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https://en.wikipedia.org/wiki/ALEKS | ALEKS (Assessment and Learning in Knowledge Spaces) is an online tutoring and assessment program that includes course material in mathematics, chemistry, introductory statistics, and business.
Rather than being based on numerical test scores, ALEKS uses the theory of knowledge spaces to develop a combinatorial understanding of the set of topics a student does or doesn't understand from the answers to its test questions. Based on this assessment, it determines the topics that the student is ready to learn and allows the student to choose from interactive learning modules for these topics.
ALEKS was initially developed at UC Irvine starting in 1994 with support from a large National Science Foundation grant. The software was granted by UC Irvine's Office of Technology Alliances to ALEKS Corporation under an exclusive, worldwide, perpetual license. In 2013, the ALEKS Corporation was acquired by McGraw-Hill Education.
Subjects covered
ALEKS is available for a variety of courses and subjects that cover K-12, higher education, and continuing education, ranging from basic arithmetic and chemistry to pre-calculus and MBA financial accounting preparation.
Notes
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External links
Virtual learning environments
Mathematics education |
https://en.wikipedia.org/wiki/Robert%20Schatten | Robert Schatten (January 28, 1911 – August 26, 1977) was an American mathematician.
Robert Schatten was born to a Jewish family in Lviv. His intellectual origins were at Lwów School of Mathematics, particularly well known for fundamental contributions to functional analysis. His entire family was murdered during World War II, he himself emigrated to the United States.
In 1933 he got magister degree at Jan Kazimierz University of Lwów, and in 1939 he got master's degree at Columbia University. Supervised by Francis Joseph Murray, he got doctorate degree in 1942 for the thesis "On the Direct Product of Banach Spaces". Shortly after being appointed to a junior professorship, he joined the United States army where during training he suffered a back injury which affected him for the remainder of his life. In 1943 he was appointed to an assistant professorship at University of Vermont. At National Research Council, by two years he worked with John von Neumann and Nelson Dunford. In 1946, he went to the University of Kansas, first as extraordinary professor until 1952 and then as ordinary professor until 1961. He stayed at Institute for Advanced Study in 1950 and 1952–1953, at University of Southern California in 1960–1961, and at State University of New York in 1961–1962. In 1962 he became professor at Hunter College, where he stayed until his death.
Schatten widely studied tensor products of Banach spaces. In functional analysis, he is the namesake of the Schatten norm and the Schatten class operators. His doctoral students included Elliott Ward Cheney, Jr. at University of Kansas, and Peter Falley and Charles Masiello at City University of New York.
Schatten died in New York City in 1977.
Further reading
A Theory of Cross-Spaces. Annals of Mathematics Studies,
Norm Ideals of Completely Continuous Operators. Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge,
References
External links
Polish mathematicians
20th-century American mathematicians
1911 births
1977 deaths
20th-century Polish Jews
University of Kansas faculty
University of Southern California faculty
State University of New York faculty
Polish emigrants to the United States |
https://en.wikipedia.org/wiki/Recurrent%20tensor | In mathematics and physics, a recurrent tensor, with respect to a connection on a manifold M, is a tensor T for which there is a one-form ω on M such that
Examples
Parallel Tensors
An example for recurrent tensors are parallel tensors which are defined by
with respect to some connection .
If we take a pseudo-Riemannian manifold then the metric g is a parallel and therefore recurrent tensor with respect to its Levi-Civita connection, which is defined via
and its property to be torsion-free.
Parallel vector fields () are examples of recurrent tensors that find importance in mathematical research. For example, if is a recurrent non-null vector field on a pseudo-Riemannian manifold satisfying
for some closed one-form , then X can be rescaled to a parallel vector field. In particular, non-parallel recurrent vector fields are null vector fields.
Metric space
Another example appears in connection with Weyl structures. Historically, Weyl structures emerged from the considerations of Hermann Weyl with regards to properties of parallel transport of vectors and their length. By demanding that a manifold have an affine parallel transport in such a way that the manifold is locally an affine space, it was shown that the induced connection had a vanishing torsion tensor
.
Additionally, he claimed that the manifold must have a particular parallel transport in which the ratio of two transported vectors is fixed. The corresponding connection which induces such a parallel transport satisfies
for some one-form . Such a metric is a recurrent tensor with respect to . As a result, Weyl called the resulting manifold with affine connection and recurrent metric a metric space. In this sense, Weyl was not just referring to one metric but to the conformal structure defined by .
Under the conformal transformation , the form transforms as . This induces a canonical map on defined by
,
where is the conformal structure. is called a Weyl structure, which more generally is defined as a map with property
.
Recurrent spacetime
One more example of a recurrent tensor is the curvature tensor on a recurrent spacetime, for which
.
References
Literature
A.G. Walker: On parallel fields of partially null vector spaces, The Quarterly Journal of Mathematics 1949, Oxford Univ. Press
E.M. Patterson: On symmetric recurrent tensors of the second order, The Quarterly Journal of Mathematics 1950, Oxford Univ. Press
J.-C. Wong: Recurrent Tensors on a Linearly Connected Differentiable Manifold, Transactions of the American Mathematical Society 1961,
G.B. Folland: Weyl Manifolds, Journal of Differential Geometry 1970
Riemannian geometry
Tensors |
https://en.wikipedia.org/wiki/Karanapaddhati | Karanapaddhati is an astronomical treatise in Sanskrit attributed to Puthumana Somayaji, an astronomer-mathematician of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the East India Company, brought this work to the attention of European scholars for the first time in a paper published in 1834. The book is divided into ten chapters and is in the form of verses in Sanskrit. The sixth chapter contains series expansions for the value of the mathematical constant π, and expansions for the trigonometric sine, cosine and inverse tangent functions.
Author and date of Karanapaddhati
Nothing definite is known about the author of Karanapaddhati. The last verse of the tenth chapter of Karanapaddhati describes the author as a Brahamin residing in a village named Sivapura. Sivapura is an area surrounding the present day Thrissur in Kerala, India.
The period in which Somayaji lived is also uncertain. There are several theories in this regard.
C.M. Whish, the first westerner to write about Karanapaddhati, based on his interpretation that certain words appearing in the final verse of Karanapaddhati denote in katapayadi system the number of days in the Kali Yuga, concluded that the book was completed in 1733 CE. Whish had also claimed that the grandson of the author of the Karanapaddhati was alive and was in his seventieth year at the time of writing his paper.
Based on reference to Puthumana Somayaji in a verse in Ganita Sucika Grantha by Govindabhatta, Raja Raja Varma placed the author of Karanapaddhati between 1375 and 1475 CE.
An internal study of Karanapaddhati suggests that the work is contemporaneous with or even antedates the Tantrasangraha of Nilakantha Somayaji (1465–1545 CE).
Synopsis of the book
A brief account of the contents of the various chapters of the book is presented below.
Chapter 1 : Rotation and revolutions of the planets in one mahayuga; the number of civil days in a mahayuga; the solar months, lunar months, intercalary months; kalpa and the four yugas and their durations, the details of Kali Yuga, calculation of the Kali era from the Malayalam Era, calculation of Kali days; the true and mean position of planets; simple methods for numerical calculations; computation of the true and mean positions of planets; the details of the orbits of planets; constants to be used for the calculation of various parameters of the different planets.
Chapter 2 : Parameters connected with Kali era, the positions of the planets, their angular motions, various parameters connected with Moon.
Chapter 3 : Mean center of Moon and various parameters of Moon based on the latitude and longitude of the same, the constants connected with Moon.
Chapter 4 : Perigee and apogee of the Mars, corrections to be given at different occasions for the Mars, constants for Mars, Mercury, Jupiter, Venus, Saturn in the respective order, the perigee and apogee of all these planets, their c |
https://en.wikipedia.org/wiki/Montgomery%20curve | In mathematics, the Montgomery curve is a form of elliptic curve introduced by Peter L. Montgomery in 1987, different from the usual Weierstrass form. It is used for certain computations, and in particular in different cryptography applications.
Definition
A Montgomery curve over a field is defined by the equation
for certain and with .
Generally this curve is considered over a finite field K (for example, over a finite field of elements, ) with characteristic different from 2 and with and , but they are also considered over the rationals with the same restrictions for and .
Montgomery arithmetic
It is possible to do some "operations" between the points of an elliptic curve: "adding" two points consists of finding a third one such that ; "doubling" a point consists of computing (For more information about operations see The group law) and below.
A point on the elliptic curve in the Montgomery form can be represented in Montgomery coordinates , where are projective coordinates and for .
Notice that this kind of representation for a point loses information: indeed, in this case, there is no distinction between the affine points and because they are both given by the point . However, with this representation it is possible to obtain multiples of points, that is, given , to compute .
Now, considering the two points and : their sum is given by the point whose coordinates are:
If , then the operation becomes a "doubling"; the coordinates of are given by the following equations:
The first operation considered above (addition) has a time-cost of 3M+2S, where M denotes the multiplication between two general elements of the field on which the elliptic curve is defined, while S denotes squaring of a general element of the field.
The second operation (doubling) has a time-cost of 2M + 2S + 1D, where D denotes the multiplication of a general element by a constant; notice that the constant is , so can be chosen in order to have a small D.
Algorithm and example
The following algorithm represents a doubling of a point on an elliptic curve in the Montgomery form.
It is assumed that . The cost of this implementation is 1M + 2S + 1*A + 3add + 1*4. Here M denotes the multiplications required, S indicates the squarings, and a refers to the multiplication by A.
Example
Let be a point on the curve .
In coordinates , with , .
Then:
The result is the point such that .
Addition
Given two points , on the Montgomery curve in affine coordinates, the point represents, geometrically the third point of intersection between and the line passing through and . It is possible to find the coordinates of , in the following way:
1) consider a generic line in the affine plane and let it pass through and (impose the condition), in this way, one obtains and ;
2) intersect the line with the curve , substituting the variable in the curve equation with ; the following equation of third degree is obtained:
As it |
https://en.wikipedia.org/wiki/Dogbone%20space | In geometric topology, the dogbone space, constructed by , is a quotient space of three-dimensional Euclidean space such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to . The name "dogbone space" refers to a fanciful resemblance between some of the diagrams of genus 2 surfaces in R. H. Bing's paper and a dog bone. showed that the product of the dogbone space with is homeomorphic to .
Although the dogbone space is not a manifold, it is a generalized homological manifold and a homotopy manifold.
See also
List of topologies
Whitehead manifold, a contractible 3-manifold not homeomorphic to .
References
Geometric topology
Topological spaces |
https://en.wikipedia.org/wiki/Dietrich%20Mahnke | Dietrich Mahnke (17 October 1884, Verden – 25 July 1939, Fürth) was a German philosopher and historian of mathematics.
From 1902–1906, Mahnke studied at Göttingen under Edmund Husserl and David Hilbert. After serving in the First World War (stationed in Lens, France), he graduated from the University of Freiburg in 1925 with a thesis on Leibniz. The thesis was later published in the Jahrbuch für Philosophie und phänomenologische Forschung as Leibnizens Synthese von Universalmathematik und Individualmetaphysik. In 1926 he habilitated at Greifswald with a thesis entitled Neue Einblicke in die Entdeckungsgeschichte der höheren Analysis. In 1927 he became a professor of philosophy at Marburg.
In 1934 he became a member of the Nazi SA.
Mahnke's work in the history of mathematics focussed primarily on Leibniz's development of the infinitesimal calculus, and his relationship to Neo-Platonism. His last book, Unendliche Sphäre und Allmittelpunkt, Beiträge zur Genealogie der mathematischen Mystik was a study of the use of mathematical symbolism, especially the notion of "infinite spheres", in religious mysticism. At the time of his death, Mahnke was editing a volume of Leibniz's mathematical correspondence. This project was then taken over by Joseph Ehrenfried Hofmann.
Mahnke was killed in a car accident.
His Nachlass is preserved at the University of Marburg.
Select Bibliography
Leibniz als Gegner der Gelehrteneinseitigkeit (1912)
Der Wille der Ewigkeit (1917)
Eine Neue Monadologie (1917)
Die Neubelebung der Leibnizschen Weltanschauung (1920)
Ewigkeit und Gegenwart, Eine Fichtische Zusammenschau (1922)
Von Hilbert zu Husserl, Erste Einführung in die Phänomenologie, besonders die formale Mathematik (1923)
Leibniz und Goethe: die Harmonie ihrer Weltansichten (1924)
Neue Einblicke in die Entdeckungsgeschichte der höheren Analysis (1926)
Ein unbekanntes Selbstzeugnis Leibnizens aus seiner Erziehertätigkeit (1931)
Unendliche Sphäre und Allmittelpunkt, Beiträge zur Genealogie der mathematischen Mystik (1937)
Die Rationalisierung der Mystik bei Leibniz und Kant (1939)
References
Joseph W. Dauben, Christoph J. Scriba (ed.): Writing the History of Mathematics – Its Historical Development. Birkhäuser, Basel 2002,
1884 births
1939 deaths
Sturmabteilung personnel
Academic staff of the University of Marburg
German historians of mathematics
20th-century German male writers
20th-century German philosophers
German Army personnel of World War I
Road incident deaths in Nazi Germany |
https://en.wikipedia.org/wiki/Donald%20Burkholder | Donald Lyman Burkholder (January 19, 1927 – April 14, 2013) was an American mathematician known for his contributions to probability theory, particularly the theory of martingales. The Burkholder–Davis–Gundy inequality is co-named after him. Burkholder spent most of his professional career as a professor in the Department of Mathematics of the University of Illinois at Urbana-Champaign. After his retirement in 1998, Donald Burkholder remained a professor emeritus in the Department of Mathematics of the University of Illinois at Urbana-Champaign and a CAS Professor Emeritus of Mathematics at the Center for Advanced Study, University of Illinois at Urbana-Champaign. He was a member of the U.S. National Academy of Sciences and a fellow of the American Mathematical Society.
Biographical data
Burkholder received a PhD in statistics in 1955 from the University of North Carolina at Chapel Hill, under the direction of Wassily Hoeffding.
He was appointed an assistant professor in the Department of Mathematics at the University of Illinois at Urbana-Champaign in 1955 where he remained until his retirement in 1998. He was promoted to associate professor in 1960, became a professor in the department in 1964 and was appointed as professor at the Center for Advanced Study at UIUC in 1978.
Burkholder delivered an invited lecture at the International Congress of Mathematicians in 1970, a Wald Lecture at the Institute of Mathematical Statistics in 1971, a Mordell Lecture at Cambridge University in 1986, and a Zygmund Lecture at the University of Chicago in 1988.
Donald Burkholder was elected a member of the U.S. National Academy of Sciences in 1992. The same year he was elected a fellow of the American Academy of Arts and Sciences. On December 20, 2010, Burkholder was elected a Fellow of the American Association for the Advancement of Science for "distinguished contributions to probability theory, particularly the theory of martingales, and his work in stochastic processes, functional analysis, and Fourier analysis."
In 2009 Burkholder was named a SIAM Fellow by the Society for Industrial and Applied Mathematics "for advances in martingale transforms and applications of probabilistic methods in analysis".
Burkholder was an editor (1964–1967) of the journal Annals of Mathematical Statistics. He served as the president of the Institute of Mathematical Statistics in 1975–76. He is a Fellow of the Institute of Mathematical Statistics.
Selected publications
Burkholder, D. L.; Gundy, Richard F., Extrapolation and interpolation of quasi-linear operators on martingales. Acta Mathematica, vol. 124 (1970), pp. 249–304
Burkholder, Donald L., Inequalities for operators on martingales. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 551–557. Gauthier-Villars, Paris, 1971.
Burkholder, Donald L., Distribution function inequalities for martingales, Annals of Probability, vol. 1 (1973), pp. 19–42
Burkholder, Donald L., A geometrical cha |
https://en.wikipedia.org/wiki/Wellington%20Phoenix%20FC%20records%20and%20statistics | Wellington Phoenix Football Club is a New Zealand professional association football club based in Wellington Central, Wellington. The club was formed in 2007 to be the second New Zealand member admitted into the A-League Men after the demise of New Zealand Knights.
The list encompasses the honours won by Wellington Phoenix, records set by the club, their managers and their players. The player records section itemises the club's leading goalscorers and those who have made most appearances in first-team competitions. Attendance records at Wellington and WIN Stadium, their temporary home ground.
All figures current as of the match played on 11 July 2023.
Honours
Domestic
A-League Pre-Season Challenge Cup
Runners-up (1): 2008
Player records
Appearances
Most A-League Men appearances: Andrew Durante, 273
Most Australia Cup appearances: Alex Rufer, 6
Youngest first-team player: Ben Waine, 17 years, 57 days (against Bentleigh Greens, FFA Cup Round of 32, 7 August 2018)
Oldest first-team player: Tony Warner, 37 years, 242 days (against Perth Glory, A-League, 8 January 2012)
Most consecutive appearances: Chris Greenacre, 68 (from 22 January 2010 to 7 April 2012)
Most appearances
Competitive matches only, includes appearances as substitute. Numbers in brackets indicate goals scored. '.a. Includes the A-League Pre-Season Challenge Cup and Australia Cup
Goalscorers
Most goals in a season: Roy Krishna, 19 goals (in 2018–19 season)
Most league goals in a season: Roy Krishna, 18 goals in the A-League, 2018–19
Youngest goalscorer: Ben Waine, 18 years, 145 days (against Melbourne City, A-League, 3 November 2019)
Oldest goalscorer: Eugene Dadi, 36 years, 205 days (against Sydney FC, A-League preliminary final, 13 March 2010)
Top goalscorersCompetitive matches only, includes appearances as substitute. Numbers in brackets indicate goals scored. Current players are in bold'.
Managerial records
First full-time manager: Ricki Herbert managed Wellington Phoenix from March 2007 to February 2013.
Longest serving manager: Ricki Herbert – (19 March 2007 to 25 February 2013)
Shortest tenure as manage: Chris Greenacre – 2 months, 23 days (26 February 2013 to 19 May 2013)
Highest win percentage: Ufuk Talay, 41.84%
Lowest win percentage: Chris Greenacre, 16.67%
Club records
Matches
Firsts
First match: Central Coast Mariners 2–0 Wellington Phoenix, A-League Pre-Season Challenge Cup, 14 July 2007
First A-League Men match: Wellington Phoenix 2–2 Melbourne Victory, 26 August 2007
First national cup match: Central Coast Mariners 2–0 Wellington Phoenix, A-League Pre-Season Challenge Cup, 14 July 2007
Record wins
Record A-League Men win:
6–0 against Gold Coast United, A-League, 25 October 2009
8–2 against Central Coast Mariners, A-League, 9 March 2019
Record national cup win: 4–0 against Devonport City, Australia Cup Round of 32, 3 August 2022
Record defeats
Record A-League Men defeat:
1–7 against Sydney FC, A-League, 19 January 2013
0–6 aga |
https://en.wikipedia.org/wiki/Johar%20Al%20Kaabi | Johar Al Kaabi (born 9 June 1988) is a Qatari footballer. He currently plays as a defender .
Al Kaabi played for Qatar at the 2005 FIFA U-17 World Championship in Peru.
Club career statistics
Statistics accurate as of 21 June 2012
1Includes Emir of Qatar Cup.
2Includes Sheikh Jassem Cup.
3Includes AFC Champions League.
References
External links
FIFA.com profile
Goalzz.com profile
Qatari men's footballers
Living people
1988 births
Al-Arabi SC (Qatar) players
Al-Wakrah SC players
Qatar Stars League players
Qatari Second Division players
Men's association football defenders |
https://en.wikipedia.org/wiki/2010%20Kelantan%20FA%20season | The 2010 season was Kelantan FA's 2nd consecutive season in the Malaysia Super League. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season. Kelantan's Super League season began with a 0–0 drawn to Terengganu FA.
Competitions
Super League
Results summary
League table
FA Cup
Aggregate 1–1. Kelantan lost on away-goal rules.
Malaysia Cup
Kelantan won on aggregate 3–0.
Kelantan won on aggregate 1–0.
Group A
Team officials
Player statistics
Squad
Last updated 23 May 2013
Key:
= Appearances,
= Goals,
= Yellow card,
= Red card
Goalscorers
Source: Competitions
Transfers
All start dates are pending confirmation.
In
Out
See also
List of Kelantan FA seasons
References
See also
List of Kelantan FA seasons
Kelantan F.C.
2010
Kelantan |
https://en.wikipedia.org/wiki/Spencer%20Bloch | Spencer Janney Bloch (born May 22, 1944; New York City) is an American mathematician known for his contributions to algebraic geometry and algebraic K-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Department of Mathematics of the University of Chicago. He is a member of the U.S. National Academy of Sciences and a Fellow of the American Academy of Arts and Sciences and of the American Mathematical Society. At the International Congress of Mathematicians, he gave an invited lecture in 1978 and a plenary lecture in 1990. He was a visiting scholar at the Institute for Advanced Study in 1981–82. He received a Humboldt Prize in 1996. He also received a 2021 Leroy P. Steele Prize for Lifetime Achievement.
See also
Bloch's formula
Bloch group
Bloch–Kato conjecture
Bloch's higher Chow group
References
James D. Lewis, and Rob de Jeu (Editors), Motives and Algebraic Cycles: A Celebration in Honour of Spencer J. Bloch. Fields Institute Communications series, 2009, American Mathematical Society.
External links
Spencer Bloch personal webpage, Department of Mathematics, University of Chicago
20th-century American mathematicians
21st-century American mathematicians
Algebraic geometers
Members of the United States National Academy of Sciences
Fellows of the American Academy of Arts and Sciences
Fellows of the American Mathematical Society
University of Chicago faculty
Columbia University alumni
Institute for Advanced Study visiting scholars
1944 births
Living people
Harvard College alumni
Mathematicians from New York (state)
Scientists from New York City |
https://en.wikipedia.org/wiki/Richard%20Kadison | Richard Vincent Kadison (July 25, 1925 – August 22, 2018) was an American mathematician known for his contributions to the study of operator algebras.
Work
Born in New York City in 1925, Kadison was a Gustave C. Kuemmerle Professor in the Department of Mathematics of the University of Pennsylvania.
Kadison was a member of the U.S. National Academy of Sciences (elected in 1996), and a foreign member of the Royal Danish Academy of Sciences and Letters and of the Norwegian Academy of Science and Letters. He was a 1969 Guggenheim Fellow.
Kadison was awarded the 1999 Leroy P. Steele Prize for Lifetime Achievement by the American Mathematical Society. In 2012 he became a fellow of the American Mathematical Society.
Personal
Kadison was a skilled gymnast with a specialty in rings, making the 1952 US Olympic Team but later withdrawing due to an injury. He married Karen M. Holm on June 5, 1956, and they had one son, Lars.
Kadison died after a short illness on August 22, 2018.
Selected publications
Books
with John Ringrose: Fundamentals of the theory of operator algebras. 2 vols., Academic Press 1983; new edition, Fundamentals of the theory of operator algebras: Elementary theory, Vol. 1, 1997 Fundamentals of the theory of operator algebras: Advanced theory, Vol. 2, 1997 AMS 1997
with John Ringrose: Fundamentals of the theory of operator algebras, III-IV. An exercise approach, Birkhäuser, Basel, III: 1991, xiv+273 pp., ; IV: 1992, xiv+586 pp.,
PNAS articles
with I. M. Singer:
with Bent Fuglede:
with Zhe Liu:
with Bent Fuglede:
References
External links
1925 births
2018 deaths
20th-century American mathematicians
21st-century American mathematicians
Members of the United States National Academy of Sciences
University of Chicago alumni
University of Pennsylvania faculty
Members of the Norwegian Academy of Science and Letters
Fellows of the American Mathematical Society
The Bronx High School of Science alumni
People from New York City |
https://en.wikipedia.org/wiki/Peter%20Graham%20%28rugby%20league%29 | Peter Graham is an Australian former professional rugby league footballer who played in the 1990s. He played for the Newcastle Knights in 1991.
External links
Statistics at rugbyleagueproject.org
Living people
Australian rugby league players
Newcastle Knights players
Year of birth missing (living people)
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/2009%20Colo-Colo%20season | The 2009 season is Club Social y Deportivo Colo-Colo's 78th season at Chilean Primera División. This article shows player statistics and all matches (official and friendly) that the club have played during the 2009 season.
Players
Squad information
Matches
Torneo Apertura
Standings
Regular stage
Results summary
Torneo Clausura
Standings
Regular stage
Results summary
Play-offs
Quarter-finals
Semi-finals
Finals
Copa Chile
Copa Libertadores
Friendlies and other matches
References
2009
Colo-Colo
Colo |
https://en.wikipedia.org/wiki/Shane%20Mackley | Shane Mackley is an Australian former professional rugby league footballer who played in the 1990s. He played for the Newcastle Knights in 1992.
External links
Statistics at rugbyleagueproject.org
Living people
Australian rugby league players
Newcastle Knights players
Rugby league centres
Rugby league wingers
Year of birth missing (living people)
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Michele%20Cipolla | Michele Cipolla (28 October 1880, Palermo – 7 September 1947, Palermo) was an Italian mathematician, mainly specializing in number theory.
He was a professor of Algebraic Analysis at the University of Catania and, later, the University of Palermo. He developed (among other things) a theory for sequences of sets and Cipolla's algorithm for finding square roots modulo a prime number. He also solved the problem of binomial congruence.
Publications
Opere (Hrsg.: Guido Zappa) Sede della Soc., Palermo 1997. XXXII, 547 S. : Ill. (Supplemento ai Rendiconti del Circolo Matematico di Palermo ; Ser. 2, No. 47)
Storia della matematica dai primordi a Leibniz. Soc. Ed. Siciliana, Mazara 1949. 174 S.
Literature
Michele Cipolla (1880–1947): la figura e l'opera ; convegno celebrativo nel cinquantenario della morte (Palermo, 8 settembre 1997) / / Associazione degli Insegnanti e dei Cultori di Matematica. - Palermo 1998. 156 S.
See also
Cipolla's algorithm: Method for taking the modular square root for a prime modulus
External links
19th-century Italian mathematicians
20th-century Italian mathematicians
1880 births
1947 deaths
Academic staff of the University of Palermo
Scientists from Palermo
Mathematicians from Sicily |
https://en.wikipedia.org/wiki/Microdata | Microdata can mean:
Microdata (statistics), a statistical term for individual response data in surveys and censuses
Microdata (HTML), a specification for semantic markup in HTML
Microdata Corporation, a California-based computer company |
https://en.wikipedia.org/wiki/Journal%20of%20Differential%20Geometry | The Journal of Differential Geometry is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called Surveys in Differential Geometry. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University.
History
The journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009.
In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 2008 Harvard held a conference dedicated to the 40th anniversary of the Journal of Differential Geometry.
Reception
In his 2005 book Mathematical Publishing: A Guidebook, Steven Krantz writes: "At some very prestigious journals, like the Annals of Mathematics or the Journal of Differential Geometry, the editorial board meets every couple of months and debates each paper in detail."
The journal is abstracted and indexed in MathSciNet, Zentralblatt MATH, Current Contents/Physical, Chemical & Earth Sciences, and the Science Citation Index. According to the Journal Citation Reports, the journal has a 2013 impact factor of 1.093.
References
External links
Surveys in Differential Geometry web page
Mathematics journals
Academic journals established in 1967
Multilingual journals
Lehigh University
Academic journals associated with universities and colleges of the United States
9 times per year journals |
https://en.wikipedia.org/wiki/Side-approximation%20theorem | In geometric topology, the side-approximation theorem was proved by . It implies that a 2-sphere in R3 can be approximated by polyhedral 2-spheres.
References
Geometric topology
Theorems in topology |
https://en.wikipedia.org/wiki/Double%20suspension%20theorem | In geometric topology, the double suspension theorem of James W. Cannon () and Robert D. Edwards states that the double suspension S2X of a homology sphere X is a topological sphere.
If X is a piecewise-linear homology sphere but not a sphere, then its double suspension S2X (with a triangulation derived by applying the double suspension operation to a triangulation of X) is an example of a triangulation of a topological sphere that is not piecewise-linear. The reason is that, unlike in piecewise-linear manifolds, the link of one of the suspension points is not a sphere.
See also
References
Steve Ferry, Geometric Topology Notes (See Chapter 26, page 166)
Geometric topology
Theorems in topology |
https://en.wikipedia.org/wiki/Bing%20shrinking | In geometric topology, a branch of mathematics, the Bing shrinking criterion, introduced by , is a method for showing that a quotient of a space is homeomorphic to the space.
References
Geometric topology |
https://en.wikipedia.org/wiki/Moise%27s%20theorem | In geometric topology, a branch of mathematics, Moise's theorem, proved by Edwin E. Moise in , states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth structure.
The analogue of Moise's theorem in dimension 4 (and above) is false: there are topological 4-manifolds with no piecewise linear structures, and others with an infinite number of inequivalent ones.
See also
Exotic sphere
References
Geometric topology |
https://en.wikipedia.org/wiki/Tobias%20Colding | Tobias Holck Colding (born 1963) is a Danish mathematician working on geometric analysis, and low-dimensional topology. He is the great grandchild of Ludwig August Colding.
Biography
He was born in Copenhagen, Denmark, to Torben Holck Colding and Benedicte Holck Colding. He received his Ph.D. in mathematics in 1992 at the University of Pennsylvania under Chris Croke. Since 2005 Colding has been a professor of mathematics at MIT. He was on the faculty at the Courant Institute of New York University in various positions from 1992 to 2008. He has also been a visiting professor at MIT (2000–01) and at Princeton University (2001–02) and a postdoctoral fellow at MSRI (1993–94).
Colding lives in Cambridge, MA, with his wife and three children.
Work
In the early stage of his career, Colding did impressive work on manifolds with bounds on Ricci curvature. In 1995 he presented this work at the Geometry Festival. He began working with Jeff Cheeger while at NYU. He gave a 45-minute invited address to the ICM on this work in 1998 in Berlin. He began coauthoring with William P. Minicozzi at this time: first on harmonic functions, later on minimal surfaces, and now on mean curvature flow.
Recognition
He gave an AMS Lecture at University of Tennessee. He also gave an invited address at the first AMS-Scandinavian International meeting in Odense, Denmark, in 2000 and an invited address at the Germany Mathematics Meeting in 2003 in Rostock. He gave the 2008 Mordell Lecture at the University of Cambridge and gave the 2010 Cantrell Lectures at University of Georgia. Since 2008 he has been a Fellow of the American Academy of Arts and Sciences, and since 2006 a foreign member of the Royal Danish Academy of Sciences and Letters, and also since 2006 an honorary professor of University of Copenhagen, Denmark.
In 2010 Tobias H. Colding received the Oswald Veblen Prize in Geometry together with William Minicozzi II for their work on minimal surfaces. In justification of the reward the American Mathematical Society wrote:
"The 2010 Veblen Prize in Geometry is awarded to Tobias H. Colding and William P. Minicozzi II for their profound work on minimal surfaces. In a series of papers they have developed a structure theory for minimal surfaces with bounded genus in 3-manifolds, which yields a remarkable global picture for an arbitrary minimal surface of bounded genus. This contribution led to the resolution of long-standing conjectures of initiated a wave of new results. Specifically, they are cited for the following joint papers, of which the first four form a series of establishing the structure theory for embedded surfaces in 3-manifolds:
"The Space of Embedded Minimal Surfaces of Fixed Genus in a 3-manifold. I. estimates off the Axis for Disks", Ann. of Math (2) 160 (2004), no. 1, 27–68.
"The Space of Embedded Minimal Surfaces of Fixed Genus in a 3-manifold. II. Multi-valued Graphs in Disks". Ann. of Math. (2) 160 (2004), no. 1, 69–92.
"The Space of Embedded M |
https://en.wikipedia.org/wiki/Crumpled%20cube | In geometric topology, a branch of mathematics, a crumpled cube is any space in R3 homeomorphic to a 2-sphere together with its interior. Lininger showed in 1965 that the union of a crumpled cube and an open 3-ball glued along their boundaries is a 3-sphere.
References
Geometric topology |
https://en.wikipedia.org/wiki/Mikhail%20Katz | Mikhail "Mischa" Gershevich Katz (born 1958, in Chișinău) is an Israeli mathematician, a professor of mathematics at Bar-Ilan University. His main interests are differential geometry, geometric topology and mathematics education; he is the author of the book Systolic Geometry and Topology, which is mainly about systolic geometry. The Katz–Sabourau inequality is named after him and Stéphane Sabourau.
Biography
Mikhail Katz was born in Chișinău in 1958. His mother was Clara Katz (née Landman). In 1976, he moved with his mother to the United States.
Katz earned a bachelor's degree in 1980 from Harvard University. He did his graduate studies at Columbia University, receiving his Ph.D. in 1984 under the joint supervision of Troels Jørgensen and Mikhael Gromov. His thesis title is Jung's Theorem in Complex Projective Geometry.
He moved to Bar-Ilan University in 1999, after previously holding positions at the University of Maryland, College Park, Stony Brook University, Indiana University Bloomington, the Institut des Hautes Études Scientifiques, the University of Rennes 1, Henri Poincaré University, and Tel Aviv University.
Work
Katz has performed research in systolic geometry in collaboration with Luigi Ambrosio, Victor Bangert, Mikhail Gromov, Steve Shnider, Shmuel Weinberger, and others. He has authored research publications appearing in journals including Communications on Pure and Applied Mathematics, Duke Mathematical Journal, Geometric and Functional Analysis, and Journal of Differential Geometry. Along with these papers, Katz was a contributor to the book "Metric Structures for Riemannian and Non-Riemannian Spaces". Marcel Berger in his article "What is... a Systole?" lists the book (Katz, 2007) as one of two books he cites in systolic geometry.
More recently Katz also contributed to the study of mathematics education including work that provides an alternative interpretation of the number 0.999....
Selected publications
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References
External links
Mikhail Katz's home page
1958 births
Living people
21st-century Israeli mathematicians
Israeli mathematicians
Differential geometers
Harvard University alumni
Columbia University alumni
University of Maryland, College Park faculty
Stony Brook University faculty
Indiana University faculty
Academic staff of Bar-Ilan University
Textbook writers
Mathematics educators |
https://en.wikipedia.org/wiki/Annulus%20theorem | In mathematics, the annulus theorem (formerly called the annulus conjecture) states roughly that the region between two well-behaved spheres is an annulus. It is closely related to the stable homeomorphism conjecture (now proved) which states that every orientation-preserving homeomorphism of Euclidean space is stable.
Statement
If S and T are topological spheres in Euclidean space, with S contained in T, then it is not true in general that the region between them is an annulus, because of the existence of wild spheres in dimension at least 3. So the annulus theorem has to be stated to exclude these examples, by adding some condition to ensure that S and T are well behaved. There are several ways to do this.
The annulus theorem states that if any homeomorphism h of Rn to itself maps the unit ball B into its interior, then B − h(interior(B)) is homeomorphic to the annulus Sn−1×[0,1].
History of proof
The annulus theorem is trivial in dimensions 0 and 1. It was proved in dimension 2 by , in dimension 3 by , in dimension 4 by , and in dimensions at least 5 by .
Torus trick
Robion Kirby's torus trick is a proof method employing an immersion of a punctured torus into , where then smooth structures can be pulled back along the immersion and be lifted to covers.
The torus trick is used in Kirby's proof of the annulus theorem in dimensions .
It was also employed in further investigations of topological manifolds with Laurent C. Siebenmann
Here is a list of some further applications of the torus trick that appeared in the literature:
Proving existence and uniqueness (up to isotopy) of smooth structures on surfaces
Proving existence and uniqueness (up to isotopy) of PL structures on 3-manifolds
The stable homeomorphism conjecture
A homeomorphism of Rn is called stable if it is a product of homeomorphisms each of which is the identity on some non-empty open set.
The stable homeomorphism conjecture states that every orientation-preserving homeomorphism of Rn is stable. previously showed that the stable homeomorphism conjecture is equivalent to the annulus conjecture, so it is true.
References
Further reading
MathOverflow discussion on the Torus trick
Video recording of interview with Robion Kirby
Topological Manifolds Seminar (University of Bonn, 2021)
Geometric topology
Theorems in topology |
https://en.wikipedia.org/wiki/Gu%C3%B0mundur%20Kristj%C3%A1nsson | Guðmundur Kristjánsson (born 1 March 1989) is an Icelandic football player, currently playing for Icelandic club Stjarnan.
Career statistics
References
External links
1989 births
Living people
Gudmundur Kristjansson
Gudmundur Kristjansson
Gudmundur Kristjansson
Gudmundur Kristjansson
Gudmundur Kristjansson
IK Start players
Gudmundur Kristjansson
Eliteserien players
Norwegian First Division players
Gudmundur Kristjansson
Gudmundur Kristjansson
Gudmundur Kristjansson
Men's association football midfielders
Men's association football central defenders |
https://en.wikipedia.org/wiki/Markus%20Fierz | Markus Eduard Fierz (20 June 1912 – 20 June 2006) was a Swiss physicist, particularly remembered for his formulation of spin–statistics theorem, and for his contributions to the development of quantum theory, particle physics, and statistical mechanics. He was awarded the Max Planck Medal in 1979 and the Albert Einstein Medal in 1989 for all his work.
Biography
Fierz's father Hans Eduard Fierz was a chemist with Geigy and later a professor at ETH Zurich, his mother was Linda Fierz-David. Fierz studied at the Realgymnasium in Zurich. In 1931 he began his studies in Göttingen, where he listened to the lectures of prolific academics including Hermann Weyl. In 1933 he returned to Zurich and studied physics at ETH under Wolfgang Pauli and Gregor Wentzel. In 1936 he earned a doctoral degree with his thesis on the infrared catastrophe in quantum electrodynamics. Afterward he went to Werner Heisenberg in Leipzig and in 1936 became an assistant to Wolfgang Pauli in Zurich. For his habilitation degree in 1939 he treated in his thesis relativistic fields with arbitrary spins (with and without mass) and proved the Spin-statistics theorem for free fields. For quantum electrodynamics the work was extended. The work on relativistic fields with arbitrary spins was later important in supergravity. In 1940 he became Privatdozent in Basel and 1943 assistant professor. From 1944 to 1959 he was a professor for theoretical physics in Basel. In 1950 he was at the Institute for Advanced Study in Princeton, where he met Res Jost. In 1959 he led the theoretical physics department at CERN in Geneva for one year and in 1960 he became the successor of his teacher Pauli at ETH. In 1977 he retired there as an emeritus professor. Fierz also worked on gravitational theory but published only one paper on the subject.
In 1940 he married Menga Biber; they became acquainted through making music (he played the violin). Their marriage produced two sons.
Publications
M. Fierz, ’Spinors’, in Proceedings of the International Conference on Relativistic Theories of Gravitation, London, July 1965, H. Bondi ed., Kings College, University of London, 1965
M. Fierz, ’Die unitären Darstellungen der homogenen Lorentzgruppe’, in Preludes in theoretical physics, in honor of V. F. Weisskopf, A. de-Shalit, H. Feshbach and L. van Hove (eds.), North Holland, Amsterdam, 1966;
M. Fierz, Vorlesungen zur Entwicklungsgeschichte der Mechanik. Springer 1972;
Notes
See also
Fierz completeness relations
Relativistic wave equations
References
External links
Family site
"Physics Today" obituary (includes a photo)
1912 births
2006 deaths
People associated with CERN
Scientists from Basel-Stadt
Swiss physicists
Theoretical physicists
Academic staff of ETH Zurich
Albert Einstein Medal recipients
Winners of the Max Planck Medal |
https://en.wikipedia.org/wiki/Pocklington%20primality%20test | In mathematics, the Pocklington–Lehmer primality test is a primality test devised by Henry Cabourn Pocklington and Derrick Henry Lehmer.
The test uses a partial factorization of to prove that an integer is prime.
It produces a primality certificate to be found with less effort than the Lucas primality test, which requires the full factorization of .
Pocklington criterion
The basic version of the test relies on the Pocklington theorem (or Pocklington criterion) which is formulated as follows:
Let be an integer, and suppose there exist natural numbers and such that
Then is prime.
Note: Equation () is simply a Fermat primality test. If we find any value of , not divisible by , such that equation () is false, we may immediately conclude that is not prime. (This divisibility condition is not explicitly stated because it is implied by equation ().) For example, let . With , we find that . This is enough to prove that is not prime.
Suppose is not prime. This means there must be a prime , where that divides .
Since , , and since is prime, .
Thus there must exist an integer , a multiplicative inverse of modulo , with the property that
and therefore, by Fermat's little theorem
This implies
, by () since
,
, by ()
This shows that divides the in (), and therefore this ; a contradiction.
Given , if and can be found which satisfy the conditions of the theorem, then is prime. Moreover, the pair (, ) constitute a primality certificate which can be quickly verified to satisfy the conditions of the theorem, confirming as prime.
The main difficulty is finding a value of which satisfies (). First, it is usually difficult to find a large prime factor of a large number. Second, for many primes , such a does not exist. For example, has no suitable because , and , which violates the inequality in (); other examples include
and .
Given , finding is not nearly as difficult. If is prime, then by Fermat's little theorem, any in the interval will satisfy () (however, the cases and are trivial and will not satisfy ()). This will satisfy () as long as ord() does not divide . Thus a randomly chosen in the interval has a good chance of working. If is a generator mod , its order is and so the method is guaranteed to work for this choice.
Generalized Pocklington test
The above version of version of Pocklington's theorem is sometimes impossible to apply because some primes are such that there is no prime dividing where . The following generalized version of Pocklington's theorem is more widely applicable.
Theorem: Factor as , where and are relatively prime, , the prime factorization of is known, but the factorization of is not necessarily known.
If for each prime factor of there exists an integer so that
then N is prime.
Let be a prime dividing and let be the maximum power of dividing .
Let be a prime factor of . For the from the corollary set
. This means
and because of also
.
This means |
https://en.wikipedia.org/wiki/Crime%20in%20Haiti | Crime in Haiti is investigated by the Haitian police.
Crime by type
Murders in Haiti
Reliable crime statistics for Haiti are difficult to come by. A comparative analysis of figures from various police/security entities operating throughout Haiti indicates that incidents of crimes tend to be inaccurately or under-reported. Thus, for example, the United Nations office on Drugs and Crime UNODC documented 1,033 murders, for a murder rate of 10.2 per 100,000 people, in 2012, and as few as 486 (5.1 per 100,000 people) in 2007.
In the 22 months after the ouster of President Aristide in 2004, the murder rate for Port-au-Prince reached an all-time high of 219 murders per 100,000 residents. In contrast, an independent study tracking a large number of households in urban areas of Haiti recorded 11 murders among 15,690 tracked residents during a 7-month period from August 2011 to February 2012.
Preliminary results of the assessment found:The number of reported homicides across all urban settings increased considerably between November 2011 and February 2012. Half of the reported murders occurred during armed robbery or attempted armed robbery. While Port-au-Prince's overall homicide is low in comparison to other Caribbean cities, this nevertheless represents a rate of 60.9 per 100,000, one of the highest recorded rates since 2004[.];All but one of the murders occurred in Port-au-Prince. See {crime in Port-au-Prince}
Sexual violence
Sexual violence in Haiti is a common phenomenon. Being raped is considered shameful in Haitian society, and victims may find themselves abandoned by loved ones or with reduced marriageability. Until 2005, rape was not legally considered a serious crime and a rapist could avoid jail by marrying his victim. Reporting a rape to police in Haiti is a difficult and convoluted process, a factor that contributes to underreporting and difficulty in obtaining accurate statistics about sexual violence. Few rapists face any punishment.
A UN Security Council study in 2006 reported 35,000 sexual assaults against women and girls between 2004 and 2006. The UN reported in 2006 that half of the women living in the capital city Port-au-Prince's slums had been raped. United Nations peacekeepers stationed in Haiti since 2004 have drawn widespread resentment after reports emerged of the soldiers raping Haitian civilians.
The 2010 Haitian earthquake caused over a million Haitians to move to refugee camps where conditions are dangerous and poor. A study by a human rights group found that 14% of Haitian households reported having at least one member who suffered sexual violence between the January 2010 earthquake and January 2012. In 2012, sexual assaults in Port-au-Prince were reported at a rate 20 times higher in the camps than elsewhere in Haiti.
A 2009 study reported that up to 225,000 Haitian children are forced to work as domestic servants and are at grave risk of rape at the hands of their captors. The children, known as restaveks, |
https://en.wikipedia.org/wiki/Wild%20arc | In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to a straight line segment. found the first example of a wild arc, and found another example called the Fox-Artin arc whose complement is not simply connected.
See also
Wild knot
Horned sphere
Further reading
Geometric topology |
https://en.wikipedia.org/wiki/Edward%20Frenkel | Edward Vladimirovich Frenkel (; born May 2, 1968) is a Russian-American mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California Berkeley, a member of the American Academy of Arts and Sciences, and author of the bestselling book Love and Math.
Biography
Edward Frenkel was born on May 2, 1968, in Kolomna, Russia, which was then part of the Soviet Union. His father is of Jewish descent and his mother is Russian. As a high school student he studied higher mathematics privately with Evgeny Evgenievich Petrov, although his initial interest was in quantum physics rather than mathematics. He was not admitted to Moscow State University because of discrimination against Jews and enrolled instead in the applied mathematics program at the Gubkin University of Oil and Gas. While a student there, he attended the seminar of Israel Gelfand and worked with Boris Feigin and Dmitry Fuchs. After receiving his degree in 1989, he was first invited to Harvard University as a visiting professor, and a year later he enrolled as a graduate student at Harvard. He received his Ph.D. at Harvard University in 1991, after one year of study, under the direction of Boris Feigin and Joseph Bernstein. He was a Junior Fellow at the Harvard Society of Fellows from 1991 to 1994, and served as an associate professor at Harvard from 1994 to 1997. He has been a professor of mathematics at University of California, Berkeley, since 1997.
Mathematical work
Jointly with Boris Feigin, Frenkel constructed the free field realizations of affine Kac–Moody algebras (these are also known as Wakimoto modules), defined the quantum Drinfeld-Sokolov reduction, and described the center of the universal enveloping algebra of an affine Kac–Moody algebra, sometimes called the Feigin–Frenkel center. The last result, often referred to as Feigin–Frenkel isomorphism, has been used by Alexander Beilinson and Vladimir Drinfeld in their work on the geometric Langlands correspondence. Together with Nicolai Reshetikhin, Frenkel introduced deformations of W-algebras and q-characters of representations of quantum affine algebras.
Frenkel's recent work has focused on the Langlands program and its connections to representation theory, integrable systems, geometry, and physics. Together with Dennis Gaitsgory and Kari Vilonen, he has proved the geometric Langlands conjecture for GL(n). His joint work with Robert Langlands and Ngô Bảo Châu suggested a new approach to the functoriality of automorphic representations and trace formulas. He has also been investigating (in particular, in a joint work with Edward Witten) connections between the geometric Langlands correspondence and dualities in quantum field theory.
Awards
Frenkel was the first recipient of the Hermann Weyl Prize in 2002. Among his other awards are Packard Fellowship for Science and Engineering (1995) and Chaire d'Excellence from .
In 2013, he became a fe |
https://en.wikipedia.org/wiki/Capped%20grope | In mathematics, a grope is a construction used in 4-dimensional topology, introduced by and named by "because of its multitudinous fingers". Capped gropes were used by as a substitute for Casson handles, that work better for non-simply-connected 4-manifolds.
A capped surface in a 4-manifold is roughly a surface together with some 2-disks, called caps, whose boundaries generate the fundamental group of the surface. A capped grope is obtained by repeatedly replacing the caps of a capped surface by another capped surface. Capped surfaces and capped gropes are studied in .
References
4-manifolds |
https://en.wikipedia.org/wiki/Markstein%20number | In combustion engineering and explosion studies, the Markstein number characterizes the effect of local heat release of a propagating flame on variations in the surface topology along the flame and the associated local flame front curvature. The dimensionless Markstein number is defined as:
where is the Markstein length, and is the characteristic laminar flame thickness. The larger the Markstein length, the greater the effect of curvature on localised burning velocity. It is named after George H. Markstein (1911—2011), who showed that thermal diffusion stabilized the curved flame front and proposed a relation between the critical wavelength for stability of the flame front, called the Markstein length, and the thermal thickness of the flame. Phenomenological Markstein numbers with respect to the combustion products are obtained by means of the comparison between the measurements of the flame radii as a function of time and the results of the analytical integration of the linear relation between the flame speed and either flame stretch rate or flame curvature. The burning velocity is obtained at zero stretch, and the effect of the flame stretch acting upon it is expressed by a Markstein length. Because both flame curvature and aerodynamic strain contribute to the flame stretch rate, there is a Markstein number associated with each of these components.
Clavin–Williams formula
The Markstein number with respect to the unburnt gas mixture for a one step reaction in the limit of large activation energy asymptotics was derived by Paul Clavin and Forman A. Williams in 1982. The Markstein number then is
where
is the heat release parameter defined with density ratio,
is the Zel'dovich number,
is the Lewis number of the deficient reactant (either fuel or oxidizer)
is the dilogarithm function.
and the Markstein number with respect to the burnt gas mixture is derived by Clavin (1985)
Second Markstein number
In general, Markstein number for the curvature effects and strain effects are not same in real flames. In that case, one defines a second Markstein number as
See also
G equation
Prandtl number
Schmidt number
References
Combustion
Dimensionless numbers of fluid mechanics
Fluid dynamics
Dimensionless numbers
Dimensionless numbers of chemistry |
https://en.wikipedia.org/wiki/Emmanuel%20Cand%C3%A8s | Emmanuel Jean Candès (born 27 April 1970) is a French statistician. He is a professor of statistics and electrical engineering (by courtesy) at Stanford University, where he is also the Barnum-Simons Chair in Mathematics and Statistics. Candès is a 2017 MacArthur Fellow.
Academic biography
Candès earned a MSc from the École Polytechnique in 1993. He did his postgraduate studies at Stanford, where he earned a PhD in statistics in 1998 under the supervision of David Donoho and immediately joined the Stanford faculty as an assistant professor of statistics. He moved to the California Institute of Technology in 2000, where in 2006 he was named the Ronald and Maxine Linde Professor of Applied and Computational Mathematics. He returned to Stanford in 2009.
Research
Candès' early research concerned nonlinear approximation theory. In his PhD thesis, he developed generalizations of wavelets called curvelets and ridgelets that were able to capture higher order structures in signals. This work has had significant impact in image processing and multiscale analysis, and earned him the Popov prize in approximation theory in 2001.
In 2006, Candès wrote a paper with Australian-American mathematician Terence Tao that spearheaded the field of compressed sensing: the recovery of sparse signals from a few carefully constructed, and seemingly random measurements. Many researchers have since contributed to this field, which has introduced the idea of a camera that can record pictures while needing only one sensor.
Awards and honors
In 2001 Candès received an Alfred P. Sloan Research Fellowship. He was awarded the James H. Wilkinson Prize in Numerical Analysis and Scientific Computing in 2005. In 2006, he received the Vasil A. Popov Prize as well as the National Science Foundation's highest honor: the Alan T. Waterman Award for research described by the NSF as "nothing short of revolutionary". In 2010 Candès and Terence Tao were awarded the George Pólya Prize. In 2011, Candès was awarded the ICIAM Collatz Prize. Candès has also received the Lagrange Prize in Continuous Optimization, awarded by the Mathematical Optimization Society (MOS) and the Society for Industrial and Applied Mathematics (SIAM). He was also presented with the Dannie Heineman Prize by the Academy of Sciences at Göttingen in 2013. In 2014 he was elected to the National Academy of Sciences. In 2015 he received the George David Birkhoff Prize of the AMS / SIAM. He is also a fellow of SIAM. In 2017 Candès received the MacArthur Fellowship for exploring the limits of signal recovery and matrix completion from incomplete data sets with implications for high-impact applications in multiple fields.
He was elected to the 2018 class of fellows of the American Mathematical Society. In 2020, Candès was awarded the Princess of Asturias Award for Technical and Scientific Research.
Personal life
Candès is married to Stanford statistician Chiara Sabatti.
References
External links
Candès' web page at Stanf |
https://en.wikipedia.org/wiki/Mark%20Eberhart | Mark Evan Eberhart is an author and a professor of chemistry and geochemistry at the Colorado School of Mines.
Education and career
Eberhart holds a BS in chemistry and in applied mathematics from the University of Colorado, an MS in physical biochemistry from the University of Colorado, and a PhD in materials science and engineering from the Massachusetts Institute of Technology, supervised by Keith H. Johnson and earned in 1983. He became a scientist in the Materials Science and Technology Division of the Los Alamos National Laboratory before moving to the Colorado School of Mines in 1992. At the School of Mines, he has been president of the faculty senate; he has also been a Jefferson Science Fellow.
Books
Eberhart has published two books: Why Things Break: Understanding the World By the Way It Comes Apart (Random House, 2003), an autobiographical book describing his education and his studies of stress and fracture and Feeding the Fire: The Lost History and Uncertain Future of Mankind's Energy Addiction (Random House, 2007).
References
Year of birth missing (living people)
Living people
21st-century American chemists
American educators
American male writers
University of Colorado Boulder alumni
Massachusetts Institute of Technology alumni
Los Alamos National Laboratory personnel
Colorado School of Mines faculty
Jefferson Science Fellows |
https://en.wikipedia.org/wiki/Demography%20of%20Liverpool | The demography of Liverpool is officially analysed by the Office for National Statistics. The Liverpool City Region is made up of Liverpool alongside the Metropolitan Boroughs of Halton, Knowsley, Sefton, St Helens, and the Wirral. With a population of around 496,784, Liverpool is the largest settlement in the region and the sixth largest in the United Kingdom.
Population change
As with other major British cities, Liverpool has a large and very diverse population. In the 2011 UK Census, the recorded population of Liverpool was 466,400, a 5.5% increase from the 435,500 recorded in the 2001 census. Liverpool's population peaked in the 1930s with 846,101 recorded in the 1931 census. Until the recent increase, the city had experienced negative population growth every decade; at its peak, over 100,000 people had left the city between 1971 and 1981. Between 2001 and 2006, the city experienced the ninth largest population percentage loss of any UK unitary authority.
In common with many cities, Liverpool's population is younger than that of England as a whole, with 42.3% of its population under the age of 30, compared to an English average of 37.4%. Those of working age make up 65.1% of the population.
Urban and metropolitan area
The Liverpool Urban Area encompasses the city of Liverpool alongside Sefton, Knowsley, St. Helens and Ashton-in-Makerfield had a population of 864,122 in 2011, which ranks sixth out of all UK conurbations. Liverpool City Region had an estimated population of 1,554,642 in 2019.
Ethnicity
While 84% of Liverpool's population is white, the city is one of the most important sites in the history of multiculturalism in the United Kingdom. Liverpool is home to Britain's oldest black community, dating to at least the 1730s, and some Liverpudlians are able to trace their black ancestors in the city back ten generations. Early black settlers in the city included seamen, the children of traders sent to be educated, and freed slaves (since slaves entering the country after 1722 were deemed free men).
The city is also home to the oldest Chinese community in Europe; the first residents of the city's Chinatown arrived as seamen in the 19th century. The gateway in Chinatown is also the largest gateway outside of China.
The city is also historically known for its large Irish and Welsh populations. The Liverpool accent (Scouse) is thought to have been influenced by the arrival of Irish and Welsh immigrants. Today, up to 50% of Liverpool's population is believed to have Irish ancestry. The influences of Irish and Welsh culture have given Liverpool's people traits usually associated with the Celtic fringes of the British Isles.
The vast majority of Liverpool's ethnic minorities live within the inner city area, particularly in and around Toxteth. According to the 2001 census, 38% of the population of Granby, 37% of Princes Park, and 27% of Central were from ethnic groups other than White British.
Census data
Per the above table, note that |
https://en.wikipedia.org/wiki/Cellular%20decomposition | In geometric topology, a cellular decomposition G of a manifold M is a decomposition of M as the disjoint union of cells (spaces homeomorphic to n-balls Bn).
The quotient space M/G has points that correspond to the cells of the decomposition. There is a natural map from M to M/G, which is given the quotient topology. A fundamental question is whether M is homeomorphic to M/G. Bing's dogbone space is an example with M (equal to R3) not homeomorphic to M/G.
Definition
Cellular decomposition of is an open cover with a function for which:
Cells are disjoint: for any distinct , .
No set gets mapped to a negative number: .
Cells look like balls: For any and for any there exists a continuous map that is an isomorphism and also .
A cell complex is a pair where is a topological space and is a cellular decomposition of .
See also
CW complex
References
Geometric topology |
https://en.wikipedia.org/wiki/Stratified%20space | In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat).
A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is also an abstract stratified space such as a Thom–Mather stratified space.
On a stratified space, a constructible sheaf can be defined as a sheaf that is locally constant on each stratum.
Among the several ideals, Grothendieck's Esquisse d’un programme considers (or proposes) a stratified space with what he calls the tame topology.
A stratified space in the sense of Mather
Mather gives the following definition of a stratified space. A prestratification on a topological space X is a partition of X into subsets (called strata) such that (a) each stratum is locally closed, (b) it is locally finite and (c) (axiom of frontier) if two strata A, B are such that the closure of A intersects B, then B lies in the closure of A. A stratification on X is a rule that assigns to a point x in X a set germ at x of a closed subset of X that satisfies the following axiom: for each point x in X, there exists a neighborhood U of x and a prestratification of U such that for each y in U, is the set germ at y of the stratum of the prestratification on U containing y.
A stratified space is then a topological space equipped with a stratification.
Pseudomanifold
In the MacPherson's stratified pseudomanifolds; the strata are the differences Xi+i-Xi between sets in the filtration. There is also a local conical condition; there must be an almost smooth atlas where locally each little open set looks like the product of two factors Rnx c(L); a euclidean factor and the topological cone of a space L. Classically, here is the point where the definitions turns to be obscure, since L is asked to be a stratified pseudomanifold. The logical problem is avoided by an inductive trick which makes different the objects L and X.
The changes of charts or cocycles have no conditions in the MacPherson's original context. Pflaum asks them to be smooth, while in the Thom-Mather context they must preserve the above decomposition, they have to be smooth in the Euclidean factor and preserve the conical radium.
See also
Equisingularity
Perverse sheaf
Stratified Morse theory
Harder–Narasimhan stratification
Footnotes
References
Appendix 1 of R. MacPherson, Intersection homology and perverse sheaves, 1990 notes
J. Mather, Stratifications and Mappings, Dynamical Systems, Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26–August 14, 1971, 1973, pages 195–232.
Markus J. Pflaum, Analytic and Geometric Study of Stratified Spaces: Contributions to Analytic and Geometric Aspects (Lecture Notes in Mathematics, 1768) ; Publisher, Springer;
Further reading
https://ncatlab.org/nlab/show/stratified+space
https://mathoverflow.net/questions/258562/correc |
https://en.wikipedia.org/wiki/Karlsruhe%20metric | In metric geometry, the Karlsruhe metric is a measure of distance that assumes travel is only possible along rays through the origin and circular arcs centered at the origin. The name alludes to the layout of the city of Karlsruhe, which has radial streets and circular avenues around a central point. This metric is also called Moscow metric.
In this metric, there are two types of shortest paths. One possibility, when the two points are on nearby rays, combines a circular arc through the nearer to the origin of the two points and a segment of a ray through the farther of the two points. Alternatively, for points on rays that are nearly opposite, it is shorter to follow one ray all the way to the origin and then follow the other ray back out. Therefore, the Karlsruhe distance between two points is the minimum of the two lengths that would be obtained for these two types of path. That is, it equals
where are the polar coordinates of and is the angular distance between the two points.
See also
Manhattan distance
Hamming distance
Notes
External links
Karlsruhe-metric Voronoi diagram, by Takashi Ohyama
Karlsruhe-Metric Voronoi Diagram, by Rashid Bin Muhammad
Metric spaces |
https://en.wikipedia.org/wiki/Twisted%20Edwards%20curve | In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein, Birkner, Joye, Lange and Peters in 2008. The curve set is named after mathematician Harold M. Edwards. Elliptic curves are important in public key cryptography and twisted Edwards curves are at the heart of an electronic signature scheme called EdDSA that offers high performance while avoiding security problems that have surfaced in other digital signature schemes.
Definition
A twisted Edwards curve over a field with characteristic not equal to 2 (that is, no element is its own additive inverse) is an affine plane curve defined by the equation:
where are distinct non-zero elements of .
Each twisted Edwards curve is a twist of an Edwards curve. The special case is untwisted, because the curve reduces to an ordinary Edwards curve.
Every twisted Edwards curve is birationally equivalent to an elliptic curve in Montgomery form and vice versa.
Group law
As for all elliptic curves, also for the twisted Edwards curve, it is possible to do some operations between its points, such as adding two of them or doubling (or tripling) one. The results of these operations are always points that belong to the curve itself. In the following sections some formulas are given to obtain the coordinates of a point resulted from an addition between two other points (addition), or the coordinates of point resulted from a doubling of a single point on a curve.
Addition on twisted Edwards curves
Let be a field with characteristic different from 2.
Let and be points on the twisted Edwards curve. The equation of twisted Edwards curve is written as;
: .
The sum of these points on is:
The neutral element is (0,1) and the negative of is
These formulas also work for doubling. If a is a square in and d is a non-square in , these formulas are complete: this means that they can be used for all pairs of points without exceptions; so they work for doubling as well, and neutral elements and negatives are accepted as inputs.
Example of addition
Given the following twisted Edwards curve with a = 3 and d = 2:
it is possible to add the points and using the formula given above. The result is a point P3 that has coordinates:
Doubling on twisted Edwards curves
Doubling can be performed with exactly the same formula as addition.
Doubling of a point on the curve is:
where
Denominators in doubling are simplified using the curve equation . This reduces the power from 4 to 2 and allows for more efficient computation.
Example of doubling
Considering the same twisted Edwards curve given in the previous example, with a=3 and d=2, it is possible to double the point . The point 2P1 obtained using the formula above has the following coordinates:
It is easy to see, with some little computations, that the point belongs to the curve .
Extended coordinates
There is another kind of coordinate system with |
https://en.wikipedia.org/wiki/John%20Micklewright | John Micklewright (born 20 June 1957) is Professor Emeritus of Economics and Social Statistics at UCL Social Research Institute, University College London.
Career
Micklewright studied at the University of Exeter (BA in Geography and Economics with First Class Honours) and then completed a PhD in Economics at the London School of Economics. He did post-doctoral work as a Prize Research Fellow at Nuffield College, Oxford. Before joining UCL, he was Professor of Social Statistics in the School of Social Sciences at the University of Southampton, head of research in the UNICEF Innocenti Research Centre, Professor of Economics at the European University Institute in Florence, and Lecturer, Reader and then Professor of Economics at Queen Mary, University of London.
He is the editor, together with Andrea Brandolini (Banca d'Italia), of Tony Atkinson's last book, published posthumously in 2019, Measuring Poverty around the World, Princeton University Press.
In 2015, he walked across France, from Normandy to the Alps, a journey described in a blog at the time ‘The Long March’ and subsequently in a book, The Opening Country: A Walk through France, Matador, 2021.
Areas
His research focuses on:
poverty, inequality and the measurement of living standards
labour market flows and behaviour
educational achievement and segregation in schools
charitable giving, especially for development.
survey methods
At UNICEF Micklewright compared living standards of children in both OECD members and the countries of Central and Eastern Europe and the former USSR. He was one of the team that started the Innocenti Report Card series on child wellbeing in the OECD.
Professional activities
He is a Research Fellow of the Institute for the Study of Labor (IZA), (Bonn).
Publications
Micklewright's publications include the following books, as well as many journal articles:
Unemployment Benefit and Unemployment Duration, STICERD Occasional Paper 6, LSE, 1985 (with A.B. Atkinson)
Economic Transformation in Eastern Europe and the Distribution of Income, Cambridge University Press, 1992 (with A.B. Atkinson)
Household Welfare in Central Asia, Macmillan, 1997 (edited with J. Falkingham, J. Klugman and S. Marnie)
The Welfare of Europe’s Children: Are EU Member States Converging? The Policy Press, 2000 (with K. Stewart)
The Dynamics of Child Poverty in Industrialised Countries, Cambridge University Press, 2001 (edited with B. Bradbury and S. Jenkins)
Inequality and Poverty Re-Examined, Oxford University Press, 2007 (edited with S. Jenkins)
The Great Recession and the Distribution of Household Income, Oxford University Press, 2012 (edited with S. Jenkins, A. Brandolini and B. Nolan)
Family Background and University Success, Oxford University Press, 2016 (with C. Crawford, L. Dearden and A. Vignoles)
The Opening Country: A Walk through France, Matador, 2021
References
British statisticians
Academics of the University of Southampton
Living people
1957 births |
https://en.wikipedia.org/wiki/Lakshminarayanan%20Mahadevan | Lakshminarayanan Mahadevan is an Indian-American scientist. He is currently the Lola England de Valpine Professor of Applied Mathematics, Organismic and Evolutionary Biology and Physics at Harvard University. His work centers around understanding the organization of matter in space and time (that is, how it is shaped and how it flows, particularly at the scale observable by the unaided senses, in both physical and biological systems). Mahadevan is a 2009 MacArthur Fellow.
Education
Mahadevan graduated from the Indian Institute of Technology, Madras, and then received an M.S. from the University of Texas at Austin, and an M.S. and Ph.D. from Stanford University in 1995.
Career and research
He started his independent career on the faculty at the Massachusetts Institute of Technology in 1996. In 2000, he was elected the inaugural Schlumberger Professor of Complex Physical Systems in the Department of Applied Mathematics and Theoretical Physics, and a professorial fellow of Trinity College, Cambridge, University of Cambridge, the first Indian to be appointed professor to the Faculty of Mathematics there.
He has been at Harvard since 2003, where he served as the chair/co-chair of Applied Mathematics from 2016–2021. Since 2017, together with A. Mahadevan, he has been the faculty dean of Mather House, one of twelve residential houses (with ~400 students) at Harvard College.
Awards
2023 American Academy of Arts and Sciences
2016 Fellow of the Royal Society
2014 Clay Senior Scholar
2009 MacArthur Fellow
2006 Guggenheim Fellowship
2007 Ig Nobel Prize for physics
2007 Visiting Miller Professor, University of California, Berkeley
2006 George Ledlie Prize, Harvard University
References
External links
"Recent Publications", The Applied Math Lab
"Google Scholar"
Living people
IIT Madras alumni
Stanford University alumni
University of Texas at Austin College of Natural Sciences alumni
Harvard University faculty
Massachusetts Institute of Technology School of Science faculty
MacArthur Fellows
Radcliffe fellows
Fellows of the Royal Society
Santa Fe Institute people
American academics of Indian descent
Year of birth missing (living people)
Fellows of the American Academy of Arts and Sciences |
https://en.wikipedia.org/wiki/Divergence%20%28statistics%29 | In information geometry, a divergence is a kind of statistical distance: a binary function which establishes the separation from one probability distribution to another on a statistical manifold.
The simplest divergence is squared Euclidean distance (SED), and divergences can be viewed as generalizations of SED. The other most important divergence is relative entropy (also called Kullback–Leibler divergence), which is central to information theory. There are numerous other specific divergences and classes of divergences, notably f-divergences and Bregman divergences (see ).
Definition
Given a differentiable manifold of dimension , a divergence on is a -function satisfying:
for all (non-negativity),
if and only if (positivity),
At every point , is a positive-definite quadratic form for infinitesimal displacements from .
In applications to statistics, the manifold is typically the space of parameters of a parametric family of probability distributions.
Condition 3 means that defines an inner product on the tangent space for every . Since is on , this defines a Riemannian metric on .
Locally at , we may construct a local coordinate chart with coordinates , then the divergence is where is a matrix of size . It is the Riemannian metric at point expressed in coordinates .
Dimensional analysis of condition 3 shows that divergence has the dimension of squared distance.
The dual divergence is defined as
When we wish to contrast against , we refer to as primal divergence.
Given any divergence , its symmetrized version is obtained by averaging it with its dual divergence:
Difference from other similar concepts
Unlike metrics, divergences are not required to be symmetric, and the asymmetry is important in applications. Accordingly, one often refers asymmetrically to the divergence "of q from p" or "from p to q", rather than "between p and q". Secondly, divergences generalize squared distance, not linear distance, and thus do not satisfy the triangle inequality, but some divergences (such as the Bregman divergence) do satisfy generalizations of the Pythagorean theorem.
In general statistics and probability, "divergence" generally refers to any kind of function , where are probability distributions or other objects under consideration, such that conditions 1, 2 are satisfied. Condition 3 is required for "divergence" as used in information geometry.
As an example, the total variation distance, a commonly used statistical divergence, does not satisfy condition 3.
Notation
Notation for divergences varies significantly between fields, though there are some conventions.
Divergences are generally notated with an uppercase 'D', as in , to distinguish them from metric distances, which are notated with a lowercase 'd'. When multiple divergences are in use, they are commonly distinguished with subscripts, as in for Kullback–Leibler divergence (KL divergence).
Often a different separator between parameters is used, particularly |
https://en.wikipedia.org/wiki/De%20Rham%20invariant | In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of – either 0 or 1. It can be thought of as the simply-connected symmetric L-group and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, ), and the Kervaire invariant, a (4k+2)-dimensional quadratic invariant
It is named for Swiss mathematician Georges de Rham, and used in surgery theory.
Definition
The de Rham invariant of a (4k+1)-dimensional manifold can be defined in various equivalent ways:
the rank of the 2-torsion in as an integer mod 2;
the Stiefel–Whitney number ;
the (squared) Wu number, where is the Wu class of the normal bundle of and is the Steenrod square ; formally, as with all characteristic numbers, this is evaluated on the fundamental class: ;
in terms of a semicharacteristic.
References
Chess, Daniel, A Poincaré-Hopf type theorem for the de Rham invariant, 1980
Geometric topology
Surgery theory |
https://en.wikipedia.org/wiki/Absolute%20scale | There is no single definition of an absolute scale. In statistics and measurement theory, it is simply a ratio scale in which the unit of measurement is fixed, and values are obtained by counting. Another definition tells us it is the count of the elements in a set, with its natural origin being zero, the empty set. Some sources tell us that even time can be measured in an absolute scale, proving year zero is measured from the beginning of the universe. How that is obtained precisely would be a matter of debate. Colloquially, the Kelvin temperature scale, where absolute zero is the temperature at which molecular energy is at a minimum, and the Rankine temperature scale are also referred to as absolute scales. In that case, an absolute scale is a system of measurement that begins at a minimum, or zero point, and progresses in only one direction. Measurement theory, however, categorizes them as ratio scales. In general, an absolute scale differs from a relative scale in having some reference point that is not arbitrarily selected.
Features
An absolute scale differs from an arbitrary, or "relative", scale, which begins at some point selected by a person and can progress in both directions. An absolute scale begins at a natural minimum, leaving only one direction in which to progress.
An absolute scale can only be applied to measurements in which a true minimum is known to exist. Time, for example, which does not have a clearly known beginning, is measured on a relative scale, with an arbitrary zero-point such as the conventional date of the birth of Jesus (see Anno Domini) or the accession of an emperor. Temperature, on the other hand, has a known minimum, absolute zero (where volume of an ideal gas becomes zero), and therefore, can be measured either in absolute terms (e.g. kelvin), or relative to a reference temperature (e.g. degree Celsius).
Uses
Absolute scales are used when precise values are needed in comparison to a natural, unchanging zero point. Measurements of length, area and volume are inherently absolute, although measurements of distance are often based on an arbitrary starting point. Measurements of weight can be absolute, such as atomic weight, but more often they are measurements of the relationship between two masses, while measurements of speed are relative to an arbitrary reference frame. (Unlike many other measurements without a known, absolute minimum, speed has a known maximum and can be measured from a purely relative scale.) Absolute scales can be used for measuring a variety of things, from the flatness of an optical flat to neuroscientific tests.
References
Measurement
Metrology |
https://en.wikipedia.org/wiki/Elliptic%20curve%20primality | In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin the same year. The algorithm was altered and improved by several collaborators subsequently, and notably by Atkin and , in 1993. The concept of using elliptic curves in factorization had been developed by H. W. Lenstra in 1985, and the implications for its use in primality testing (and proving) followed quickly.
Primality testing is a field that has been around since the time of Fermat, in whose time most algorithms were based on factoring, which become unwieldy with large input; modern algorithms treat the problems of determining whether a number is prime and what its factors are separately. It became of practical importance with the advent of modern cryptography. Although many current tests result in a probabilistic output (N is either shown composite, or probably prime, such as with the Baillie–PSW primality test or the Miller–Rabin test), the elliptic curve test proves primality (or compositeness) with a quickly verifiable certificate.
Previously-known prime-proving methods such as the Pocklington primality test required at least partial factorization of in order to prove that is prime. As a result, these methods required some luck and are generally slow in practice.
Elliptic curve primality proving
It is a general-purpose algorithm, meaning it does not depend on the number being of a special form. ECPP is currently in practice the fastest known algorithm for testing the primality of general numbers, but the worst-case execution time is not known. ECPP heuristically runs in time:
for some . This exponent may be decreased to for some versions by heuristic arguments. ECPP works the same way as most other primality tests do, finding a group and showing its size is such that is prime. For ECPP the group is an elliptic curve over a finite set of quadratic forms such that is trivial to factor over the group.
ECPP generates an Atkin–Goldwasser–Kilian–Morain certificate of primality by recursion and then
attempts to verify the certificate. The step that takes the most CPU time is the certificate generation, because factoring over a class field must be performed. The certificate can be verified quickly, allowing a check of operation to take very little time.
, the largest prime that has been proved with ECPP method is . The certification was performed by Andreas Enge using his fastECPP software CM.
Proposition
The elliptic curve primality tests are based on criteria analogous to the Pocklington criterion, on which that test is based, where the group
is replaced by and E is a properly chosen elliptic curve. We will now state a proposition on which to base our test, which is analogous to the Pocklington criterion, and gives rise to the Goldwa |
https://en.wikipedia.org/wiki/Daniel%20L.%20Stein | Daniel L. Stein (born August 19, 1953) is an American physicist and Professor of Physics and Mathematics at New York University. From 2006 to 2012 he served as the NYU Dean of Science.
He has contributed to a wide range of scientific fields. His early research covered diverse topics, including theoretical work on protein biophysics, biological evolution, amorphous semiconductors, quantum liquids, topology of order parameter spaces, liquid crystals, neutron stars, and the interface between particle physics and cosmology. His
primary focus, however, has been on quenched randomness in condensed matter and on stochastic processes in both irreversible and extended systems. His research on these topics was cited by the American Association for the Advancement of Science as "pioneering work on the statistical mechanics of disordered and noisy systems".
He is best known for work on hierarchical dynamics (in collaboration with Elihu Abrahams, Philip Warren Anderson, and Richard Palmer); for observing that protein fluctuational conformations can be modeled using spin glass techniques; for constructing a theory of fluctuation-driven transitions in the absence of detailed balance (in collaboration with Robert Maier); for applying stochastic methods to determine lifetimes, stability, and decay of nanowires and nanomagnets (with a variety of collaborators); and for a series of rigorous and analytical results (largely with Charles M. Newman) on
short-range spin glasses, including the introduction of the Newman-Stein metastate as a general mathematical tool for analyzing the thermodynamic properties of disordered systems.
Education and early career
Stein graduated from Brown University in 1975 with degrees in physics and mathematics. He received his Ph.D. in Physics from Princeton University in 1979, under the thesis supervision of Philip Warren Anderson. He stayed on as a faculty member in the Princeton Physics Department until 1987, when he moved to the University of Arizona Physics Department, where he served as Department Head from 1995 to 2005. During that period he also served as the first Director of the Complex Systems Summer School in Santa Fe (1988, 1990–1998). In 2005 he moved to New York University as Professor of Physics and Mathematics and as Provost Faculty Fellow. He became the NYU Dean of Science in September 2006, serving until 2012.
Honors
He currently serves as co-chair of the Santa Fe Institute Science Board and is a General Member of the Aspen Center for Physics. From 2008 through 2012 he served on the Air Force Scientific Advisory Board. His awards include a Sloan Foundation Fellowship (1985–1989), election to Fellowship of the American Physical Society (1999), University of Arizona Commission on the
Status of Women Vision 2000 Award, election to Fellowship of the American Association for the Advancement of Science (2008), the Exemplary Civilian Service Medal of the U.S. Air Force (2012), and a John Simon Guggenheim Fellowship ( |
https://en.wikipedia.org/wiki/Demonic%20composition | In mathematics, demonic composition is an operation on binary relations that is similar to the ordinary composition of relations but is robust to refinement of the relations into (partial) functions or injective relations.
Unlike ordinary composition of relations, demonic composition is not associative.
Definition
Suppose is a binary relation between and and is a relation between and Their is a relation between and Its graph is defined as
Conversely, their is defined by
References
.
Algebraic logic
Binary operations
Mathematical relations |
https://en.wikipedia.org/wiki/Set%20function | In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and
A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
Definitions
If is a family of sets over (meaning that where denotes the powerset) then a is a function with domain and codomain or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures.
The domain of a set function may have any number properties; the commonly encountered properties and categories of families are listed in the table below.
In general, it is typically assumed that is always well-defined for all or equivalently, that does not take on both and as values. This article will henceforth assume this; although alternatively, all definitions below could instead be qualified by statements such as "whenever the sum/series is defined". This is sometimes done with subtraction, such as with the following result, which holds whenever is finitely additive:
: is defined with satisfying and
Null sets
A set is called a (with respect to ) or simply if
Whenever is not identically equal to either or then it is typically also assumed that:
: if
Variation and mass
The is
where denotes the absolute value (or more generally, it denotes the norm or seminorm if is vector-valued in a (semi)normed space).
Assuming that then is called the of and is called the of
A set function is called if for every the value is (which by definition means that and ; an is one that is equal to or ).
Every finite set function must have a finite mass.
Common properties of set functions
A set function on is said to be
if it is valued in
if for all pairwise disjoint finite sequences such that
If is closed under binary unions then is finitely additive if and only if for all disjoint pairs
If is finitely additive and if then taking shows that which is only possible if or where in the latter case, for every (so only the case is useful).
or if in addition to being finitely additive, for all pairwise disjoint sequences in such that all of the following hold:
The series on the left hand side is defined in the usual way as the limit
As a consequence, if is any permutation/bijection then this is because and applying this condition (a) twice guarantees that both and hold. By definition, a convergent series with this property is said to be unconditionally convergent. Stated in plain English, this means that rearranging/relabeling the sets to the new order does not affect the sum of their measures. This is desirable since just |
https://en.wikipedia.org/wiki/Oveis%20Kordjahan | Oveis Kordjahan (born April 13, 1985) is an Iranian footballer who plays for ُSahrdari Mahshahr in Division 2.
Club career
In 2007, Kordjahan joined Pas Hamedan.
Club career statistics
Assist Goals
References
1985 births
People from Sari, Iran
Living people
PAS Tehran F.C. players
Iranian men's footballers
Shahid Ghandi Yazd F.C. players
Aluminium Hormozgan F.C. players
Mes Sarcheshme players
Sportspeople from Sari, Iran
Footballers from Mazandaran province
Men's association football defenders |
https://en.wikipedia.org/wiki/Idaho%20Standards%20Achievement%20Test | The Idaho Standards Achievement Tests (ISAT) is the state achievement test for Idaho It is administered for reading, English language use, and mathematics in grades 3-8 and once in grade 11. Science is additionally assessed in grades 5 and 7. The ISAT is used to monitor golas state, district, and school monitoring. At this time, Idaho does not use these tests for graduation purposes at any level.
References
Education in Idaho
Standardized tests in the United States
2004 establishments in Idaho |
https://en.wikipedia.org/wiki/Finite%20pointset%20method | In applied mathematics, the name finite pointset method is a general approach for the numerical solution of problems in continuum mechanics, such as the simulation of fluid flows. In this approach (often abbreviated as FPM) the medium is represented by a finite set of points, each endowed with the relevant local properties of the medium such as density, velocity, pressure, and temperature.
The sampling points can move with the medium, as in the Lagrangian approach to fluid dynamics or they may be fixed in space while the medium flows through them, as in the Eulerian approach. A mixed Lagrangian-Eulerian approach may also be used. The Lagrangian approach is also known (especially in the computer graphics field) as particle method.
Finite pointset methods are meshfree methods and therefore are easily adapted to domains with complex and/or time-evolving geometries and moving phase boundaries (such as a liquid splashing into a container, or the blowing of a glass bottle) without the software complexity that would be required to handle those features with topological data structures. They can be useful in non-linear problems involving viscous fluids, heat and mass transfer, linear and non-linear elastic or plastic deformations, etc.
Description
In the simplest implementations, the finite point set is stored as an unstructured list of points in the medium. In the Lagrangian approach the points move with the medium, and points may be added or deleted in order to maintain a prescribed sampling density. The point density is usually prescribed by a smoothing length defined locally. In the Eulerian approach the points are fixed in space, but new points may be added where there is need for increased accuracy. So, in both approaches the nearest neighbors of a point are not fixed, and are determined again at each time step.
Advantages
This method has various advantages over grid-based techniques; for example, it can handle fluid domains, which change naturally, whereas grid based techniques require additional computational effort. The finite points have to completely cover the whole flow domain, i.e. the point cloud has to fulfill certain quality criteria (finite points are not allowed to form “holes” which means finite points have to find sufficiently numerous neighbours; also, finite points are not allowed to cluster; etc.).
The finite point cloud is a geometrical basis, which allows for a numerical formulation making FPM a general finite difference idea applied to continuum mechanics. That especially means, if the point reduced to a regular cubic point grid, then FPM would reduce to a classical finite difference method. The idea of general finite differences also means that FPM is not based on a weak formulation like Galerkin's approach. Rather, FPM is a strong formulation which models differential equations by direct approximation of the occurring differential operators. The method used is a moving least squares idea which was especially develop |
https://en.wikipedia.org/wiki/Bob%20Glassey | Robert John Glassey (13 August 1914 – 1984) was a footballer who played in the Football League for Liverpool and Mansfield Town.
Career statistics
Source:
References
1914 births
1984 deaths
Footballers from Chester-le-Street
Men's association football forwards
English men's footballers
Darlington Town F.C. players
Liverpool F.C. players
Stoke City F.C. players
Mansfield Town F.C. players
Stockton F.C. players
English Football League players |
https://en.wikipedia.org/wiki/Free%20Polish%20University | Free Polish University (), founded in 1918 in Warsaw, was a private high school with different departments: mathematics and natural sciences, humanities, political sciences and social pedagogy.
From 1929, its degrees were equivalent to those of university.
In the years 1919–1939 the institution employed 70–80 professors. In the academic year 1938/39 educated about 3000 students. The university conducted clandestine courses during the German occupation, but after the war, its activities were not resumed.
The university was disbanded in 1952.
See also
Education in Poland
List of modern universities in Europe (1801–1945)
References
Universities and colleges in Warsaw
1918 establishments in Poland
Universities and colleges established in 1918
Educational institutions disestablished in 1952 |
https://en.wikipedia.org/wiki/Small%20complex%20icosidodecahedron | In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron.
A small complex icosidodecahedron can be constructed from a number of different vertex figures.
A very similar figure emerges as a geometrical truncation of the great stellated dodecahedron, where the pentagram faces become doubly-wound pentagons ({5/2} --> {10/2}), making the internal pentagonal planes, and the three meeting at each vertex become triangles, making the external triangular planes.
As a compound
The small complex icosidodecahedron can be seen as a compound of the icosahedron {3,5} and the great dodecahedron {5,5/2} where all vertices are precise and edges coincide. The small complex icosidodecahedron resembles an icosahedron, because the great dodecahedron is completely contained inside the icosahedron.
Its two-dimensional analogue would be the compound of a regular pentagon, {5}, representing the icosahedron as the n-dimensional pentagonal polytope, and regular pentagram, {5/2}, as the n-dimensional star. These shapes would share vertices, similarly to how its 3D equivalent shares edges.
See also
Great complex icosidodecahedron
Small complex rhombicosidodecahedron
Complex rhombidodecadodecahedron
Great complex rhombicosidodecahedron
References
(Table 6, degenerate cases)
Polyhedra
Polyhedral compounds |
https://en.wikipedia.org/wiki/Great%20complex%20icosidodecahedron | In geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron. It has 12 vertices, and 60 (doubled) edges, and 32 faces, 12 pentagrams and 20 triangles. All edges are doubled (making it degenerate), sharing 4 faces, but are considered as two overlapping edges as topological polyhedron.
It can be constructed from a number of different vertex figures.
As a compound
The great complex icosidodecahedron can be considered a compound of the small stellated dodecahedron, {5/2,5}, and great icosahedron, {3,5/2}, sharing the same vertices and edges, while the second is hidden, being completely contained inside the first.
{|
|
See also
Small complex icosidodecahedron
Small complex rhombicosidodecahedron
Complex rhombidodecadodecahedron
Great complex rhombicosidodecahedron
References
(Table 6, degenerate cases)
Polyhedra
Polyhedral compounds |
https://en.wikipedia.org/wiki/Lim%20Keun-jae | Lim Keun-Jae (born November 5, 1969) is a South Korean football manager. He played for FC Seoul and Pohang Steelers then known as 'LG Cheetahs' and 'Pohang Atoms'.
Club career statistics
Honours
Player
LG Cheeths
K-League Cup Runners-up (1) : 1994
Manager
Seoul United
K3 League Winners (1) : 2007
Individual
K-League Top Scorer : 1992
External links
1969 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korea men's international footballers
FC Seoul players
Pohang Steelers players
K League 1 players |
https://en.wikipedia.org/wiki/Kang%20Deuk-soo | Kang Deuk-Soo (Korean: 강득수) (born on January 1, 1961) is a South Korean football player and manager.
Club career statistics
Honours
Club
Lucky-Goldstar Hwangso
K League (1) : 1985
Korean National Football Championship (1) : 1988
Individual
K League Best XI : 1985
K League Top Assists Award : 1986
References
Legends of K-League : 강득수 - 어시스트의 달인, 찬스메이커
External links
1961 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korea men's international footballers
FC Seoul players
Ulsan Hyundai FC players
K League 1 players
1986 FIFA World Cup players
Asian Games medalists in football
Footballers at the 1986 Asian Games
Asian Games gold medalists for South Korea
Medalists at the 1986 Asian Games |
https://en.wikipedia.org/wiki/Prince%20Rupert%27s%20cube | In geometry, Prince Rupert's cube is the largest cube that can pass through a hole cut through a unit cube without splitting it into two pieces. Its side length is approximately 1.06, 6% larger than the side length 1 of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution.
Prince Rupert's cube is named after Prince Rupert of the Rhine, who asked whether a cube could be passed through a hole made in another cube of the same size without splitting the cube into two pieces. A positive answer was given by John Wallis. Approximately 100 years later, Pieter Nieuwland found the largest possible cube that can pass through a hole in a unit cube.
Many other convex polyhedra, including all five Platonic solids, have been shown to have the Rupert property: a copy of the polyhedron, of the same or larger shape, can be passed through a hole in the polyhedron. It is unknown whether this is true for all convex polyhedra.
Solution
Place two points on two adjacent edges of a unit cube, each at a distance of 3/4 from the point where the two edges meet, and two more points symmetrically on the opposite face of the cube. Then these four points form a square with side length
One way to see this is to first observe that these four points form a rectangle, by the symmetries of their construction. The lengths of all four sides of this rectangle equal , by the Pythagorean theorem or (equivalently) the formula for Euclidean distance in three dimensions. For instance, the first two points, together with the third point where their two edges meet, form an isosceles right triangle with legs of length , and the distance between the first two points is the hypotenuse of the triangle. As a rectangle with four equal sides, the shape formed by these four points is a square. Extruding the square in both directions perpendicularly to itself forms the hole through which a cube larger than the original one, up to side length , may pass.
The parts of the unit cube that remain, after emptying this hole, form two triangular prisms and two irregular tetrahedra, connected by thin bridges at the four vertices of the square.
Each prism has as its six vertices two adjacent vertices of the cube, and four points along the edges of the cube at distance 1/4 from these cube vertices. Each tetrahedron has as its four vertices one vertex of the cube, two points at distance 3/4 from it on two of the adjacent edges, and one point at distance 3/16 from the cube vertex along the third adjacent edge.
History
Prince Rupert's cube is named after Prince Rupert of the Rhine. According to a story recounted in 1693 by English mathematician John Wallis, Prince Rupert wagered that a hole could be cut through a cube, large enough to let another cube of the same size pass through it. Wallis showed that in fact such a hole was possible (with some errors that were not corrected until much later), an |
https://en.wikipedia.org/wiki/Octagrammic%20antiprism | In geometry, the octagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two octagrams.
See also
Prismatic uniform polyhedron
Octagrammic crossed-antiprism
External links
Paper models of prisms and antiprisms
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/Octagrammic%20crossed-antiprism | In geometry, the octagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two octagrams.
See also
Prismatic uniform polyhedron
Octagrammic antiprism
External links
Paper models of prisms and antiprisms
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/Decagrammic%20antiprism | In geometry, the decagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two decagrams.
See also
Prismatic uniform polyhedron
External links
Paper models of prisms and antiprisms
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/Plane%20of%20rotation | In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space.
The main use for planes of rotation is in describing more complex rotations in four-dimensional space and higher dimensions, where they can be used to break down the rotations into simpler parts. This can be done using geometric algebra, with the planes of rotations associated with simple bivectors in the algebra.
Planes of rotation are not used much in two and three dimensions, as in two dimensions there is only one plane (so, identifying the plane of rotation is trivial and rarely done), while in three dimensions the axis of rotation serves the same purpose and is the more established approach.
Mathematically such planes can be described in a number of ways. They can be described in terms of planes and angles of rotation. They can be associated with bivectors from geometric algebra. They are related to the eigenvalues and eigenvectors of a rotation matrix. And in particular dimensions they are related to other algebraic and geometric properties, which can then be generalised to other dimensions.
Definitions
Plane
For this article, all planes are planes through the origin, that is they contain the zero vector. Such a plane in -dimensional space is a two-dimensional linear subspace of the space. It is completely specified by any two non-zero and non-parallel vectors that lie in the plane, that is by any two vectors and , such that
where is the exterior product from exterior algebra or geometric algebra (in three dimensions the cross product can be used). More precisely, the quantity is the bivector associated with the plane specified by and , and has magnitude , where is the angle between the vectors; hence the requirement that the vectors be nonzero and nonparallel.
If the bivector is written , then the condition that a point lies on the plane associated with is simply
This is true in all dimensions, and can be taken as the definition on the plane. In particular, from the properties of the exterior product it is satisfied by both and , and so by any vector of the form
with and real numbers. As and range over all real numbers, ranges over the whole plane, so this can be taken as another definition of the plane.
Plane of rotation
A plane of rotation for a particular rotation is a plane that is mapped to itself by the rotation. The plane is not fixed, but all vectors in the plane are mapped to other vectors in the same plane by the rotation. This transformation of the plane to itself is always a rotation about the origin, through an angle which is the angle of rotation for the plane.
Every rotation except for the identity rotation (with matrix the identity matrix) has at least one plane of rotation, and up to
planes of rotation, where is the dimension. The maximum number of planes up to eight dimensions is shown in this table:
{| class="wikitable" border="1"
! Dimension
| 2 || 3 || 4 || 5 || 6 || 7 || 8
|-
|
https://en.wikipedia.org/wiki/Fuzzy%20retrieval | Fuzzy retrieval techniques are based on the Extended Boolean model and the Fuzzy set theory. There are two classical fuzzy retrieval models: Mixed Min and Max (MMM) and the Paice model. Both models do not provide a way of evaluating query weights, however this is considered by the P-norms algorithm.
Mixed Min and Max model (MMM)
In fuzzy-set theory, an element has a varying degree of membership, say dA, to a given set A instead of the traditional membership choice (is an element/is not an element).
In MMM each index term has a fuzzy set associated with it. A document's weight with respect to an index term A is considered to be the degree of membership of the document in the fuzzy set associated with A. The degree of membership for union and intersection are defined as follows in Fuzzy set theory:
According to this, documents that should be retrieved for a query of the form A or B, should be in the fuzzy set associated with the union of the two sets A and B. Similarly, the documents that should be retrieved for a query of the form A and B, should be in the fuzzy set associated with the intersection of the two sets. Hence, it is possible to define the similarity of a document to the or query to be max(dA, dB) and the similarity of the document to the and query to be min(dA, dB). The MMM model tries to soften the Boolean operators by considering the query-document similarity to be a linear combination of the min and max document weights.
Given a document D with index-term weights dA1, dA2, ..., dAn for terms A1, A2, ..., An, and the queries:
Qor = (A1 or A2 or ... or An)
Qand = (A1 and A2 and ... and An)
the query-document similarity in the MMM model is computed as follows:
SlM(Qor, D) = Cor1 * max(dA1, dA2, ..., dAn) + Cor2 * min(dA1, dA2, ..., dAn)
SlM(Qand, D) = Cand1 * min(dA1, dA2, ..., dAn) + Cand2 * max(dA1, dA2 ..., dAn)
where Cor1, Cor2 are "softness" coefficients for the or operator, and Cand1, Cand2 are softness coefficients for the and operator. Since we would like to give the maximum of the document weights more importance while considering an or query and the minimum more importance while considering an and query, generally we have Cor1 > Cor2 and Cand1 > Cand2. For simplicity it is generally assumed that Cor1 = 1 - Cor2 and Cand1 = 1 - Cand2.
Lee and Fox experiments indicate that the best performance usually occurs with Cand1 in the range [0.5, 0.8] and with Cor1 > 0.2. In general, the computational cost of MMM is low, and retrieval effectiveness is much better than with the Standard Boolean model.
Paice model
The Paice model is a general extension to the MMM model. In comparison to the MMM model that considers only the minimum and maximum weights for the index terms, the Paice model incorporates all of the term weights when calculating the similarity:
where r is a constant coefficient and wdi is arranged in ascending order for and queries and descending order for or queries. When n = 2 the Paice model shows the same behav |
https://en.wikipedia.org/wiki/Census%20and%20Statistics%20Department%20%28Hong%20Kong%29 | The Census and Statistics Department (C&SD; ) is the provider of major social and economic official statistics in Hong Kong. It is also responsible for conducting Population Census and By-census in Hong Kong since 1971. Its head office is in the Wanchai Tower in Wan Chai.
Antecedent
The history of population censuses in Hong Kong can be traced back to the 1840s. According to early government records, the first set of census results were published in the 2nd issue of H.K. Govt. Gazette (1841 May). Regular population censuses have been taken ever since, except for the main gap between 1931 and 1961. In addition to population censuses, other statistics like number of ships entered, trade tonnage, public revenue and expenditure, death rate for European and American residents, number of schools, school attendance, number of prisoners and police strength were collected through various government departments in a scattered fashion.
In 1947, a Department of Statistics was set up under W. G. Wormal to organize a statistical system, working on such matters as retail price index and trade statistics. With the abortion of the idea of a population census scheduled for 1948 due to great fluctuations in the population in those few years, the Department of Statistics was disbanded in 1952. In its place a Statistics Branch was set up in the then Commerce and Industry Department headed by C. T. Stratton. Its work mainly concentrated on economic statistics, in particular trade statistics.
In 1959, following the decision to hold a population census in 1961, a temporary Census Department was set up with K. M. A. Barnett as Commissioner. It was disbanded in 1962 after the completion of the census operation. In 1963, Barnett was appointed Commissioner of Census and Statistical Planning in an office forming part of the then Colonial Secretariat with the immediate task of preparing a report on the statistics in Hong Kong and subsequently the further task of conducting a by-census in 1966. It was only following recommendations made by Barnett that the Census and Statistics Department was formally established in 1967 December.
Major historical events
Organisation and Management
Government Statistical Service
C&SD together with statistical units established in various government departments and bureaux form the Government Statistical Service (GSS). The latter are generally called the "outposted statistical units".
Broadly speaking, most general-purpose statistics come under the responsibility of C&SD. The statistical units in various government departments and bureaux will take care of specific-purpose statistics (for dedicated use in their respective work) and provide necessary support in the application of statistics.
The Commissioner for Census and Statistics is the Government's principal adviser on all statistical matters and the head of the GSS. On the one hand, he fulfills his responsibilities by being the head of C&SD and, on the other hand, co-ordinates t |
https://en.wikipedia.org/wiki/Aleksandr%20Kirov | Aleksandr Kirov (; born 4 September 1984) is a retired Kazakh footballer who primarily played left back.
Career statistics
International
Statistics accurate as of match played 4 June 2013
References
External links
Living people
1984 births
Kazakhstani men's footballers
Men's association football defenders
Kazakhstan men's under-21 international footballers
Kazakhstan men's international footballers
Kazakhstan Premier League players
FC Astana players
FC Aktobe players
FC Shakhter Karagandy players
FC Zhetysu players
FC Taraz players
Sportspeople from Astana |
https://en.wikipedia.org/wiki/Sankara%20Variar | Shankara Variyar (; ) was an astronomer-mathematician of the Kerala school of astronomy and mathematics. His family were employed as temple-assistants in the temple at near modern Ottapalam.
Mathematical lineage
He was taught mainly by Nilakantha Somayaji (1444–1544), the author of the Tantrasamgraha and Jyesthadeva (1500–1575), the author of Yuktibhāṣā. Other teachers of Shankara include Netranarayana, the patron of Nilakantha Somayaji and Chitrabhanu, the author of an astronomical treaties dated to 1530 and a small work with solutions and proofs for algebraic equations.
Works
The known works of Shankara Variyar are the following:
Yukti-dipika - an extensive commentary in verse on Tantrasamgraha based on Yuktibhāṣā.
Laghu-vivrti - a short commentary in prose on Tantrasamgraha.
Kriya-kramakari - a lengthy prose commentary on Lilavati of Bhaskara II.
An astronomical commentary dated 1529 CE.
An astronomical handbook completed around 1554 CE.
See also
List of astronomers and mathematicians of the Kerala school
References
K. V. Sarma (1997), "Sankara Variar", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures edited by Helaine Selin, Springer, </ref>
Indian Hindus
Kerala school of astronomy and mathematics
Year of death unknown
Year of birth uncertain
16th-century Indian mathematicians
People from Ottapalam
Scientists from Kerala
16th-century Indian astronomers |
https://en.wikipedia.org/wiki/Serre%27s%20conjecture%20II%20%28algebra%29 | In mathematics, Jean-Pierre Serre conjectured the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group. Namely, he conjectured that if G is such a group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H1(F, G) is zero.
A converse of the conjecture holds: if the field F is perfect and if the cohomology set H1(F, G) is zero for every semisimple simply connected algebraic group G then the p-cohomological dimension of F is at most 2 for every prime p.
The conjecture holds in the case where F is a local field (such as p-adic field) or a global field with no real embeddings (such as Q()). This is a special case of the Kneser–Harder–Chernousov Hasse principle for algebraic groups over global fields. (Note that such fields do indeed have cohomological dimension at most 2.)
The conjecture also holds when F is finitely generated over the complex numbers and has transcendence degree at most 2.
The conjecture is also known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Suslin theorem. Building on this result, the conjecture holds if G is a classical group. The conjecture also holds if G is one of certain kinds of exceptional group.
References
External links
Philippe Gille's survey of the conjecture
Field (mathematics)
Algebraic number theory
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Pentellated%206-simplexes | In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.
There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-simplex is also called an expanded 6-simplex, constructed by an expansion operation applied to the regular 6-simplex. The highest form, the pentisteriruncicantitruncated 6-simplex, is called an omnitruncated 6-simplex with all of the nodes ringed.
Pentellated 6-simplex
Alternate names
Expanded 6-simplex
Small terated tetradecapeton (Acronym: staf) (Jonathan Bowers)
Coordinates
The vertices of the pentellated 6-simplex can be positioned in 7-space as permutations of (0,1,1,1,1,1,2). This construction is based on facets of the pentellated 7-orthoplex.
A second construction in 7-space, from the center of a rectified 7-orthoplex is given by coordinate permutations of:
(1,-1,0,0,0,0,0)
Root vectors
Its 42 vertices represent the root vectors of the simple Lie group A6. It is the vertex figure of the 6-simplex honeycomb.
Images
Configuration
This configuration matrix represents the expanded 6-simplex, with 12 permutations of elements. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole polytope. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Pentitruncated 6-simplex
Alternate names
Teracellated heptapeton (Acronym: tocal) (Jonathan Bowers)
Coordinates
The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.
Images
Penticantellated 6-simplex
Alternate names
Teriprismated heptapeton (Acronym: topal) (Jonathan Bowers)
Coordinates
The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the penticantellated 7-orthoplex.
Images
Penticantitruncated 6-simplex
Alternate names
Terigreatorhombated heptapeton (Acronym: togral) (Jonathan Bowers)
Coordinates
The vertices of the penticantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 7-orthoplex.
Images
Pentiruncitruncated 6-simplex
Alternate names
Tericellirhombated heptapeton (Acronym: tocral) (Jonathan Bowers)
Coordinates
The vertices of the pentiruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the pentiruncitruncated 7-orthoplex.
Images
Pentiruncicantellated 6-simplex
Alternate names
Teriprismatorhombated tetradecapeton (Acronym: taporf) (Jonathan Bowers)
Coordinates
The |
https://en.wikipedia.org/wiki/Conformal%20radius | In mathematics, the conformal radius is a way to measure the size of a simply connected planar domain D viewed from a point z in it. As opposed to notions using Euclidean distance (say, the radius of the largest inscribed disk with center z), this notion is well-suited to use in complex analysis, in particular in conformal maps and conformal geometry.
A closely related notion is the transfinite diameter or (logarithmic) capacity of a compact simply connected set D, which can be considered as the inverse of the conformal radius of the complement E = Dc viewed from infinity.
Definition
Given a simply connected domain D ⊂ C, and a point z ∈ D, by the Riemann mapping theorem there exists a unique conformal map f : D → D onto the unit disk (usually referred to as the uniformizing map) with f(z) = 0 ∈ D and f′(z) ∈ R+. The conformal radius of D from z is then defined as
The simplest example is that the conformal radius of the disk of radius r viewed from its center is also r, shown by the uniformizing map x ↦ x/r. See below for more examples.
One reason for the usefulness of this notion is that it behaves well under conformal maps: if φ : D → D′ is a conformal bijection and z in D, then .
The conformal radius can also be expressed as where is the harmonic extension of from to .
A special case: the upper-half plane
Let K ⊂ H be a subset of the upper half-plane such that D := H\K is connected and simply connected, and let z ∈ D be a point. (This is a usual scenario, say, in the Schramm-Loewner evolution). By the Riemann mapping theorem, there is a conformal bijection g : D → H. Then, for any such map g, a simple computation gives that
For example, when K = ∅ and z = i, then g can be the identity map, and we get rad(i, H) = 2. Checking that this agrees with the original definition: the uniformizing map f : H → D is
and then the derivative can be easily calculated.
Relation to inradius
That it is a good measure of radius is shown by the following immediate consequence of the Schwarz lemma and the Koebe 1/4 theorem: for z ∈ D ⊂ C,
where dist(z, ∂D) denotes the Euclidean distance between z and the boundary of D, or in other words, the radius of the largest inscribed disk with center z.
Both inequalities are best possible:
The upper bound is clearly attained by taking D = D and z = 0.
The lower bound is attained by the following “slit domain”: D = C\R+ and z = −r ∈ R−. The square root map φ takes D onto the upper half-plane H, with and derivative . The above formula for the upper half-plane gives , and then the formula for transformation under conformal maps gives rad(−r, D) = 4r, while, of course, dist(−r, ∂D) = r.
Version from infinity: transfinite diameter and logarithmic capacity
When D ⊂ C is a connected, simply connected compact set, then its complement E = Dc is a connected, simply connected domain in the Riemann sphere that contains ∞, and one can define
where f : C\D → E is the unique bijective conformal map with f(∞) = |
https://en.wikipedia.org/wiki/Bivariate%20von%20Mises%20distribution | In probability theory and statistics, the bivariate von Mises distribution is a probability distribution describing values on a torus. It may be thought of as an analogue on the torus of the bivariate normal distribution. The distribution belongs to the field of directional statistics. The general bivariate von Mises distribution was first proposed by Kanti Mardia in 1975. One of its variants is today used in the field of bioinformatics to formulate a probabilistic model of protein structure in atomic detail, such as backbone-dependent rotamer libraries.
Definition
The bivariate von Mises distribution is a probability distribution defined on the torus, in .
The probability density function of the general bivariate von Mises distribution for the angles is given by
where and are the means for and , and their concentration and the matrix is related to their correlation.
Two commonly used variants of the bivariate von Mises distribution are the sine and cosine variant.
The cosine variant of the bivariate von Mises distribution has the probability density function
where and are the means for and , and their concentration and is related to their correlation. is the normalization constant. This distribution with =0 has been used for kernel density estimates of the distribution of the protein dihedral angles and .
The sine variant has the probability density function
where the parameters have the same interpretation.
See also
Von Mises distribution, a similar distribution on the one-dimensional unit circle
Kent distribution, a related distribution on the two-dimensional unit sphere
von Mises–Fisher distribution
Directional statistics
References
Continuous distributions
Directional statistics
Multivariate continuous distributions |
https://en.wikipedia.org/wiki/Sankara%20Varman | Sankara Varman (1774–1839) was an astronomer-mathematician belonging to the Kerala school of astronomy and mathematics. He is best known as the author of Sadratnamala, a treatise on astronomy and mathematics, composed in 1819. Sankara Varman is considered as the last notable figure in the long line of illustrious astronomers and mathematicians in the Kerala school of astronomy and mathematics beginning with Madhava of Sangamagrama. Sadratnamala was composed in the traditional style followed by members of the Kerala school at a time when India had been introduced to the western style of mathematics and of writing books in mathematics. One of Varman's contribution to mathematics was his computation of the value of the mathematical constant π correct to 17 decimal places.
Biographical sketch
Varman was born as a younger prince in the principality of Katattanad, in North Malabar, Kerala, in the year 1774. He had two elder brothers, the eldest being Raja Udaya varma, the ruler of the principality, and second one being Rama Varma, the crown prince. Local people referred to Varman as Appu Thampuran. Katattanad principality was under the suzerainty of the Zamorin of Calicut. The family deity of Sankara Varman was Goddess Parvati installed in the temple of Lokamalayarkavu and he was also a devotee of Lord Krishna.
Varman completed the composition of Sadratnamala in 1819. He also wrote a Malayalam commentary on Sadratnamala. Two versions of the text of Sadratnamala and their commentaries have been discovered. A critical examination of the manuscripts suggests that both are written by Varman, one being a revision of the other.
During the invasion of Malabar by Tipu Sultan/Hyder Ali during 1766 - 1781, many people including members of royal families fled Malabar and took shelter in Travancore. This brought the principality of Katattanad also into close contact with the Travancore Royal Family. Varman was especially attached to Maharaja Swati Tirunal (1813–47).
Varman was an astute astrologer. There is a legend that he predicted the time of his own death as being in 1839, the year in which he died.
Whish's acquaintance
Varman was a close acquaintance of C.M. Whish a civil servant of East India Company attached to its Madras establishment. Whish spoke of him and his work thus: "The author of Sadratnamalah is SANCARA VARMA, the younger brother of the present Raja of Cadattanada near Tellicherry, a very intelligent man and acute mathematician. This work, which is a complete system of Hindu astronomy, is comprehended in two hundred and eleven verses of different measures, and abounds with fluxional forms and series, to be found in no work of foreign or other Indian countries." Whish was the first to bring to the notice of the western mathematical scholarship the achievements of the Kerala school of astronomy and mathematics.
See also
List of astronomers and mathematicians of the Kerala school
References
Kerala school of astronomy and mathematics
Hindu |
https://en.wikipedia.org/wiki/Lifting%20theory | In mathematics, lifting theory was first introduced by John von Neumann in a pioneering paper from 1931, in which he answered a question raised by Alfréd Haar. The theory was further developed by Dorothy Maharam (1958) and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961). Lifting theory was motivated to a large extent by its striking applications. Its development up to 1969 was described in a monograph of the Ionescu Tulceas. Lifting theory continued to develop since then, yielding new results and applications.
Definitions
A lifting on a measure space is a linear and multiplicative operator
which is a right inverse of the quotient map
where is the seminormed Lp space of measurable functions and is its usual normed quotient. In other words, a lifting picks from every equivalence class of bounded measurable functions modulo negligible functions a representative— which is henceforth written or or simply — in such a way that and for all and all
Liftings are used to produce disintegrations of measures, for instance conditional probability distributions given continuous random variables, and fibrations of Lebesgue measure on the level sets of a function.
Existence of liftings
Theorem. Suppose is complete. Then admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in whose union is
In particular, if is the completion of a σ-finite measure or of an inner regular Borel measure on a locally compact space, then admits a lifting.
The proof consists in extending a lifting to ever larger sub-σ-algebras, applying Doob's martingale convergence theorem if one encounters a countable chain in the process.
Strong liftings
Suppose is complete and is equipped with a completely regular Hausdorff topology such that the union of any collection of negligible open sets is again negligible – this is the case if is σ-finite or comes from a Radon measure. Then the support of can be defined as the complement of the largest negligible open subset, and the collection of bounded continuous functions belongs to
A strong lifting for is a lifting
such that on for all in This is the same as requiring that for all open sets in
Theorem. If is σ-finite and complete and has a countable basis then admits a strong lifting.
Proof. Let be a lifting for and a countable basis for For any point in the negligible set
let be any character on that extends the character of Then for in and in define:
is the desired strong lifting.
Application: disintegration of a measure
Suppose and are σ-finite measure spaces ( positive) and is a measurable map. A disintegration of along with respect to is a slew of positive σ-additive measures on such that
is carried by the fiber of over , i.e. and for almost all
for every -integrable function in the sense that, for -almost all in is -integrable, the function is -integrable, and the displayed equality holds.
Disintegrati |
https://en.wikipedia.org/wiki/Crime%20statistics%20in%20the%20United%20Kingdom | Crime statistics in the United Kingdom refers to the data collected in the United Kingdom, and that collected by the individual areas, England and Wales, Scotland and Northern Ireland, which operate separate judicial systems. It covers data related to crime in the United Kingdom. As with crime statistics elsewhere, they are broadly divided into victim studies and police statistics. More recently, third-party reporting is used to quantify specific under-reported issues, for example, hate crime.
Crime surveys
The Crime Survey for England and Wales is an attempt to measure both the amount of crime, and the impact of crime on England and Wales. The original survey (carried out in 1982, to cover the 1981 year) covered all three judicial areas of the UK, and was therefore referred to as the British Crime Survey, but now it only covers England and Wales. In Scotland and Northern Ireland, similar surveys, namely the Scottish Crime and Victimisation Survey and Northern Ireland Crime Survey have similar purposes. These surveys collect information about the victims of crime, the circumstances surrounding the crime, and the behaviour of the perpetrators. They are used to plan, and measure the results of, crime reduction or perception measures. In addition, they collect data about the perception of issues such as antisocial behaviour and the criminal justice system.
Other crime surveys include the Commercial Victimisation Survey, which covers small and medium-sized businesses, and the Offending, Crime and Justice Survey, with a particular focus on young people.
Background and counting rules
Until the late 1990s crime figures for varying crime types were not released to the general public at individual police force level. The annual publication 'Crime in England & Wales' produced by the Home Office began to break the figures down to a smaller area in 1996. Crime figures in England & Wales during the late 1990s and early 2000s were often misinterpreted in the media and scrutinised because of frequent changes in the way crimes were counted and recorded that lead to rises in the crime category 'Violence Against the Person'.
Commenting on figures from 1 April 1998 onwards, the then-Home Secretary Jack Straw said "changes in the way crime statistics are compiled are in line with recommendations by senior police officers. They are intended to give a more accurate picture of the level of offences". The largest increases were recorded in the "Violence Against the Person" category owing to the inclusion of common assault figures to accompany other offence types within this category that include assault occasioning actual bodily harm, grievous bodily harm, harassment, murder, possession of offensive weapons and a selection of other low volume violent crimes grouped together by the Metropolitan Police as 'other violence'.
The change in counting rules, and the significant impact it had on violence against the person figures, was often misconstrued by the media as r |
https://en.wikipedia.org/wiki/1964%E2%80%9365%20Cincinnati%20Royals%20season | The 1964–65 season was the Royals' 19th season in the NBA and eighth in Cincinnati. By the end of the season, Oscar Robertson's career statistics for the first five years of his career averaged out to a triple double: 30.3 points per game, 10.4 rebounds per game, and 10.6 assists per game.
The season began with high hopes as the Royals had played well the previous season against Boston and were improving as a team. In addition to Robertson, second-year big man Jerry Lucas rose to superstar status this season. He averaged 21 points and 20 rebounds over 66 games played. He joined Robertson on the All-NBA First Team named at the season's conclusion.
Injuries, though, were a big factor this season. Key guard Arlen Bockhorn was lost to a career-ending injury in November. The other four opening-day starters, Robertson, Lucas, Jack Twyman and Wayne Embry, were each lost for several games or more also.
Lucas was named MVP of the 1965 NBA All-Star Game. But the same day's events saw superstar Wilt Chamberlain traded to the rival Philadelphia 76ers. Now Cincinnati had two strong title contenders to deal with in their own division. Philadelphia later defeated the Royals in the 1965 playoffs.
Draft picks
Roster
Regular season
Season standings
Record vs. opponents
Game log
Playoffs
|- align="center" bgcolor="#ffcccc"
| 1
| March 24
| Philadelphia
| L 117–119 (OT)
| Jack Twyman (25)
| Jerry Lucas (27)
| Oscar Robertson (13)
| Cincinnati Gardens6,422
| 0–1
|- align="center" bgcolor="#ccffcc"
| 2
| March 26
| @ Philadelphia
| W 121–120
| Oscar Robertson (40)
| Jerry Lucas (21)
| Oscar Robertson (13)
| Municipal Auditorium5,801
| 1–1
|- align="center" bgcolor="#ffcccc"
| 3
| March 28
| Philadelphia
| L 94–108
| Oscar Robertson (27)
| Jerry Lucas (17)
| Oscar Robertson (12)
| Cincinnati Gardens6,289
| 1–2
|- align="center" bgcolor="#ffcccc"
| 4
| March 31
| @ Philadelphia
| L 112–119
| Jerry Lucas (35)
| Jerry Lucas (19)
| Oscar Robertson (10)
| Municipal Auditorium7,451
| 1–3
|-
Player Statistics
Regular season
Playoffs
Awards and honors
Oscar Robertson, All-NBA First Team
Jerry Lucas, All-NBA First Team
References
External links
1964–65 Royals on Basketball Reference
Sacramento Kings seasons
Cincinnati
Cincinnati
Cincinnati |
https://en.wikipedia.org/wiki/William%20Vernon%20Skiles | William Vernon Skiles (April 23, 1879 in Troy Grove, Illinois - September 10, 1947 in Atlanta, Georgia) was a professor of mathematics and dean at the Georgia Institute of Technology. He helped create what is now the Georgia Tech Research Institute.
Education
Skiles possessed a Bachelor of Science degree from the University of Chicago, a Master of Arts degree from Harvard University and an honorary Doctor of Science degree from the University of Georgia. He was a member of Beta Theta Pi, Phi Beta Kappa, Phi Kappa Phi, and the Georgia Academy of Science.
Georgia Tech
After Skiles' retirement in December 1945, the faculty were requested to donate to a fund to "give him a first class dinner and a gift." So much was donated that the remainder was put into the Dean Skiles Fund, which "provides the faculty expenses for flowers, gifts, entertainment" that could not be paid for with funds from the State of Georgia.
Legacy
The Skiles Classroom Building has been home to the Department of Mathematics since 1958; its previous home was the second Shop Building. The nearby Skiles Walkway is the primary east-west pedestrian corridor through campus, connecting the Georgia Tech Library to the Georgia Tech Student Center.
See also
History of Georgia Tech
References
Georgia Tech faculty
1879 births
1947 deaths
University of Chicago alumni
Harvard University alumni |
https://en.wikipedia.org/wiki/List%20of%20Shamrock%20Rovers%20F.C.%20records%20and%20statistics | Shamrock Rovers Football Club are a football club from Dublin, Ireland. They compete in the League of Ireland and are the most successful club in the history of football in the Republic of Ireland, having won 21 League of Ireland titles and 25 FAI Cups.
They have also won the League of Ireland Shield on 18 occasions and the League of Ireland Cup once. Shamrock Rovers have supplied more players to the Republic of Ireland national football team (64) than any other single club. This list comprises the major honours won by Shamrock Rovers and the records set by the players and managers of the club.
Honours
National titles
League of Ireland: 21 (record)
1922–23, 1924–25, 1926–27, 1931–32, 1937–38, 1938–39, 1953–54, 1956–57, 1958–59, 1963–64 1983–84, 1984–85, 1985–86, 1986–87, 1993–94, 2010, 2011, 2020, 2021, 2022 league of Ireland premier division 2023
FAI Cup: 25 (record)
1925, 1929, 1930, 1931, 1932, 1933, 1936, 1940, 1944, 1945, 1948, 1955, 1956, 1962, 1964, 1965, 1966, 1967, 1968, 1969, 1978, 1985, 1986, 1987, 2019
League of Ireland Shield: 18
1924–25, 1926–27, 1931–32, 1932–33, 1934–35, 1937–38, 1941–42, 1949–50, 1951–52, 1954–55, 1955–56, 1956–57, 1957–58, 1962–63, 1963–64, 1964–65, 1965–66, 1967–68.
League of Ireland Cup: 2
1976–77
2013
League of Ireland First Division: 1
2006
Setanta Sports Cup: 2
2011 Setanta Sports Cup
2013 Setanta Sports Cup
Regional titles
Leinster Senior Cup: 18
1923, 1927, 1929, 1930, 1933, 1938, 1953, 1955, 1956, 1957, 1958, 1964, 1969, 1982, 1985, 1997, 2012, 2013
Complete list of honours
European record
Shamrock Rovers have a long history in European competition. They were the first League of Ireland side to enter European competition, and featured regularly in the 1960s and 1980s. The club has had some relative success with victories in the Intertoto-Cup and the Europa League. Throughout their participation Rovers have beaten teams from Luxembourg, Cyprus, Iceland and Germany, and were the first Irish club to beat teams from Turkey, Poland, Israel, Serbia Slovakia, Albania and Hungary. Their first victory in the UEFA Champions League came in a 1–0 victory in the 2011–12 qualifying phase against FC Flora Tallinn at Tallaght Stadium .
Their biggest win was a 7–0 aggregate victory (3–0 away, 4–0 home) over Fram Reykjavik in the UEFA Cup first round in September 1982, which remains a record for League of Ireland clubs in European competition.
On 25 August 2011, they became the first Irish team to qualify for the UEFA Europa League group stage when they defeated Partizan Belgrade 2–1 after extra-time in Serbia, for a 3–2 aggregate victory.
Rovers qualified for the group stages of the 2022–23 UEFA Europa Conference League
Some of their more notable European performances include:
a 2–2 draw and 1–0 defeat to defending champions and finalists, Valencia CF in Inter-Cities Fairs Cup 1963–64
a 1–1 draw and 2–1 defeat to finalists, Real Zaragoza in the Inter-Cities Fairs Cup 1965–66
a 1–1 |
https://en.wikipedia.org/wiki/Rectified%205-orthoplexes | In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.
There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex.
Rectified 5-orthoplex
Its 40 vertices represent the root vectors of the simple Lie group D5. The vertices can be seen in 3 hyperplanes, with the 10 vertices rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex, these vertices represent the 50 root vectors of the B5 and C5 simple Lie groups.
E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr51 as a first rectification of a 5-dimensional cross polytope.
Alternate names
rectified pentacross
rectified triacontiditeron (32-faceted 5-polytope)
Construction
There are two Coxeter groups associated with the rectified pentacross, one with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 16-cell facets, alternating, with the D5 or [32,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified pentacross, centered at the origin, edge length are all permutations of:
(±1,±1,0,0,0)
Images
Related polytopes
The rectified 5-orthoplex is the vertex figure for the 5-demicube honeycomb:
or
This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
o3x3o3o4o - rat
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/Rectified%207-orthoplexes | In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.
There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex.
Rectified 7-orthoplex
The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb.
or
Alternate names
rectified heptacross
rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon
Images
Construction
There are two Coxeter groups associated with the rectified heptacross, one with the C7 or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D7 or [34,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified heptacross, centered at the origin, edge length are all permutations of:
(±1,±1,0,0,0,0,0)
Root vectors
Its 84 vertices represent the root vectors of the simple Lie group D7. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B7 and C7 simple Lie groups.
Birectified 7-orthoplex
Alternate names
Birectified heptacross
Birectified hecatonicosoctaexon (Acronym barz) (Jonathan Bowers) - birectified 128-faceted polyexon
Images
Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length are all permutations of:
(±1,±1,±1,0,0,0,0)
Trirectified 7-orthoplex
A trirectified 7-orthoplex is the same as a trirectified 7-cube.
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
o3x3o3o3o3o4o - rez, o3o3x3o3o3o4o - barz
External links
Polytopes of Various Dimensions
Mu |
https://en.wikipedia.org/wiki/Rectified%208-orthoplexes | In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.
Rectified 8-orthoplex
The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups.
Related polytopes
The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.
or
Alternate names
rectified octacross
rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)
Construction
There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or [4,36] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or [35,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length are all permutations of:
(±1,±1,0,0,0,0,0,0)
Images
Birectified 8-orthoplex
Alternate names
birectified octacross
birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)
Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length are all permutations of:
(±1,±1,±1,0,0,0,0,0)
Images
Trirectified 8-orthoplex
The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.
Alternate names
trirectified octacross
trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)
Cartesian coordinates
Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length are all permutations of:
(±1,±1,±1,±1,0,0,0,0)
Images
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Po |
https://en.wikipedia.org/wiki/Focal%20subgroup%20theorem | In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and fusion such as described in . Various applications of these ideas include local criteria for p-nilpotence and various non-simplicity criteria focussing on showing that a finite group has a normal subgroup of index p.
Background
The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index a power of p, the transfer homomorphism, and fusion of elements.
Subgroups
The following three normal subgroups of index a power of p are naturally defined, and arise as the smallest normal subgroups such that the quotient is (a certain kind of) p-group. Formally, they are kernels of the reflection onto the reflective subcategory of p-groups (respectively, elementary abelian p-groups, abelian p-groups).
Ep(G) is the intersection of all index p normal subgroups; G/Ep(G) is an elementary abelian group, and is the largest elementary abelian p-group onto which G surjects.
Ap(G) (notation from ) is the intersection of all normal subgroups K such that G/K is an abelian p-group (i.e., K is an index normal subgroup that contains the derived group ): G/Ap(G) is the largest abelian p-group (not necessarily elementary) onto which G surjects.
Op(G) is the intersection of all normal subgroups K of G such that G/K is a (possibly non-abelian) p-group (i.e., K is an index normal subgroup): G/Op(G) is the largest p-group (not necessarily abelian) onto which G surjects. Op(G) is also known as the p-residual subgroup.
Firstly, as these are weaker conditions on the groups K, one obtains the containments These are further related as:
Ap(G) = Op(G)[G,G].
Op(G) has the following alternative characterization as the subgroup generated by all Sylow q-subgroups of G as q≠p ranges over the prime divisors of the order of G distinct from p.
Op(G) is used to define the lower p-series of G, similarly to the upper p-series described in p-core.
Transfer homomorphism
The transfer homomorphism is a homomorphism that can be defined from any group G to the abelian group H/[H,H] defined by a subgroup H ≤ G of finite index, that is [G:H] < ∞. The transfer map from a finite group G into its Sylow p-subgroup has a kernel that is easy to describe:
The kernel of the transfer homomorphism from a finite group G into its Sylow p-subgroup P has Ap(G) as its kernel, .
In other words, the "obvious" homomorphism onto an abelian p-group is in fact the most general such homomorphism.
Fusion
The fusion pattern of a subgroup H in G is the equivalence relation on the elements of H where two elements h, k of H are fused if they are G-conjugate, that is, if there is some g in G such that h = kg. The normal structure of G has an effect on the fusion patte |
https://en.wikipedia.org/wiki/Rectified%206-orthoplexes | In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.
There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the 6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.
Rectified 6-orthoplex
The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb.
or
Alternate names
rectified hexacross
rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)
Construction
There are two Coxeter groups associated with the rectified hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or [33,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of:
(±1,±1,0,0,0,0)
Images
Root vectors
The 60 vertices represent the root vectors of the simple Lie group D6. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplices cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B6 and C6 simple Lie groups.
The 60 roots of D6 can be geometrically folded into H3 (Icosahedral symmetry), as to , creating 2 copies of 30-vertex icosidodecahedra, with the Golden ratio between their radii:
Birectified 6-orthoplex
The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb.
Alternate names
birectified hexacross
birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of:
(±1,±1,±1,0,0,0)
Images
It can also be projected into 3D-dimensions as --> , a dodecahedron envelope.
Related polytopes
These polytopes are a part a family of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Hon |
https://en.wikipedia.org/wiki/Resolvent%20cubic | In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four:
In each case:
The coefficients of the resolvent cubic can be obtained from the coefficients of using only sums, subtractions and multiplications.
Knowing the roots of the resolvent cubic of is useful for finding the roots of itself. Hence the name “resolvent cubic”.
The polynomial has a multiple root if and only if its resolvent cubic has a multiple root.
Definitions
Suppose that the coefficients of belong to a field whose characteristic is different from . In other words, we are working in a field in which . Whenever roots of are mentioned, they belong to some extension of such that factors into linear factors in . If is the field of rational numbers, then can be the field of complex numbers or the field of algebraic numbers.
In some cases, the concept of resolvent cubic is defined only when is a quartic in depressed form—that is, when .
Note that the fourth and fifth definitions below also make sense and that the relationship between these resolvent cubics and are still valid if the characteristic of is equal to .
First definition
Suppose that is a depressed quartic—that is, that . A possible definition of the resolvent cubic of is:
The origin of this definition lies in applying Ferrari's method to find the roots of . To be more precise:
Add a new unknown, , to . Now you have:
If this expression is a square, it can only be the square of
But the equality
is equivalent to
and this is the same thing as the assertion that = 0.
If is a root of , then it is a consequence of the computations made above that the roots of are the roots of the polynomial
together with the roots of the polynomial
Of course, this makes no sense if , but since the constant term of is , is a root of if and only if , and in this case the roots of can be found using the quadratic formula.
Second definition
Another possible definition (still supposing that is a depressed quartic) is
The origin of this definition is similar to the previous one. This time, we start by doing:
and a computation similar to the previous one shows that this last expression is a square if and only if
A simple computation shows that
Third definition
Another possible definition (again, supposing that is a depressed quartic) is
The origin of this definition lies in another method of solving quartic equations, namely Descartes' method. If you try to find the roots of by expressing it as a product of two monic quadratic polynomials and , then
If there is a solution of this system with (note that if , then this is automatically true for any solution), the previous system is equivalent to
It is a consequence of the first two equations that then
and
After replacing, in the third equation, and by these values one gets that
and this is equivalent to the assertion that is a root of . So, again, knowing the roots |
https://en.wikipedia.org/wiki/Multiplicity-one%20theorem | In the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group. The multiplicity in question is the number of times a given abstract group representation is realised in a certain space, of square-integrable functions, given in a concrete way.
A multiplicity one theorem may also refer to a result about the restriction of a representation of a group G to a subgroup H. In that context, the pair (G, H) is called a strong Gelfand pair.
Definition
Let G be a reductive algebraic group over a number field K and let A denote the adeles of K. Let Z denote the centre of G and let be a continuous unitary character from Z(K)\Z(A)× to C×. Let L20(G(K)/G(A), ) denote the space of cusp forms with central character ω on G(A). This space decomposes into a direct sum of Hilbert spaces
where the sum is over irreducible subrepresentations and m are non-negative integers.
The group of adelic points of G, G(A), is said to satisfy the multiplicity-one property if any smooth irreducible admissible representation of G(A) occurs with multiplicity at most one in the space of cusp forms of central character , i.e. m is 0 or 1 for all such .
Results
The fact that the general linear group, GL(n), has the multiplicity-one property was proved by for n = 2 and independently by and for n > 2 using the uniqueness of the Whittaker model. Multiplicity-one also holds for SL(2), but not for SL(n) for n > 2 .
Strong multiplicity one theorem
The strong multiplicity one theorem of and states that two cuspidal automorphic representations of the general linear group are isomorphic if their local components are isomorphic for all but a finite number of places.
See also
Gan-Gross-Prasad conjecture
References
Representation theory of groups
Automorphic forms
Theorems in number theory
Theorems in representation theory |
https://en.wikipedia.org/wiki/Polynomial%20identity | Polynomial identity may refer to:
Algebraic identities of polynomials (see Factorization)
Polynomial identity ring
Polynomial identity testing |
https://en.wikipedia.org/wiki/Matrix%20consimilarity | In linear algebra, two n-by-n matrices A and B are called consimilar if
for some invertible matrix , where denotes the elementwise complex conjugation. So for real matrices similar by some real matrix , consimilarity is the same as matrix similarity.
Like ordinary similarity, consimilarity is an equivalence relation on the set of matrices, and it is reasonable to ask what properties it preserves.
The theory of ordinary similarity arises as a result of studying linear transformations referred to different bases. Consimilarity arises as a result of studying antilinear transformations referred to different bases.
A matrix is consimilar to itself, its complex conjugate, its transpose and its adjoint matrix. Every matrix is consimilar to a real matrix and to a Hermitian matrix. There is a standard form for the consimilarity class, analogous to the Jordan normal form.
References
(sections 4.5 and 4.6 discuss consimilarity)
Matrices |
https://en.wikipedia.org/wiki/Double%20groupoid | In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension.
Definition
A double groupoid D is a higher-dimensional groupoid involving a relationship for both `horizontal' and `vertical' groupoid structures. (A double groupoid can also be considered as a generalization of certain higher-dimensional groups.) The geometry of squares and their compositions leads to a common representation of a double groupoid in the following diagram:
where M is a set of 'points', H and V are, respectively, 'horizontal' and 'vertical' groupoids, and S is a set of 'squares' with two compositions. The composition laws for a double groupoid D make it also describable as a groupoid internal to the category of groupoids.
Given two groupoids H and V over a set M, there is a double groupoid with H,V as horizontal and vertical edge groupoids, and squares given by quadruples
for which one assumes always that h, h′ are in H and v, v′ are in V, and that the initial and final points of these edges match in M as suggested by the notation; that is for example sh = sv, th = sv', ..., etc. The compositions are to be inherited from those of H,V; that is:
and
This construction is the right adjoint to the forgetful functor which takes the double groupoid as above, to the pair of groupoids H,V over M.
Other related constructions are that of a double groupoid with connection and homotopy double groupoids. The homotopy double groupoid of a pair of pointed spaces is a key element of the proof of a two-dimensional Seifert-van Kampen Theorem, first proved by Brown and Higgins in 1978, and given an extensive treatment in the book.
Examples
An easy class of examples can be cooked up by considering crossed modules, or equivalently the data of a morphism of groupswhich has an equivalent description as the groupoid internal to the category of groupswhereare the structure morphisms for this groupoid. Since groups embed in the category of groupoids sending a group to the category with a single object and morphisms giving the group , the structure above gives a double groupoid. Let's give an explicit example: from the group extensionand the embedding of , there is an associated double groupoid from the two term complex of groupswith kernel is and cokernel is given by . This gives an associated homotopy type with and Its postnikov invariant can be determined by the class of in the group cohomology group . Because this is not a trivial crossed-module, it's postnikov invariant is , giving a homotopy type which is not equivalent to the geometric realization of a simplicial abelian group.
Homotopy double groupoid
A generalisation to dimension 2 of the fundamental groupoid on a set of base was given by Brown and Higgins in 1978 as follows. Let be a triple of spaces, i.e. . Define to be the set of homotopy classes rel vertices of maps of a square into X which take the edges into |
https://en.wikipedia.org/wiki/Stationary%20source | The term "stationary source" may refer to one of the following:
A source of data produced by a stationary process, in the mathematical theory of probability and stochastic processes
A source of pollutant emissions that has a fixed location, such as a major stationary source, in pollution and air quality terminology |
https://en.wikipedia.org/wiki/First-order | In mathematics and other formal sciences, first-order or first order most often means either:
"linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of higher degree", or
"without self-reference", as in first-order logic and other logic uses, where it is contrasted with "allowing some self-reference" (higher-order logic)
In detail, it may refer to:
Mathematics
First-order approximation
First-order arithmetic
First-order condition
First-order hold, a mathematical model of the practical reconstruction of sampled signals
First-order inclusion probability
First Order Inductive Learner, a rule-based learning algorithm
First-order reduction, a very weak type of reduction between two computational problems
First-order resolution
First-order stochastic dominance
First order stream
Differential equations
Exact first-order ordinary differential equation
First-order differential equation
First-order differential operator
First-order linear differential equation
First-order non-singular perturbation theory
First-order partial differential equation, a partial differential equation that involves only first derivatives of the unknown function of n variables
Order of accuracy
Logic
First-order language
First-order logic, a formal logical system used in mathematics, philosophy, linguistics, and computer science
First-order predicate, a predicate that takes only individual(s) constants or variables as argument(s)
First-order predicate calculus
First-order theorem provers
First-order theory
Monadic first-order logic
Chemistry
First-order fluid, another name for a power-law fluid with exponential dependence of viscosity on temperature
First-order reaction a first-order chemical reaction
First-order transition
Computer science
First-order abstract syntax
First-order function
First-order query
Other uses
First-order desire
First-order election, in political science, the relative importance of certain elections
First order Fresnel lens
See also
First Order (disambiguation)
Original order, the first ordering
First (disambiguation)
Order (disambiguation) |
https://en.wikipedia.org/wiki/R-algebroid | In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie algebras in the theory of Lie groups. (Thus, a Lie algebroid can be thought of as 'a Lie algebra with many objects ').
Definition
An R-algebroid, , is constructed from a groupoid as follows. The object set of is the same as that of and is the free R-module on the set , with composition given by the usual bilinear rule, extending the composition of .
R-category
A groupoid can be regarded as a category with invertible morphisms.
Then an R-category is defined as an extension of the R-algebroid concept by replacing the groupoid in this construction with a general category C that does not have all morphisms invertible.
R-algebroids via convolution products
One can also define the R-algebroid, , to be the set of functions with finite support, and with the convolution product defined as follows:
.
Only this second construction is natural for the topological case, when one needs to replace 'function' by 'continuous function with compact support', and in this case .
Examples
Every Lie algebra is a Lie algebroid over the one point manifold.
The Lie algebroid associated to a Lie groupoid.
See also
References
Sources
Algebras
Algebraic topology
Category theory
Lie algebras
Lie groupoids |
https://en.wikipedia.org/wiki/Rigid%20cohomology | In mathematics, rigid cohomology is a p-adic cohomology theory introduced by . It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky–Washnitzer cohomology to non-affine varieties. For a scheme X of finite type over a perfect field k, there are rigid cohomology groups H(X/K) which are finite dimensional vector spaces over the field K of fractions of the ring of Witt vectors of k. More generally one can define rigid cohomology with compact supports, or with support on a closed subscheme, or with coefficients in an overconvergent isocrystal.
If X is smooth and proper over k the rigid cohomology groups are the same as the crystalline cohomology groups.
The name "rigid cohomology" comes from its relation to rigid analytic spaces.
used rigid cohomology to give a new proof of the Weil conjectures.
References
External links
Arithmetic geometry
Cohomology theories |
https://en.wikipedia.org/wiki/Fisher%20distribution | Fisher distribution may refer to any of several probability distributions named after Ronald Fisher:
Behrens–Fisher distribution
Fisher's noncentral hypergeometric distribution
Fisher's z-distribution
Fisher's fiducial distribution
Fisher–Bingham distribution
F-distribution, also called Fisher–Snedecor distribution or Fisher F-distribution
Fisher–Tippett distribution
Von Mises–Fisher distribution on a sphere |
https://en.wikipedia.org/wiki/Monsky%E2%80%93Washnitzer%20cohomology | In algebraic geometry, Monsky–Washnitzer cohomology is a p-adic cohomology theory defined for non-singular affine varieties over fields of positive characteristic p introduced by , who were motivated by the work of . The idea is to lift the variety to characteristic 0, and then take a suitable subalgebra of the algebraic de Rham cohomology of . The construction was simplified by . Its extension to more general varieties is called rigid cohomology.
References
(letter to Atiyah, Oct. 14 1963)
Algebraic geometry
Cohomology theories
Homological algebra |
https://en.wikipedia.org/wiki/Evasive%20Boolean%20function | In mathematics, an evasive Boolean function ƒ (of n variables) is a Boolean function for which every decision tree algorithm has running time of exactly n. Consequently, every decision tree algorithm that represents the function has, at worst case, a running time of n.
Examples
An example for a non-evasive boolean function
The following is a Boolean function on the three variables x, y, z:
(where is the bitwise "and", is the bitwise "or", and is the bitwise "not").
This function is not evasive, because there is a decision tree that solves it by checking exactly two variables: The algorithm first checks the value of x. If x is true, the algorithm checks the value of y and returns it.
( )
If x is false, the algorithm checks the value of z and returns it.
A simple example for an evasive boolean function
Consider this simple "and" function on three variables:
A worst-case input (for every algorithm) is 1, 1, 1. In every order we choose to check the variables, we have to check all of them. (Note that in general there could be a different worst-case input for every decision tree algorithm.) Hence the functions: "and", "or" (on n variables) are evasive.
Binary zero-sum games
For the case of binary zero-sum games, every evaluation function is evasive.
In every zero-sum game, the value of the game is achieved by the minimax algorithm (player 1 tries to maximize the profit, and player 2 tries to minimize the cost).
In the binary case, the max function equals the bitwise "or", and the min function equals the bitwise "and".
A decision tree for this game will be of this form:
every leaf will have value in {0, 1}.
every node is connected to one of {"and", "or"}
For every such tree with n leaves, the running time in the worst case is n (meaning that the algorithm must check all the leaves):
We will exhibit an adversary that produces a worst-case input – for every leaf that the algorithm checks, the adversary will answer 0 if the leaf's parent is an Or node, and 1 if the parent is an And node.
This input (0 for all Or nodes' children, and 1 for all And nodes' children) forces the algorithm to check all nodes:
As in the second example
in order to calculate the Or result, if all children are 0 we must check them all.
In order to calculate the And result, if all children are 1 we must check them all.
See also
Aanderaa–Karp–Rosenberg conjecture, the conjecture that every nontrivial monotone graph property is evasive.
Boolean algebra |
https://en.wikipedia.org/wiki/Yemeni%20Football%20Records | Records and statistics of football in Yemen.
Most Successful Teams
Successful Teams
Football in Yemen |
https://en.wikipedia.org/wiki/Hilbert%20operator | Hilbert operator may refer to:
The epsilon operator in Hilbert's epsilon calculus
The Hilbert–Schmidt operators on a Hilbert space
Hilbert–Schmidt integral operators
Generally, any operator on a Hilbert space |
https://en.wikipedia.org/wiki/SDIC | SDIC may refer to the following:
San Domingo Improvement Company, an entity formed to assume control of Dominican Republic railroads in its colonial period; see
In mathematics, Sensitive dependency on initial conditions, also called the butterfly effect
Singapore Deposit Insurance Corporation, see
Sodium dichloroisocyanurate
South Dakota Intercollegiate Conference, a former athletic conference affiliated in the National Association of Intercollegiate Athletics (NAIA)
Spatial Data Interest Community, or Spatial Data Infrastructure Community, a community with interests in spatial data as defined by INSPIRE
State Development & Investment Corporation, a state-owned investment holding company in China |
https://en.wikipedia.org/wiki/Harold%20Edwards%20%28mathematician%29 | Harold Mortimer Edwards, Jr. (August 6, 1936 – November 10, 2020) was an American mathematician working in number theory, algebra, and the history and philosophy of mathematics.
He was one of the co-founding editors, with Bruce Chandler, of The Mathematical Intelligencer.
He is the author of expository books on the Riemann zeta function, on Galois theory, and on Fermat's Last Theorem. He wrote a book on Leopold Kronecker's work on divisor theory providing a systematic exposition of that work—a task that Kronecker never completed. He wrote textbooks on linear algebra, calculus, and number theory. He also wrote a book of essays on constructive mathematics.
Edwards graduated from the University of Wisconsin–Madison in 1956, received a Master of Arts from Columbia University in 1957, and a Ph.D from Harvard University in 1961, under the supervision of Raoul Bott.
He taught at Harvard and Columbia University; he joined the faculty at New York University in 1966, and was an emeritus professor starting in 2002.
In 1980, Edwards won the Leroy P. Steele Prize for Mathematical Exposition of the American Mathematical Society, for his books on the Riemann zeta function and Fermat's Last Theorem. For his contribution in the field of the history of mathematics he was awarded the Albert Leon Whiteman Memorial Prize by the AMS in 2005. In 2012 he became a fellow of the American Mathematical Society.
Edwards was married to Betty Rollin, a former NBC News correspondent, author, and breast cancer survivor. Edwards died on November 10, 2020, of colon cancer.
Books
Higher Arithmetic: An Algorithmic Introduction to Number Theory (2008)An extension of Edwards' work in Essays in Constructive Mathematics, this textbook covers the material of a typical undergraduate number theory course, but follows a constructivist viewpoint in focusing on algorithms for solving problems rather than allowing purely existential solutions. The constructions are intended to be simple and straightforward, rather than efficient, so, unlike works on algorithmic number theory, there is no analysis of how efficient they are in terms of their running time.
Essays in Constructive Mathematics (2005)Although motivated in part by the history and philosophy of mathematics, the main goal of this book is to show that advanced mathematics such as the fundamental theorem of algebra, the theory of binary quadratic forms, and the Riemann–Roch theorem can be handled in a constructivist framework. The second edition (2022) adds a new set of essays that reflect and expand upon the first. This was Edwards' final book, finished shortly before his death.
Linear Algebra, Birkhäuser, (1995)
Divisor Theory (1990)Algebraic divisors were introduced by Kronecker as an alternative to the theory of ideals. According to the citation for Edwards' Whiteman Prize, this book completes the work of Kronecker by providing "the sort of systematic and coherent exposition of divisor theory that Kronecker himself was nev |
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