingrid Glad and has two children, Simon and Ada.
References
External links
Frigessi's home page
Google scholar
BiGInsight's web page
(SFI)^2, Statistics for Innovation
(sfi)2 Statistics for Innovation – The experience of the Oslo centre in industrial statistics
Apollon article
1959 births
Living people
Italian statisticians
Italian expatriates in Norway
Academic staff of the University of Oslo
Members of the Norwegian Academy of Science and Letters
Members of the Norwegian Academy of Technological Sciences
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https://en.wikipedia.org/wiki/Marshall%20Hall%20%28mathematician%29
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Marshall Hall Jr. (17 September 1910 – 4 July 1990) was an American mathematician who made significant contributions to group theory and combinatorics.
Education and career
Hall studied mathematics at Yale University, graduating in 1932. He studied for a year at Cambridge University under a Henry Fellowship working with G. H. Hardy. He returned to Yale to take his Ph.D. in 1936 under the supervision of Øystein Ore.
He worked in Naval Intelligence during World War II, including six months in 1944 at Bletchley Park, the center of British wartime code breaking. In 1946 he took a position at Ohio State University. In 1959 he moved to the California Institute of Technology where, in 1973, he was named the first IBM Professor at Caltech, the first named chair in mathematics. After retiring from Caltech in 1981, he accepted a post at Emory University in 1985.
Hall died in 1990 in London on his way to a conference to mark his 80th birthday.
Contributions
He wrote a number of papers of fundamental importance in group theory, including his solution of Burnside's problem for groups of exponent 6, showing that a finitely generated group in which the order of every element divides 6 must be finite.
His work in combinatorics includes an important paper of 1943 on projective planes, which for many years was one of the most cited mathematics research papers. In this paper he constructed a family of non-Desarguesian planes which are known today as Hall planes. He also worked on block designs and coding theory.
His classic book on group theory was well received when it came out and is still useful today. His book Combinatorial Theory came out in a second edition in 1986, published by John Wiley & Sons.
He proposed Hall's conjecture on the differences between perfect squares and perfect cubes, which remains an open problem as of 2015.
Publications
1943: "Projective Planes", Transactions of the American Mathematical Society 54(2): 229–77
1959: The Theory of Groups, Macmillan
Wilhelm Magnus (1960) Review: Marshall Hall, Jr. Theory of Groups Bulletin of the American Mathematical Society 66(3): 144–6.
1964: (with James K. Senior) The Groups of Order 2n n ≤ 6), Macmillan
Preface: "An exhaustive catalog of the 340 groups of order dividing 64 with detailed tables of defining relations, constants, and lattice presentations of each group in the notation the text defines. "Of enduring value to those interested in finite groups".
1967: Combinatorial Theory, Blaisdell
Notes
References
External links
1910 births
1990 deaths
20th-century American mathematicians
Mathematicians from Missouri
Scientists from St. Louis
Algebraists
Group theorists
Combinatorialists
Yale Graduate School of Arts and Sciences alumni
Yale College alumni
Emory University faculty
California Institute of Technology faculty
Ohio State University faculty
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https://en.wikipedia.org/wiki/Otero%20Mesa
|
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Otero Mesa is a plateau in the Trans-Pecos. The plateau extends north from Hudspeth County, Texas into Otero County, New Mexico. Otero Mesa is the dominant landform in Hudspeth County, composing 70% of its land area. Otero Mesa has a more limited extant in Otero County, NM. Overall, 2/3 of Otero Mesa are in Texas, but the colloquial usage of "Otero Mesa" is restricted to the component of the plateau in New Mexico. This is only a political distinction; Otero Mesa is physiographically continuous across the NM-TX state line.
In the center of Otero Mesa, the plateau is interrupted by the Cornudas Mountains, a cluster of buttes that jut almost 2,000' above the plateau. The Cornudas Mountains include Wind Mountain, the highest point on Otero Mesa at 7,282'. The range is peppered with thousands of petroglyphs, complementing the well-known Hueco Tanks site farther west.
Otero Mesa is the northernmost part of the Chihuahuan Desert at its longitude. While the Chihuahuan Desert extends another 200 miles north along the Pecos and Rio Grande River Valleys, the high backslopes of the Sacramento, White, and Manzano Mountains between the basins are too mesic to support Chihuahuan Desert vegetative sites. These areas are instead classified as Southwestern Tablelands.
Grassland is the predominant landcover on Otero Mesa. These semi-arid grasslands are a remnant of a much larger network of Chihuahuan Desert steppes that carpeted uplands and bajadas 150 years ago. Overgrazing and fire suppression has degraded large swaths of this ecoregion into scrubland. Consequently, conservation organizations have recognized Otero Mesa as a significant ecosystem deserving protection.
In Texas, Otero Mesa is divided into private ranches
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https://en.wikipedia.org/wiki/Edmond%20Laforest
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Edmond Laforest (20 June 1876 – 17 October 1915) was a Haitian poet.
Life and works
Born in Jérémie, Laforest was a teacher of French and mathematics. Some of his most noted works are Poèmes Mélancoliques (1901), Sonnets-Médaillons (1909), and Cendres et Flammes.
He killed himself by tying a Larousse dictionary around his neck and jumping off a bridge, to expose how the French language, imposed upon him by colonists, had killed him artistically.
References
1876 births
1915 deaths
20th-century Haitian poets
20th-century male writers
1915 suicides
Haitian educators
Haitian male poets
Suicides by drowning
Suicides in Haiti
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https://en.wikipedia.org/wiki/Cheeger%20constant%20%28graph%20theory%29
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In mathematics, the Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph has a "bottleneck". The Cheeger constant as a measure of "bottleneckedness" is of great interest in many areas: for example, constructing well-connected networks of computers, card shuffling. The graph theoretical notion originated after the Cheeger isoperimetric constant of a compact Riemannian manifold.
The Cheeger constant is named after the mathematician Jeff Cheeger.
Definition
Let be an undirected finite graph with vertex set and edge set . For a collection of vertices , let denote the collection of all edges going from a vertex in to a vertex outside of (sometimes called the edge boundary of ):
Note that the edges are unordered, i.e., . The Cheeger constant of , denoted , is defined by
The Cheeger constant is strictly positive if and only if is a connected graph. Intuitively, if the Cheeger constant is small but positive, then there exists a "bottleneck", in the sense that there are two "large" sets of vertices with "few" links (edges) between them. The Cheeger constant is "large" if any possible division of the vertex set into two subsets has "many" links between those two subsets.
Example: computer networking
In applications to theoretical computer science, one wishes to devise network configurations for which the Cheeger constant is high (at least, bounded away from zero) even when (the number of computers in the network) is large.
For example, consider a ring network of computers, thought of as a graph . Number the computers clockwise around the ring. Mathematically, the vertex set and the edge set are given by:
Take to be a collection of of these computers in a connected chain:
So,
and
This example provides an upper bound for the Cheeger constant , which also tends to zero as . Consequently, we would regard a ring network as highly "bottlenecked" for large , and this is highly undesirable in practical terms. We would only need one of the computers on the ring to fail, and network performance would be greatly reduced. If two non-adjacent computers were to fail, the network would split into two disconnected components.
Cheeger Inequalities
The Cheeger constant is especially important in the context of expander graphs as it is a way to measure the edge expansion of a graph. The so-called Cheeger inequalities relate the eigenvalue gap of a graph with its Cheeger constant. More explicitly
in which is the maximum degree for the nodes in and is the spectral gap of the Laplacian matrix of the graph. The Cheeger inequality is a fundamental result and motivation for spectral graph theory.
See also
Spectral graph theory
Algebraic connectivity
Cheeger bound
Conductance (graph)
Connectivity (graph theory)
Expander graph
References
Computer network analysis
Graph invariants
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https://en.wikipedia.org/wiki/Fiber%20bundle%20construction%20theorem
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In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic. The theorem is important in the associated bundle construction where one starts with a given bundle and surgically replaces the fiber with a new space while keeping all other data the same.
Formal statement
Let X and F be topological spaces and let G be a topological group with a continuous left action on F. Given an open cover {Ui} of X and a set of continuous functions
defined on each nonempty overlap, such that the cocycle condition
holds, there exists a fiber bundle E → X with fiber F and structure group G that is trivializable over {Ui} with transition functions tij.
Let E′ be another fiber bundle with the same base space, fiber, structure group, and trivializing neighborhoods, but transition functions t′ij. If the action of G on F is faithful, then E′ and E are isomorphic if and only if there exist functions
such that
Taking ti to be constant functions to the identity in G, we see that two fiber bundles with the same base, fiber, structure group, trivializing neighborhoods, and transition functions are isomorphic.
A similar theorem holds in the smooth category, where X and Y are smooth manifolds, G is a Lie group with a smooth left action on Y and the maps tij are all smooth.
Construction
The proof of the theorem is constructive. That is, it actually constructs a fiber bundle with the given properties. One starts by taking the disjoint union of the product spaces Ui × F
and then forms the quotient by the equivalence relation
The total space E of the bundle is T/~ and the projection π : E → X is the map which sends the equivalence class of (i, x, y) to x. The local trivializations
are then defined by
Associated bundle
Let E → X a fiber bundle with fiber F and structure group G, and let F′ be another left G-space. One can form an associated bundle E′ → X with a fiber F′ and structure group G by taking any local trivialization of E and replacing F by F′ in the construction theorem. If one takes F′ to be G with the action of left multiplication then one obtains the associated principal bundle.
References
See Part I, §2.10 and §3.
Fiber bundles
Theorems in topology
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https://en.wikipedia.org/wiki/5-manifold
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In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure.
Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected compact 5-manifolds were first classified by Stephen Smale and then in full generality by Dennis Barden, while another proof was later given by Aleksey V. Zhubr. This turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Thurston geometrisation conjecture, and the 4-dimensional case was solved by Michael Freedman (1982) in the topological case, but is a very hard unsolved problem in the smooth case.
In dimension 5, the smooth classification of simply connected manifolds is governed by classical algebraic topology. Namely, two simply connected, smooth 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving the linking form and the second Stiefel–Whitney class. Moreover, any such isomorphism in second homology is induced by some diffeomorphism. It is undecidable if a given 5-manifold is homeomorphic to , the 5-sphere.
Examples
Here are some examples of smooth, closed, simply connected 5-manifolds:
, the 5-sphere.
, the product of a 2-sphere with a 3-sphere.
, the total space of the non-trivial -bundle over .
, the homogeneous space obtained as the quotient of the special unitary group SU(3) by the rotation subgroup SO(3).
References
External links
Geometric topology
Manifolds
|
https://en.wikipedia.org/wiki/Borsuk%27s%20conjecture
|
The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's conjecture, is a question in discrete geometry. It is named after Karol Borsuk.
Problem
In 1932, Karol Borsuk showed that an ordinary 3-dimensional ball in Euclidean space can be easily dissected into 4 solids, each of which has a smaller diameter than the ball, and generally -dimensional ball can be covered with compact sets of diameters smaller than the ball. At the same time he proved that subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question:
The question was answered in the positive in the following cases:
— which is the original result by Karol Borsuk (1932).
— shown by Julian Perkal (1947), and independently, 8 years later, by H. G. Eggleston (1955). A simple proof was found later by Branko Grünbaum and Aladár Heppes.
For all for smooth convex bodies — shown by Hugo Hadwiger (1946).
For all for centrally-symmetric bodies — shown by A.S. Riesling (1971).
For all for bodies of revolution — shown by Boris Dekster (1995).
The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai, who showed that the general answer to Borsuk's question is . They claim that their construction shows that pieces do not suffice for and for each . However, as pointed out by Bernulf Weißbach, the first part of this claim is in fact false. But after improving a suboptimal conclusion within the corresponding derivation, one can indeed verify one of the constructed point sets as a counterexample for (as well as all higher dimensions up to 1560).
Their result was improved in 2003 by Hinrichs and Richter, who constructed finite sets for , which cannot be partitioned into parts of smaller diameter.
In 2013, Andriy V. Bondarenko had shown that Borsuk's conjecture is false for all . Shortly after, Thomas Jenrich derived a 64-dimensional counterexample from Bondarenko's construction, giving the best bound up to now.
Apart from finding the minimum number of dimensions such that the number of pieces , mathematicians are interested in finding the general behavior of the function . Kahn and Kalai show that in general (that is, for sufficiently large), one needs many pieces. They also quote the upper bound by Oded Schramm, who showed that for every , if is sufficiently large, . The correct order of magnitude of is still unknown. However, it is conjectured that there is a constant such that for all .
See also
Hadwiger's conjecture on covering convex bodies with smaller copies of themselves
Kahn–Kalai conjecture
Note
References
Further reading
Oleg Pikhurko, Algebraic Methods in Combinatorics, course notes.
Andrei M. Raigorodskii, The Borsuk partition problem: the seventieth anniversary, Mathematical Intelligencer 26 (2004), no. 3, 4–12.
External links
Disproved conjectures
Discrete geometry
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https://en.wikipedia.org/wiki/T-theory
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T-theory is a branch of discrete mathematics dealing with analysis of trees and discrete metric spaces.
General history
T-theory originated from a question raised by Manfred Eigen in the late 1970s. He was trying to fit twenty distinct t-RNA molecules of the Escherichia coli bacterium into a tree.
An important concept of T-theory is the tight span of a metric space. If X is a metric space, the tight span T(X) of X is, up to isomorphism, the unique minimal injective metric space that contains X. John Isbell was the first to discover the tight span in 1964, which he called the injective envelope. Andreas Dress independently constructed the same construct, which he called the tight span.
Application areas
Phylogenetic analysis, which is used to create phylogenetic trees.
Online algorithms - k-server problem
Recent developments
Bernd Sturmfels, Professor of Mathematics and Computer Science at Berkeley, and Josephine Yu classified six-point metrics using T-theory.
References
Metric geometry
Trees (data structures)
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https://en.wikipedia.org/wiki/Conference%20matrix
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In mathematics, a conference matrix (also called a C-matrix) is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix I. Thus, if the matrix has order n, CTC = (n−1)I.
Some authors use a more general definition, which requires there to be a single 0 in each row and column but not necessarily on the diagonal.
Conference matrices first arose in connection with a problem in telephony. They were first described by Vitold Belevitch, who also gave them their name. Belevitch was interested in constructing ideal telephone conference networks from ideal transformers and discovered that such networks were represented by conference matrices, hence the name. Other applications are in statistics, and another is in elliptic geometry.
For n > 1, there are two kinds of conference matrix. Let us normalize C by, first (if the more general definition is used), rearranging the rows so that all the zeros are on the diagonal, and then negating any row or column whose first entry is negative. (These operations do not change whether a matrix is a conference matrix.)
Thus, a normalized conference matrix has all 1's in its first row and column, except for a 0 in the top left corner, and is 0 on the diagonal. Let S be the matrix that remains when the first row and column of C are removed. Then either n is evenly even (a multiple of 4), and S is antisymmetric (as is the normalized C if its first row is negated), or n is oddly even (congruent to 2 modulo 4) and S is symmetric (as is the normalized C).
Symmetric conference matrices
If C is a symmetric conference matrix of order n > 1, then not only must n be congruent to 2 (mod 4) but also n − 1 must be a sum of two square integers; there is a clever proof by elementary matrix theory in van Lint and Seidel. n will always be the sum of two squares if n − 1 is a prime power.
Given a symmetric conference matrix, the matrix S can be viewed as the Seidel adjacency matrix of a graph. The graph has n − 1 vertices, corresponding to the rows and columns of S, and two vertices are adjacent if the corresponding entry in S is negative. This graph is strongly regular of the type called (after the matrix) a conference graph.
The existence of conference matrices of orders n allowed by the above restrictions is known only for some values of n. For instance, if n = q + 1 where q is a prime power congruent to 1 (mod 4), then the Paley graphs provide examples of symmetric conference matrices of order n, by taking S to be the Seidel matrix of the Paley graph.
The first few possible orders of a symmetric conference matrix are n = 2, 6, 10, 14, 18, (not 22, since 21 is not a sum of two squares), 26, 30, (not 34 since 33 is not a sum of two squares), 38, 42, 46, 50, 54, (not 58), 62 ; for every one of these, it is known that a symmetric conference matrix of that order exists. Order 66 seems to be an open problem.
Example
The essentially unique conference
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https://en.wikipedia.org/wiki/Seidel%20adjacency%20matrix
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In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non-adjacent vertices.
It is also called the Seidel matrix or—its original name—the (−1,1,0)-adjacency matrix.
It can be interpreted as the result of subtracting the adjacency matrix of G from the adjacency matrix of the complement of G.
The multiset of eigenvalues of this matrix is called the Seidel spectrum.
The Seidel matrix was introduced by J. H. van Lint and in 1966 and extensively exploited by Seidel and coauthors.
The Seidel matrix of G is also the adjacency matrix of a signed complete graph KG in which the edges of G are negative and the edges not in G are positive. It is also the adjacency matrix of the two-graph associated with G and KG.
The eigenvalue properties of the Seidel matrix are valuable in the study of strongly regular graphs.
References
van Lint, J. H., and Seidel, J. J. (1966), Equilateral point sets in elliptic geometry. Indagationes Mathematicae, vol. 28 (= Proc. Kon. Ned. Aka. Wet. Ser. A, vol. 69), pp. 335–348.
Seidel, J. J. (1976), A survey of two-graphs. In: Colloquio Internazionale sulle Teorie Combinatorie (Proceedings, Rome, 1973), vol. I, pp. 481–511. Atti dei Convegni Lincei, No. 17. Accademia Nazionale dei Lincei, Rome.
Seidel, J. J. (1991), ed. D.G. Corneil and R. Mathon, Geometry and Combinatorics: Selected Works of J. J. Seidel. Boston: Academic Press. Many of the articles involve the Seidel matrix.
Seidel, J. J. (1968), Strongly Regular Graphs with (−1,1,0) Adjacency Matrix Having Eigenvalue 3. Linear Algebra and its Applications 1, 281–298.
Algebraic graph theory
Matrices
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https://en.wikipedia.org/wiki/Birch%E2%80%93Tate%20conjecture
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The Birch–Tate conjecture is a conjecture in mathematics (more specifically in algebraic K-theory) proposed by both Bryan John Birch and John Tate.
Statement
In algebraic K-theory, the group K2 is defined as the center of the Steinberg group of the ring of integers of a number field F. K2 is also known as the tame kernel of F. The Birch–Tate conjecture relates the order of this group (its number of elements) to the value of the Dedekind zeta function . More specifically, let F be a totally real number field and let N be the largest natural number such that the extension of F by the Nth root of unity has an elementary abelian 2-group as its Galois group. Then the conjecture states that
Status
Progress on this conjecture has been made as a consequence of work on Iwasawa theory, and in particular of the proofs given for the so-called "main conjecture of Iwasawa theory."
References
J. T. Tate, Symbols in Arithmetic, Actes, Congrès Intern. Math., Nice, 1970, Tome 1, Gauthier–Villars(1971), 201–211
External links
Conjectures
K-theory
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/Encyclopedia%20of%20Statistical%20Sciences
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The Encyclopedia of Statistical Sciences is an encyclopaedia of statistics published by John Wiley & Sons.
The first edition, in nine volumes, was published in 1982; it was edited by Norman Lloyd Johnson and Samuel Kotz. The second edition, in 16 volumes, was published in 2006; the senior editor was Samuel Kotz.
See also
International Encyclopedia of Statistical Science
References
External links
Wiley page
Statistical Sciences
Statistics books
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https://en.wikipedia.org/wiki/John%20D.%20O%27Bryant%20School%20of%20Mathematics%20%26%20Science
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The John D. O'Bryant School of Mathematics and Science (abbreviated as O'B), formerly known as Boston Technical High School is a college preparatory public exam school along with Boston Latin School and Boston Latin Academy. The O’Bryant specializes in science, technology, engineering and mathematics ("STEM") in the city of Boston, Massachusetts, and is named for one of Boston's prominent African-American educators John D. O'Bryant. The school is currently located on 55 Malcolm X Boulevard in the neighborhood of Roxbury, Massachusetts. With a student body of 1,500 7th–12th graders, this school is part of the Boston Public Schools.
History
Now over one hundred years old, the O'Bryant began as the Mechanic Arts High School in 1893. Until the early 1970s, it was an all-boys school. In 1944, the school became Boston Technical High School. The original building containing the various shops, woodworking, machine shop, forge shop and drafting rooms was built around 1900 and was located on the corner of Dalton and Belvidere Streets in the Back Bay. The Hilton Hotel is located there today. In 1909 the five-story class room, chemistry and physics labs building was completed on Scotia Street adjacent to the older building. Later, the school moved to the building that originally housed Roxbury Memorial High School (1930 to 1960) at 205 Townsend Street in Roxbury, Massachusetts. That school building is now the home of Boston Latin Academy. Boston Technical High School remained there until 1987 when it relocated to a new building at 55 New Dudley Street (now Malcolm X Boulevard). In 1989, Boston Technical High School and Mario Umana Technical High School merged but still kept the name of Boston Technical High School. In 1994, the school graduated the first class for the school renamed after Boston educator John D. O'Bryant.
Notable alumni
Liz Miranda (Class of 1998), Massachusetts State Senator
Fred Ahern (Class of 1970), former National Hockey League player
Harry Barnes (Class of 1964), NBA player
William Bratton (Class of 1965), former Chief of Police for the LAPD, NYPD, and BPD
Alvin Campbell, member of the Campbell brothers criminal duo
Richard Egan (Class of 1953), co-founder of EMC Corporation and former United States Ambassador to Ireland
Arthur Gajarsa (Class of 1958), federal judge in the United States Court of Appeals for the Federal Circuit
Wayne Selden Jr. (left in 2010 after his freshman year), basketball player in the Israeli Basketball Premier League
Dan Sullivan (Class of 1957), former National Football League player
N.C. Wyeth (Class of 1899), artist and illustrator
Charles Yancey (Class of 1965), Boston City Councillor
References
External links
John D. O'Bryant School official website
The John D. O'Bryant Boston Tech Alumni Association
High schools in Boston
Educational institutions established in 1893
Magnet schools in Massachusetts
Public high schools in Massachusetts
Public middle schools in Massachusetts
1893 establishm
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https://en.wikipedia.org/wiki/Joint%20Mathematical%20Council
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The Joint Mathematical Council (JMC) of the United Kingdom was formed in 1963 to "provide co-ordination between the Constituent Societies and generally to promote the advancement of mathematics and the improvement of the teaching of mathematics".
The JMC serves as a forum for discussion between societies and for making representations to government and other bodies and responses to their enquiries. It is concerned with all aspects of mathematics at all levels from primary to higher education.
Members
The participating bodies are
Adults Learning Mathematics
Association of Teachers of Mathematics
Association of Mathematics Education Teachers
British Society for the History of Mathematics
British Society for Research into Learning Mathematics
HoDoMS
Edinburgh Mathematical Society
Institute of Mathematics and its Applications
London Mathematical Society
Mathematical Association
Mathematics in Education and Industry
National Association for Numeracy and Mathematics in Colleges
National Association of Mathematics Advisers
National Numeracy
STEM Learning
NRICH
Operational Research Society
Royal Academy of Engineering
Royal Statistical Society
Scottish Mathematical Council
United Kingdom Mathematics Trust
The observing bodies are
Advisory Committee on Mathematics Education
Department for Education (England)
Department of Education (Northern Ireland)
Education Scotland
National Centre for Excellence in Teaching Mathematics
Office for Standards in Education
The Office of Qualifications and Examinations Regulation
The Royal Society
Scottish Qualifications Authority
Welsh Government Education Directorate
Leadership
The Chair of the JMC is Andy Noyes, Professor of Education at the University of Nottingham and is a member of the Royal Society Advisory Committee on Mathematics Education.
References
External links
Web site
1963 establishments in the United Kingdom
Mathematics education in the United Kingdom
Mathematical societies
Learned societies of the United Kingdom
Professional associations based in the United Kingdom
Royal Statistical Society
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https://en.wikipedia.org/wiki/Roscommon%20High%20School
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Roscommon High School is located in Roscommon, Michigan. It is the secondary institution for the Roscommon Area Public School District. Roscommon, or RHS, is a class C/Division 3 School.
Statistics
2009
Enrollment: 440
ACT Average Score: 21.0
ACT Participation Rate: 44%
2005
Drop Out Rate: 3.43%
Graduation Rate: 87.22%
Demographics (2009)
White: 97.9%
Hispanic: 0.95%
Asian/Pacific Islander: 1.13%
Black: 0.38%
American Indian/Alaskan: 0.95%
Two or more races: 0%
Athletics
Highlights
2013 MHSAA Class B Volleyball Districts Runner-Up
2013 Jack Pine Conference Volleyball Champions
2006 MHSAA Division 3 Boys' Soccer State Runner-Up
2006 MHSAA Division 3 Wrestling State Runner-Up
1989 MHSAA Class C Boys Basketball Regional Champions
1989 MHSAA Clas C Baseball State Semi-Finalists
1988 MHSAA Class C Boys' Basketball State Runner-Up
2001 Division 3 District Wrestling Champions
2002 Division 3 District Wrestling Champions
2003 Division 3 District Wrestling Champions
2004 Division 3 District Wrestling Champions
2005 Division 3 District Wrestling Champions
2006 Division 3 District Wrestling Champions
2007 Division 3 District Wrestling Champions
2008 Division 3 District Wrestling Champions
2009 Division 3 District Wrestling Champions
2010 Division 3 District Wrestling Champions
2011 Division 3 District Wrestling Champions
2012 Division 3 District Wrestling Champions
2013 Division 3 District Wrestling Champions
2012 Division 3 Regional Wrestling Champions
Teams
Boys' Sports
Football
Wrestling- Coached by Trevor Tyler
Basketball- Coached by MJ Ewald
Soccer
Track/Cross Country- Coach by Todd Hofer
Baseball- Smitz
Girls' Sports
Girls' Soccer
Softball- Mark Sisco
Track/Cross Country- Todd Hofer
Volleyball- Heather Compton
Basketball- Scott Mires
Conference Affiliation
Football, wrestling, basketball, baseball, softball, golf, running, and cheerleading compete in the Jack Pine Conference. The soccer teams compete in the Northern Michigan Soccer League.
Other Teams
Forensics (Acting)
Quiz Bowl
Active Athletes in Action
Youth Advisory Committee (YAC)
Interact
Band
Marching
Steel
Jazz
Symphonic Wind Ensemble
The Roscommon High School Band Program is one of the largest band programs in Northern Michigan. There are currently 88 members, ranging from grade 9 to 12. The Band makes regular appearances at band festivals for competitive band. Director Seth Kilbourn has been with the band for 18 years. In 1989 under the direction of Larry Summerix the RHS Jazz band played on the main stage at Carnegie Hall in New York City.
Choir
Women's
Men's
Honor's
The Roscommon Choirs have been under the direction of Emerick Dee since 2008. They have been previously directed by Doug Armstead. Throughout the years, this choir program has attended several Choral Festivals and received high enough scores to receive State qualifying rating.
Musical
(From 2008-2013)
Sound of Music
Once Upon A Mattress
Into the Woods
Anything Goes
Bye Bye Birdie
Cinderella
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https://en.wikipedia.org/wiki/Goa%2C%20Botswana
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Goa is a small town in Botswana. It lies near the Namibian border, near the Caprivi Strip, and about 11 kilometres from Shakawe which is also the nearest airport.
Statistics
Elevation = 999m
References
North-West District (Botswana)
Villages in Botswana
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https://en.wikipedia.org/wiki/Quantum%20cohomology
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In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former. In each, the choice of coefficient ring (typically a Novikov ring, described below) significantly affects its structure, as well.
While the cup product of ordinary cohomology describes how submanifolds of the manifold intersect each other, the quantum cup product of quantum cohomology describes how subspaces intersect in a "fuzzy", "quantum" way. More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov–Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product.
Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has important implications for enumerative geometry. It also connects to many ideas in mathematical physics and mirror symmetry. In particular, it is ring-isomorphic to symplectic Floer homology.
Throughout this article, X is a closed symplectic manifold with symplectic form ω.
Novikov ring
Various choices of coefficient ring for the quantum cohomology of X are possible. Usually a ring is chosen that encodes information about the second homology of X. This allows the quantum cup product, defined below, to record information about pseudoholomorphic curves in X. For example, let
be the second homology modulo its torsion. Let R be any commutative ring with unit and Λ the ring of formal power series of the form
where
the coefficients come from R,
the are formal variables subject to the relation ,
for every real number C, only finitely many A with ω(A) less than or equal to C have nonzero coefficients .
The variable is considered to be of degree , where is the first Chern class of the tangent bundle TX, regarded as a complex vector bundle by choosing any almost complex structure compatible with ω. Thus Λ is a graded ring, called the Novikov ring for ω. (Alternative definitions are common.)
Small quantum cohomology
Let
be the cohomology of X modulo torsion. Define the small quantum cohomology with coefficients in Λ to be
Its elements are finite sums of the form
The small quantum cohomology is a graded R-module with
The ordinary cohomology H*(X) embeds into QH*(X, Λ) via , and QH*(X, Λ) is generated as a Λ-module by H*(X).
For any two cohomology classes a, b in H*(X) of pure degree, and for any A in , define (a∗b)A to be the unique element of H*(X) such that
(The right-hand side is a genus-0, 3-point Gromov–Witten invariant.) Then define
This extends by linearity to a well-defined Λ-bilinear map
called the small quantum cup product.
Geometric interpretation
The only pseudoholomorphic curves in class A = 0 are constant maps, whose images are points. It follows that
in oth
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https://en.wikipedia.org/wiki/Novikov%20ring
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In mathematics, given an additive subgroup , the Novikov ring of is the subring of consisting of formal sums such that and . The notion was introduced by Sergei Novikov in the papers that initiated the generalization of Morse theory using a closed one-form instead of a function. The notion is used in quantum cohomology, among the others.
The Novikov ring is a principal ideal domain. Let S be the subset of consisting of those with leading term 1. Since the elements of S are unit elements of , the localization of with respect to S is a subring of called the "rational part" of ; it is also a principal ideal domain.
Novikov numbers
Given a smooth function f on a smooth manifold with nondegenerate critical points, the usual Morse theory constructs a free chain complex such that the (integral) rank of is the number of critical points of f of index p (called the Morse number). It computes the (integral) homology of (cf. Morse homology):
In an analogy with this, one can define "Novikov numbers". Let X be a connected polyhedron with a base point. Each cohomology class may be viewed as a linear functional on the first homology group ; when composed with the Hurewicz homomorphism, it can be viewed as a group homomorphism . By the universal property, this map in turns gives a ring homomorphism,
,
making a module over . Since X is a connected polyhedron, a local coefficient system over it corresponds one-to-one to a -module. Let be a local coefficient system corresponding to with module structure given by . The homology group is a finitely generated module over which is, by the structure theorem, the direct sum of its free part and its torsion part. The rank of the free part is called the Novikov Betti number and is denoted by . The number of cyclic modules in the torsion part is denoted by . If , is trivial and is the usual Betti number of X.
The analog of Morse inequalities holds for Novikov numbers as well (cf. the reference for now.)
Notes
References
S. P. Novikov, Multi-valued functions and functionals: An analogue of Morse theory. Soviet Mathematics - Doklady 24 (1981), 222–226.
S. P. Novikov: The Hamiltonian formalism and a multi-valued analogue of Morse theory. Russian Mathematical Surveys 35:5 (1982), 1–56.
External links
Different definitions of Novikov ring?
Commutative algebra
Ring theory
Morse theory
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https://en.wikipedia.org/wiki/Rado%20graph
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In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of . The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one vertex at a time. The Rado graph is uniquely defined, among countable graphs, by an extension property that guarantees the correctness of this algorithm: no matter which vertices have already been chosen to form part of the induced subgraph, and no matter what pattern of adjacencies is needed to extend the subgraph by one more vertex, there will always exist another vertex with that pattern of adjacencies that the greedy algorithm can choose.
The Rado graph is highly symmetric: any isomorphism of its finite induced subgraphs can be extended to a symmetry of the whole graph.
The first-order logic sentences that are true of the Rado graph are also true of almost all random finite graphs, and the sentences that are false for the Rado graph are also false for almost all finite graphs. In model theory, the Rado graph is an example of the unique countable model of an ω-categorical theory.
History
The Rado graph was first constructed by in two ways, with vertices either the hereditarily finite sets or the natural numbers. (Strictly speaking Ackermann described a directed graph, and the Rado graph is the corresponding undirected graph given by forgetting the directions on the edges.) constructed the Rado graph as the random graph on a countable number of points. They proved that it has infinitely many automorphisms, and their argument also shows that it is unique though they did not mention this explicitly. rediscovered the Rado graph as a universal graph, and gave an explicit construction of it with vertex set the natural numbers. Rado's construction is essentially equivalent to one of Ackermann's constructions.
Constructions
Binary numbers
and constructed the Rado graph using the BIT predicate as follows. They identified the vertices of the graph with the natural numbers 0, 1, 2, ...
An edge connects vertices and in the graph (where ) whenever the th bit of the binary representation of is nonzero. Thus, for instance, the neighbors of vertex 0 consist of all odd-numbered vertices
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https://en.wikipedia.org/wiki/Association%20of%20Teachers%20of%20Mathematics
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The Association of Teachers of Mathematics (ATM) was established by Caleb Gattegno in 1950 to encourage the development of mathematics education to be more closely related to the needs of the learner. ATM is a membership organisation representing a community of students, nursery, infant, primary, secondary and tertiary teachers, numeracy consultants, overseas teachers, academics and anybody interested in mathematics education.
Aims
The stated aims of the Association of Teachers of Mathematics are to support the teaching and learning of mathematics by:
encouraging increased understanding and enjoyment of mathematics
encouraging increased understanding of how people learn mathematics
encouraging the sharing and evaluation of teaching and learning strategies and practices
promoting the exploration of new ideas and possibilities
initiating and contributing to discussion of and developments in mathematics education at all levels
Guiding principles
ATM lists as its guiding principles:
The ability to operate mathematically is an aspect of human functioning which is as universal as language itself. Attention needs constantly to be drawn to this fact. Any possibility of intimidating with mathematical expertise is to be avoided.
The power to learn rests with the learner. Teaching has a subordinate role. The teacher has a duty to seek out ways to engage the power of the learner.
It is important to examine critically approaches to teaching and to explore new possibilities, whether deriving from research, from technological developments or from the imaginative and insightful ideas of others.
Teaching and learning are cooperative activities. Encouraging a questioning approach and giving due attention to the ideas of others are attitudes to be encouraged. Influence is best sought by building networks of contacts in professional circles.
Structure
There are about 3500 members, mainly teachers in primary and secondary schools. It is a registered charity and all profits from subscriptions and trading are re-invested. Its head office is located in central Derby.
Branches
Working within the aims and guiding principles of the Association of Teachers of Mathematics, ATM Branches provide the opportunity for professionals to share ideas and experiences in their own areas.
Publications
ATM publishes Mathematics Teaching, a non-refereed journal with articles of interest to those involved in mathematics education. The journal is sent to all registered members. There are some free 'open access' journals available to all on the ATM website. ATM also publishes a range of resources suitable for teachers at all levels of teaching.
See also
Association for Science Education
Science, Technology, Engineering and Mathematics Network
Science Learning Centres - based at the University of York
References
External links
Web site
Easter Professional Development Conference
Mathematics Teaching journal
News items
Learning of maths plateaus in December 2007
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https://en.wikipedia.org/wiki/Generalized%20linear%20array%20model
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In statistics, the generalized linear array model (GLAM) is used for analyzing data sets with array structures. It based on the generalized linear model with the design matrix written as a Kronecker product.
Overview
The generalized linear array model or GLAM was introduced in 2006. Such models provide a structure and a computational procedure for fitting generalized linear models or GLMs whose model matrix can be written as a Kronecker product and whose data can be written as an array. In a large GLM, the GLAM approach gives very substantial savings in both storage and computational time over the usual GLM algorithm.
Suppose that the data is arranged in a -dimensional array with size ; thus, the corresponding data vector has size . Suppose also that the design matrix is of the form
The standard analysis of a GLM with data vector and design matrix proceeds by repeated evaluation of the scoring algorithm
where represents the approximate solution of , and is the improved value of it; is the diagonal weight matrix with elements
and
is the working variable.
Computationally, GLAM provides array algorithms to calculate the linear predictor,
and the weighted inner product
without evaluation of the model matrix
Example
In 2 dimensions, let , then the linear predictor is written where is the matrix of coefficients; the weighted inner product is obtained from and is the matrix of weights; here is the row tensor function of the matrix given by
where means element by element multiplication and is a vector of 1's of length .
On the other hand, the row tensor function of the matrix is the example of Face-splitting product of matrices, which was proposed by Vadym Slyusar in 1996:
where means Face-splitting product.
These low storage high speed formulae extend to -dimensions.
Applications
GLAM is designed to be used in -dimensional smoothing problems where the data are arranged in an array and the smoothing matrix is constructed as a Kronecker product of one-dimensional smoothing matrices.
References
Regression models
Array model
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https://en.wikipedia.org/wiki/Signature%20%28logic%29
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In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.
Definition
Formally, a (single-sorted) signature can be defined as a 4-tuple where and are disjoint sets not containing any other basic logical symbols, called respectively
function symbols (examples: ),
s or predicates (examples: ),
constant symbols (examples: ),
and a function which assigns a natural number called arity to every function or relation symbol. A function or relation symbol is called -ary if its arity is Some authors define a nullary (-ary) function symbol as constant symbol, otherwise constant symbols are defined separately.
A signature with no function symbols is called a , and a signature with no relation symbols is called an .
A is a signature such that and are finite. More generally, the cardinality of a signature is defined as
The is the set of all well formed sentences built from the symbols in that signature together with the symbols in the logical system.
Other conventions
In universal algebra the word or is often used as a synonym for "signature". In model theory, a signature is often called a , or identified with the (first-order) language to which it provides the non-logical symbols. However, the cardinality of the language will always be infinite; if is finite then will be .
As the formal definition is inconvenient for everyday use, the definition of a specific signature is often abbreviated in an informal way, as in:
"The standard signature for abelian groups is where is a unary operator."
Sometimes an algebraic signature is regarded as just a list of arities, as in:
"The similarity type for abelian groups is "
Formally this would define the function symbols of the signature as something like (which is binary), (which is unary) and (which is nullary), but in reality the usual names are used even in connection with this convention.
In mathematical logic, very often symbols are not allowed to be nullary, so that constant symbols must be treated separately rather than as nullary function symbols. They form a set disjoint from on which the arity function is not defined. However, this only serves to complicate matters, especially in proofs by induction over the structure of a formula, where an additional case must be considered. Any nullary relation symbol, which is also not allowed under such a definition, can be emulated by a unary relation symbol together with a sentence expressing that its value is the same for all elements. This translation fails only for empty structures (which are often excluded by convention). If nullary symbols are allowed, then every formula of propositional logic is also a formula of first-order logic.
An example f
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https://en.wikipedia.org/wiki/Donald%20Rubin
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Donald Bruce Rubin (born December 22, 1943) is an Emeritus Professor of Statistics at Harvard University, where he chaired the department of Statistics for 13 years. He also works at Tsinghua University in China and at Temple University in Philadelphia.
He is most well known for the Rubin causal model, a set of methods designed for causal inference with observational data, and for his methods for dealing with missing data.
In 1977 he was elected as a Fellow of the American Statistical Association.
Biography
Rubin was born in Washington, D.C. into a family of lawyers. As an undergraduate Rubin attended the accelerated Princeton University PhD program where he was one of a cohort of 20 students mentored by the physicist John Wheeler (the intention of the program was to confer degrees within 5 years of freshman matriculation). He switched to psychology and graduated in 1965. He began graduate school in psychology at Harvard with a National Science Foundation fellowship, but because his statistics background was considered insufficient, he was asked to take introductory statistics courses.
Rubin became a PhD student again, this time in Statistics under William Cochran at the Harvard Statistics Department. After graduating from Harvard in 1970, he began working at the Educational Testing Service in 1971, and served as a visiting faculty member at Princeton's new statistics department. He published his major papers on the Rubin causal model in 1974–1980, seminal papers on propensity score matching in the early 1980s with Paul Rosenbaum, and a textbook on the subject with Nobel prize winning econometrician Guido Imbens in 2015.
References
External links
Rubin's page on Harvard University Statistics Department website
American statisticians
Survey methodologists
Harvard University faculty
Temple University faculty
Quantitative psychologists
Princeton University alumni
1943 births
Living people
Bayesian statisticians
Fellows of the American Statistical Association
Members of the United States National Academy of Sciences
Harvard John A. Paulson School of Engineering and Applied Sciences alumni
Corresponding Fellows of the British Academy
American educational psychologists
Mathematical statisticians
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https://en.wikipedia.org/wiki/British%20Society%20for%20the%20History%20of%20Mathematics
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The British Society for the History of Mathematics (BSHM) was founded in 1971 to promote research into the history of mathematics at all levels and to further the use of the history of mathematics in education.
The BSHM is concerned with all periods and cultures, and with all aspects of mathematics. It participates in the Joint Mathematical Council of the United Kingdom.
The Society's journal, the British Journal for the History of Mathematics, is published on behalf of BSHM by Taylor & Francis.
Neumann Prize
The Neumann prize is awarded biennially by the BSHM for "a book in English (including books in translation) dealing with the history of mathematics and aimed at a broad audience." The prize was named in honour of Peter M. Neumann, who was a longstanding supporter of and contributor to the society. It carries an award of £600.The previous winners are:
2021: The Flying Mathematicians of World War I, Tony Royle
2019: Going Underground, Martin Beech
2017: A Mind at Play, Jimmy Soni & Rob Goodman
2015: The Thrilling Adventures of Lovelace and Babbage, Sydney Padua.
2013: The History of Mathematics: A Very Short Introduction, Jacqueline Stedall.
2011: The Math Book, Clifford A. Pickover.
2009: The Archimedes Codex, Reviel Netz and William Noel.
Other prizes
HiMEd Awards
LMS-BSHM Hirst Prize (jointly awarded by the London Mathematical Society and the BSHM)
Schools Prize
Taylor and Francis Early Career Research Prize
Undergraduate Essay Prize
Past Presidents of the BSHM
1971–1973: Gerald Whitrow
1974–1976: Clive Kilmister
1977–1979: John Dubbey
1980–1982: Graham Flegg
1983–1985: Frank Smithies
1986–1988: Ivor Grattan-Guinness
1989–1991: Eric Aiton
1992–1994: John Fauvel
1995–1996: Steve Russ
1997–1999: Judith V. Field
2000–2002: Peter Neumann
2003–2005: June Barrow-Green
2006–2008: Raymond Flood
2009–2011: Tony Mann
2012–2014: Robin Wilson
2015 - 2017: Philip Beeley
2018 - 2020: Mark McCartney
2021 - 2023: Sarah B. Hart
References
External links
BSHM website
Organizations established in 1971
Mathematical societies
History of mathematics
Learned societies of the United Kingdom
1971 establishments in the United Kingdom
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https://en.wikipedia.org/wiki/BSHM
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BSHM may refer to:
British Society for the History of Mathematics
British Society for the History of Medicine
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https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3%20R%C3%A1tz
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László Rátz (9 April 1863 in Sopron – 30 September 1930 in Budapest) was a Hungarian mathematics high school teacher best known for educating such people as John von Neumann and Nobel laureate Eugene Wigner. He was a legendary teacher of "Budapest-Fasori Evangélikus Gimnázium", the Budapest Lutheran Gymnasium, a famous secondary school in Budapest in Hungary.
Biography
He was born on 9 April 1863 in Sopron, a city in Hungary on the Austrian border, near the Lake Neusiedl/Lake Fertő. His father, Ágost Rátz, was a hardware merchant and ironmonger, and his mother was Emma Töpler of Danube Swabian origin. He graduated from the Lutheran Grammar School of Sopron in 1882.
The courses of study for elementary and middle school the first two years are not available. He was a student in the Hungarian royal state grammar school, "Főreáliskola in Sopron" between 1875 and 1880, now Széchenyi István Gimnázium (Sopron). From 1880 to 1882 he studied at the Sopron Lutheran High School and graduated in 1882. From 1883 to 1887 he was a student at the University of Budapest.
Then he attended the Science University of Budapest from 1883 to 1887. His university studies were at the Academy of Science in Budapest until 1887. He also studied philosophy at Berlin University between 4 October 1887 and 7 August 1888, and natural science at Strasbourg University from 31 October 1888. He worked as a practicing teacher in the Main Practising Secondary School of Budapest Science University from September 1889. He took his university degree specializing in mathematics and physics on 28 November 1890.
From 1890 he was a mathematics professor at the "Budapest-Fasori Evangélikus Gimnázium", a German-speaking Lutheran High School in Városligeti fasor 17–21 in Budapest. Beginning 1 September 1890 he was employed as a substitute teacher. From 1 September 1892 until 1925, he tenured as a regular teacher.
From 1909 to 1914, he served as director of the Gimnázium. One of his successors in this role was Sándor Mikola, who was school principal from 1928–1935.
Between the years of 1912–1921, he taught several students who became excellent mathematicians, physicians and chemists, including Nobel prize-winning physicist Eugene Wigner (Jenő Wigner) and mathematician and polymath John von Neumann (János Neumann). At the age of 11, Eugene Wigner developed an interest in mathematical problems. From 1915 through 1919 Wigner studied at the Gimnázium, where he and Von Neumann were taught by Rátz. von Neumann entered the Gimnázium in 1911. Although his father insisted he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude.
Lászlo Rátz died on 30 September 1930 in Grünwald Sanatorium, a nursing home, in Budapest.
Commemorative plaque
An embossed marble tablet commemorates him on the wall of Budapest Lutheran Gymnasium, Budapest-Fasori Evangélikus Gimnázium. The commemorat
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https://en.wikipedia.org/wiki/Gambling%20mathematics
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The mathematics of gambling is a collection of probability applications encountered in games of chance and can get included in game theory. From a mathematical point of view, the games of chance are experiments generating various types of aleatory events, and it is possible to calculate by using the properties of probability on a finite space of possibilities.
Experiments, events, and probability spaces
The technical processes of a game stand for experiments that generate aleatory events. Here are a few examples:
The occurrences could be defined; however, when formulating a probability problem, they must be done extremely carefully. From a mathematical point of view, the events are nothing more than subsets, and the space of events is a Boolean algebra. We find elementary and compound events, exclusive and nonexclusive events, and independent and non-independent events.
In the experiment of rolling a die:
Event {3, 5} (whose literal definition is the occurrence of 3 or 5) is compound because {3, 5}= {3} U {5};
Events {1}, {2}, {3}, {4}, {5}, {6} are elementary;
Events {3, 5} and {4} are incompatible or exclusive because their intersection is empty; that is, they cannot occur simultaneously;
Events {1, 2, 5} and {2, 5} are nonexclusive, because their intersection is not empty;
In the experiment of rolling two dice one after another, the events obtaining "3" on the first die and obtaining "5" on the second die are independent because the occurrence of the first does not influence the occurrence of the second event, and vice versa.
In the experiment of dealing the pocket cards in Texas Hold'em Poker:
The event of dealing (3♣, 3♦) to a player is an elementary event;
The event of dealing two 3's to a player is compound because it is the union of events (3♣, 3♠), (3♣, 3♥), (3♣, 3♦), (3♠, 3♥), (3♠, 3♦) and (3♥, 3♦);
The events "player 1 is dealt a pair of kings" and "player 2 is dealt a pair of kings" are nonexclusive (they can both occur);
The events player 1 is dealt two connectors of hearts higher than J and player 2 is dealt two connectors of hearts higher than J are exclusive (only one can occur);
The events player 1 is dealt (7, K) and player 2 is dealt (4, Q) are non-independent (the occurrence of the second depends on the happening of the first, while the same deck is in use).
These are a few examples of gambling events whose properties of compoundness, exclusiveness, and independency are readily observable. These properties are fundamental in practical probability calculus.
Combinations
Games of chance are also good examples of combinations, permutations, and arrangements, which are met at every step: combinations of cards in a player's hand, on the table, or expected in any card game; combinations of numbers when rolling several dice once; combinations of numbers in lottery and Bingo; combinations of symbols in slots; permutations and arrangements in a race to be bet on and the like. Combinatorial calculus is an integral pa
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https://en.wikipedia.org/wiki/Robert%20Maskell%20Patterson
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Robert Maskell Patterson (March 23, 1787 – September 5, 1854) was an American chemist, mathematician, and physician. He was a professor of mathematics, chemistry and natural philosophy at the University of Pennsylvania and professor of natural philosophy at the University of Virginia. He also served as director of the United States Mint and as president of the American Philosophical Society (elected in 1809).
Biography
Born in Philadelphia, to Robert Patterson, a professor at the University of Pennsylvania, and director of the Mint from 1805 to 1824, who had emigrated to the British North American colonies from Ireland. His mother was Amy Hunter Ewing and he was one of eight children of that marriage. Patterson attended the University of Pennsylvania graduating in 1804 with a B.A., followed by his M.D. in 1808. He journeyed to France that year, where he studied with Haüy, Vauquelin, Legendre and Poisson. In 1811, Patterson went to England where he studied with Humphry Davy. Returning to the United States in 1812, he was appointed a professor at the University of Pennsylvania. Patterson remained at Penn until 1828 when he joined the faculty of the University of Virginia. He was elected an Associate Fellow of the American Academy of Arts and Sciences in 1834. Patterson returned to Philadelphia in 1835 to become director of the U.S. Mint. He was asked by a committee of the American Philosophical Society in 1836 to write a brief report on recommendations for astronomical and physics observations to be carried out by the United States Exploring Expedition, which sailed in 1838. Ill health forced him to resign from the Mint in 1851. Patterson died in Philadelphia on September 5, 1854. He is buried at Laurel Hill Cemetery in Philadelphia in Section H, Plot 13, 20, 25 & 26. He was married to Helen Hamilton Leiper (April 20, 1792 – December 17, 1874), daughter of Thomas Leiper.
Cryptology
Patterson was interested in ciphers and regularly exchanged coded correspondence with Thomas Jefferson. One of Patterson's ciphers included in a December 19, 1801 dated letter to Jefferson was decoded in 2007 by Lawren Smithline.
The cipher consists of 7 digit pairs and is decoded by decrypting 7 blocks at a time.
The cipher was of the Declaration of Independence, of which Jefferson was the primary author. Patterson called it his "perfect cipher" and Jefferson considered adopting it for government use.
American Philosophical Society
Patterson was the youngest person elected to the American Philosophical Society at 22 in 1809. Four years later he was elected a secretary, the a vice-president in 1825. He became president in 1849 and served in that capacity until his death.
References
Frederik Nebeker, Astronomy and the Geophysical Tradition in the United States in the Nineteenth Century: A Guide to Manuscript Sources in the Library of the American Philosophical Society, APS Publication No. 16 (Philadelphia, 1991), p. 75–76.
Obituary: Proceedings of the American P
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https://en.wikipedia.org/wiki/Basu%27s%20theorem
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In statistics, Basu's theorem states that any boundedly complete minimal sufficient statistic is independent of any ancillary statistic. This is a 1955 result of Debabrata Basu.
It is often used in statistics as a tool to prove independence of two statistics, by first demonstrating one is complete sufficient and the other is ancillary, then appealing to the theorem. An example of this is to show that the sample mean and sample variance of a normal distribution are independent statistics, which is done in the Example section below. This property (independence of sample mean and sample variance) characterizes normal distributions.
Statement
Let be a family of distributions on a measurable space and a statistic maps from to some measurable space . If is a boundedly complete sufficient statistic for , and is ancillary to , then conditional on , is independent of . That is, .
Proof
Let and be the marginal distributions of and respectively.
Denote by the preimage of a set under the map . For any measurable set we have
The distribution does not depend on because is ancillary. Likewise, does not depend on because is sufficient. Therefore
Note the integrand (the function inside the integral) is a function of and not . Therefore, since is boundedly complete the function
is zero for almost all values of and thus
for almost all . Therefore, is independent of .
Example
Independence of sample mean and sample variance of a normal distribution
Let X1, X2, ..., Xn be independent, identically distributed normal random variables with mean μ and variance σ2.
Then with respect to the parameter μ, one can show that
the sample mean, is a complete and sufficient statistic – it is all the information one can derive to estimate μ, and no more – and
the sample variance, is an ancillary statistic – its distribution does not depend on μ.
Therefore, from Basu's theorem it follows that these statistics are independent conditional on , conditional on .
This independence result can also be proven by Cochran's theorem.
Further, this property (that the sample mean and sample variance of the normal distribution are independent) characterizes the normal distribution – no other distribution has this property.
Notes
References
Mukhopadhyay, Nitis (2000). Probability and Statistical Inference. Statistics: A Series of Textbooks and Monographs. 162. Florida: CRC Press USA. .
Theorems in statistics
Independence (probability theory)
Articles containing proofs
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https://en.wikipedia.org/wiki/Countably%20generated%20space
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In mathematics, a topological space is called countably generated if the topology of is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) is determined by the convergent sequences.
The countably generated spaces are precisely the spaces having countable tightness—therefore the name is used as well.
Definition
A topological space is called if for every subset is closed in whenever for each countable subspace of the set is closed in . Equivalently, is countably generated if and only if the closure of any equals the union of closures of all countable subsets of
Countable fan tightness
A topological space has if for every point and every sequence of subsets of the space such that there are finite set such that
A topological space has if for every point and every sequence of subsets of the space such that there are points such that Every strong Fréchet–Urysohn space has strong countable fan tightness.
Properties
A quotient of a countably generated space is again countably generated. Similarly, a topological sum of countably generated spaces is countably generated. Therefore, the countably generated spaces form a coreflective subcategory of the category of topological spaces. They are the coreflective hull of all countable spaces.
Any subspace of a countably generated space is again countably generated.
Examples
Every sequential space (in particular, every metrizable space) is countably generated.
An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of Arens–Fort space.
See also
− Tightness is a cardinal function related to countably generated spaces and their generalizations.
References
External links
A Glossary of Definitions from General Topology
https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf
General topology
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https://en.wikipedia.org/wiki/Bianchi%20classification
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In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism). The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized family of Lie algebras. (Sometimes two of the groups are included in the infinite families, giving 9 instead of 11 classes.) The classification is important in geometry and physics, because the associated Lie groups serve as symmetry groups of 3-dimensional Riemannian manifolds. It is named for Luigi Bianchi, who worked it out in 1898.
The term "Bianchi classification" is also used for similar classifications in other dimensions and for classifications of complex Lie algebras.
Classification in dimension less than 3
Dimension 0: The only Lie algebra is the abelian Lie algebra R0.
Dimension 1: The only Lie algebra is the abelian Lie algebra R1, with outer automorphism group the multiplicative group of non-zero real numbers.
Dimension 2: There are two Lie algebras:
(1) The abelian Lie algebra R2, with outer automorphism group GL2(R).
(2) The solvable Lie algebra of 2×2 upper triangular matrices of trace 0. It has trivial center and trivial outer automorphism group. The associated simply connected Lie group is the affine group of the line.
Classification in dimension 3
All the 3-dimensional Lie algebras other than types VIII and IX can be constructed as a semidirect product of R2 and R, with R acting on R2 by some 2 by 2 matrix M. The different types correspond to different types of matrices M, as described below.
Type I: This is the abelian and unimodular Lie algebra R3. The simply connected group has center R3 and outer automorphism group GL3(R). This is the case when M is 0.
Type II: The Heisenberg algebra, which is nilpotent and unimodular. The simply connected group has center R and outer automorphism group GL2(R). This is the case when M is nilpotent but not 0 (eigenvalues all 0).
Type III: This algebra is a product of R and the 2-dimensional non-abelian Lie algebra. (It is a limiting case of type VI, where one eigenvalue becomes zero.) It is solvable and not unimodular. The simply connected group has center R and outer automorphism group the group of non-zero real numbers. The matrix M has one zero and one non-zero eigenvalue.
Type IV: The algebra generated by [y,z] = 0, [x,y] = y, [x, z] = y + z. It is solvable and not unimodular. The simply connected group has trivial center and outer automorphism group the product of the reals and a group of order 2. The matrix M has two equal non-zero eigenvalues, but is not diagonalizable.
Type V: [y,z] = 0, [x,y] = y, [x, z] = z. Solvable and not unimodular. (A limiting case of type VI where both eigenvalues are equal.) The simply connected group has trivial center and outer automorphism group the elements of GL2(R) of determinant +1 or −1. The matrix M has two equal eigenvalues, and is diagonalizable.
Type VI: An infinite family: semidirect produc
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https://en.wikipedia.org/wiki/Bernd%20Sturmfels
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Bernd Sturmfels (born March 28, 1962 in Kassel, West Germany) is a Professor of Mathematics and Computer Science at the University of California, Berkeley and is a director of the Max Planck Institute for Mathematics in the Sciences in Leipzig since 2017.
Education and career
He received his PhD in 1987 from the University of Washington and the Technische Universität Darmstadt. After two postdoctoral years at the Institute for Mathematics and its Applications in Minneapolis, Minnesota, and the Research Institute for Symbolic Computation in Linz, Austria, he taught at Cornell University, before joining University of California, Berkeley in 1995. His Ph.D. students include Melody Chan, Jesús A. De Loera, Mike Develin, Diane Maclagan, Rekha R. Thomas, Caroline Uhler, and Cynthia Vinzant.
Contributions
Bernd Sturmfels has made contributions to a variety of areas of mathematics, including algebraic geometry, commutative algebra, discrete geometry, Gröbner bases, toric varieties, tropical geometry, algebraic statistics, and computational biology. He has written several highly cited papers in algebra with Dave Bayer.
He has authored or co-authored multiple books including Introduction to tropical geometry with Diane Maclagan.
Awards and honors
Sturmfels' honors include a National Young Investigator Fellowship, an Alfred P. Sloan Fellowship, and a David and Lucile Packard Fellowship. In 1999 he received a Lester R. Ford Award for his expository article Polynomial equations and convex polytopes. He was awarded a Miller Research Professorship at the University of California Berkeley for 2000–2001. In 2018, he was awarded the George David Birkhoff Prize in Applied Mathematics.
In 2012, he became a fellow of the American Mathematical Society.
References
Further reading
External links
Homepage at Berkeley
1962 births
Living people
Scientists from Kassel
20th-century German mathematicians
20th-century American mathematicians
21st-century American mathematicians
University of Washington alumni
Fellows of the American Mathematical Society
UC Berkeley College of Engineering faculty
Fellows of the Society for Industrial and Applied Mathematics
Mathematics popularizers
Technische Universität Darmstadt alumni
Algebraic geometers
Combinatorialists
Sloan Research Fellows
Algebraists
Cornell University faculty
21st-century German mathematicians
Academic staff of Max Planck Society
Max Planck Institute directors
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https://en.wikipedia.org/wiki/Paradoxical%20set
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In set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is two families of disjoint subsets, along with appropriate group actions that act on some universe (of which the set in question is a subset), such that each partition can be mapped back onto the entire set using only finitely many distinct functions (or compositions thereof) to accomplish the mapping. A set that admits such a paradoxical decomposition where the actions belong to a group is called -paradoxical or paradoxical with respect to .
Paradoxical sets exist as a consequence of the Axiom of Infinity. Admitting infinite classes as sets is sufficient to allow paradoxical sets.
Definition
Suppose a group acts on a set . Then is -paradoxical if there exists some disjoint subsets and some group elements such that:
and
Examples
Free group
The Free group F on two generators a,b has the decomposition where e is the identity word and is the collection of all (reduced) words that start with the letter i. This is a paradoxical decomposition because
Banach–Tarski paradox
The most famous example of paradoxical sets is the Banach–Tarski paradox, which divides the sphere into paradoxical sets for the special orthogonal group. This result depends on the axiom of choice.
See also
Pathological (mathematics)
References
Set theory
Geometric dissection
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https://en.wikipedia.org/wiki/Rhisiart%20ap%20Rhys
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Rhisiart ap Rhys (fl. c. 1495 – c. 1510) was a Welsh-language poet from the cwmwd of Tir Iarll, Glamorgan.
He was the son of Rhys Brydydd and nephew, in all probability, to the poet Gwilym Tew. 36 of his poems are extant.
Bibliography
Eurys I. Roland (ed.), Gwaith Rhys Brydydd a Rhisiart ap Rhys (Cardiff, 1976)
Year of birth uncertain
Year of death unknown
15th-century Welsh poets
16th-century Welsh poets
Welsh male poets
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https://en.wikipedia.org/wiki/Full%20measure
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Full measure or Full Measure may refer to:
"Full Measure" (Breaking Bad), a 2010 episode of Breaking Bad
Full measure (mathematics), a set whose complement is of measure zero
Full Measure (TV series), a 2015 series hosted by Sharyl Attkisson
Full Measure, a 1929 novel by Hans Otto Storm
"Full Measure", a 1966 song by The Lovin' Spoonful from Hums of the Lovin' Spoonful
Full Measure, a 2014 novel by T. Jefferson Parker
See also
The Last Full Measure (disambiguation)
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https://en.wikipedia.org/wiki/Isomorphism%20%28disambiguation%29
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Isomorphism or isomorph may refer to:
Isomorphism, in mathematics, logic, philosophy, and information theory, a mapping that preserves the structure of the mapped entities, in particular:
Graph isomorphism a mapping that preserves the edges and vertices of a graph
Group isomorphism a mapping that preserves the group structure
Order isomorphism a mapping that preserves the comparabilities of a partially ordered set.
Ring isomorphism a mapping that preserves both the additive and multiplicative structure of a ring
Isomorphism theorems theorems that assert that some homomorphisms involving quotients and subobjects are isomorphisms
Isomorphism (sociology), a similarity of the processes or structure of one organization to those of another
Isomorphism (crystallography), a similarity of crystal form
Isomorphism (Gestalt psychology), a correspondence between a stimulus array and the brain state created by that stimulus
Cybernetic isomorphism, a recursive property of viable systems, as defined in Stafford Beer's viable system model
Isomorph (gene), a classification of gene mutation
See also
Isomorph Records, a British music label
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https://en.wikipedia.org/wiki/Tight%20span
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In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes known as the injective envelope or hyperconvex hull of M. It has also been called the injective hull, but should not be confused with the injective hull of a module in algebra, a concept with a similar description relative to the category of R-modules rather than metric spaces.
The tight span was first described by , and it was studied and applied by Holsztyński in the 1960s. It was later independently rediscovered by and ; see for this history. The tight span is one of the central constructions of T-theory.
Definition
The tight span of a metric space can be defined as follows. Let (X,d) be a metric space, and let T(X) be the set of extremal functions on X, where we say an extremal function on X to mean a function f from X to R such that
For any x, y in X, d(x,y) ≤ f(x) + f(y), and
For each x in X, f(x) = sup{d(x,y) - f(y):y in X}.
In particular (taking x = y in property 1 above) f(x) ≥ 0 for all x. One way to interpret the first requirement above is that f defines a set of possible distances from some new point to the points in X that must satisfy the triangle inequality together with the distances in (X,d). The second requirement states that none of these distances can be reduced without violating the triangle inequality.
The tight span of (X,d) is the metric space (T(X),δ), where
is analogous to the metric induced by the norm. (If d is bounded, then δ is the subspace metric induced by the metric induced by the norm. If d is not bounded, then every extremal function on X is unbounded and so Regardless, it will be true that for any f,g in T(X), the difference belongs to , i.e., is bounded.)
Equivalent definitions of extremal functions
For a function f from X to R satisfying the first requirement, the following versions of the second requirement are equivalent:
For each x in X, f(x) = sup{d(x,y) - f(y):y in X}.
f is pointwise minimal with respect to the aforementioned first requirement, i.e., for any function g from X to R such that d(x,y) ≤ g(x) + g(y) for all x,y in X, if g≤f pointwise, then f=g.
Basic properties and examples
For all x in X,
For each x in X, is extremal. (Proof: Use symmetry and the triangle inequality.)
If X is finite, then for any function f from X to R that satisfies the first requirement, the second requirement is equivalent to the condition that for each x in X, there exists y in X such that f(x) + f(y) = d(x,y). (If then both conditions are true. If then the supremum is achieved, and the first requirement implies the equivalence.)
Say |X|=2, and choose distinct a, b such that X={a,b}. Then is the convex hull of {{(a,1),(b,0)},{(a,0),(b,1)}}. [Add a picture. Caption: If X={0,1}, then is the convex hull o
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https://en.wikipedia.org/wiki/Nice%20name
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In set theory, a nice name is used in forcing to impose an upper bound on the number of subsets in the generic model. It is used in the context of forcing to prove independence results in set theory such as Easton's theorem.
Formal definition
Let ZFC be transitive, a forcing notion in , and suppose is generic over .
Then for any -name in , we say that is a nice name for a subset of if is a -name satisfying the following properties:
(1)
(2) For all -names , forms an antichain.
(3) (Natural addition): If , then there exists in such that .
References
Forcing (mathematics)
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https://en.wikipedia.org/wiki/Selangau%20District
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The Selangau District is a district in Sarawak, Malaysia.
History
Selangau was declared a district on 1 March 2002.
Demographics
According to the Department of Statistics Malaysia, Selangau has a population of 26,100.
References
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https://en.wikipedia.org/wiki/Singular%20distribution
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In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero.
Other names
These distributions are sometimes called singular continuous distributions, since their cumulative distribution functions are singular and continuous.
Properties
Such distributions are not absolutely continuous with respect to Lebesgue measure.
A singular distribution is not a discrete probability distribution because each discrete point has a zero probability. On the other hand, neither does it have a probability density function, since the Lebesgue integral of any such function would be zero.
In general, distributions can be described as a discrete distribution (with a probability mass function), an absolutely continuous distribution (with a probability density), a singular distribution (with neither), or can be decomposed into a mixture of these.
Example
An example is the Cantor distribution; its cumulative distribution function is a devil's staircase. Less curious examples appear in higher dimensions. For example, the upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
See also
Singular measure
Lebesgue's decomposition theorem
External links
Singular distribution in the Encyclopedia of Mathematics
Types of probability distributions
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https://en.wikipedia.org/wiki/Distributive%20law%20between%20monads
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In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other one.
Suppose that and are two monads on a category C. In general, there is no natural monad structure on the composite functor ST. However, there is a natural monad structure on the functor ST if there is a distributive law of the monad S over the monad T.
Formally, a distributive law of the monad S over the monad T is a natural transformation
such that the diagrams
commute.
This law induces a composite monad ST with
as multiplication: ,
as unit: .
See also
distributive law
References
Adjoint functors
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https://en.wikipedia.org/wiki/Lipschitz%20domain
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In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The term is named after the German mathematician Rudolf Lipschitz.
Definition
Let . Let be a domain of and let denote the boundary of . Then is called a Lipschitz domain if for every point there exists a hyperplane of dimension through , a Lipschitz-continuous function over that hyperplane, and reals and such that
where
is a unit vector that is normal to
is the open ball of radius ,
In other words, at each point of its boundary, is locally the set of points located above the graph of some Lipschitz function.
Generalization
A more general notion is that of weakly Lipschitz domains, which are domains whose boundary is locally flattable by a bilipschitz mapping. Lipschitz domains in the sense above are sometimes called strongly Lipschitz by contrast with weakly Lipschitz domains.
A domain is weakly Lipschitz if for every point there exists a radius and a map such that
is a bijection;
and are both Lipschitz continuous functions;
where denotes the unit ball in and
A (strongly) Lipschitz domain is always a weakly Lipschitz domain but the converse is not true. An example of weakly Lipschitz domains that fails to be a strongly Lipschitz domain is given by the two-bricks domain
Applications of Lipschitz domains
Many of the Sobolev embedding theorems require that the domain of study be a Lipschitz domain. Consequently, many partial differential equations and variational problems are defined on Lipschitz domains.
References
Geometry
Lipschitz maps
Sobolev spaces
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https://en.wikipedia.org/wiki/Tetrated%20dodecahedron
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In geometry, the tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. It was then independently rediscovered in 2003, and named, by Robert Austin.
It has 28 faces: twelve regular pentagons arranged in four panels of three pentagons each, four equilateral triangles (shown in blue), and six pairs of isosceles triangles (shown in yellow). All edges of the tetrated dodecahedron have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times as long as the other edges. This polyhedron has tetrahedral symmetry.
Topologically, as a near-miss Johnson solid, the four triangles corresponding to the face planes of a tetrahedron are always equilateral, while the pentagons and the other triangles only have reflection symmetry.
Related polyhedra
See also
Tetrahedrally diminished dodecahedron
Notes
Polyhedra
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https://en.wikipedia.org/wiki/Hexlet
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Hexlet may refer to:
Soddy's hexlet, in geometry a chain of six spheres, each of which is tangent to both of its neighbors and also to three mutually tangent given spheres
Hexlet (computing), a group of 128 bits in computing
See also
Hextet
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https://en.wikipedia.org/wiki/Ross%E2%80%93Littlewood%20paradox
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The Ross–Littlewood paradox (also known as the balls and vase problem or the ping pong ball problem) is a hypothetical problem in abstract mathematics and logic designed to illustrate the paradoxical, or at least non-intuitive, nature of infinity. More specifically, like the Thomson's lamp paradox, the Ross–Littlewood paradox tries to illustrate the conceptual difficulties with the notion of a supertask, in which an infinite number of tasks are completed sequentially. The problem was originally described by mathematician John E. Littlewood in his 1953 book Littlewood's Miscellany, and was later expanded upon by Sheldon Ross in his 1988 book A First Course in Probability.
The problem starts with an empty vase and an infinite supply of balls. An infinite number of steps are then performed, such that at each step 10 balls are added to the vase and 1 ball removed from it. The question is then posed: How many balls are in the vase when the task is finished?
To complete an infinite number of steps, it is assumed that the vase is empty at one minute before noon, and that the following steps are performed:
The first step is performed at 30 seconds before noon.
The second step is performed at 15 seconds before noon.
Each subsequent step is performed in half the time of the previous step, i.e., step n is performed at 2 minutes before noon.
This guarantees that a countably infinite number of steps is performed by noon. Since each subsequent step takes half as much time as the previous step, an infinite number of steps is performed by the time one minute has passed. The question is then: How many balls are in the vase at noon?
Solutions
Answers to the puzzle fall into several categories.
Vase contains infinitely many balls
The most intuitive answer seems to be that the vase contains an infinite number of balls by noon, since at every step along the way more balls are being added than removed. By definition, at each step, there will be a greater number of balls than at the previous step. There is no step, in fact, where the number of balls is decreased from the previous step. If the number of balls increases each time, then after infinite steps there will be an infinite number of balls.
Vase is empty
Suppose that the balls of the infinite supply of balls were numbered, and that at step 1 balls 1 through 10 are inserted into the vase, and ball number 1 is then removed. At step 2, balls 11 through 20 are inserted, and ball 2 is then removed. This means that by noon, every ball labeled n that is inserted into the vase is eventually removed in a subsequent step (namely, at step n). Hence, the vase is empty at noon. This is the solution favored by mathematicians Allis and Koetsier. It is the juxtaposition of this argument that the vase is empty at noon, together with the more intuitive answer that the vase should have infinitely many balls, that has warranted this problem to be named the Ross–Littlewood paradox.
Ross's probabilistic version of the prob
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https://en.wikipedia.org/wiki/Dual%20bundle
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In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces.
Definition
The dual bundle of a vector bundle is the vector bundle whose fibers are the dual spaces to the fibers of .
Equivalently, can be defined as the Hom bundle that is, the vector bundle of morphisms from to the trivial line bundle
Constructions and examples
Given a local trivialization of with transition functions a local trivialization of is given by the same open cover of with transition functions (the inverse of the transpose). The dual bundle is then constructed using the fiber bundle construction theorem. As particular cases:
The dual bundle of an associated bundle is the bundle associated to the dual representation of the structure group.
The dual bundle of the tangent bundle of a differentiable manifold is its cotangent bundle.
Properties
If the base space is paracompact and Hausdorff then a real, finite-rank vector bundle and its dual are isomorphic as vector bundles. However, just as for vector spaces, there is no natural choice of isomorphism unless is equipped with an inner product.
This is not true in the case of complex vector bundles: for example, the tautological line bundle over the Riemann sphere is not isomorphic to its dual. The dual of a complex vector bundle is indeed isomorphic to the conjugate bundle but the choice of isomorphism is non-canonical unless is equipped with a hermitian product.
The Hom bundle of two vector bundles is canonically isomorphic to the tensor product bundle
Given a morphism of vector bundles over the same space, there is a morphism between their dual bundles (in the converse order), defined fibrewise as the transpose of each linear map Accordingly, the dual bundle operation defines a contravariant functor from the category of vector bundles and their morphisms to itself.
References
Vector bundles
Geometry
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https://en.wikipedia.org/wiki/Computer-based%20mathematics%20education
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Computer-based mathematics education (CBME) is an approach to teaching mathematics that emphasizes the use of computers.
Computers in math education
Computers are used in education in a number of ways, such as interactive tutorials, hypermedia, simulations and educational games. Tutorials are types of software that present information, check learning by question/answer method, judge responses, and provide feedback. Educational games are more like simulations and are used from the elementary to college level. E learning systems can deliver math lessons and exercises and manage homework assignments.
See also
ALEKS, a computer-based education system that includes mathematics among its curricula
Computer-Based Math, a project aimed at using computers for computational tasks and spending more classroom time on applications
Mathletics (educational software), a popular K-12 mathematics learning program from 3P Learning
Mathspace, a similar program for students aged 7-18, founded in Australia in 2010
Sokikom, a team-based math learning game
References
Mathematics education
Educational math software
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https://en.wikipedia.org/wiki/Income%20in%20the%20United%20States
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Income in the United States is measured by the various federal agencies including the Internal Revenue Service, Bureau of Labor Statistics, US Department of Commerce, and the US Census Bureau. Additionally, various agencies, including the Congressional Budget Office compile reports on income statistics. The primary classifications are by household or individual. The top quintile in personal income in 2019 was $103,012 (included in the chart below). The differences between household and personal income are considerable, since 61% of households now have two or more income earners.
Median personal income in 2020 was $56,287 for full time workers.
This difference becomes very apparent when comparing the percentage of households with six figure incomes to that of individuals. Overall, including all households/individuals regardless of employment status, the median household income was $67,521 in 2020 while the median personal income (including individuals aged 15 and over) was $35,805.
While wages for women have increased greatly, median earnings of male wage earners have remained stagnant since the late 1970s. Household income, however, has risen due to the increasing number of households with more than one income earner and women's increased presence in the labor force.
Income Percentiles 2019
Inflation adjusted US Dollars - see IRS for further reading IRS.GOV income statistics
Income at a glance
See also
Compensation in the United States
Economy of the United States
Income inequality in the United States
Socio-economic mobility in the United States
Unemployment in the United States
List of United States counties by per capita income
References
Further reading
INCOME STATISTICS AT IRS.GOV
External links
Savings rate vs Fed rate from 1954 Historical relationship between the savings rate and the Fed rate - since 1954
Social class in the United States
United States
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https://en.wikipedia.org/wiki/Grandi%27s%20series
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In mathematics, the infinite series , also written
is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it does not have a sum.
However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. For example, the Cesàro summation and the Ramanujan summation of this series is 1/2.
Unrigorous methods
One obvious method to find the sum of the series
1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + ...
is to treat it like a telescoping series and perform the subtractions in place:
(1 − 1) + (1 − 1) + (1 − 1) + ... = 0 + 0 + 0 + ... = 0.
On the other hand, a similar bracketing procedure leads to the apparently contradictory result
1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... = 1 + 0 + 0 + 0 + ... = 1.
Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value". (Variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebra.). By taking the average of these two "values", one can justify that the series converges to .
Treating Grandi's series as a divergent geometric series and using the same algebraic methods that evaluate convergent geometric series to obtain a third value:
S = 1 − 1 + 1 − 1 + ..., so
1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S'
1 − S = S1 = 2S,
resulting in S = . The same conclusion results from calculating −S, subtracting the result from S, and solving 2S = 1.
The above manipulations do not consider what the sum of a series actually means and how said algebraic methods can be applied to divergent geometric series. Still, to the extent that it is important to be able to bracket series at will, and that it is more important to be able to perform arithmetic with them, one can arrive at two conclusions:
The series 1 − 1 + 1 − 1 + ... has no sum.Davis p.152
...but its sum should be .
In fact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts that arose in the 19th century. After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians.Knopp p.457
Relation to the geometric series
For any number in the interval , the sum to infinity of a geometric series can be evaluated via
For any , one thus finds
and so the limit of series evaluations is
However, as mentioned, the series obtained by switching the limits,
is divergent.
In the terms of complex analysis, is thus seen to be the value at of the analytic continuation of the series , which is only defined on the complex unit disk, .
Early ideas
Divergence
In modern mathematics
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https://en.wikipedia.org/wiki/Pascal%20matrix
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In mathematics, particularly matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × 5 matrices are:
There are other ways in which Pascal's triangle can be put into matrix form, but these are not easily extended to infinity.
Definition
The non-zero elements of a Pascal matrix are given by the binomial coefficients:
such that the indices i, j start at 0, and ! denotes the factorial.
Properties
The matrices have the pleasing relationship Sn = LnUn. From this it is easily seen that all three matrices have determinant 1, as the determinant of a triangular matrix is simply the product of its diagonal elements, which are all 1 for both Ln and Un. In other words, matrices Sn, Ln, and Un are unimodular, with Ln and Un having trace n.
The trace of Sn is given by
with the first few terms given by the sequence 1, 3, 9, 29, 99, 351, 1275, … .
Construction
A Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix. The example below constructs a 7 × 7 Pascal matrix, but the method works for any desired n × n Pascal matrices. The dots in the following matrices represent zero elements.
It is important to note that one cannot simply assume exp(A) exp(B) = exp(A + B), for n × n matrices A and B; this equality is only true when AB = BA (i.e. when the matrices A and B commute). In the construction of symmetric Pascal matrices like that above, the sub- and superdiagonal matrices do not commute, so the (perhaps) tempting simplification involving the addition of the matrices cannot be made.
A useful property of the sub- and superdiagonal matrices used for the construction is that both are nilpotent; that is, when raised to a sufficiently great integer power, they degenerate into the zero matrix. (See shift matrix for further details.) As the n × n generalised shift matrices we are using become zero when raised to power n, when calculating the matrix exponential we need only consider the first n + 1 terms of the infinite series to obtain an exact result.
Variants
Interesting variants can be obtained by obvious modification of the matrix-logarithm PL7 and then application of the matrix exponential.
The first example below uses the squares of the values of the log-matrix and constructs a 7 × 7 "Laguerre"- matrix (or matrix of coefficients of Laguerre polynomials
The Laguerre-matrix is actually used with some other scaling and/or the scheme of alternating signs.
(Literature about generalizations to higher powers is not found yet)
The second example below uses the products v(v + 1) of the values of the log-matrix and constructs a 7 × 7 "Lah"- matrix (or matrix of coefficients of Lah numbers)
Using v(v − 1)
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https://en.wikipedia.org/wiki/Combinatorial%20explosion
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In mathematics, a combinatorial explosion is the rapid growth of the complexity of a problem due to how the combinatorics of the problem is affected by the input, constraints, and bounds of the problem. Combinatorial explosion is sometimes used to justify the intractability of certain problems. Examples of such problems include certain mathematical functions, the analysis of some puzzles and games, and some pathological examples which can be modelled as the Ackermann function.
Examples
Latin squares
A Latin square of order is an array with entries from a set of elements with the property that each element of the set occurs exactly once in each row and each column of the array. An example of a Latin square of order three is given by,
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
|-
| 1|| 2 || 3
|-
| 2 || 3 || 1
|-
| 3 || 1 || 2
|}
A common example of a Latin square would be a completed Sudoku puzzle. A Latin square is a combinatorial object (as opposed to an algebraic object) since only the arrangement of entries matters and not what the entries actually are. The number of Latin squares as a function of the order (independent of the set from which the entries are drawn) provides an example of combinatorial explosion as illustrated by the following table.
Sudoku
A combinatorial explosion can also occur in some puzzles played on a grid, such as Sudoku. A Sudoku is a type of Latin square with the additional property that each element occurs exactly once in sub-sections of size (called boxes). Combinatorial explosion occurs as increases, creating limits to the properties of Sudokus that can be constructed, analyzed, and solved, as illustrated in the following table.
Games
One example in a game where combinatorial complexity leads to a solvability limit is in solving chess (a game with 64 squares and 32 pieces). Chess is not a solved game. In 2005 all chess game endings with six pieces or fewer were solved, showing the result of each position if played perfectly. It took ten more years to complete the tablebase with one more chess piece added, thus completing a 7-piece tablebase. Adding one more piece to a chess ending (thus making an 8-piece tablebase) is considered intractable due to the added combinatorial complexity.
Furthermore, the prospect of solving larger chess-like games becomes more difficult as the board-size is increased, such as in large chess variants, and infinite chess.
Computing
Combinatorial explosion can occur in computing environments in a way analogous to communications and multi-dimensional space. Imagine a simple system with only one variable, a boolean called A. The system has two possible states, A = true or A = false. Adding another boolean variable B will give the system four possible states, A = true and B = true, A = true and B = false, A = false and B = true, A = false and B = false. A system with n booleans has 2n possible states
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https://en.wikipedia.org/wiki/Smooth%20morphism
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In algebraic geometry, a morphism between schemes is said to be smooth if
(i) it is locally of finite presentation
(ii) it is flat, and
(iii) for every geometric point the fiber is regular.
(iii) means that each geometric fiber of f is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.
If S is the spectrum of an algebraically closed field and f is of finite type, then one recovers the definition of a nonsingular variety.
Equivalent definitions
There are many equivalent definitions of a smooth morphism. Let be locally of finite presentation. Then the following are equivalent.
f is smooth.
f is formally smooth (see below).
f is flat and the sheaf of relative differentials is locally free of rank equal to the relative dimension of .
For any , there exists a neighborhood of x and a neighborhood of such that and the ideal generated by the m-by-m minors of is B.
Locally, f factors into where g is étale.
A morphism of finite type is étale if and only if it is smooth and quasi-finite.
A smooth morphism is stable under base change and composition.
A smooth morphism is universally locally acyclic.
Examples
Smooth morphisms are supposed to geometrically correspond to smooth submersions in differential geometry; that is, they are smooth locally trivial fibrations over some base space (by Ehresmann's theorem).
Smooth Morphism to a Point
Let be the morphism of schemes
It is smooth because of the Jacobian condition: the Jacobian matrix
vanishes at the points which has an empty intersection with the polynomial, since
which are both non-zero.
Trivial Fibrations
Given a smooth scheme the projection morphism
is smooth.
Vector Bundles
Every vector bundle over a scheme is a smooth morphism. For example, it can be shown that the associated vector bundle of over is the weighted projective space minus a point
sending
Notice that the direct sum bundles can be constructed using the fiber product
Separable Field Extensions
Recall that a field extension is called separable iff given a presentation
we have that . We can reinterpret this definition in terms of Kähler differentials as follows: the field extension is separable iff
Notice that this includes every perfect field: finite fields and fields of characteristic 0.
Non-Examples
Singular Varieties
If we consider of the underlying algebra for a projective variety , called the affine cone of , then the point at the origin is always singular. For example, consider the affine cone of a quintic -fold given by
Then the Jacobian matrix is given by
which vanishes at the origin, hence the cone is singular. Affine hypersurfaces like these are popular in singularity theory because of their relatively simple algebra but rich underlying structures.
Another example of a singular variety is the projective cone of a smooth variety: given a smooth projective variety its projective cone is the union o
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https://en.wikipedia.org/wiki/Inter-rater%20reliability
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In statistics, inter-rater reliability (also called by various similar names, such as inter-rater agreement, inter-rater concordance, inter-observer reliability, inter-coder reliability, and so on) is the degree of agreement among independent observers who rate, code, or assess the same phenomenon.
Assessment tools that rely on ratings must exhibit good inter-rater reliability, otherwise they are not valid tests.
There are a number of statistics that can be used to determine inter-rater reliability. Different statistics are appropriate for different types of measurement. Some options are joint-probability of agreement, such as Cohen's kappa, Scott's pi and Fleiss' kappa; or inter-rater correlation, concordance correlation coefficient, intra-class correlation, and Krippendorff's alpha.
Concept
There are several operational definitions of "inter-rater reliability," reflecting different viewpoints about what is a reliable agreement between raters. There are three operational definitions of agreement:
Reliable raters agree with the "official" rating of a performance.
Reliable raters agree with each other about the exact ratings to be awarded.
Reliable raters agree about which performance is better and which is worse.
These combine with two operational definitions of behavior:
Statistics
Joint probability of agreement
The joint-probability of agreement is the simplest and the least robust measure. It is estimated as the percentage of the time the raters agree in a nominal or categorical rating system. It does not take into account the fact that agreement may happen solely based on chance. There is some question whether or not there is a need to 'correct' for chance agreement; some suggest that, in any case, any such adjustment should be based on an explicit model of how chance and error affect raters' decisions.
When the number of categories being used is small (e.g. 2 or 3), the likelihood for 2 raters to agree by pure chance increases dramatically. This is because both raters must confine themselves to the limited number of options available, which impacts the overall agreement rate, and not necessarily their propensity for "intrinsic" agreement (an agreement is considered "intrinsic" if it is not due to chance).
Therefore, the joint probability of agreement will remain high even in the absence of any "intrinsic" agreement among raters. A useful inter-rater reliability coefficient is expected (a) to be close to 0 when there is no "intrinsic" agreement and (b) to increase as the "intrinsic" agreement rate improves. Most chance-corrected agreement coefficients achieve the first objective. However, the second objective is not achieved by many known chance-corrected measures.
Kappa statistics
Kappa is a way of measuring agreement or reliability, correcting for how often ratings might agree by chance. Cohen's kappa, which works for two raters, and Fleiss' kappa, an adaptation that works for any fixed number of raters, improve upon the joint
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https://en.wikipedia.org/wiki/%C5%A0ar%C5%ABnas%20Raudys
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Šarūnas Raudys is head of the Data Analysis Department at the Institute of Mathematics and Informatics in Vilnius, Lithuania. Within the department, he is guiding the data mining and artificial neural networks group. His group's research interests include multivariate analysis, statistical pattern recognition, artificial neural networks, data mining methods and biological information processing systems with applications to analysis of technological, economical and biological problems.
Education
USSR Doctor of Sciences, Institute of Electronics and Computer science, Riga, 1978.
Ph.D. Computer science, Institute Physics and Mathematics, 1969.
M.S. Electrical and Computer Engineering, Kaunas University of Technology, 1963.
Panevezys, the first secondary school, 1958.
Selected publications
S. Raudys. (2001) Statistical and Neural Classifiers: An integrated approach to design. Springer. London. 312 pages.
S. Raudys and Jain K. (1991). Small sample size problems in designing Artificial Neural Networks. - Artificial Neural Networks and Statistical Pattern Recognition, Old and New Connections, I.K. Sethi and A.K. Jain (Eds), Elsevier Science Publishers B.V, 33-50.
S. Raudys. (1984) Statistical Pattern Recognition: Small design sample problems. A monograph, (a manuscript) Institute of Mathematics and Cybernetics, Vilnius, 480 pages, 30 copies distributed around the world.
S. Raudys, (1978) Optimization of nonparametric classification algorithm. Adaptive systems and applications. Nauka, Novosibirsk, (A.Medvedev Ed.), 57-62.
S. Raudys. (1976) Limitation of Sample Size in Classification Problems, Inst. of Physics and Mathematics Press, Vilnius. 186 pages.
References
Lithuanian schoolteachers
Living people
Kaunas University of Technology alumni
Academic staff of Vilnius University
21st-century Lithuanian educators
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Z-matrix%20%28mathematics%29
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In mathematics, the class of Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form:
Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix, thus the term quasinegative matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made.
The Jacobian of a competitive dynamical system is a Z-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is J, then (−J) is a Z-matrix.
Related classes are L-matrices, M-matrices, P-matrices, Hurwitz matrices and Metzler matrices. L-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a Z-matrix is an M-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both Z-matrices and P-matrices are nonsingular M-matrices.
In the context of quantum complexity theory, these are referred to as stoquastic operators.
See also
Hurwitz matrix
M-matrix
Metzler matrix
P-matrix
References
Matrices
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https://en.wikipedia.org/wiki/Matrix%20equivalence
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In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if
for some invertible n-by-n matrix P and some invertible m-by-m matrix Q. Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.
The notion of equivalence should not be confused with that of similarity, which is only defined for square matrices, and is much more restrictive (similar matrices are certainly equivalent, but equivalent square matrices need not be similar). That notion corresponds to matrices representing the same endomorphism V → V under two different choices of a single basis of V, used both for initial vectors and their images.
Properties
Matrix equivalence is an equivalence relation on the space of rectangular matrices.
For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions
The matrices can be transformed into one another by a combination of elementary row and column operations.
Two matrices are equivalent if and only if they have the same rank.
Canonical form
The rank property yields an intuitive canonical form for matrices of the equivalence class of rank as
,
where the number of s on the diagonal is equal to . This is a special case of the Smith normal form, which generalizes this concept on vector spaces to free modules over principal ideal domains.
See also
Row equivalence
Matrix congruence
Matrices
Equivalence (mathematics)
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https://en.wikipedia.org/wiki/Bonferroni%20correction
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In statistics, the Bonferroni correction is a method to counteract the multiple comparisons problem.
Background
The method is named for its use of the Bonferroni inequalities.
An extension of the method to confidence intervals was proposed by Olive Jean Dunn.
Statistical hypothesis testing is based on rejecting the null hypothesis if the likelihood of the observed data under the null hypotheses is low. If multiple hypotheses are tested, the probability of observing a rare event increases, and therefore, the likelihood of incorrectly rejecting a null hypothesis (i.e., making a Type I error) increases.
The Bonferroni correction compensates for that increase by testing each individual hypothesis at a significance level of , where is the desired overall alpha level and is the number of hypotheses. For example, if a trial is testing hypotheses with a desired , then the Bonferroni correction would test each individual hypothesis at . Likewise, when constructing multiple confidence intervals the same phenomenon appears.
Definition
Let be a family of hypotheses and their corresponding p-values. Let be the total number of null hypotheses, and let be the number of true null hypotheses (which is presumably unknown to the researcher). The family-wise error rate (FWER) is the probability of rejecting at least one true , that is, of making at least one type I error. The Bonferroni correction rejects the null hypothesis for each , thereby controlling the FWER at . Proof of this control follows from Boole's inequality, as follows:
This control does not require any assumptions about dependence among the p-values or about how many of the null hypotheses are true.
Extensions
Generalization
Rather than testing each hypothesis at the level, the hypotheses may be tested at any other combination of levels that add up to , provided that the level of each test is decided before looking at the data. For example, for two hypothesis tests, an overall of 0.05 could be maintained by conducting one test at 0.04 and the other at 0.01.
Confidence intervals
The procedure proposed by Dunn can be used to adjust confidence intervals. If one establishes confidence intervals, and wishes to have an overall confidence level of , each individual confidence interval can be adjusted to the level of .
Continuous problems
When searching for a signal in a continuous parameter space there can also be a problem of multiple comparisons, or look-elsewhere effect. For example, a physicist might be looking to discover a particle of unknown mass by considering a large range of masses; this was the case during the Nobel Prize winning detection of the Higgs boson. In such cases, one can apply a continuous generalization of the Bonferroni correction by employing Bayesian logic to relate the effective number of trials, , to the prior-to-posterior volume ratio.
Alternatives
There are alternative ways to control the family-wise error rate. For example, the Holm–Bonferroni method an
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