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https://en.wikipedia.org/wiki/Quilleco | Quilleco (literally means "water of tears") is a Chilean town and commune located in the Bío Bío Province, Bío Bío Region.
Demographics
According to the 2002 census of the National Statistics Institute, Quilleco spans an area of and has 10,428 inhabitants (5,378 men and 5,050 women). Of these, 5,486 (52.6%) lived in urban areas and 4,942 (47.4%) in rural areas. Between the 1992 and 2002 censuses, the population fell by 0.6% (64 persons).
Economy
The principal economic activity of the area is agriculture with traditional crops such as beets and wheat in addition with vast plantations of exotic radiata pine and eucalyptus. Another important activity is the generation of electric energy with two electric grids at Rucúe and Quilleco, the latter of which was inaugurated 23 July 2007 by former president Michelle Bachelet. It treats and utilizes the flowing waters of the Laja River and its tributaries. Rucúe has a 160 MW installed potential, and its average annual generation is 1.190 GWh. Quilleco has a 70 MW installed potential, and it generates an average annual of 450 GWh.
Administration
As a commune, Quilleco is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Rodrigo Tapia Avello (ILE). The municipal council has the following members:
Sergio Daniel Espinoza Almendras (PDC)
Claudio Velosos Vallejos (ILE)
Miriam Soto Quezada (PRSD)
Omar Parra Lagos (ILA)
German Sepulveda Mellado (PS)
Eduardo Bartholomaus Strick (UDI)
Within the electoral divisions of Chile, Quilleco is represented in the Chamber of Deputies by Juan Lobos (UDI) and José Pérez (PRSD) as part of the 47th electoral district, together with Los Ángeles, Tucapel, Antuco, Santa Bárbara, Quilaco, Mulchén, Negrete, Nacimiento, San Rosendo, Laja and Alto Bío Bío. The commune is represented in the Senate by Victor Pérez Varela (UDI) and Mariano Ruiz-Esquide Jara (PDC) as part of the 13th senatorial constituency (Biobío-Coast).
References
External links
Municipality of Quilleco
Communes of Chile
Populated places in Bío Bío Province |
https://en.wikipedia.org/wiki/G%C3%BCnter%20Pilz | Günter Pilz (born 1945 in Bad Hall, Upper Austria) is professor of mathematics at the Johannes Kepler University (JKU) Linz. Until his retirement in 2013 he was the head of the Institute of Algebra.
Vita
After studying mathematics and physics at the University of Vienna (1963–1967) and his PhD (1967), Günter Pilz was assistant professor at several institutions: at the department of mathematics of the University of Vienna (1966–1968), at the department of statistics at the University of Technology of Vienna (1968–1969), as research associate at the department of mathematics, University of Arizona, United States (1969–1970) and at the department of mathematics at the University of Linz (1970–1974). In 1971, he received his Habilitation. In 1974, he was promoted to a professor of mathematics at the JKU. He was head of the department of mathematics in Linz (1980–1983 und 1987–1993) and head of the Institute of Algebra (since 1996).
From 1996 to 2000, Günter Pilz was dean of studies at the Faculty of Science and Technology and from 2000 to 2007 vice rector for research. Also, he was the chairman in the “Forum Research” of the Austrian Conference of Rectors from 2001 to 2005 and was chosen in 2003 to be the Austrian representative for Research Integrity in the corresponding OECD group.
His main area of research are the theories and applications of algebraic structures. He is honorary professor at the Shandong Univ. of Technology in China and Honorary Doctor from the Ural State University in Ekaterinburg, Russia. He was also a member in the Council for Research and Technology in Upper Austria (2003–2008).
Main areas of research
Theory and applications of near-rings: these are „collections of objects with which one can calculate almost as well as with numbers“. These near-rings have numerous applications, for instance in the construction of optimal designs for statistical experiments.
Prizes
Honorary medal in silver, by the County of Upper Austria (2007/08)
External links
The Institute of Algebra
Publications by Günter Pilz
1945 births
Living people
Austrian mathematicians
Academic staff of Johannes Kepler University Linz |
https://en.wikipedia.org/wiki/Los%20Sauces | Los Sauces () is a Chilean town and commune in the Malleco Province, Araucanía Region.
Demographics
According to the 2002 census of the National Statistics Institute, Los Sauces spans an area of and has 7,581 inhabitants (3,847 men and 3,734 women). Of these, 3,638 (48%) lived in urban areas and 3,943 (52%) in rural areas. Between the 1992 and 2002 censuses, the population fell by 15.7% (1,414 persons).
Administration
As a commune, Los Sauces is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Ramón Vilches Álvarez (PDC), and the council has the following members:
Jaime Badilla E. (Ind./RN)
Nelda Gallegos (RN)
Pedro Carrillo F. (UDI)
Eva Calderón M. (PDC)
Victor Gonzalez (UDI)
Jose Ortega E. (PDC)
Within the electoral divisions of Chile, Los Sauces is represented in the Chamber of Deputies by Gonzalo Arenas (UDI) and Mario Venegas (PDC) as part of the 48th electoral district, together with Angol, Renaico, Collipulli, Ercilla, Purén, Lumaco and Traiguén. The commune is represented in the Senate by Alberto Espina Otero (RN) and Jaime Quintana Leal (PPD) as part of the 14th senatorial constituency (Araucanía-North).
References
External links
Municipality of Los Sauces
Communes of Chile
Populated places in Malleco Province |
https://en.wikipedia.org/wiki/Traigu%C3%A9n | Traiguén () is a Chilean city and commune in the Malleco Province, Araucanía Region.
Demographics
According to the 2002 census of the National Statistics Institute, Traiguén spans an area of and has 19,534 inhabitants (9,734 men and 9,800 women). Of these, 14,140 (72.4%) lived in urban areas and 5,394 (27.6%) in rural areas. Between the 1992 and 2002 censuses, the population fell by 5.3% (1,088 persons).
Administration
As a commune, Traiguén is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2012-2016 alcalde is Luis Alvarez (Ind.).And The municipal council 2012-2016 has the following members:
Ricardo Sanhueza Pirce PPD
Essio Guidotti Vallejos PPD
Eliecer Cerda Soto IND
Pablo Mena Osses PRS
Roberto Weidmann Ramirez UDI
Rosanna Rathgeb Fuentes RN
Within the electoral divisions of Chile, Traiguén is represented in the Chamber of Deputies by Gonzalo Arenas (UDI) and Mario Venegas (PDC) as part of the 48th electoral district, together with Angol, Renaico, Collipulli, Ercilla, Los Sauces, Purén and Lumaco. The commune is represented in the Senate by Alberto Espina Otero (RN) and Jaime Quintana Leal (PPD) as part of the 14th senatorial constituency (Araucanía-North).
Climate
Education
Previously the area had a German school, the Deutsche Schule Traiguén.
References
External links
Municipality of Traiguén
Portal web of traiguen city : www.traiguencity.cl
Communes of Chile
Populated places in Malleco Province |
https://en.wikipedia.org/wiki/Lars%20Kolind | Lars Kolind (born 5 May 1947) is a Danish businessman. Kolind holds an M.Sc. in Mathematics from Aarhus University from 1972 and a B.Comm. from the Copenhagen Business School from 1977. He is adjunct professor of leadership and strategy at Aarhus University Business School (Aarhus School of Business) since 2000. He serves as Detao Master of Leadership and Strategy since 2012.
Career
Kolind was executive vice president of Risø National Laboratory from 1981 to 1984 and chief operating officer of Radiometer (company) from 1984 to 1988. Kolind served as CEO of Demant A/S (1988-98), which owns hearing aid manufacturer Oticon. He introduced a non-hierarchical organization (the Spaghetti Organization) in 1991 and took the company public on Nasdaq Copenhagen in 1995. Kolind and the Spaghetti organization were featured by Tom Peters in several books including Liberation management in 1992. Kolind’s transformation of Oticon has been subject of management cases from IMD.
Kolind carried through a financial turnaround of Oticon 1988–90, and in 1991 Kolind changed the company by designing and implementing the so-called "Spaghetti Organization", which has been featured as one of the first knowledge-based, almost paperless organisations in the world. Kolind left William Demant Holding in 1998. Kolind's work in Oticon has been featured in articles and books including Tom Peters' Liberation Management and Per Thygesen Poulsen's Think the Unthinkable (in Danish: Tænk det Utænkelige), both in 1993. Upon Kolind's departure from Oticon, professor Mette Morsing from the Copenhagen Business School co-edited the book Managing the Unmanagable for a Decade (with Kristian Eiberg) in 1998, which also discusses Kolind's works.
Since 1998 Kolind has served as non-executive board member of K. J. Jacobs AG, Grundfos, Poul Due Jensens Fond, Unimerco Group, Zealand Pharma, LinKS and Wemind. In 2000 he started PreVenture A/S, a venture capital firm, which was managed by BankInvest. PreVenture owned Retail Internet A/S, Yellowtel A/S, Isabella Smith A/S and KeepFocus A/S and was dissolved in 2009. Today, Kolind is majority shareholder in KeepFocus A/S through his personal holding company Kolind A/S. Other investments are Spiir, Impero A/S, and Bookanaut Aps. The Kolind family owned the Løndal and Addithus Estates in Western Denmark 1996-2017 and has owned the Sømer Skov Forest Estate on the island of Sjælland since 2011. Kolind is chairman of the supervisory board of Kristeligt Dagblad A/S and a member of the advisory board of Danske Bank. Kolind is also chairman of the board of Jacob Jensen Holding ApS, and LinKS.
Kolind co-founded Impero A/S, a global compliance platform, in 2012, and took the company public on Nasdaq First North Copenhagen in 2021.
In 2011 Lars Kolind became Chairman of Jacob Jensen Design, and in 2018 became a majority shareholder. Jacob Jensen Design has since been experiencing financial losses. In 2020, Salling Bank and Vækstfonden intervened to help pr |
https://en.wikipedia.org/wiki/Rotation%20map | In mathematics, a rotation map is a function that represents an undirected edge-labeled graph, where each vertex enumerates its outgoing neighbors. Rotation maps were first introduced by Reingold, Vadhan and Wigderson (“Entropy waves, the zig-zag graph product, and new constant-degree expanders”, 2002) in order to conveniently define the zig-zag product and prove its properties.
Given a vertex and an edge label , the rotation map returns the 'th neighbor of and the edge label that would lead back to .
Definition
For a D-regular graph G, the rotation map is defined as follows: if the i th edge leaving v leads to w, and the j th edge leaving w leads to v.
Basic properties
From the definition we see that is a permutation, and moreover is the identity map ( is an involution).
Special cases and properties
A rotation map is consistently labeled if all the edges leaving each vertex are labeled in such a way that at each vertex, the labels of the incoming edges are all distinct. Every regular graph has some consistent labeling.
A consistent rotation map can be used to encode a coined discrete time quantum walk on a (regular) graph.
A rotation map is -consistent if . From the definition, a -consistent rotation map is consistently labeled.
See also
Zig-zag product
Rotation system
References
Extensions and generalizations of graphs
Graph operations |
https://en.wikipedia.org/wiki/Pytkeev%20space | In mathematics, and especially topology, a Pytkeev space is a topological space that satisfies qualities more subtle than a convergence of a sequence. They are named after E. G. Pytkeev, who proved in 1983 that sequential spaces have this property.
Definitions
Let X be a topological space. For a subset S of X let S denote the closure of S. Then a point x is called a Pytkeev point if for every set A with , there is a countable -net of infinite subsets of A. A Pytkeev space is a space in which every point is a Pytkeev point.
Examples
Every sequential space is also a Pytkeev space. This is because, if then there exists a sequence {ak} that converges to x. So take the countable π-net of infinite subsets of A to be }.
If X is a Pytkeev space, then it is also a Weakly Fréchet–Urysohn space.
References
Further reading
Topology |
https://en.wikipedia.org/wiki/Voronoi%20pole | In geometry, the positive and negative Voronoi poles of a cell in a Voronoi diagram are certain vertices of the diagram.
Definition
Let be the Voronoi diagram for a set of sites , and let be the Voronoi cell of corresponding to a site . If is bounded, then its positive pole is the vertex of the boundary of that has maximal distance to the point . If the cell is unbounded, then a positive pole is not defined.
Furthermore, let be the vector from to the positive pole, or, if the cell is unbounded, let be a vector in the average direction of all unbounded Voronoi edges of the cell. The negative pole is then the Voronoi vertex in with the largest distance to such that the vector and the vector from to make an angle larger than .
References
Computational geometry |
https://en.wikipedia.org/wiki/Vera%20Zvonareva%20career%20statistics | This is a list of the main career statistics of Russian professional tennis player Vera Zvonareva. She has won 12 WTA Tour singles titles and reached the finals of the 2008 WTA Tour Championships, 2010 Wimbledon Championships and 2010 US Open. She also was a bronze medalist at the 2008 Beijing Olympics.
Performance timelines
Only Main Draw results in WTA Tour, Grand Slam tournaments, Fed Cup/Billie Jean King Cup and Olympic Games are included in win–loss records.
Singles
Current through the 2023 Jiangxi Open.
Doubles
Current through the 2023 Jiangxi Open.
Mixed doubles
Significant finals
Grand Slam finals
Singles: 2 (2 runner–ups)
Doubles: 5 (3 titles, 2 runner–ups)
Mixed doubles: 2 (2 titles)
WTA Finals finals
Singles: 1 (1 runner–up)
WTA 1000 finals
Singles: 7 (1 title, 6 runner–ups)
Doubles: 5 (4 titles, 1 runner–up)
Olympic medal matches
Singles: 1 (bronze medal)
WTA career finals
Singles: 30 (12 titles, 18 runner–ups)
Doubles: 22 (15 titles, 7 runner–ups)
WTA Challenger finals
Singles: 1 (1 runner–up)
Doubles: 4 (2 titles, 2 runner-ups)
ITF career finals
Singles: 4 (3 titles, 1 runner-up)
Doubles: 3 (1 title, 2 runner–ups)
Team competition: 2 (2 titles)
Grand Slam tournament seedings
The tournaments won by Zvonareva are in boldface, and advanced into finals by Zvonareva are in italics.
WTA Tour career earnings
Correct as of the end of the 2021
Head-to-head records
Zvonareva's record against players who have been ranked in the top 10 at some point in their careers. Active players are in boldface.
Statistics correct .
Wins over top-10 players
Wins over reigning world No. 1's
Notes
References
Zvonareva, Vera |
https://en.wikipedia.org/wiki/Kamnitsa | Kamnitsa is a village in Võru Parish, Võru County, in southeastern Estonia.
References
Kamnitsa Has a population of 8 according to a study in 2011 done by StatisticsEstonia
Villages in Võru County |
https://en.wikipedia.org/wiki/Positive%20polynomial | In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively non-negative) on that set. Precisely, Let p be a polynomial in n variables with real coefficients and let S be a subset of the n-dimensional Euclidean space ℝn. We say that:
p is positive on S if p(x) > 0 for every x in S.
p is non-negative on S if p(x) ≥ 0 for every x in S.
Positivstellensatz (and nichtnegativstellensatz)
For certain sets S, there exist algebraic descriptions of all polynomials that are positive (resp. non-negative) on S. Such a description is a positivstellensatz (resp. nichtnegativstellensatz). The importance of Positivstellensatz theorems in computation arises from its ability to transform problems of polynomial optimization into semidefinite programming problems, which can be efficiently solved using convex optimization techniques.
Examples of positivstellensatz (and nichtnegativstellensatz)
Globally positive polynomials and sum of squares decomposition.
Every real polynomial in one variable is non-negative on ℝ if and only if it is a sum of two squares of real polynomials in one variable. This equivalence does not generalize for polynomial with more than one variable: for instance, the Motzkin polynomial X 4Y 2 + X 2Y 4 − 3X 2Y 2 + 1 is non-negative on ℝ2 but is not a sum of squares of elements from ℝ[X, Y].
A real polynomial in n variables is non-negative on ℝn if and only if it is a sum of squares of real rational functions in n variables (see Hilbert's seventeenth problem and Artin's solution).
Suppose that p ∈ ℝ[X1, ..., Xn] is homogeneous of even degree. If it is positive on ℝn \ {0}, then there exists an integer m such that (X12 + ... + Xn2)m p is a sum of squares of elements from ℝ[X1, ..., Xn].
Polynomials positive on polytopes.
For polynomials of degree ≤ 1 we have the following variant of Farkas lemma: If f, g1, ..., gk have degree ≤ 1 and f(x) ≥ 0 for every x ∈ ℝn satisfying g1(x) ≥ 0, ..., gk(x) ≥ 0, then there exist non-negative real numbers c0, c1, ..., ck such that f = c0 + c1g1 + ... + ckgk.
Pólya's theorem: If p ∈ ℝ[X1, ..., Xn] is homogeneous and p is positive on the set {x ∈ ℝn | x1 ≥ 0, ..., xn ≥ 0, x1 + ... + xn ≠ 0}, then there exists an integer m such that (x1 + ... + xn)m p has non-negative coefficients.
Handelman's theorem: If K is a compact polytope in Euclidean d-space, defined by linear inequalities gi ≥ 0, and if f is a polynomial in d variables that is positive on K, then f can be expressed as a linear combination with non-negative coefficients of products of members of {gi}.
Polynomials positive on semialgebraic sets.
The most general result is Stengle's Positivstellensatz.
For compact semialgebraic sets we have Schmüdgen's positivstellensatz, Putinar's positivstellensatz and Vasilescu's positivstellensatz. The point here is that no denominators are needed.
For nice compact semialgebraic sets of low dimension, there exists |
https://en.wikipedia.org/wiki/Equilateral%20dimension | In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of a -dimensional Euclidean space is , achieved by a regular simplex, and the equilateral dimension of a -dimensional vector space with the Chebyshev distance ( norm) is , achieved by a hypercube. However, the equilateral dimension of a space with the Manhattan distance ( norm) is not known; Kusner's conjecture, named after Robert B. Kusner, states that it is exactly , achieved by a cross polytope.
Lebesgue spaces
The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional normed vector spaces with the norm
The equilateral dimension of spaces of dimension behaves differently depending on the value of :
For , the norm gives rise to Manhattan distance. In this case, it is possible to find equidistant points, the vertices of an axis-aligned cross polytope. The equilateral dimension is known to be exactly for , and to be upper bounded by for all . Robert B. Kusner suggested in 1983 that the equilateral dimension for this case should be exactly ; this suggestion (together with a related suggestion for the equilateral dimension when ) has come to be known as Kusner's conjecture.
For , the equilateral dimension is at least where is a constant that depends on .
For , the norm is the familiar Euclidean distance. The equilateral dimension of -dimensional Euclidean space is : the vertices of an equilateral triangle, regular tetrahedron, or higher-dimensional regular simplex form an equilateral set, and every equilateral set must have this form.
For , the equilateral dimension is at least : for instance the basis vectors of the vector space together with another vector of the form for a suitable choice of form an equilateral set. Kusner's conjecture states that in these cases the equilateral dimension is exactly . Kusner's conjecture has been proven for the special case that . When is an odd integer the equilateral dimension is upper bounded by .
For (the limiting case of the norm for finite values of , in the limit as grows to infinity) the norm becomes the Chebyshev distance, the maximum absolute value of the differences of the coordinates. For a -dimensional vector space with the Chebyshev distance, the equilateral dimension is : the vertices of an axis-aligned hypercube are at equal distances from each other, and no larger equilateral set is possible.
Normed vector spaces
Equilateral dimension has also been considered for normed vector spaces with norms other than the norms. The problem of determining the equilateral dimension for a given norm is closely related to the kissing number problem: the kissing number in a normed space is the maximum number of disjoint translates of |
https://en.wikipedia.org/wiki/College%20Football%20Data%20Warehouse | College Football Data Warehouse was an American college football statistics website that was established in 2000. The site compiled the yearly team records, game-by-game results, championships, and statistics of college football teams, conferences, and head coaches at the NCAA Division I FBS and Division I FCS levels, as well as those of some NCAA Division II, NCAA Division III, NAIA, NJCAA, and discontinued programs. The site listed as its references annual editions of Spalding's Official Football Guide, Street and Smith's Football Yearbooks, NCAA, NAIA, and NJCAA record books and guides, and historical college football texts.
College Football Data Warehouse was administered by Gary "Tex" Noel and David DeLassus. Noel lived in Bedford, Indiana and, prior to his death, in Fresno, California, and was the executive director of Intercollegiate Football Researchers Association, a college football historian, statistician, and author.
The website has been cited as a source by The New York Sun, The Fort Worth Star-Telegram, The Columbus Ledger-Enquirer, The State, and The Lawrentian. It has also widely cited in books on college football history, and in scholarly journals such as the Journal of Sports Economics, the Utah Law Review, the Tulsa Law Review, the Oklahoma Law Review, and Sports Law.
The website was shut down sometime after February 19, 2017, but in September 2017 it was relaunched, although complete only through the 2015 season. The site yet again shutdown in 2020.
College Football Data Warehouse recognized national champions (1869–2015)
College Football Data Warehouse (CFBDW) is an online resource and database that has collected and researched information on college football and national championship selections. It provides a comprehensive list of national championship selectors and has itself recognized selectors that it has deemed to be the most acceptable throughout history. These include the National Championship Foundation (1869–1882), the Helms Athletic Foundation (1883–1935), the College Football Researchers Association (1919–1935), the Associated Press Poll (1936–2015), and the Coaches Poll (1950–2015). From its research, it has compiled a list of Recognized National Championships for each season. Some years include recognition of multiple teams for a particular season. Some universities claim championships not recognized by CFBDW or do not claim championships that are recognized by CFBDW. The table of National championship claims by school or individual team articles and websites may include additional or alternative national championship claims.
Below is a list of all of the CFBDW recognized national championships from 1869 to 2015.
References
External links
American football websites
Internet properties established in 2000 |
https://en.wikipedia.org/wiki/List%20of%20Burnley%20F.C.%20records%20and%20statistics | Burnley Football Club is an English professional association football club based in the town of Burnley, Lancashire. Founded on 18 May 1882, the club was one of the first to become professional (in 1883), putting pressure on the Football Association (FA) to permit payments to players. In 1885, the FA legalised professionalism, so the team entered the FA Cup for the first time in 1885–86, and were one of the twelve founding members of the Football League in 1888–89. Burnley have played in all four professional divisions of English football from 1888 to the present day. The team have been champions of England twice, in 1920–21 and 1959–60, have won the FA Cup once, in 1913–14, and have won the FA Charity Shield twice, in 1960 and 1973. Burnley are one of only five teams to have won all four professional divisions of English football, along with Wolverhampton Wanderers, Preston North End, Sheffield United and Portsmouth. They were the second to achieve this by winning the Fourth Division in the 1991–92 season.
The record for most games played for the club is held by Jerry Dawson, who made 569 appearances between 1907 and 1928. George Beel scored 188 goals during his Burnley career and is the club's record goalscorer. Jimmy McIlroy made 51 appearances for Northern Ireland and so is the player who has gained the most caps while with Burnley. The highest transfer fee paid by the club is the £16.1 million paid to FC Basel for Zeki Amdouni in 2023; the highest fees received are the £25 million paid by Everton and Newcastle United for Michael Keane and Chris Wood in 2017 and 2022 respectively. The highest attendance recorded at home ground Turf Moor was 54,775 for the visit of Huddersfield Town in a third round FA Cup match in 1924.
All records and statistics are correct as of the 2022–23 season.
Honours and achievements
Burnley won their first honour in 1883, when the team won the Dr Dean's Cup, a knockout competition between amateur clubs in the Burnley area. The club turned professional by the end of 1883, and was one of the twelve founder members of the Football League in 1888. Burnley reached their first major final in 1914, when they reached the FA Cup Final and beat Liverpool 1–0. Burnley have been champions of England two times, in 1920–21 and 1959–60, and have won the Charity Shield twice, in 1960 (shared with Wolverhampton Wanderers) and 1973. The side have competed in one of the four professional levels of English football from 1888 to the present day, and are one of only five teams—and were the second—to have won all four tiers, along with Wolverhampton Wanderers, Preston North End, Sheffield United and Portsmouth. Burnley's honours include the following:
League
First Division (Tier 1)
Winners: 1920–21, 1959–60
Runners–up: 1919–20, 1961–62
Second Division/Championship (Tier 2)
Winners: 1897–98, 1972–73, 2015–16, 2022–23
Promoted: 1912–13, 1946–47, 2013–14
Play–off winners: 2008–09
Third Division/Second Division (Tier 3)
Winners: 1981–82
P |
https://en.wikipedia.org/wiki/Yvan%20Arpa | Y. Arpa is a Swiss watch designer. He started his career as a mathematics professor who, after participating in professional martial art combats in Thailand and crossing Papua New Guinea by foot, came back to his roots in Switzerland.
In 1997, he joined the Richemont Group for Baume & Mercier as the Managing Director for Switzerland and as Sales Director for Europe and Asia.
From 2002 to 2006 he acted as the Managing Director for Hublot participating in the launch of the Big Bang.
From 2006 to 2009 he was the CEO of Romain Jerome introducing watches with rusted steel from the Titanic or real Moon dust.
In 2009 Yvan Arpa created his own company called Luxury Artpieces for which he launched the brands below :
-Black Belt Watch is inspired by the seven virtues, the samurai code of conduct, associated with the Bushido warrior.
The brand proposes also a watch only for people who defends a black belt in Martial Arts.
-ArtyA : The brand is characterized by the unconventional raw materials it uses as butterfly wings, real bullets, fossilized dinosaur droppings, real spiders, boxes struck by lightning, tobacco leaves etc.
From 2010 to 2011, he acted as COO of Jacob&Co.
At Basel World 2013, he launched Spero Lucem, named in honor of his birth city Geneva.
For this brand, Yvan Arpa also created some pens and some knives.
At last, Yvan Arpa personalized a motorbike, which took more than 1'000 hours to sculpt and customize, with heavy artillery incrusted at the back, in the shape of a XXL shaped bullet belt.
In 2016, Yvan Arpa designed the smartwatch Gear S3 for Samsung, which is the first smartwatch using high-end watchmaking codes.
He also presents the Samsung Gear S3 in Berlin, at its International Launch Event.
According to hodinkee and due to his incredible life history "Yvan Arpa is a strong candidate to be the Most interesting Man in the World, Swiss Edition"
In 2017, Yvan Arpa joined Pascal Meyer, director of the website Qoqa. From their union, the watch Magma was born, entirely “swiss made” respecting the codes of fine watchmaking.
Particularity: they possess a plate issued of 5 iconics watches from swiss watchmaking.
References
Living people
Swiss chief executives
Chief operating officers
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Kerem%20%C4%B0nan | Kerem İnan (born 25 March 1980) is a Turkish professional football goalkeeper who plays for Erokspor.
Career statistics
Honours
Galatasaray
Turkish League: 2 (1999–00, 2001–02)
Turkish Cup: 2 (1998–99, 1999–00)
UEFA Cup: 1 (1999–00)
UEFA Super Cup: 1 (2000)
References
External links
1980 births
Living people
Footballers from Istanbul
Turkish men's footballers
Turkey men's under-21 international footballers
Men's association football goalkeepers
Süper Lig players
Galatasaray U21 footballers
Galatasaray S.K. footballers
Çaykur Rizespor footballers
Samsunspor footballers
Turanspor footballers
Karşıyaka S.K. footballers
Mersin Talim Yurdu footballers
Turkey men's youth international footballers
UEFA Cup winning players |
https://en.wikipedia.org/wiki/Moon%20Kwang-eun | Moon Kwang-eun (; born November 9, 1987 in Gwangju, South Korea) is a South Korean pitcher who plays for the LG Twins of the KBO League.
External links
Career statistics and player information from Korea Baseball Organization
Moon Kwang-eun at SK Wyverns Baseball Club
1987 births
Living people
South Korean baseball players
SSG Landers players
KBO League pitchers
Baseball players from Gwangju |
https://en.wikipedia.org/wiki/Wente%20torus | In differential geometry, a Wente torus is an immersed torus in of constant mean curvature, discovered by . It is a counterexample to the conjecture of Heinz Hopf that every closed, compact, constant-mean-curvature surface is a sphere (though this is true if the surface is embedded). There are similar examples known for every positive genus.
References
The Wente torus, University of Toledo Mathematics Department, retrieved 2013-09-01.
External links
Visualization of the Wente torus
Differential geometry of surfaces |
https://en.wikipedia.org/wiki/Shun%20Ito | is a Japanese football player currently playing for Roasso Kumamoto.
Career statistics
Updated to 23 February 2018.
1Includes J1 Promotion Playoffs.
References
External links
Profile at Roasso Kumamoto
Profile at Kyoto Sanga
1987 births
Living people
Kokushikan University alumni
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Montedio Yamagata players
Ehime FC players
Kyoto Sanga FC players
Roasso Kumamoto players
Men's association football midfielders
Association football people from Sapporo |
https://en.wikipedia.org/wiki/Ciprian%20Foias | Ciprian Ilie Foiaș (20 July 1933 – 22 March 2020) was a Romanian-American mathematician. He was awarded the Norbert Wiener Prize in Applied Mathematics in 1995, for his contributions in operator theory.
Education and career
Born in Reșița, Romania, Foias studied mathematics at the University of Bucharest. He completed his dissertation in 1957, but was not allowed to defend his thesis by the Communist government until 1962. He received his doctorate in 1962 under supervision of Miron Nicolescu.
Foias defected to France following his lecture at the International Congress of Mathematicians in 1978. He later emigrated to the United States.
Foias taught at his alma mater (1966–1979), Paris-Sud 11 University (1979–1983), and Indiana University (1983 until retirement). Beginning in 2000, he was a teacher and researcher at Texas A&M University, where he was a Distinguished Professor.
The Foias constant is named after him. Foias is listed as an ISI highly cited researcher.
Together with Sz-Nagy, Foias proved the celebrated commutant lifting theorem.
He died in Tempe, Arizona on March 22, 2020.
Publications
with Béla Szőkefalvi-Nagy: Harmonic analysis of operators on Hilbert Space. North Holland 1970 (Translated from the French; first edition: Masson 1967).
with Roger Temam, Oscar Manley, and Ricardo Rosa: Navier Stokes equations and Turbulence. Cambridge University Press, 2001.
with Peter Constantin, Roger Temam: Attractors representing turbulent flows. American Mathematical Society, 1985.
with Peter Constantin: Navier Stokes Equations. University of Chicago Press, 1988.
with Hitay Özbay, Allen Tannenbaum: Robust control of infinite dimensional systems. Springer, 1995.
with Hari Bercovici, Carl Pearcy: Dual algebras with applications to invariant subspaces and dilation theory. American Mathematical Society, 1985.
with Ion Colojoară: Theory of generalized spectral operators. Gordon and Breach, 1968.
with Peter Constantin, Roger Temam, and Basil Nicolaenko: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Springer-Verlag, Applied Mathematical Sciences Series, volume 70, 1988.
See also
Attractor
Evolution equation
References
External links
1933 births
2020 deaths
People from Reșița
Romanian emigrants to the United States
University of Bucharest alumni
Indiana University faculty
Texas A&M University faculty
20th-century Romanian mathematicians
21st-century Romanian mathematicians
Academic staff of Paris-Sud University
Operator theorists |
https://en.wikipedia.org/wiki/Virbhadra%E2%80%93Ellis%20lens%20equation | The Virbhadra-Ellis lens equation in astronomy and mathematics relates to the angular positions of an unlensed source , the image , the Einstein bending angle of light , and the angular diameter lens-source and observer-source distances.
.
This lens equation is useful for studying gravitational lensing in a strong gravitational field.
References
Gravitational lensing
Astrophysics
Equations of astronomy |
https://en.wikipedia.org/wiki/Collider%20%28statistics%29 | In statistics and causal graphs, a variable is a collider when it is causally influenced by two or more variables. The name "collider" reflects the fact that in graphical models, the arrow heads from variables that lead into the collider appear to "collide" on the node that is the collider. They are sometimes also referred to as inverted forks.
The causal variables influencing the collider are themselves not necessarily associated. If they are not adjacent, the collider is unshielded. Otherwise, the collider is shielded and part of a triangle.
The result of having a collider in the path is that the collider blocks the association between the variables that influence it. Thus, the collider does not generate an unconditional association between the variables that determine it.
Conditioning on the collider via regression analysis, stratification, experimental design, or sample selection based on values of the collider creates a non-causal association between X and Y (Berkson's paradox). In the terminology of causal graphs, conditioning on the collider opens the path between X and Y. This will introduce bias when estimating the causal association between X and Y, potentially introducing associations where there are none. Colliders can therefore undermine attempts to test causal theories.
Colliders are sometimes confused with confounder variables. Unlike colliders, confounder variables should be controlled for when estimating causal associations.
See also
Causality
Causal graphs
Confounding
Directed acyclic graph
Selection bias
Path analysis
Bad control
References
Epidemiology
Independence (probability theory)
Graphical models
Causal inference |
https://en.wikipedia.org/wiki/Klein%20polyhedron | In the geometry of numbers, the Klein polyhedron, named after Felix Klein, is used to generalize the concept of continued fractions to higher dimensions.
Definition
Let be a closed simplicial cone in Euclidean space . The Klein polyhedron of is the convex hull of the non-zero points of .
Relation to continued fractions
Suppose is an irrational number. In , the cones generated by and by give rise to two Klein polyhedra, each of which is bounded by a sequence of adjoining line segments. Define the integer length of a line segment to be one less than the size of its intersection with Then the integer lengths of the edges of these two Klein polyhedra encode the continued-fraction expansion of , one matching the even terms and the other matching the odd terms.
Graphs associated with the Klein polyhedron
Suppose is generated by a basis of (so that ), and let be the dual basis (so that ). Write for the line generated by the vector , and for the hyperplane orthogonal to .
Call the vector irrational if ; and call the cone irrational if all the vectors and are irrational.
The boundary of a Klein polyhedron is called a sail. Associated with the sail of an irrational cone are two graphs:
the graph whose vertices are vertices of , two vertices being joined if they are endpoints of a (one-dimensional) edge of ;
the graph whose vertices are -dimensional faces (chambers) of , two chambers being joined if they share an -dimensional face.
Both of these graphs are structurally related to the directed graph whose set of vertices is , where vertex is joined to vertex if and only if is of the form where
(with , ) and is a permutation matrix. Assuming that has been triangulated, the vertices of each of the graphs and can be described in terms of the graph :
Given any path in , one can find a path in such that , where is the vector .
Given any path in , one can find a path in such that , where is the -dimensional standard simplex in .
Generalization of Lagrange's theorem
Lagrange proved that for an irrational real number , the continued-fraction expansion of is periodic if and only if is a quadratic irrational. Klein polyhedra allow us to generalize this result.
Let be a totally real algebraic number field of degree , and let be the real embeddings of . The simplicial cone is said to be split over if where is a basis for over .
Given a path in , let . The path is called periodic, with period , if for all . The period matrix of such a path is defined to be . A path in or associated with such a path is also said to be periodic, with the same period matrix.
The generalized Lagrange theorem states that for an irrational simplicial cone , with generators and as above and with sail , the following three conditions are equivalent:
is split over some totally real algebraic number field of degree .
For each of the there is periodic path of vertices in such that the asymptotically approach the |
https://en.wikipedia.org/wiki/Fr%C3%A9chet%E2%80%93Urysohn%20space | In the field of topology, a Fréchet–Urysohn space is a topological space with the property that for every subset the closure of in is identical to the sequential closure of in
Fréchet–Urysohn spaces are a special type of sequential space.
Fréchet–Urysohn spaces are the most general class of spaces for which sequences suffice to determine all topological properties of subsets of the space.
That is, Fréchet–Urysohn spaces are exactly those spaces for which knowledge of which sequences converge to which limits (and which sequences do not) suffices to completely determine the space's topology.
Every Fréchet–Urysohn space is a sequential space but not conversely.
The space is named after Maurice Fréchet and Pavel Urysohn.
Definitions
Let be a topological space.
The of in is the set:
where or may be written if clarity is needed.
A topological space is said to be a if
for every subset where denotes the closure of in
Sequentially open/closed sets
Suppose that is any subset of
A sequence is if there exists a positive integer such that for all indices
The set is called if every sequence in that converges to a point of is eventually in ;
Typically, if is understood then is written in place of
The set is called if or equivalently, if whenever is a sequence in converging to then must also be in
The complement of a sequentially open set is a sequentially closed set, and vice versa.
Let
denote the set of all sequentially open subsets of where this may be denoted by is the topology is understood.
The set is a topology on that is finer than the original topology
Every open (resp. closed) subset of is sequentially open (resp. sequentially closed), which implies that
Strong Fréchet–Urysohn space
A topological space is a if for every point and every sequence of subsets of the space such that there exist a sequence in such that for every and in
The above properties can be expressed as selection principles.
Contrast to sequential spaces
Every open subset of is sequentially open and every closed set is sequentially closed.
However, the converses are in general not true.
The spaces for which the converses are true are called ;
that is, a sequential space is a topological space in which every sequentially open subset is necessarily open, or equivalently, it is a space in which every sequentially closed subset is necessarily closed.
Every Fréchet-Urysohn space is a sequential space but there are sequential spaces that are not Fréchet-Urysohn spaces.
Sequential spaces (respectively, Fréchet-Urysohn spaces) can be viewed/interpreted as exactly those spaces where for any single given subset knowledge of which sequences in converge to which point(s) of (and which do not) is sufficient to is closed in (respectively, is sufficient to of in ).
Thus sequential spaces are those spaces for which sequences in can be used as a "test" to determine whether or not any given |
https://en.wikipedia.org/wiki/Pisan%20calendar | The Pisan calendar, also referred to as the stile pisano ("Pisan style") or the calculus Pisanus ("Pisan calculation"), was the calendar used in the Republic of Pisa in Italy during the Middle Ages, which differed from the traditional Julian calendar.
Beginning of the year
The Pisan year began on 25 March, and not on 1 January, with the apparent year lying ahead of the traditional Julian calendar. Thus, 24 March 1200 was followed by 25 March 1201 (not 1200, as it would remain in the Julian calendar), and 31 December 1201 would then be followed by 1 January 1201, which was the point at which the two calendars synchronised. This is the reason that most dates have an apparent discrepancy of one year, as the two calendars differ for just over nine months of each cycle. For example, a birth date of 10 September 1552 in Pisan reckoning translates to 10 September 1551 in the Julian calendar.
Beginning the year on a date other than 1 January was common during the mediaeval period. The first day of the year falling on 25 March meant that the Pisan calendar was in the stile dell'Annunciazione ("style of the Annunciation") or stile dell'Incarnazione ("style of the Incarnation") - also styled in Latin as ab [Dominica] incarnatione ("by the [Lord's] Incarnation") - by reference to the Solemnity of the Annunciation, and similar calendars saw use in Cortona and Pistoia. The Florentine, Sienese, English and Scottish calendars were also in this style, but confusingly ran behind the Julian calendar rather than ahead, resulting in them lying exactly one year behind of the Pisan calendar. By contrast, calendars in the stile della Natività ("style of the Nativity") as in Arezzo, Assisi and Perugia began on the Solemnity of the Nativity of the Lord (Christmas) on 25 December, the Venetian calendar began on 1 March until the Fall of the Venetian Republic, and the French year on Easter day until 1564. The traditional Julian calendar was sometimes said to be in the stile della Circoncisione ("style of the Circumcision"). See beginning of the year.
End of use
Italy was one of the few regions to immediately convert from the Julian calendar to the Gregorian: 4 October 1582 was followed by 15 October 1582, the latter being the first day of the new Gregorian calendar. Not until 1749, however, were the ancient calendars definitively outlawed in Tuscany: in that year the recently appointed Grand Duke and Holy Roman Emperor, Francis I, ordered that, starting from 1750, the first of January should become the first day of the year, thus having the "peoples of Tuscia" conform to all the others. A plaque in Latin commemorating the grand ducal/imperial decree is affixed to the west wall of the Loggia dei Lanzi, in Piazza della Signoria in Florence.
Notes
References
Further reading
.
Obsolete calendars
History of Pisa
Medieval Italy
Time in Italy |
https://en.wikipedia.org/wiki/Robert%20Lisjak | Robert Lisjak (born 5 February 1978) is a Croatian retired football goalkeeper who played for NK Funtana.
Career statistics
References
1978 births
Living people
Sportspeople from Koprivnica
Men's association football goalkeepers
Croatian men's footballers
NK Slaven Belupo players
NK Istra 1961 players
HNK Rijeka players
NK Osijek players
NK Istra players
Croatian Football League players |
https://en.wikipedia.org/wiki/Pedal%20equation | In Euclidean geometry, for a plane curve and a given fixed point , the pedal equation of the curve is a relation between and where is the distance from to a point on and is the perpendicular distance from to the tangent line to at the point. The point is called the pedal point and the values and are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of to the normal (the contrapedal coordinate) even though it is not an independent quantity and it relates to as
Some curves have particularly simple pedal equations and knowing the pedal equation of a curve may simplify the calculation of certain of its properties such as curvature. These coordinates are also well suited for solving certain type of force problems in classical mechanics and celestial mechanics.
Equations
Cartesian coordinates
For C given in rectangular coordinates by f(x, y) = 0, and with O taken to be the origin, the pedal coordinates of the point (x, y) are given by:
The pedal equation can be found by eliminating x and y from these equations and the equation of the curve.
The expression for p may be simplified if the equation of the curve is written in homogeneous coordinates by introducing a variable z, so that the equation of the curve is g(x, y, z) = 0. The value of p is then given by
where the result is evaluated at z=1
Polar coordinates
For C given in polar coordinates by r = f(θ), then
where is the polar tangential angle given by
The pedal equation can be found by eliminating θ from these equations.
Alternatively, from the above we can find that
where is the "contrapedal" coordinate, i.e. distance to the normal. This implies that if a curve satisfies an autonomous differential equation in polar coordinates of the form:
its pedal equation becomes
Example
As an example take the logarithmic spiral with the spiral angle α:
Differentiating with respect to we obtain
hence
and thus in pedal coordinates we get
or using the fact that we obtain
This approach can be generalized to include autonomous differential equations of any order as follows: A curve C which a solution of an n-th order autonomous differential equation () in polar coordinates
is the pedal curve of a curve given in pedal coordinates by
where the differentiation is done with respect to .
Force problems
Solutions to some force problems of classical mechanics can be surprisingly easily obtained in pedal coordinates.
Consider a dynamical system:
describing an evolution of a test particle (with position and velocity ) in the plane in the presence of central and Lorentz like potential. The quantities:
are conserved in this system.
Then the curve traced by is given in pedal coordinates by
with the pedal point at the origin. This fact was discovered by P. Blaschke in 2017.
Example
As an example consider the so-called Kepler problem, i.e. central force problem, where the force varies inversely |
https://en.wikipedia.org/wiki/Lorenz%20%28disambiguation%29 | Lorenz is a German name.
Lorenz may also refer to:
Mathematics and science
Lorenz system, a system of equations notable for having chaotic solutions
Lorenz gauge condition, in electromagnetism
Lorenz curve, an income distribution curve
Lorenz cipher, a German cryptography machine
Other uses
St. Lorenz, Nuremberg, Germany
St. Lorenz Basilica in Kempten im Allgäu, Germany
C. Lorenz AG, a German electrical and electronics firm
Lorenz beam, a radio-navigation system
Lorenz Publishing, American music publisher
Lorenz rifle, an Austrian rifle designed in 1854
Lorenz Snack-World, a German food company
See also
Lorentz (disambiguation) |
https://en.wikipedia.org/wiki/Martin%20J.%20Beckmann | Martin Joseph Beckmann (5 July 1924, Ratingen, Germany – 11 April 2017) was a professor for Economics and Applied Mathematics. He was professor at the University of Chicago, Yale University and Brown University, as well as the University of Bonn and Technische Universität München. He received honorary degrees from the University of Karlsruhe, the Umeå University and the University of the Bundeswehr Hamburg. He was president of the European Regional Science Association and received the Regional Science Founders Medal in 1983. His research spans a wide field in spatial analysis and regional economics, with a special focus on transport economics.
Martin Beckmann established basic principles for user behavior on congested transportation networks, as well as for optimal network vehicle flows, when user choices are respectively uncoordinated or coordinated. Beckmann’s contribution launched the new subfield of transportation network economics.
Further reading
References
External links
Founder's medal: Martin J. Beckmann
1924 births
2017 deaths
German economists
Brown University faculty
Regional economists
Fellows of the Econometric Society
Helmut Schmidt University alumni
German expatriates in the United States
Yale University faculty
University of Chicago faculty
People from Ratingen |
https://en.wikipedia.org/wiki/Mohammad%20Hassan%20Rajabzadeh | Mohammad Hassan Rajabzadeh (born June 10, 1983) is an Iranian footballer.
Club career
Club career statistics
Honours
2nd Hazfi Cup 2004–05 with Aboomoslem
Promoted to Persian Gulf League Azadegan League 2007–08 with Payam
Persian Gulf Cup 2010–11 with Sepahan
Persian Gulf Cup 2011–12 with Sepahan
Azadegan League 2013–14 with Padideh
Promoted to Persian Gulf League Azadegan League 2013–14 with Padideh
Azadegan League 2015–16 with Paykan
Promoted to Persian Gulf League Azadegan League 2015–16 with Paykan
Coaching Honours
2nd Azadegan League 2020–21 with Havadar
Promoted to Persian Gulf League Azadegan League 2020–21 with Havadar
2nd 2023 WAFF U-23 Championship with Iran U23
References
1983 births
Living people
F.C. Aboomoslem players
Sepahan S.C. footballers
Payam Khorasan F.C. players
Malavan F.C. players
Rah Ahan Tehran F.C. players
Shahr Khodro F.C. players
Paykan F.C. players
Iranian men's footballers
Men's association football midfielders
Footballers from Mashhad |
https://en.wikipedia.org/wiki/RPMC | RPMC may refer to:
Computers and mathematics
Read-only memory, a type of storage media that is used in computers and other electronic devices
ICAO airport code
Naval Air Station Cubi Point
Cotabato Airport |
https://en.wikipedia.org/wiki/Stratifold | In differential topology, a branch of mathematics, a stratifold is a generalization of a differentiable manifold where certain kinds of singularities are allowed. More specifically a stratifold is stratified into differentiable manifolds of (possibly) different dimensions. Stratifolds can be used to construct new homology theories. For example, they provide a new geometric model for ordinary homology. The concept of stratifolds was invented by Matthias Kreck. The basic idea is similar to that of a topologically stratified space, but adapted to differential topology.
Definitions
Before we come to stratifolds, we define a preliminary notion, which captures the minimal notion for a smooth structure on a space: A differential space (in the sense of Sikorski) is a pair where X is a topological space and C is a subalgebra of the continuous functions such that a function is in C if it is locally in C and is in C for smooth and A simple example takes for X a smooth manifold and for C just the smooth functions.
For a general differential space and a point x in X we can define as in the case of manifolds a tangent space as the vector space of all derivations of function germs at x. Define strata has dimension i For an n-dimensional manifold M we have that and all other strata are empty. We are now ready for the definition of a stratifold, where more than one stratum may be non-empty:
A k-dimensional stratifold is a differential space where S is a locally compact Hausdorff space with countable base of topology. All skeleta should be closed. In addition we assume:
The are i-dimensional smooth manifolds.
For all x in S, restriction defines an isomorphism of stalks
All tangent spaces have dimension ≤ k.
For each x in S and every neighbourhood U of x, there exists a function with and (a bump function).
A n-dimensional stratifold is called oriented if its (n − 1)-stratum is empty and its top stratum is oriented. One can also define stratifolds with boundary, the so-called c-stratifolds. One defines them as a pair of topological spaces such that is an n-dimensional stratifold and is an (n − 1)-dimensional stratifold, together with an equivalence class of collars.
An important subclass of stratifolds are the regular stratifolds, which can be roughly characterized as looking locally around a point in the i-stratum like the i-stratum times a (n − i)-dimensional stratifold. This is a condition which is fulfilled in most stratifold one usually encounters.
Examples
There are plenty of examples of stratifolds. The first example to consider is the open cone over a manifold M. We define a continuous function from S to the reals to be in C if and only if it is smooth on and it is locally constant around the cone point. The last condition is automatic by point 2 in the definition of a stratifold. We can substitute M by a stratifold S in this construction. The cone is oriented if and only if S is oriented and not zero-dimensional. If we consid |
https://en.wikipedia.org/wiki/Ra%C3%BAl%20Ch%C3%A1vez%20Sarmiento | Raúl Arturo Chávez Sarmiento (born 24 October 1997) is a Peruvian child prodigy in mathematics. At the age of , he won a bronze medal at the 2009 International Mathematical Olympiad, making him the second youngest medalist in IMO history, behind Terence Tao who won bronze in 1986 at the age of 10.
He won a silver medal at the 2010 IMO at age 12 years, 263 days, a gold medal (6th ranked overall) at the 2011 IMO, and again a silver medal at the 2012 IMO.
Chávez Sarmiento is currently a graduate student in mathematics at Harvard University.
See also
List of child prodigies
List of International Mathematical Olympiad participants
References
External links
1997 births
Living people
International Mathematical Olympiad participants
Peruvian mathematicians
21st-century mathematicians |
https://en.wikipedia.org/wiki/Ostrowski%20%28disambiguation%29 | Ostrowski is a Polish surname
Ostrowski may also refer to:
Ostrowski Prize, mathematics award
Ostrowski's theorem, mathematical theorem
Two counties in Poland with the native name powiat ostrowski:
Ostrów Mazowiecka County
Ostrów Wielkopolski County
See also
Ostrovsky (disambiguation) |
https://en.wikipedia.org/wiki/HJM | HJM may refer to:
Heath–Jarrow–Morton framework, a financial model
Herzog–Jackson Motorsports, a NASCAR team
Higman-Janko-McKay group, in group theory
Hans-Joachim Marseille, German Fighter Ace |
https://en.wikipedia.org/wiki/Permutation%20representation%20%28disambiguation%29 | In mathematics, permutation representation may refer to:
A group action, see also Permutation representation
A representation of a symmetric group (see Representation theory of the symmetric group) |
https://en.wikipedia.org/wiki/Hamid%20Neshatjoo | Hamid Neshatjoo (born January 13, 1979) is an Iranian footballer who plays for Steel Azin F.C. in the IPL.
Career
Neshatjoo has spent his entire career with Steel Azin F.C.
Club Career Statistics
Last Update 13 December 2010
References
1979 births
Living people
Futsal goalkeepers
Iranian men's footballers
Iranian men's futsal players
Steel Azin F.C. players
Men's association football goalkeepers
21st-century Iranian people
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/HNK%20Hajduk%20Split%20statistics%20and%20records | Hrvatski nogometni klub Hajduk Split is a Croatian football club founded based in the city of Split, that competes in Prva HNL, top football league in the country. The club was founded on 13 February 1911. in Prague, and played its first competitive match on 11 June 1911 against Calcio Spalato, winning 9–0. The first to score for Hajduk was Šime Raunig. The first official game was played on 28 March 1920 in Split Championship against Borac Split, winning 8–0. This article lists various records and statistics related to the club and individual players and managers.
All records and statistics are accurate as of 4 March 2021
Individual records and statistics
Current players and manager are in bold/italics.
Appearances
Most appearances:
All fixtures – 739, Frane Matošić (1935–39, 1940–41, 1944–55).
Official matches – 390, Vedran Rožić (1972–84).
In Yugoslav First League – 310, Ivica Hlevnjak (1962–73).
In Prva HNL – 199, Srđan Andrić (1999–2004, 2007–12).
In football cups – 44, Ante Miše (1985–94, 1997–2003).
In European competitions – 49, Vedran Rožić (1972–84).
Foreign player with most appearances in all competitions – 214, Josip Skoko (1995–99, 2008–10).
Most appearances in one season:
All official matches – 49, Vedran Rožić (1983–84).
All official matches (1992–) – 46, Mirsad Hibić (1994–95).
In Yugoslav First League – 34, Ivica Hlevnjak (1968–69, 1970–71), Dragan Holcer (1968–69), Vilson Džoni (1971–72), Ivica Šurjak (1973–74), Ivan Buljan (1974–75), Nenad Šalov, Boro Primorac (1980–81), Zlatko Vujović (1984–85), Dragutin Čelić (1987–88).
In Prva HNL – 34, Goran Milović (2013–14), Marko Livaja (2021–22).
In Yugoslav Cup – 8, Dragutin Čelić (1986–87, 1989–90), Stipe Andrijašević, Dragi Setinov, Jerko Tipurić (1986–87), Nikola Jerkan (1989–90), Robert Jarni (1990–91).
In Croatian Cup – 9, Ante Miše (1992–93) and Nenad Pralija (1994–95).
In European competitions – 10, Vedran Rožić and Zoran Simović (1983–84), Aljoša Asanović (1994–95), Senijad Ibričić and Jurica Buljat (2010–11).
Numerous seasons appearances:
Longest-serving player – 11 years, 6 months and 25 days, Vedran Rožić (from 5 November 1972 to 30 May 1984).
Most major trophies with Hajduk Split – 10, Luka Peruzović.
Other records in the Croatian First Football League
Youngest player – 16 years, 0 month, 2 days, Luka Vušković (26 February 2023 v Dinamo).
Oldest player – 37 years, 7 months, 11 days, Vladimir Balić (10 May 2008 v Zagreb).
Oldest débutante – 36 years, 3 months, 5 days, Đovani Roso (22 February 2009 v Dinamo).
Most substituted player – 53, Srđan Andrić (1999–2004, 2007–12).
Most substituted player (one season) – 18, Ante Erceg (2016–17).
Most used substitute – 57, Zvonimir Deranja (1996–2004, 2005–06).
Most used substitute (one season) – 19, Mirko Oremuš (2008–09).
Appearances in most seasons – 10, Tomislav Erceg (1991–95, 1997–98, 1999–2000, 2001–02, 2006), Vik Lalić (1993–2001, 2002–03), Srđan |
https://en.wikipedia.org/wiki/List%20of%20Sporting%20de%20Gij%C3%B3n%20records%20and%20statistics | Sporting de Gijón is a football club from Gijón, Spain. They have finished runner-up in La Liga one occasion, and were beaten finalists in the Copa del Rey on two occasions. Real Sporting also won the Segunda División five times.
Honours
National titles
La Liga
Runners-up: 1978–79
Copa del Rey:
Runners-up: 1981, 1982
Segunda División: (5) 1943–44, 1950–51, 1956–57, 1969–70, 1976–77
Runners-up: 1929–30, 2014–15
Regional titles
Asturian Championship: 1916–17, 1917–18, 1918–19, 1919–20, 1920–21, 1921–22, 1922–23, 1923–24, 1925–26, 1926–27, 1929–30, 1930–31, 1939–40
Friendly tournaments
Ramón de Carranza Trophy: 1984
Trofeo Villa de Gijón: 1996, 2001, 2002, 2008, 2014, 2015, 2016
Trofeo Costa Verde: 1962, 1963, 1965, 1970, 1972, 1975–1979, 1984–1986, 1988–1992
Trofeo Principado: 1988, 1991, 1993, 1994, 2006
Trofeo Ciudad de Pamplona: 1979
Torneo Ciudad de León: 1997
Trofeo Presidente - Ciudad de Oviedo: 1980, 1986
Trofeo Ciudad del Cid: 1981
Trofeo Ibérico: 1972, 1979
Trofeo Conde de Fontao: 1969-1971, 1974, 1977, 1989
Trofeo Emma Cuervo: 1957, 1960, 1963, 1967, 1972, 2007, 2012
Trofeo Ramón Losada: 1997, 1998, 1999, 2001, 2004, 2008, 2009, 2010
Trofeo Concepción Arenal: 1970, 1971
Trofeo Luis Otero: 2003, 2008
Youth football
Copa de Campeones: 2004
Runners-up: 2005, 2018
División de Honor Juvenil de Fútbol (group): (4) 2003–04, 2004–05, 2011–12, 2017–18
Copa del Rey Juvenil de Fútbol:
Runners-up: 2005
Under-16 Spanish Championship: 2000
Under-10 Spanish Championship: 2010, 2012, 2018, 2019
Individual honours
Pichichi Trophy
In Spanish football, the Pichichi Trophy is awarded by Spanish sports newspaper 'Marca' to the top goalscorer for each league season.
Primera División
Segunda División
Zamora Trophy
In Spanish football, the Zamora Trophy is awarded by Spanish sports newspaper Marca to the goalkeeper who has the lowest "goals-to-games" ratio for each league season.
Primera División
Segunda División
Segunda División Player of the Month
Segunda División Manager of the Month
Players
Appearances
Youngest first-team player (in national competition): Eloy Olaya at (0–4 win v Turón, Copa del Rey, 28 November 1979)
Youngest La Liga players1:
Emilio Blanco: (1–1 v Athletic Bilbao, La Liga, 31 October 1982)
Juan Muñiz: (0–2 loss v Racing Santander, La Liga, 16 May 2010)
Sergio Álvarez: (0–2 loss v Racing Santander, La Liga, 16 May 2010)
Oldest La Liga player: Molinucu at (3–5 loss v Racing Santander, La Liga, 26 April 1954)
(1) Not considering players from youth ranks called up during the footballers' strikes of 1981 and 1984.
Most appearances in La Liga
Source
Top goalscorers in La Liga
As of August 2010
Source
Player of the season
The following players have been awarded 'El Molinón de Plata' since its creation in 1967, given to the best player of the season by the Federation of Sporting Gijón Supporters Clubs:
Đurđević
(*) Following the disastrous 1997–98 campaign, the prize was not given to any player.
Team sta |
https://en.wikipedia.org/wiki/Eric%20Aparicio | Eric Daniel Aparicio (born 25 January 1990 in Glew, Buenos Aires) is an Argentine footballer who plays as a striker.
External links
Argentine Primera statistics
1990 births
Living people
Argentine men's footballers
Argentine expatriate men's footballers
Men's association football forwards
Club Atlético Lanús footballers
Tiro Federal footballers
C.D. Jorge Wilstermann players
Club Atlético Brown footballers
San Martín de San Juan footballers
Defensa y Justicia footballers
Barracas Central players
Club Atlético San Telmo footballers
Chacarita Juniors footballers
Club Atlético Los Andes footballers
Argentine Primera División players
Primera Nacional players
Primera B Metropolitana players
Bolivian Primera División players
Footballers from Buenos Aires
Expatriate men's footballers in Bolivia
Argentine expatriate sportspeople in Bolivia |
https://en.wikipedia.org/wiki/T%20pad | The T pad is a specific type of attenuator circuit in electronics whereby the topology of the circuit is formed in the shape of the letter "T".
Attenuators are used in electronics to reduce the level of a signal. They are also referred to as pads due to their effect of padding down a signal by analogy with acoustics. Attenuators have a flat frequency response attenuating all frequencies equally in the band they are intended to operate. The attenuator has the opposite task of an amplifier. The topology of an attenuator circuit will usually follow one of the simple filter sections. However, there is no need for more complex circuitry, as there is with filters, due to the simplicity of the frequency response required.
Circuits are required to be balanced or unbalanced depending on the geometry of the transmission lines they are to be used with. For radio frequency applications, the format is often unbalanced, such as coaxial. For audio and telecommunications, balanced circuits are usually required, such as with the twisted pair format. The T pad is intrinsically an unbalanced circuit. However, it can be converted to a balanced circuit by placing half the series resistances in the return path. Such a circuit is called an H-section, or else an I section because the circuit is formed in the shape of a serifed letter "I".
Terminology
An attenuator is a form of a two-port network with a generator connected to one port and a load connected to the other. In all of the circuits given below it is assumed that the generator and load impedances are purely resistive (though not necessarily equal) and that the attenuator circuit is required to perfectly match to these. The symbols used for these impedances are;
the impedance of the generator
the impedance of the load
Popular values of impedance are 600Ω in telecommucations and audio, 75Ω for video and dipole antennae, 50Ω for RF
The voltage transfer function, A, is,
While the inverse of this is the loss, L, of the attenuator,
The value of attenuation is normally marked on the attenuator as its loss, LdB, in decibels (dB). The relationship with L is;
Popular values of attenuator are 3dB, 6dB, 10dB, 20dB and 40dB.
However, it is often more convenient to express the loss in nepers,
where is the attenuation in nepers (one neper is approximately 8.7 dB).
Impedance and loss
The values of resistance of the attenuator's elements can be calculated using image parameter theory. The starting point here is the image impedances of the L section in figure 2. The image impedance of the input is,
and the image admittance of the output is,
The loss of the L section when terminated in its image impedances is,
where the image parameter transmission function, γL is given by,
The loss of this L section in the reverse direction is given by,
For an attenuator, Z and Y are simple resistors and γ becomes the image parameter attenuation (that is, the attenuation when terminated with the image impeda |
https://en.wikipedia.org/wiki/List%20of%20S.L.%20Benfica%20records%20and%20statistics | Sport Lisboa e Benfica, commonly known as Benfica, is a Portuguese professional football club based in Lisbon. Founded on 28 February 1904 as Sport Lisboa, it merged with Grupo Sport Benfica in 1908, thus being renamed Sport Lisboa e Benfica. The club's records and statistics here gathered only concern competitive, professional matches – no exhibition games are considered. These records and statistics include data on honours, players, transfers, managers, and team records, respectively.
Honours
Benfica is the most decorated club in Portugal, holding the records for most Primeira Liga, Taça de Portugal, and Taça da Liga titles, three of the four most important competitions in the country (the other being Supertaça Cândido de Oliveira). Moreover, Benfica is the only club to have won four consecutive editions of Taça de Portugal, from 1948–49 to 1952–53 (in 1949–50, the cup was not held), and four consecutive editions of Taça da Liga, from 2008–09 to 2011–12. On the international stage, Benfica have reached a record ten European finals, winning back-to-back European Cups in 1960–61 and 1961–62.
National titles
Primeira Liga
Winners (38) – record: 1935–36, 1936–37, 1937–38, 1941–42, 1942–43, 1944–45, 1949–50, 1954–55, 1956–57, 1959–60, 1960–61, 1962–63, 1963–64, 1964–65, 1966–67, 1967–68, 1968–69, 1970–71, 1971–72, 1972–73, 1974–75, 1975–76, 1976–77, 1980–81, 1982–83, 1983–84, 1986–87, 1988–89, 1990–91, 1993–94, 2004–05, 2009–10, 2013–14, 2014–15, 2015–16, 2016–17, 2018–19, 2022–23
Taça de Portugal
Winners (26) – record: 1939–40, 1942–43, 1943–44, 1948–49, 1950–51, 1951–52, 1952–53, 1954–55, 1956–57, 1958–59, 1961–62, 1963–64, 1968–69, 1969–70, 1971–72, 1979–80, 1980–81, 1982–83, 1984–85, 1985–86, 1986–87, 1992–93, 1995–96, 2003–04, 2013–14, 2016–17
Supertaça Cândido de Oliveira
Winners (9): 1980, 1985, 1989, 2005, 2014, 2016, 2017, 2019, 2023
Taça da Liga
Winners (7) – record: 2008–09, 2009–10, 2010–11, 2011–12, 2013–14, 2014–15, 2015–16
Campeonato de Portugal
Winners (3): 1929–30, 1930–31, 1934–35
UEFA titles
European Cup
Winners (2): 1960–61, 1961–62
Doubles
Primeira Liga and Taça de Portugal
11 – record: 1942–43, 1954–55, 1956–57, 1963–64, 1968–69, 1971–72, 1980–81, 1982–83, 1986–87, 2013–14, 2016–17
Primeira Liga and Taça da Liga
4 – record: 2009–10, 2013–14, 2014–15, 2015–16
Taça de Portugal and Taça da Liga
1 – shared record: 2013–14
European Double
1 – shared national record: 1960–61
European Cup Double
1 – national record: 1961–62
Trebles
Primeira Liga, Taça de Portugal and Supertaça Cândido de Oliveira
2 – record: 1980–81, 2016–17
Primeira Liga, Taça de Portugal and Taça da Liga
1 – record: 2013–14
Primeira Liga, Taça da Liga and Supertaça Cândido de Oliveira
1 – record: 2014–15
Players
Benfica's record appearance-maker is Nené, who made 575 appearances over the course of his career. Ten players have made more than 400 appearances, including four members of the 1961 European Cup-winning team. Eusébio |
https://en.wikipedia.org/wiki/Mamokgethi%20Phakeng | Rosina Mamokgethi Phakeng (née Mmutlana, born 1 November 1966) is a South African professor of mathematics education who in 2018 became a vice-chancellor of the University of Cape Town (UCT). She has been the vice principal of research and innovation, at the University of South Africa and acting executive dean of the College of Science, Engineering and Technology at UNISA. In 2018 she was an invited speaker at the International Congresses of Mathematicians. In February 2023 it was announced that she would leave her position as vice-chancellor of UCT and take early retirement. She was succeeded by Professor Daya Reddy on 13 March 2023
Early life
Phakeng was born in Eastwood, Pretoria, to Frank and Wendy Mmutlana (née Thipe). Her mother went back to school after having her three children to complete Form 3 as entry to gaining a Primary Teachers Certificate to practice as a teacher. Her father was one of the first black radio announcers at the South African Broadcasting Corporation (SABC).
Phakeng started school in 1972 at Ikageleng Primary in Marapyane village and then Ikageng Primary in Ga-Rankuwa. She attended Tsela-tshweu higher primary; Tswelelang Higher Primary; Thuto-Thebe Middle School; Odi High School and Hebron. She completed her matric with University Exemption in 1983 (Grade 12) in the village of Hebron's College of Education.
Higher education
Phakeng achieved a Bachelor of Education in mathematics education at the University of North-West, and a M.ed in mathematics education at the University of the Witwatersrand, and in 2002 became the first black female South African to obtain a PhD in mathematics education. In September 2022, Phakeng won the first Africa Education Medal for her commitment to promoting education in Africa, particularly for her research on language practices in multilingual mathematics classrooms.
Career accomplishments
Phakeng has won awards for excellence in service. These honors include:
Doctor of Science, honoris causa, University of Bristol
The Order of the Baobab (Silver) for her excellent contribution in the field of science and representing South Africa on the international stage through her outstanding research work presented to her by former president of South Africa Jacob Zuma. (April 2016)
Order of Ikhamanga in gold
CEO Magazine award for being the most influential woman in education and training in South Africa (August 2013):
NSTF award for being the most outstanding Senior Black Female Researcher over the last 5 to 10 years in recognition of her innovative, quality research on teaching and learning mathematics in multilingual classrooms. (May 2011)
Golden key International Society Honorary life membership (May 2009)
Association of Mathematics Education of South Africa (AMESA) Honorary life membership (July 2009)
Amstel Salute to Success finalist (2005)
Dr. T. W. Khambule Research Award for being the most outstanding young female black researcher for 2003: Conferred by the NSTF (May 2004)
Out |
https://en.wikipedia.org/wiki/Triangular%20array | In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index. That is, the ith row contains only i elements.
Examples
Notable particular examples include these:
The Bell triangle, whose numbers count the partitions of a set in which a given element is the largest singleton
Catalan's triangle, which counts strings of parentheses in which no close parenthesis is unmatched
Euler's triangle, which counts permutations with a given number of ascents
Floyd's triangle, whose entries are all of the integers in order
Hosoya's triangle, based on the Fibonacci numbers
Lozanić's triangle, used in the mathematics of chemical compounds
Narayana triangle, counting strings of balanced parentheses with a given number of distinct nestings
Pascal's triangle, whose entries are the binomial coefficients
Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called generalized Pascal triangles; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers.
Generalizations
Triangular arrays may list mathematical values other than numbers; for instance the Bell polynomials form a triangular array in which each array entry is a polynomial.
Arrays in which the length of each row grows as a linear function of the row number (rather than being equal to the row number) have also been considered.
Applications
Apart from the representation of triangular matrices, triangular arrays are used in several algorithms. One example is the CYK algorithm for parsing context-free grammars, an example of dynamic programming.
Romberg's method can be used to estimate the value of a definite integral by completing the values in a triangle of numbers.
The Boustrophedon transform uses a triangular array to transform one integer sequence into another.
See also
Triangular number, the number of entries in such an array up to some particular row
References
External links |
https://en.wikipedia.org/wiki/Poisson%20binomial%20distribution | In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The concept is named after Siméon Denis Poisson.
In other words, it is the probability distribution of the
number of successes in a collection of n independent yes/no experiments with success probabilities . The ordinary binomial distribution is a special case of the Poisson binomial distribution, when all success probabilities are the same, that is .
Definitions
Probability Mass Function
The probability of having k successful trials out of a total of n can be written as the sum
where is the set of all subsets of k integers that can be selected from {1,2,3,...,n}. For example, if n = 3, then . is the complement of , i.e. .
will contain elements, the sum over which is infeasible to compute in practice unless the number of trials n is small (e.g. if n = 30, contains over 1020 elements). However, there are other, more efficient ways to calculate .
As long as none of the success probabilities are equal to one, one can calculate the probability of k successes using the recursive formula
where
The recursive formula is not numerically stable, and should be avoided if is greater than approximately 20. An alternative is to use a divide-and-conquer algorithm: if we assume is a power of two, denoting by the Poisson binomial of and the convolution operator, we have .
Another possibility is using the discrete Fourier transform.
where and .
Still other methods are described in "Statistical Applications of the Poisson-Binomial and conditional Bernoulli distributions" by Chen and Liu.
Properties
Mean and Variance
Since a Poisson binomial distributed variable is a sum of n independent Bernoulli distributed variables, its mean and variance will simply be sums of the mean and variance of the n Bernoulli distributions:
For fixed values of the mean () and size (n), the variance is maximal when all success probabilities are equal and we have a binomial distribution. When the mean is fixed, the variance is bounded from above by the variance of the Poisson distribution with the same mean which is attained asymptotically as n tends to infinity.
Entropy
There is no simple formula for the entropy of a Poisson binomial distribution, but the entropy is bounded above by the entropy of a binomial distribution with the same number parameter and the same mean. Therefore, the entropy is also bounded above by the entropy of a Poisson distribution with the same mean.
The Shepp–Olkin concavity conjecture, due to Lawrence Shepp and Ingram Olkin in 1981, states that the entropy of a Poisson binomial distribution is a concave function of the success probabilities . This conjecture was proved by Erwan Hillion and Oliver Johnson in 2015. The Shepp-Olkin monotonicity conjecture, also from the same 1981 paper, is that the entropy is monotone incre |
https://en.wikipedia.org/wiki/Rouch%C3%A9%E2%80%93Capelli%20theorem | In linear algebra, the Rouché–Capelli theorem determines the number of solutions for a system of linear equations, given the rank of its augmented matrix and coefficient matrix. The theorem is variously known as the:
Rouché–Capelli theorem in English speaking countries, Italy and Brazil;
Kronecker–Capelli theorem in Austria, Poland, Croatia, Romania, Serbia and Russia;
Rouché–Fontené theorem in France;
Rouché–Frobenius theorem in Spain and many countries in Latin America;
Frobenius theorem in the Czech Republic and in Slovakia.
Formal statement
A system of linear equations with n variables has a solution if and only if the rank of its coefficient matrix A is equal to the rank of its augmented matrix [A|b]. If there are solutions, they form an affine subspace of of dimension n − rank(A). In particular:
if n = rank(A), the solution is unique,
otherwise there are infinitely many solutions.
Example
Consider the system of equations
x + y + 2z = 3,
x + y + z = 1,
2x + 2y + 2z = 2.
The coefficient matrix is
and the augmented matrix is
Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are infinitely many solutions.
In contrast, consider the system
x + y + 2z = 3,
x + y + z = 1,
2x + 2y + 2z = 5.
The coefficient matrix is
and the augmented matrix is
In this example the coefficient matrix has rank 2, while the augmented matrix has rank 3; so this system of equations has no solution. Indeed, an increase in the number of linearly independent columns has made the system of equations inconsistent.
See also
Cramer's rule
Gaussian elimination
References
External links
Kronecker-Capelli Theorem at Wikibooks
Kronecker-Capelli's Theorem - YouTube video with a proof
Kronecker-Capelli theorem in the Encyclopaedia of Mathematics
Theorems in linear algebra
Matrix theory |
https://en.wikipedia.org/wiki/Statistics%20Greenland | Statistics Greenland (, ) is a central statistical organization in Greenland, operating under the auspices of the Government of Greenland, working in cooperation with the Ministry for Finance. Based in Nuuk, the capital of Greenland, the organization was founded on 19 July 1989 by the Government of Greenland.
See also
List of national and international statistical services
References
External links
Statistics Greenland home page (English version)
Nuuk
Greenland
Organisations based in Greenland |
https://en.wikipedia.org/wiki/Lie%20conformal%20algebra | A Lie conformal algebra is in some sense a generalization of a Lie algebra in that it too is a "Lie algebra," though in a different pseudo-tensor category. Lie conformal algebras are very closely related to vertex algebras and have many applications in other areas of algebra and integrable systems.
Definition and relation to Lie algebras
A Lie algebra is defined to be a vector space with a skew symmetric bilinear multiplication which satisfies the Jacobi identity. More generally, a Lie algebra is an object, in the category of vector spaces (read: -modules) with a morphism
that is skew-symmetric and satisfies the Jacobi identity. A Lie conformal algebra, then, is an object in the category of -modules with morphism
called the lambda bracket, which satisfies modified versions of bilinearity, skew-symmetry and the Jacobi identity:
One can see that removing all the lambda's, mu's and partials from the brackets, one simply has the definition of a Lie algebra.
Examples of Lie conformal algebras
A simple and very important example of a Lie conformal algebra is the Virasoro conformal algebra. Over it is generated by a single element with lambda bracket given by
In fact, it has been shown by Wakimoto that any Lie conformal algebra with lambda bracket satisfying the Jacobi identity on one generator is actually the Virasoro conformal algebra.
Classification
It has been shown that any finitely generated (as a -module) simple Lie conformal algebra is isomorphic to either the Virasoro conformal algebra, a current conformal algebra or a semi-direct product of the two.
There are also partial classifications of infinite subalgebras of and .
Generalizations
Use in integrable systems and relation to the calculus of variations
References
Victor Kac, "Vertex algebras for beginners". University Lecture Series, 10. American Mathematical Society, 1998. viii+141 pp.
Non-associative algebra
Lie algebras
Conformal field theory |
https://en.wikipedia.org/wiki/Gerald%20Teschl | Gerald Teschl (born 12 May 1970 in Graz) is an Austrian mathematical physicist and professor of mathematics.
He works in the area of mathematical physics; in particular direct and inverse spectral theory with application to completely integrable partial differential equations (soliton equations).
Career
After studying physics at the Graz University of Technology (diploma thesis 1993), he continued with a PhD in mathematics at the University of Missouri. The title of his thesis supervised by Fritz Gesztesy was Spectral Theory for Jacobi Operators (1995). After a postdoctoral position at the Rheinisch-Westfälischen Technische Hochschule Aachen (1996/97), he moved to Vienna, where he received his Habilitation at the University of Vienna in May 1998. Since then he has been a professor of mathematics there.
In 1997 he received the Ludwig Boltzmann Prize from the Austrian Physical Society, 1999 the Prize of the Austrian Mathematical Society. In 2006 he was awarded with the prestigious START-Preis by the Austrian Science Fund (FWF). In 2011 he became a member of the Austrian Academy of Sciences (ÖAW).
His most important contributions are to the fields of Sturm–Liouville theory, Jacobi operators and the Toda lattice. He also works in biomathematics, in particular in the novel area of breath gas analysis, and has written a successful undergraduate textbook (Mathematics for Computer Science, in German) with his wife Susanne Teschl.
Selected publications
Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, Graduate Studies in Mathematics, Volume 140, 2012,
with Julian King, Helin Koc, Karl Unterkofler, Pawel Mochalski, Alexander Kupferthaler, Susanne Teschl, Hartmann Hinterhuber, Anton Amann: Physiological modeling of isoprene dynamics in exhaled breath, J. Theoret. Biol. 267 (2010), 626–637.
Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, American Mathematical Society, Graduate Studies in Mathematics, Volume 99, 2009,
with Susanne Teschl: Mathematik für Informatiker, 2 Bände, Springer Verlag, Bd. 1 (Diskrete Mathematik und Lineare Algebra), 3. Auflage 2008, , Bd. 2 (Analysis und Statistik), 2. Auflage 2007,
with Fritz Gesztesy, Helge Holden and Johanna Michor: Soliton Equations and their Algebro-Geometric Solutions, Volume 2 (1+1 dimensional discrete models), Cambridge Studies in Advanced Mathematics Bd.114, Cambridge University Press 2008,
with Spyridon Kamvissis: Stability of periodic soliton equations under short range perturbations, Phys. Lett. A 364 (2007), 480–483.
Jacobi Operators and Completely Integrable Nonlinear Lattices, American Mathematical Society, Mathematical Surveys and Monographs, Volume 72, 2000,
with Fritz Gesztesy and Barry Simon: Zeros of the Wronskian and renormalized oscillation theory, Am. J. Math. 118 (1996) 571–594.
External links
Prof. Teschl's Homepage
Portrait at the website of the Austrian Science Fund (FWF)
1970 births
Li |
https://en.wikipedia.org/wiki/Quasi-category | In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. The study of such generalizations is known as higher category theory.
Quasi-categories were introduced by .
André Joyal has much advanced the study of quasi-categories showing that most of the usual basic category theory and some of the advanced notions and theorems have their analogues for quasi-categories. An elaborate treatise of the theory of quasi-categories has been expounded by .
Quasi-categories are certain simplicial sets. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories, the composition of two morphisms need not be uniquely defined. All the morphisms that can serve as composition of two given morphisms are related to each other by higher order invertible morphisms (2-simplices thought of as "homotopies"). These higher order morphisms can also be composed, but again the composition is well-defined only up to still higher order invertible morphisms, etc.
The idea of higher category theory (at least, higher category theory when higher morphisms are invertible) is that, as opposed to the standard notion of a category, there should be a mapping space (rather than a mapping set) between two objects. This suggests that a higher category should simply be a topologically enriched category. The model of quasi-categories is, however, better suited to applications than that of topologically enriched categories, though it has been proved by Lurie that the two have natural model structures that are Quillen equivalent.
Definition
By definition, a quasi-category C is a simplicial set satisfying the inner Kan conditions (also called weak Kan condition): every inner horn in C, namely a map of simplicial sets where , has a filler, that is, an extension to a map . (See Kan fibration#Definitions for a definition of the simplicial sets and .)
The idea is that 2-simplices are supposed to represent commutative triangles (at least up to homotopy). A map represents a composable pair. Thus, in a quasi-category, one cannot define a composition law on morphisms, since one can choose many ways to compose maps.
One consequence of the definition is that is a trivial Kan fibration. In other words, while the composition law is not uniquely defined, it is unique up to a contractible choice.
The homotopy category
Given a quasi-category C, one can associate to it an ordinary category hC, called the homotopy category of C. The homotopy category has as objects the vertices of C. The morphisms are given by homotopy classes of edges between vertices. Composition is given using the horn filler condition for n = 2.
For a general simplicial set there is a functor from sSet to Cat, known as the fundamental category |
https://en.wikipedia.org/wiki/Mohammed%20Mubarak | Mohammed Mubarak (born 11 August 1984) is a Qatari footballer who is a goalkeeper . His brother, Meshal Mubarak, is also a footballer and has represented the national team.
Club career statistics
Statistics accurate as of 21 August 2011
1Includes Emir of Qatar Cup.
2Includes Sheikh Jassem Cup.
3Includes AFC Champions League.
References
External links
1984 births
Living people
Qatari men's footballers
Qatar SC players
Al-Arabi SC (Qatar) players
Al Ahli SC (Doha) players
Al-Shamal SC players
Al-Waab SC players
Qatar men's international footballers
2011 AFC Asian Cup players
Men's association football goalkeepers
Qatar Stars League players
Qatari Second Division players |
https://en.wikipedia.org/wiki/Hironori%20Ishikawa | is a former Japanese footballer who last played for SC Sagamihara on loan from Thespakusatsu Gunma in the J3 League.
Career statistics
Updated to 23 February 2017.
1Includes Japanese Super Cup and FIFA Club World Cup.
Honours
Club
Sanfrecce Hiroshima
J. League Division 1 (2) : 2012, 2013
Japanese Super Cup (1): 2013
References
External links
Profile at SC Sagamihara
1988 births
Living people
People from Edogawa, Tokyo
Ryutsu Keizai University alumni
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Mito HollyHock players
Sanfrecce Hiroshima players
Vegalta Sendai players
Oita Trinita players
Thespakusatsu Gunma players
SC Sagamihara players
Men's association football defenders |
https://en.wikipedia.org/wiki/Juraj%20Hovan%C4%8D%C3%ADk | Juraj Hovančík (born 22 November 1990) is a Slovak football midfielder who currently plays for Odeva Lipany.
Career statistics
External links
Futbalnet profile
MFK Košice profile
1990 births
Living people
Footballers from Prešov
Men's association football midfielders
Slovak men's footballers
FC VSS Košice players
FK Železiarne Podbrezová players
1. FC Tatran Prešov players
FC Lokomotíva Košice players
ŠK Odeva Lipany players
Slovak First Football League players
2. Liga (Slovakia) players
3. Liga (Slovakia) players |
https://en.wikipedia.org/wiki/D%C3%A1vid%20Heged%C5%B1s | Dávid Hegedűs (born 6 June 1985) is a Hungarian football player who plays for Eger.
Club statistics
Updated to games played as of 17 June 2020.
References
Player profile at HLSZ
1985 births
Footballers from Eger
Living people
Hungarian men's footballers
Men's association football defenders
Marcali VFC footballers
Jászapáti VSE footballers
Bőcs KSC footballers
Kazincbarcikai SC footballers
Egri FC players
Szolnoki MÁV FC footballers
Kaposvári Rákóczi FC players
Mezőkövesdi SE footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players
21st-century Hungarian people |
https://en.wikipedia.org/wiki/2010%E2%80%9311%20NK%20Osijek%20season | This article shows statistics of individual players for the Osijek football club. It also lists all matches that Osijek played in the 2010–11 season.
First-team squad
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Results by opponent
Source: 2010–11 Prva HNL article
Matches
Pre-season
Prva HNL
Croatian Cup
Last updated 6 May 2011Sources: Prva-HNL.hr, Sportnet.hr
Player seasonal records
Competitive matches only. Updated to games played 21 May 2011.
Goalscorers
Source: Competitive matches
Disciplinary record
Includes all competitive matches. Players with 1 card or more included only.
Source: Prva-HNL.hr
Appearances and goals
Source: Prva-HNL.hr
Transfers
In
Out
Loans out
Sources: nogometni-magazin.com
References
External links
Official website
Croatian football clubs 2010–11 season
2010 |
https://en.wikipedia.org/wiki/Cantellated%206-simplexes | In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.
There are unique 4 degrees of cantellation for the 6-simplex, including truncations.
Cantellated 6-simplex
Alternate names
Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)
Coordinates
The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.
Images
Bicantellated 6-simplex
Alternate names
Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)
Coordinates
The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.
Images
Cantitruncated 6-simplex
Alternate names
Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)
Coordinates
The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.
Images
Bicantitruncated 6-simplex
Alternate names
Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)
Coordinates
The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.
Images
Related uniform 6-polytopes
The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
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.
x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes |
https://en.wikipedia.org/wiki/Runcinated%206-simplexes | In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.
There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.
Runcinated 6-simplex
Alternate names
Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)
Coordinates
The vertices of the runcinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,1,2). This construction is based on facets of the runcinated 7-orthoplex.
Images
Biruncinated 6-simplex
Alternate names
Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)
Coordinates
The vertices of the biruncinted 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 7-orthoplex.
Images
Runcitruncated 6-simplex
Alternate names
Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)
Coordinates
The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.
Images
Biruncitruncated 6-simplex
Alternate names
Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)
Coordinates
The vertices of the biruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 7-orthoplex.
Images
Runcicantellated 6-simplex
Alternate names
Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)
Coordinates
The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 7-orthoplex.
Images
Runcicantitruncated 6-simplex
Alternate names
Runcicantitruncated heptapeton
Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)
Coordinates
The vertices of the runcicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 7-orthoplex.
Images
Biruncicantitruncated 6-simplex
Alternate names
Biruncicantitruncated heptapeton
Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)
Coordinates
The vertices of the biruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 7-orthoplex.
Images
Related uniform 6-polytopes
The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
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 b |
https://en.wikipedia.org/wiki/Stericated%206-simplexes | In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.
There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.
Stericated 6-simplex
Alternate names
Small cellated heptapeton (Acronym: scal) (Jonathan Bowers)
Coordinates
The vertices of the stericated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,1,2). This construction is based on facets of the stericated 7-orthoplex.
Images
Steritruncated 6-simplex
Alternate names
Cellitruncated heptapeton (Acronym: catal) (Jonathan Bowers)
Coordinates
The vertices of the steritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 7-orthoplex.
Images
Stericantellated 6-simplex
Alternate names
Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers)
Coordinates
The vertices of the stericantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 7-orthoplex.
Images
Stericantitruncated 6-simplex
Alternate names
Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers)
Coordinates
The vertices of the stericanttruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the stericantitruncated 7-orthoplex.
Images
Steriruncinated 6-simplex
Alternate names
Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers)
Coordinates
The vertices of the steriruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,2,3,3). This construction is based on facets of the steriruncinated 7-orthoplex.
Images
Steriruncitruncated 6-simplex
Alternate names
Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers)
Coordinates
The vertices of the steriruncittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncitruncated 7-orthoplex.
Images
Steriruncicantellated 6-simplex
Alternate names
Bistericantitruncated 6-simplex as t1,2,3,5{3,3,3,3,3}
Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers)
Coordinates
The vertices of the steriruncitcantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncicantellated 7-orthoplex.
Images
Steriruncicantitruncated 6-simplex
Alternate names
Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers)
Coordinates
The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 7-orthoplex.
Ima |
https://en.wikipedia.org/wiki/Square-integrable%20function | In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows.
One may also speak of quadratic integrability over bounded intervals such as for .
An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part.
The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the space with Among the spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the spaces are complete under their respective -norms.
Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.
Properties
The square integrable functions (in the sense mentioned in which a "function" actually means an equivalence class of functions that are equal almost everywhere) form an inner product space with inner product given by
where
and are square integrable functions,
is the complex conjugate of
is the set over which one integrates—in the first definition (given in the introduction above), is , in the second, is .
Since , square integrability is the same as saying
It can be shown that square integrable functions form a complete metric space under the metric induced by the inner product defined above.
A complete metric space is also called a Cauchy space, because sequences in such metric spaces converge if and only if they are Cauchy.
A space that is complete under the metric induced by a norm is a Banach space.
Therefore, the space of square integrable functions is a Banach space, under the metric induced by the norm, which in turn is induced by the inner product.
As we have the additional property of the inner product, this is specifically a Hilbert space, because the space is complete under the metric induced by the inner product.
This inner product space is conventionally denoted by and many times abbreviated as
Note that denotes the set of square integrable functions, but no selection of metric, norm or inner product are specified by this notation.
The set, together with the specific inner product specify the inner product space.
The space of square integrable functions is the space in which
Examples
The function defined on is in for but not for
The function defined on is square-integrable.
Bounded functions, defined on are square-integr |
https://en.wikipedia.org/wiki/Keita%20Goto%20%28footballer%29 | is a Japanese football player who plays for SC Sagamihara.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Matsumoto Yamaga
1986 births
Living people
Association football people from Ibaraki Prefecture
Japanese men's footballers
J1 League players
J2 League players
Kashima Antlers players
Fagiano Okayama players
Matsumoto Yamaga FC players
SC Sagamihara players
Men's association football defenders |
https://en.wikipedia.org/wiki/Landweber%20exact%20functor%20theorem | In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal group law. The Landweber exact functor theorem (or LEFT for short) can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.
Statement
The coefficient ring of complex cobordism is , where the degree of is . This is isomorphic to the graded Lazard ring . This means that giving a formal group law F (of degree ) over a graded ring is equivalent to giving a graded ring morphism . Multiplication by an integer is defined inductively as a power series, by
and
Let now F be a formal group law over a ring . Define for a topological space X
Here gets its -algebra structure via F. The question is: is E a homology theory? It is obviously a homotopy invariant functor, which fulfills excision. The problem is that tensoring in general does not preserve exact sequences. One could demand that be flat over , but that would be too strong in practice. Peter Landweber found another criterion:
Theorem (Landweber exact functor theorem)
For every prime p, there are elements such that we have the following: Suppose that is a graded -module and the sequence is regular for , for every p and n. Then
is a homology theory on CW-complexes.
In particular, every formal group law F over a ring yields a module over since we get via F a ring morphism .
Remarks
There is also a version for Brown–Peterson cohomology BP. The spectrum BP is a direct summand of with coefficients . The statement of the LEFT stays true if one fixes a prime p and substitutes BP for MU.
The classical proof of the LEFT uses the Landweber–Morava invariant ideal theorem: the only prime ideals of which are invariant under coaction of are the . This allows to check flatness only against the (see Landweber, 1976).
The LEFT can be strengthened as follows: let be the (homotopy) category of Landweber exact -modules and the category of MU-module spectra M such that is Landweber exact. Then the functor is an equivalence of categories. The inverse functor (given by the LEFT) takes -algebras to (homotopy) MU-algebra spectra (see Hovey, Strickland, 1999, Thm 2.7).
Examples
The archetypical and first known (non-trivial) example is complex K-theory K. Complex K-theory is complex oriented and has as formal group law . The corresponding morphism is also known as the Todd genus. We have then an isomorphism
called the Conner–Floyd isomorphism.
While complex K-theory was constructed before by geometric means, many homology theories were first constructed via the Landweber exact functor theorem. This includes elliptic homology, the Johnson–Wilson theories and the Lubin–Tate spectra .
While homology with rational coefficients is Landweber exact, homology with integer coefficients is not Landweber exact. Furthermore, Morava K-theory K(n) is not Landwebe |
https://en.wikipedia.org/wiki/Cantellated%207-simplexes | In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.
There are unique 6 degrees of cantellation for the 7-simplex, including truncations.
Cantellated 7-simplex
Alternate names
Small rhombated octaexon (acronym: saro) (Jonathan Bowers)
Coordinates
The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.
Images
Bicantellated 7-simplex
Alternate names
Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)
Coordinates
The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.
Images
Tricantellated 7-simplex
Alternate names
Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)
Coordinates
The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.
Images
Cantitruncated 7-simplex
Alternate names
Great rhombated octaexon (acronym: garo) (Jonathan Bowers)
Coordinates
The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.
Images
Bicantitruncated 7-simplex
Alternate names
Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)
Coordinates
The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.
Images
Tricantitruncated 7-simplex
Alternate names
Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)
Coordinates
The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.
Images
Related polytopes
This polytope is one of 71 uniform 7-polytopes with A7 symmetry.
See also
List of A7 polytopes
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.
x3o3x3o3o3o3o |
https://en.wikipedia.org/wiki/Runcinated%207-simplexes | In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.
There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.
Runcinated 7-simplex
Alternate names
Small prismated octaexon (acronym: spo) (Jonathan Bowers)
Coordinates
The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex.
Images
Biruncinated 7-simplex
Alternate names
Small biprismated octaexon (sibpo) (Jonathan Bowers)
Coordinates
The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex.
Images
Runcitruncated 7-simplex
Alternate names
Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)
Coordinates
The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex.
Images
Biruncitruncated 7-simplex
Alternate names
Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)
Coordinates
The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex.
Images
Runcicantellated 7-simplex
Alternate names
Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)
Coordinates
The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex.
Images
Biruncicantellated 7-simplex
Alternate names
Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)
Coordinates
The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex.
Images
Runcicantitruncated 7-simplex
Alternate names
Great prismated octaexon (acronym: gapo) (Jonathan Bowers)
Coordinates
The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex.
Images
Biruncicantitruncated 7-simplex
Alternate names
Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)
Coordinates
The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex.
Images
Related polytopes
These polytopes are among 71 uniform 7-polytopes with A7 symmetry.
Notes
References
H.S.M. Coxeter: |
https://en.wikipedia.org/wiki/Stericated%207-simplexes | In seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-simplex.
There are 14 unique sterication for the 7-simplex with permutations of truncations, cantellations, and runcinations.
Stericated 7-simplex
Alternate names
Small cellated octaexon (acronym: sco) (Jonathan Bowers)
Coordinates
The vertices of the stericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 8-orthoplex.
Images
Bistericated 7-simplex
Alternate names
Small bicellated hexadecaexon (acronym: sabach) (Jonathan Bowers)
Coordinates
The vertices of the bistericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 8-orthoplex.
Images
Steritruncated 7-simplex
Alternate names
Cellitruncated octaexon (acronym: cato) (Jonathan Bowers)
Coordinates
The vertices of the steritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 8-orthoplex.
Images
Bisteritruncated 7-simplex
Alternate names
Bicellitruncated octaexon (acronym: bacto) (Jonathan Bowers)
Coordinates
The vertices of the bisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,3). This construction is based on facets of the bisteritruncated 8-orthoplex.
Images
Stericantellated 7-simplex
Alternate names
Cellirhombated octaexon (acronym: caro) (Jonathan Bowers)
Coordinates
The vertices of the stericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 8-orthoplex.
Images
Bistericantellated 7-simplex
Alternate names
Bicellirhombihexadecaexon (acronym: bacroh) (Jonathan Bowers)
Coordinates
The vertices of the bistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,3). This construction is based on facets of the stericantellated 8-orthoplex.
Images
Stericantitruncated 7-simplex
Alternate names
Celligreatorhombated octaexon (acronym: cagro) (Jonathan Bowers)
Coordinates
The vertices of the stericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,4). This construction is based on facets of the stericantitruncated 8-orthoplex.
Images
Bistericantitruncated 7-simplex
Alternate names
Bicelligreatorhombated octaexon (acronym: bacogro) (Jonathan Bowers)
Coordinates
The vertices of the bistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,4). This construction is based on facets of the bistericantitruncated 8-orthoplex.
Images
Steriruncinated 7-simplex
Alternate names
Celliprismated octaexon (acronym: cepo) (Jonathan Bo |
https://en.wikipedia.org/wiki/Pentellated%207-simplexes | In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-simplex.
There are 16 unique pentellations of the 7-simplex with permutations of truncations, cantellations, runcinations, and sterications.
Pentellated 7-simplex
Alternate names
Small terated octaexon (acronym: seto) (Jonathan Bowers)
Coordinates
The vertices of the pentellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,1,2). This construction is based on facets of the pentellated 8-orthoplex.
Images
Pentitruncated 7-simplex
Alternate names
Teritruncated octaexon (acronym: teto) (Jonathan Bowers)
Coordinates
The vertices of the pentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,3). This construction is based on facets of the pentitruncated 8-orthoplex.
Images
Penticantellated 7-simplex
Alternate names
Terirhombated octaexon (acronym: tero) (Jonathan Bowers)
Coordinates
The vertices of the penticantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,2,3). This construction is based on facets of the penticantellated 8-orthoplex.
Images
Penticantitruncated 7-simplex
Alternate names
Terigreatorhombated octaexon (acronym: tegro) (Jonathan Bowers)
Coordinates
The vertices of the penticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 8-orthoplex.
Pentiruncinated 7-simplex
Alternate names
Teriprismated octaexon (acronym: tepo) (Jonathan Bowers)
Coordinates
The vertices of the pentiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,2,3). This construction is based on facets of the pentiruncinated 8-orthoplex.
Images
Pentiruncitruncated 7-simplex
Alternate names
Teriprismatotruncated octaexon (acronym: tapto) (Jonathan Bowers)
Coordinates
The vertices of the pentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,4). This construction is based on facets of the pentiruncitruncated 8-orthoplex.
Images
Pentiruncicantellated 7-simplex
Alternate names
Teriprismatorhombated octaexon (acronym: tapro) (Jonathan Bowers)
Coordinates
The vertices of the pentiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,3,4). This construction is based on facets of the pentiruncicantellated 8-orthoplex.
Images
Pentiruncicantitruncated 7-simplex
Alternate names
Terigreatoprismated octaexon (acronym: tegapo) (Jonathan Bowers)
Coordinates
The vertices of the pentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,5). This construction is based on facets of the pentiruncicantitruncated 8-orthoplex.
Images
Pentistericated 7-simplex
Alternate names
Teri |
https://en.wikipedia.org/wiki/Hexicated%207-simplexes | In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.
There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations.
The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called a omnitruncated 7-simplex with all of the nodes ringed.
Hexicated 7-simplex
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.
Root vectors
Its 56 vertices represent the root vectors of the simple Lie group A7.
Alternate names
Expanded 7-simplex
Small petated hexadecaexon (acronym: suph) (Jonathan Bowers)
Coordinates
The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, .
A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of:
(1,-1,0,0,0,0,0,0)
Images
Hexitruncated 7-simplex
Alternate names
Petitruncated octaexon (acronym: puto) (Jonathan Bowers)
Coordinates
The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .
Images
Hexicantellated 7-simplex
Alternate names
Petirhombated octaexon (acronym: puro) (Jonathan Bowers)
Coordinates
The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .
Images
Hexiruncinated 7-simplex
Alternate names
Petiprismated hexadecaexon (acronym: puph) (Jonathan Bowers)
Coordinates
The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .
Images
Hexicantitruncated 7-simplex
Alternate names
Petigreatorhombated octaexon (acronym: pugro) (Jonathan Bowers)
Coordinates
The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .
Images
Hexiruncitruncated 7-simplex
Alternate names
Petiprismatotruncated octaexon (acronym: pupato) (Jonathan Bowers)
Coordinates
The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on |
https://en.wikipedia.org/wiki/SCGP | SCGP may refer to:
Simons Center for Geometry and Physics
Supplier Credit Guarantee Program |
https://en.wikipedia.org/wiki/List%20of%20Indonesian%20provinces%20by%20Human%20Development%20Index |
Methodology
The figures come from the Indonesia Human Development Report, published by Statistics Indonesia and Human Development Index (by UN Method) of Indonesian provinces since 1990 (2021 revision).
By Statistics Indonesia in 2022
Source by statistic Indonesia published by Statistics Indonesia
Cities and regencies
List of cities of Indonesia with very high HDI (2022)
According to BPS Indonesia 2020 data, the total population of all cities with very high HDI is 37,570,879 people.
List of regencies of Indonesia with very high HDI (2022)
According to BPS Indonesia 2020 data, the total population of all regencies with very high HDI is 4,693,525 people.
By UNDP reports
Trends by Statistics Indonesia
Trends by UNDP reports
Notes
See also
Economy of Indonesia
List of Indonesian provinces by GDP
List of Indonesian provinces by GRP per capita
List of Indonesian cities by GDP
References
Indonesia
Human Development Index
Provinces by HDI
Human Development Index
Provinces of Indonesia by HDI
Indonesia |
https://en.wikipedia.org/wiki/Ibn%20Ghazi%20al-Miknasi | Abu Abdallah Muhammad b. Ahmad b. Muhammad Ibn Ghazi al-'Utmani al-Miknasi () (1437–1513) was a Moroccan scholar in the field of history, Islamic law, Arabic philology and mathematics. He was born in Meknes from Banu Uthman, a clan in the Berber kutama tribe, but spent his life in Fez. Ibn Ghazi wrote a three-volume history of Meknes and a commentary to the treatise of Ibn al-Banna, Munyat al-hussab. For an explanation of his work, Ibn Ghazi wrote another treatise (about 300 pages long) titled Bughyat al-tulab fi sharh munyat al-hussab ("The desire of students for an explanation of the calculator's craving"). He included sections on arithmetic and algebraic methods.
He is also the author of Kulliyat, a short work on legal questions and judgements in the Maliki madhab.
References
Further reading
Al-Hamza MH, Ibn Ghazi al-Fasi al-Miknasi and his treatise “The purpose of studying in explaining the desire of calculators" . Academic year-long conference IHST Academy of Sciences. M. 2005. pp. 299–301.
1437 births
1513 deaths
15th-century Berber people
16th-century Berber people
15th-century mathematicians
15th-century Moroccan historians
16th-century Moroccan historians
Berber historians
Berber scholars
Berber scientists
Berber writers
Kutama
Linguists from Morocco
Moroccan Maliki scholars
Medieval Moroccan mathematicians
Moroccan scientists
People from Fez, Morocco
People from Meknes
15th-century jurists
16th-century jurists |
https://en.wikipedia.org/wiki/School%20of%20Mathematics%20and%20Physics | The School of Mathematics and Physics (SMP) is in the Faculty of Science at The University of Queensland in Brisbane, Australia.
History
Mathematics and Physics was a foundation department of The University of Queensland when it opened in 1911. Professor Henry James Priestley was the foundation Professor of Mathematics and Physics. In 1922 the departments separated and Professor Thomas Parnell became head of physics. The two departments were joined again in 2000, along with Earth Sciences, as the School of Physical Sciences until Earth Sciences was moved out of the school in 2009 when it was renamed as Mathematics and Physics.
The school occupies several buildings on the St Lucia campus including Parnell Building on the Great Court, the Physics Annex and Priestley Building.
Research
The school has a long and successful history of research. Its most famous experiment is Parnell's Pitch Drop Experiment which is the longest running scientific experiment in the world. The school has achieved international standing in research areas including applied mathematics (particularly in biological sciences), quantum computing, quantum atom optics, engineered quantum systems, astrophysics and soft condensed matter physics.
The school is one of the strongest performing research schools in the country with a number of high-profile researchers and centres. Recent research successes include:
Australian Research Council Centres of Excellence
Australian Centre for Quantum Atom Optics (ACQAO)
Centre for Engineered Quantum Systems (EQUS)
Centre for Mathematics and Statistics of Complex Systems (MASCOS)
Centre for Quantum Computing and Communication Technology (CQCCT) previously known as Centre for Quantum Computer Technology (CQCT)
Australian Research Council Federation Fellows
Professor Kevin Burrage (mathematics)
Professor Gerard Milburn (physics)
Professor Michael Nielsen (physics)
Professor Hugh Possingham (mathematics)
Professor Guifre Vidal (physics)
Professor Andrew White (physics)
Teaching
Teaching staff from the School of Mathematics and Physics regularly win national teaching awards and grants. Recent winners include:
Professor Peter Adams (mathematics)
Mr Michael Jennings (mathematics)
Dr Margaret Wegener (physics)
Outreach
The school has an active outreach program including the highly popular science lecture series BrisScience, physics and mathematics colloquia series, the demo troupe (undergraduate students perform physics shows aimed at high school students and general audiences) and the Physics Museum curated by Professor Norman Heckenberg.
The demo troupe regularly performs at schools around Brisbane and also travels to regional and remote Queensland to promote science.
Alumni
The school has an active alumni network and holds several alumni reunion events each year.
Senior School Staff
Professor Halina Rubinsztein-Dunlop, Head of School
Professor Joe Grotowski, Head of Mathematics
Professor Michael Drinkwater, Head of Physics
Mr Chr |
https://en.wikipedia.org/wiki/Humberston%20Academy | Humberston Academy (formerly Humberston Maths and Computing College) is a secondary school with academy status (DRET) based in Humberston (near Grimsby), North East Lincolnshire, England.
Admissions
It does not have a sixth form. It is situated on Humberston Avenue (B1219 - off the A1031) in the west of Humberston. Further to the west along the B1219 is the Tollbar Academy, and the school is less than one mile north of Lincolnshire (East Lindsey - Tetney and Holton-le-Clay, which has grammar schools). Directly to the west is the Humberston Country Club golf club.
History
The school opened in 1977 as Humberston Comprehensive School.
Until 1996 it was administered by Humberside Education Committee, based in Beverley. It gained specialist status in 2006.
In January 2009 it was placed in special measures. From September 2009, the school name changed from Humberston School to Humberston Maths and Computing College.
The school converted to academy status in 2011 and was renamed Humberston Academy. the school is now sponsored by the David Ross Education Trust.
Academic performance
In the last round of GCSE exams, the school gained the third best GCSE results for North East Lincolnshire.
Notable former pupils
Keeley Donovan, BBC weather presenter
Max Wright (English footballer), Grimsby Town footballer: Max Wright
Harry Clifton (footballer, born 1998), Grimsby Town footballer: Harry Clifton
References
External links
School website
EduBase
Schools in Grimsby
Educational institutions established in 1977
Secondary schools in the Borough of North East Lincolnshire
1977 establishments in England
Academies in the Borough of North East Lincolnshire |
https://en.wikipedia.org/wiki/Georges%20Darmois | Georges Darmois (24 June 1888 – 3 January 1960) was a French mathematician and statistician. He pioneered in the theory of sufficiency, in stellar statistics, and in factor analysis. He was also one of the first French mathematicians to teach British mathematical statistics.
He is one of the eponyms of the Koopman–Pitman–Darmois theorem and sufficient statistics and exponential families.
Biography
Darmois was born on 24 June 1888 in Éply. He was admitted to École normale supérieure in 1906 and passed subsequently the agrégation de mathematiques in 1909. From 1911 to 1914, he was a qualified assistant (agrégé préparateur) at the École normale supérieure, where his scientific activities were directed by Émile Borel who rapidly appreciated his talent.
Darmois earned his doctorate from the University of Paris in 1921. He defended a thesis on algebraic curves and partial differential equations before the jury consisting of Émile Picard and Édouard Goursat. In 1949, he succeeded Maurice René Fréchet as the Chair of Calculus of Probabilities and Mathematical Physics at the University of Paris, who had himself succeeded Emile Borel. His research spanned several fields of pure and applied mathematics, including geometry, general relativity, physics, statistics, time series, and econometrics.
He was elected fellow of the Econometric Society in 1952.
In 1955 he was elected as a Fellow of the American Statistical Association. He was also the president of International Statistical Institute from 1953 until 1960.
Contributions
His scientific contributions include:
Giving the first rigorous proof of Fréchet–Darmois–Cramér–Rao inequality, also known as Cramér–Rao bound, in 1945 independently from Rao and Cramér.
Developing the notion of Koopman–Darmois family of distributions also known as the exponential family of distributions.
Establishing Koopman–Pitman–Darmois theorem
Characterizing Gaussian distributions with Darmois–Skitovich theorem
References
External links
Correspondence between Ronald Fisher and Georges Darmois
Laurent Mazliak (2010) Borel, Fréchet, Darmois: La découverte des statistiques par les probabilistes français. Journal Electronique d'Histoire des Probabilités et de la Statistique December.
John Aldrich (2010) Tales of Two Societies: London and Paris 1860-1940 Journal Electronique d'Histoire des Probabilités et de la Statistique December.
1888 births
1960 deaths
French mathematicians
20th-century American mathematicians
University of Paris alumni
Academic staff of the University of Paris
Presidents of the International Statistical Institute
Members of the French Academy of Sciences
Fellows of the American Statistical Association
Fellows of the Econometric Society |
https://en.wikipedia.org/wiki/Bryan%20Shader | Bryan Lynn Shader (born 17 December 1961) is a professor of mathematics at the University of Wyoming. He received his Ph.D. from University of Wisconsin-Madison in 1990, his advisor was Professor Richard Brualdi. Shader is the Editor-in-chief of the Electronic Journal of Linear Algebra. He is also Associate Editor of other two journals, Linear Algebra and its Applications (since 2003) and Linear & Multilinear Algebra (since 2009). He is one of the most active mathematicians working on Combinatorial Matrix Theory. He is also noted for his monograph on matrices of sign-solvable linear systems. Besides organizing many workshops, he is a co-principal investigator of Math Teacher Leadership Program, a National Science Foundation project (2009–2014). Shader is special assistant to the Vice-President of Research of University of Wyoming.
Shader received the 2005 Burton W. Jones Distinguished Teaching Award from the Rocky Mountain Section of the Mathematical Association of America.
Personal life
Bryan Shader has a daughter named Sarah Shader who graduated from the Massachusetts Institute of Technology with a degree in computer science.
Books
(With Richard Brualdi) Matrices of sign-solvable linear systems. Cambridge Tracts in Mathematics, 116, 1995
(With Richard Brualdi) Graph and Matrices, Chapter 3 in Topics in Algebraic Graph Theory, Encyclopedia of Mathematics and its Applications, 102, Cambridge University Press, 2005
Bipartite Graph and Matrices, Chapter 30, Handbook of Linear Algebra, CRC Press, 2007
References
Notes
Bryan Shader at University of Wyoming website
Math Sci-Net
External links
Bryan Shader at the Mathematics Genealogy Project
20th-century American mathematicians
21st-century American mathematicians
Combinatorialists
University of Wisconsin–Madison alumni
University of Wyoming faculty
1961 births
Living people
Electronic Journal of Linear Algebra editors |
https://en.wikipedia.org/wiki/Habel%20Satya | Habel Satya (born 12 September 1987 in Wamena, Papua) is an Indonesian former footballer who plays as a midfielder. He is known for his acceleration, pace, and agility in the field.
Career statistics
As of 27 June 2012.
References
External links
Profile Habel Satya at Liga-Indonesia.co.id
Profile Habel Satya at Persiwa-mania.blogspot.com
Indonesian men's footballers
1987 births
Living people
Papuan people
Indonesian Christians
People from Wamena
Indonesian Premier Division players
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
Persiwa Wamena players
Persigubin Pegunungan Bintang players
Yahukimo F.C. players
Men's association football midfielders
Indonesia men's youth international footballers
Footballers from Papua |
https://en.wikipedia.org/wiki/String%20topology | String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by .
Motivation
While the singular cohomology of a space has always a product structure, this is not true for the singular homology of a space. Nevertheless, it is possible to construct such a structure for an oriented manifold of dimension . This is the so-called intersection product. Intuitively, one can describe it as follows: given classes and , take their product and make it transversal to the diagonal . The intersection is then a class in , the intersection product of and . One way to make this construction rigorous is to use stratifolds.
Another case, where the homology of a space has a product, is the (based) loop space of a space . Here the space itself has a product
by going first through the first loop and then through the second one. There is no analogous product structure for the free loop space of all maps from to since the two loops need not have a common point. A substitute for the map is the map
where is the subspace of , where the value of the two loops coincides at 0 and is defined again by composing the loops.
The Chas–Sullivan product
The idea of the Chas–Sullivan product is to now combine the product structures above. Consider two classes and . Their product lies in . We need a map
One way to construct this is to use stratifolds (or another geometric definition of homology) to do transversal intersection (after interpreting as an inclusion of Hilbert manifolds). Another approach starts with the collapse map from to the Thom space of the normal bundle of . Composing the induced map in homology with the Thom isomorphism, we get the map we want.
Now we can compose with the induced map of to get a class in , the Chas–Sullivan product of and (see e.g. ).
Remarks
As in the case of the intersection product, there are different sign conventions concerning the Chas–Sullivan product. In some convention, it is graded commutative, in some it is not.
The same construction works if we replace by another multiplicative homology theory if is oriented with respect to .
Furthermore, we can replace by . By an easy variation of the above construction, we get that is a module over if is a manifold of dimensions .
The Serre spectral sequence is compatible with the above algebraic structures for both the fiber bundle with fiber and the fiber bundle for a fiber bundle , which is important for computations (see and ).
The Batalin–Vilkovisky structure
There is an action by rotation, which induces a map
.
Plugging in the fundamental class , gives an operator
of degree 1. One can show that this operator interacts nicely with the Chas–Sullivan product in the sense that they form together the structure of a Batalin–Vilkovisky algebra on . This operator tends to be difficult to compute in general. The defining identities of a Batalin-Vilkovisky algebra were checked in the or |
https://en.wikipedia.org/wiki/Erwin%20Schr%C3%B6dinger%20International%20Institute%20for%20Mathematics%20and%20Physics | The Erwin Schrödinger International Institute for Mathematics and Physics (ESI) is a visitors oriented research institute in Vienna, Austria. It is located close to the city center in the remodeled historical premises of a seminary in Boltzmanngasse 9 in Vienna's ninth district.
The Institute was founded upon the initiatives of Peter W. Michor and Walter Thirring and opened on 20 April 1993. It was run by the private ESI association under the auspices of the Austrian Ministry of Science until 31 December 2010. Since 1 June 2011 the ESI has been embedded into the University of Vienna.
It has close connections with the Faculty of Physics and the Faculty of Mathematics of the University of Vienna.
The ESI hosts high-profile thematic programs, workshops, summer and winter schools, junior and senior research fellowships, as well as a program for research in teams.
The ESI publishes Scientific Reports with detailed records of activities, guests, and related preprints. In 2019, the year before the COVID-19 pandemic, the ESI recorded an all-time high of 1116 participants and 171 preprints related to its activities.
The ESI is managed by the ESI Director, who is supported by two Deputy Directors and reports directly to the Rectorate of the University of Vienna. The Director and the Deputy Directors are part of the ESI Kollegium, which is composed of three professors from the Faculty of Mathematics and three from the Faculty of Physics. The Kollegium oversees the operation of the Institute and decides over proposals for potential future workshops, schools, fellowships, and research in teams at the ESI based on a peer review process and available resources. Decisions on future thematic programs at the ESI are made by a Scientific Advisory Board that is composed of six to eight international peers.
The current ESI Director is Christoph Dellago; the Scientific Advisory Board is chaired by Sandra Di Rocco.
As of 2020, the Institute awards the annual Medal of the Erwin Schrödinger Institute for Mathematics and Physics (or ESI Medal, for short) to celebrate recent breakthroughs in any area of mathematics or physics. The selection is made by the Scientific Advisory Board based on nominations from previous recipients of the ESI Medal, organizers of thematic programs at the ESI, former ESI Directors, former members of the ESI Scientific Advisory Board, and the president of the ESI Association. The recipients of the ESI Medal to date are
Anton Alekseev (2020)
Elliott Lieb (2021)
Martin Hairer (2022)
Isabelle Gallagher (2023)
History
The private association Erwin Schrödinger International Institute for Mathematical Physics was founded in April 1992 under the auspices of the Austrian Federal Minister of Science, Erhard Busek, and upon the initiatives of Peter W. Michor and Walter Thirring.
Walter Thirring was elected president of the association at the first general assembly on 27 May 1992, with deputies Wolfgang Reiter (to represent the Ministry of |
https://en.wikipedia.org/wiki/Leelavati%20Award | The Leelavati Award is an award for outstanding contribution to public outreach in mathematics. It is named after the 12th-century mathematical treatise "Lilavati" devoted to arithmetic and algebra written by the Indian mathematician Bhāskara II, also known as Bhaskara Achārya. In the book the author posed, in verse form, a series of problems in (elementary) arithmetic to one Leelavati (perhaps his daughter) and followed them up with hints to solutions. This work appears to have been the main source of learning arithmetic and algebra in medieval India. The work was also translated into Persian and was influential in West Asia.
History
The Leelavati Prize was handed out for the first time at the closing ceremony of the International Congress of Mathematicians (ICM) 2010 in Hyderabad, India. Established by the Executive Organising Committee (EOC) of the ICM with the endorsement of the IMU Executive Committee (EC), the Leelavati Prize was initiated as a one-time international award for outstanding public outreach work for mathematics. The award was so well received at the conference and in the mathematical press that the IMU decided to turn the prize into a recurring four-yearly award and the award ceremony a regular feature of every ICM closing ceremony.
The Leelavati prize is not intended to reward mathematical research but rather outreach activities in the broadest possible sense. It carries a cash prize of 1,000,000 Indian Rupees ( US dollars) together with a citation, and is sponsored by Infosys since 2014.
Laureates
See also
List of mathematics awards
References
External links
Leelavati Prize - IMU website
Awards of the International Mathematical Union
Awards established in 2010 |
https://en.wikipedia.org/wiki/Anders%20Wiman | Anders Wiman (11 February 1865 – 13 August 1959) was a Swedish mathematician. He is known for his work in algebraic geometry and applications of group theory.
Life
Wiman was born to well-off land-owing farmer family in Hammarlöv, Sweden in 1865. He attended school in Lund, and graduated from secondary school in 1885. In the autumn of the same year, Wiman went to Lund University studying Mathematics, Botany and Latin. He attained Bachelor's degree in 1887 and his Licentiate in 1891. He continued his study in the same university under supervision of Carl Fabian Björling and was awarded doctorate in 1892, with thesis Klassifikation af regelytorna af sjette graden (Classification of regular surfaces of degree 6).
In 1892, Wiman was appointed as a docent (equivalent of assistant professor) in Lund University. There, his work on the classification of finite geometrical groups in the last few years of the 19th century was seen impressive. He classified all algebraic curves of genus 3, 4, 5 and 6 which have non-trivial algebraic automorphisms. In a paper for approximations to small denominators in 1900, Wiman applied measure theory to the probabilistic problem, and became the first person to do so.
In 1901, Wiman accepted an extraordinary professorship in algebra and number theory at Uppsala University. Topics he studied here included solubility of algebraic equations, Galois group of soluble equations of prime degree, and entire functions. Wiman was promoted to an ordinary professor in 1906, and held the chair at Uppsala until 1929. In that year, Wiman was made professor emeritus, but he remained active in teaching.
In 1904 Wiman was an Invited Speaker of the ICM in Heidelberg. Wiman was from 1908 an editor of Acta Mathematica.
Wiman returned to Lund in his final year, and stayed there until he died.
Awards and honours
Membership of the Royal Physiographic Society in Lund (1900);
Membership of the Royal Society of Sciences in Uppsala (1905);
Membership of Royal Swedish Academy of Sciences (1905);
Membership of the Royal Society of Arts and Sciences in Gothenburg (1920);
Honorary membership of the Royal Society of Sciences in Uppsala (1938).
Work
The main focus of Wiman's research was algebraic geometry and applications of group theory to geometry and function theory. He proved that for n > 7, in less than n–2 dimensions, there are no groups of collineations that are isomorphic to the symmetric or alternating group on n symbols. He also determined all finite groups of birational transformations of the plane. Wiman wrote the article on finite groups of linear transformations for Klein's encyclopedia.
In function theory he did important work on entire functions. From 1914 to 1916 he introduced what is now called Wiman-Valiron theory (named after him and Georges Valiron). Wiman's generalization of a theorem of Hadamard is known as Wiman's theorem. Wiman's theorems for quasiregular mappings shows that an entire holomorphic function of order less |
https://en.wikipedia.org/wiki/Algebraic%20number%20field | In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension).
Thus is a field that contains and has finite dimension when considered as a vector space over
The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind usual rational numbers, by using algebraic methods.
Definition
Prerequisites
The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication.
Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples)
(x1, x2, …)
whose entries are elements of a fixed field, such as the field Any two such sequences can be added by adding the corresponding entries. Furthermore, any sequence can be multiplied by a single element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "infinite-dimensional", that is to say that the sequences constituting the vector spaces are of infinite length. If, however, the vector space consists of finite sequences
(x1, x2, …, xn),
the vector space is said to be of finite dimension, n.
Definition
An algebraic number field (or simply number field) is a finite-degree field extension of the field of rational numbers. Here degree means the dimension of the field as a vector space over
Examples
The smallest and most basic number field is the field of rational numbers. Many properties of general number fields are modeled after the properties of At the same time, many other properties of algebraic number fields are substantially different from the properties of rational numbers - one notable example is that the ring of algebraic integers of a number field is not a principal ideal domain, in general.
The Gaussian rationals, denoted (read as " adjoined "), form the first (historically) non-trivial example of a number field. Its elements are elements of the form where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity Explicitly, Non-zero Gaussian rational numbers are invertible, which can be seen from the identity It follows that the Gaussian rationals form a number field which is two-dimensional as a vec |
https://en.wikipedia.org/wiki/Nikita%20Nekrasov | Nikita Alexandrovich Nekrasov (; born 10 April 1973) is a mathematical and theoretical physicist at the Simons Center for Geometry and Physics and C.N.Yang Institute for Theoretical Physics at Stony Brook University in New York, and a Professor of the Russian Academy of Sciences.
Career
Nekrasov studied at the Moscow State 57th School in 1986–1989. He graduated with honors from the Moscow Institute of Physics and Technology in 1995, and joined the theory division of the Institute for Theoretical and Experimental Physics. In parallel, in 1994–1996 Nekrasov did his graduate work at Princeton University, under the supervision of David Gross. His Ph.D. thesis on Four Dimensional Holomorphic Theories was defended in 1996.
He joined Harvard Society of Fellows at Harvard University as a Junior Fellow 1996–1999.
He was then a Robert. H. Dicke Fellow at Princeton University from 1999 to 2000. In 2000 he moved to France as a permanent professor at the Institut des Hautes Études Scientifiques. Since 2013, he is a full professor at the Simons Center for Geometry and Physics and C. N. Yang Institute for Theoretical Physics at Stony Brook University in New York.
Research
Nekrasov is mostly known for his work on supersymmetric gauge theory, quantum integrability, and string theory. The Nekrasov partition function, which he introduced in his 2002 paper, relates in an intricate way the instantons in gauge theory, integrable systems, and representation theory of infinite-dimensional algebras.
Honours and awards
For his discovery of noncommutative instantons together with Albert Schwarz in 1998, noncommutative monopoles and monopole strings with David Gross in 2000 and for his work with Alexander S. Gorsky on the relations between gauge theories and many-body systems he was awarded the of the French Academy of Sciences in 2004. For his contributions to topological string theory and the ADHM construction he received the Hermann Weyl Prize in 2004. In 2008 together with Davesh Maulik, Andrei Okounkov and Rahul Pandharipande he formulated a set of conjectures relating Gromov–Witten theory and Donaldson–Thomas theory, for which the four authors were awarded the Compositio Prize in 2009.
In late 2022, it was announced that Nekrasov was awarded the 2023 Dannie Heineman Prize for Mathematical Physics by the American Institute of Physics (AIP) and the American Physical Society (APS). The prize was awarded for Nekrasov's “elegant application of powerful mathematical techniques to extract exact results for quantum field theories, as well as shedding light on integrable systems and non-commutative geometry.”
References
Living people
1973 births
Russian string theorists
French physicists
Jewish physicists
Mathematical physicists
Princeton University alumni
Theoretical physicists
Moscow Institute of Physics and Technology alumni
Stony Brook University faculty |
https://en.wikipedia.org/wiki/Carley%20Garner | Carley Garner (born 1977) is an American commodity market strategist and futures and options broker and the author of Trading Commodity Options with Creativity, Higher Probability Commodity Trading, and A Trader's First Book on Commodities, published by DT publishing an imprint of Wyatt-MacKenzie. She has also previously written four books published by FT Press, Currency Trading in the FOREX and Futures Markets, A Trader's First Book on Commodities (two editions), and Commodity Options. Commodity Options was named one of the "Top 10 Investing & Trading Books of 2009" by SFO Magazine. A Trader's First Book on Commodities (2nd Ed.) was named as the best futures trading book in 2021 by a UK financial education website. Garner was also featured in FT Press' e-book series entitled "Insights for the Agile Investor", and an educational video series. Carley can also be found at TradersEXPOs and MoneyShows throughout the country.
Carley Garner, considered an expert in commodity market technical analysis, is a regular contributor to the Mad Money television show hosted by Jim Cramer on CNBC. She is also a regular guest on Bloomberg Television's Options Insight segment with Abigail Doolittle. and appears weekly on RFD-TV's Cow Guy Close with Scott Shellady. Garner also writes content for RealMoney, the premium service of TheStreet.com.
She has been featured by Technical Analysis of Stocks & Commodities magazine as a monthly columnist since early 2008 and is a RealMoney.com contributor to TheStreet.com.
Early life and education
After Garner graduated from Monticello High School in Monticello, Utah in 1996, she attended Snow College where she earned an associate's degree. In 2003, she graduated from the University of Nevada, Las Vegas with bachelor's degrees in accounting and finance with Magna Cum Laude accolades.
Career
During her time in the commodity industry, Garner has been an active member of the financial media. Along with regular television appearances, she has participated in several radio interviews and is often quoted and referenced in business news publications such as The Wall Street Journal and monthly print publications such as Futures Magazine. In addition to a monthly column in Stocks & Commodities Magazine, her work is also published in multiple national magazines and on several industry trading education websites, including TheStreet.com, Barchart.com, and many others.
Garner's chart analysis is frequently featured on the Mad Money TV show on CNBC. On the show, Jim Cramer discusses the commodity market analysis of Carley Garner on markets such as gold, crude oil, natural gas, and Treasuries.
Carley Garner appears on RFD-TV's Cow Guy Close program hosted by Scott Shellady every Wednesday afternoon to discuss current events in the commodity and financial markets. She also joins the Market Day Report on the same network each Friday to discuss grain market fundamentals and news.
Carley Garner has appeared on various programs on Bloo |
https://en.wikipedia.org/wiki/Spectrahedron | In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of positive semidefinite matrices forms a convex cone in , and a spectrahedron is a shape that can be formed by intersecting this cone with a affine subspace.
Spectrahedra are the feasible regions of semidefinite programs. The images of spectrahedra under linear or affine transformations are called projected spectrahedra or spectrahedral shadows. Every spectrahedral shadow is a convex set that is also semialgebraic, but the converse (conjectured to be true until 2017) is false.
An example of a spectrahedron is the spectraplex, defined as
,
where is the set of positive semidefinite matrices and is the trace of the matrix . The spectraplex is a compact set, and can be thought of as the "semidefinite" analog of the simplex.
See also
N-ellipse - a special case of spectrahedra.
References
Real algebraic geometry |
https://en.wikipedia.org/wiki/Spherical%20shell | In geometry, a spherical shell is a generalization of an annulus to three dimensions. It is the region of a ball between two concentric spheres of differing radii.
Volume
The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere:
where is the radius of the inner sphere and is the radius of the outer sphere.
Approximation
An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell:
when is very small compared to ().
The total surface area of the spherical shell is .
See also
Spherical pressure vessel
Ball
Solid torus
Bubble
Sphere
References
Elementary geometry
Geometric shapes
Spherical geometry |
https://en.wikipedia.org/wiki/List%20of%20countries%20by%20cremation%20rate | This article is a list of countries by cremation rate. Cremation rates vary widely across the world. As of 2019, international statistics report that countries with large Buddhist and Hindu populations like Bhutan, Cambodia, Hong Kong, Japan, Myanmar, Nepal, Tibet, Sri Lanka, South Korea, Thailand and India have a cremation rate ranging from 80% to 99%, while Roman Catholic majority-countries like Italy, France, Ireland, Latvia, Poland, Spain, and Portugal report much lower rates. Factors include both culture and religion; for example, the cremation rate in Muslim, Eastern Orthodox, Oriental Orthodox, and Roman Catholic majority-countries is much lower due to religious sanctions on the practice of cremation, whereas for Hindu, Jain, and Buddhist majority-countries the cremation rate is much higher. However, economic factors such as cemetery fees, prices on coffins and funerals greatly impel towards the choice of cremation. Also, in many countries where population following Hindu lifestyle is on rise, the number of people performing cremation of their dead ones is also increasing.
Africa
Zimbabwe
Cremation is still considered taboo in Zimbabwe, but the practice is not forbidden. The Bulawayo City Council, the second largest city in the country, planned mandatory cremation for those that died before the age of 25. However, this plan was cancelled after many protests from Pentecostal Christian groups.
South Africa
The rate of cremation is about 12% in Cape Town, which has a significant White population, but it is lower in other parts of the country.
Ghana
There is a crematorium in Accra, the capital of Ghana, but the cremation rate is low. Pentecostal Christians, which constitute the largest religious group in the country, are officially against cremation.
Kenya
Nairobi has the only crematorium in Kenya. Since Kenya is a Christian-majority country, the opposition against cremation largerly derives from Christian beliefs about the practice.
Nigeria
Cremation is legal in the Lagos State of Nigeria.
Asia
China
The People's Republic of China reported 4,534,000 cremations out of 9,348,453 deaths (a 48.50% rate) in 2008. The cremation rate was 45.6% for 2014, according to the Chinese Ministry of Civil Affairs.
India
Almost all people adhering to Sanatana Dharma also known as Hinduism which is in majority and it’s other sects like Buddhism, Jainism, and Sikhism choose cremation as the traditional method of disposal of the dead, which makes the Republic of India one of the countries with the highest cremation rate in the world around 80.1% and is increasing day by day. However, different Indigenous peoples of India follow various funerary practices, which include cremation, inhumation, or both methods.
Japan
Japan has one of the highest cremation rates in the world, with the country reporting a cremation rate of 99.97% in 2019.
Nepal
Almost everyone adhering to Hinduism and Buddhism cremates their dead, which makes Nepal one of the countries wi |
https://en.wikipedia.org/wiki/Charuchandra%20College | Charuchandra College is an undergraduate college of science, arts and commerce with 8 science departments including statistics NAAC grade B++ and of 2.59 in Kolkata, India. It is affiliated to the University of Calcutta.
Notable alumni
Dolon Roy, actress
Pronay Halder, footballer
Swadesh Bharati, poet
Swapna Barman, athlete
Partho Mukherjee, actor
See also
List of colleges affiliated to the University of Calcutta
Education in India
Education in West Bengal
References
External links
University of Calcutta affiliates
Universities and colleges established in 1947
1947 establishments in India |
https://en.wikipedia.org/wiki/1/3%E2%80%932/3%20conjecture | In order theory, a branch of mathematics, the 1/3–2/3 conjecture states that, if one is comparison sorting a set of items then, no matter what comparisons may have already been performed, it is always possible to choose the next comparison in such a way that it will reduce the number of possible sorted orders by a factor of 2/3 or better. Equivalently, in every finite partially ordered set that is not totally ordered, there exists a pair of elements x and y with the property that at least 1/3 and at most 2/3 of the linear extensions of the partial order place x earlier than y.
Example
The partial order formed by three elements a, b, and c with a single comparability relationship, has three linear extensions, and In all three of these extensions, a is earlier than b. However, a is earlier than c in only two of them, and later than c in the third. Therefore, the pair of a and c have the desired property, showing that this partial order obeys the 1/3–2/3 conjecture.
This example shows that the constants 1/3 and 2/3 in the conjecture are tight; if q is any fraction strictly between 1/3 and 2/3, then there would not exist a pair x, y in which x is earlier than y in a number of partial orderings that is between q and times the total number of partial orderings.
More generally, let P be any series composition of three-element partial orders and of one-element partial orders, such as the one in the figure. Then P forms an extreme case for the 1/3–2/3 conjecture in the sense that, for each pair x, y of elements, one of the two elements occurs earlier than the other in at most 1/3 of the linear extensions of P. Partial orders with this structure are necessarily series-parallel semiorders; they are the only known extreme cases for the conjecture and can be proven to be the only extreme cases with width two.
Definitions
A partially ordered set is a set X together with a binary relation ≤ that is reflexive, antisymmetric, and transitive. A total order is a partial order in which every pair of elements is comparable. A linear extension of a finite partial order is a sequential ordering of the elements of X, with the property that if x ≤ y in the partial order, then x must come before y in the linear extension. In other words, it is a total order compatible with the partial order. If a finite partially ordered set is totally ordered, then it has only one linear extension, but otherwise it will have more than one. The 1/3–2/3 conjecture states that one can choose two elements x and y such that, among this set of possible linear extensions, between 1/3 and 2/3 of them place x earlier than y, and symmetrically between 1/3 and 2/3 of them place y earlier
There is an alternative and equivalent statement of the 1/3–2/3 conjecture in the language of probability theory.
One may define a uniform probability distribution on the linear extensions in which each possible linear extension is equally likely to be chosen. The 1/3–2/3 conjecture states that, under th |
https://en.wikipedia.org/wiki/Spectral%20abscissa | In mathematics, the spectral abscissa of a matrix or a bounded linear operator is the greatest real part of the matrix's spectrum (its set of eigenvalues). It is sometimes denoted . As a transformation , the spectral abscissa maps a square matrix onto its largest real eigenvalue.
Matrices
Let λ1, ..., λs be the (real or complex) eigenvalues of a matrix A ∈ Cn × n. Then its spectral abscissa is defined as:
In stability theory, a continuous system represented by matrix is said to be stable if all real parts of its eigenvalues are negative, i.e. . Analogously, in control theory, the solution to the differential equation is stable under the same condition .
See also
Spectral radius
References
Spectral theory
Matrix theory |
https://en.wikipedia.org/wiki/%C5%A0umadija%20and%20Western%20Serbia | The Šumadija and Western Serbia () is one of the five statistical regions of Serbia. It is also a level-2 statistical region according to the Nomenclature of Territorial Units for Statistics (NUTS). It was formed in 2010. As of 2022 census, the regions has a total of 1,819,318 inhabitants.
Formation
In July 2009, the Serbian parliament adopted a new law in which Serbia was divided into seven statistical regions. According to the law, territory of present-day Šumadija and Western Serbia was divided into two statistical regions – Western Region () and Central Region ().
However, in May 2010, the law was modified and Western and Central region were merged into a single statistical region whose name is Šumadija and Western Serbia.
Districts
The statistical region of Šumadija and Western Serbia is composed of 8 administrative districts:
Demographic structure
Cities and towns
The largest cities and towns of the region are:
References
External links
Usvojene izmene i dopune Zakona o regionalnom-razvoju (in Serbian)
Statistical regions of Serbia |
https://en.wikipedia.org/wiki/Jimena%2C%20Spain | Jimena is a city the Province of Jaén, Andalusia, Spain, with 1489 inhabitants according to the National Statistics Institute of Spain in 2007.
Geography and heritage
It is in the northwest area of Sierra Magina, which portion is mountainous terrain southeast and included in the natural park of Sierra Magina.
The rest of the field is the predominant crop in the olive grove. Economic activity is based on olive growing and processing industry, complemented by the cultivation of the fig tree and use of the fig and fig.
The Canavan pine (forest of ancient pines and exceptional size) has been declared a Natural Monument for its ecological and scenic value.
His Feasts of 7 to 10 September, where we can enjoy the procession and their carriages, to which numerous people come from other villages around.
History
Jimena has a rich legacy of prehistoric times, a group of rock paintings in the Cave of the Rook, which highlight the culture of groups of pastors between the fourth and third millennia BC populated the Southern Sierras. In this phase have been recorded several settlements in the municipality, among which the Cerro Alcalá, fundamental reference for the prehistory and early history of this town as well as medieval times.
In the Iberian stage Cerro Alcalá is one of the oppida in the sixth century BC which will held in the following centuries until the Roman era. Some researchers have linked Alcala to Cerro Ossigi referring to the written sources.
Epigraphic and constructive Numerous findings show that this settlement would hold some sort of Roman status as Municipium.
In the Arab period there was an intense occupation of the town of Jimena by small or rural farmsteads. The town had several forts for shelter, Fountain of the Moor, Hill Alcalá or mentioned in the chronicles as San Istibin or San Astabin, a name that has been in a spot close to Jimena, Santisteban. Reports suggest that this was one of the castles in which the Banu Business rebelled against the power of the emir of Córdoba. Jimena (Xemena) could be another of these fortified castles or a farm after the Christian conquest.
Jimena was conquered by Ferdinand III the day of Santiago, 1234 and integrated into the land of the Council of Baeza. In 1284 it became the property of Don Pedro Ruy de Berrio. During the fourteenth and fifteenth centuries was a small manor until in 1434 King John II gave to the Order of Calatrava, which was the task of Torres, Canena, inheriting from Jimena and pray.
Jimena In the sixteenth century, in line with Baeza, commoner in the conflict in the Castle of Charles V, even served as an outpost hidden from community members. Completion of these events was sold by the Emperor Charles V to his secretary, Francisco de los Cobos. From this date Jimena was the domain of Don Francisco de los Cobos and his descendants after the Marquis of Camarasa, until the extinction of the noble privileges in 1812.
Jimena Patron
Our Lady of the Remedies.
The popular voice |
https://en.wikipedia.org/wiki/List%20of%20Hilbert%20systems | This article contains a list of sample Hilbert-style deductive systems for propositional logics.
Classical propositional calculus systems
Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent and its main property is that it is strongly complete, otherwise said that whenever a formula semantically follows from a set of premises, it also follows from that set syntactically. Many different equivalent complete axiom systems have been formulated. They differ in the choice of basic connectives used, which in all cases have to be functionally complete (i.e. able to express by composition all n-ary truth tables), and in the exact complete choice of axioms over the chosen basis of connectives.
Implication and negation
The formulations here use implication and negation as functionally complete set of basic connectives. Every logic system requires at least one non-nullary rule of inference. Classical propositional calculus typically uses the rule of modus ponens:
We assume this rule is included in all systems below unless stated otherwise.
Frege's axiom system:
Hilbert's axiom system:
Łukasiewicz's axiom systems:
First:
Second:
Third:
Fourth:
Łukasiewicz and Tarski's axiom system:
Meredith's axiom system:
Mendelson's axiom system:
Russell's axiom system:
Sobociński's axiom systems:
First:
Second:
Implication and falsum
Instead of negation, classical logic can also be formulated using the functionally complete set of connectives.
Tarski–Bernays–Wajsberg axiom system:
Church's axiom system:
Meredith's axiom systems:
First:
Second:
Negation and disjunction
Instead of implication, classical logic can also be formulated using the functionally complete set of connectives. These formulations use the following rule of inference;
Russell–Bernays axiom system:
Meredith's axiom systems:
First:
Second:
Third:
Dually, classical propositional logic can be defined using only conjunction and negation.
Conjunction and negation
Rosser J. Barkley created a system based on conjunction and negation , with the modus ponens as inference rule. In his book, he used the implication to present his axiom schemes. "" is an abbreviation for "":
1)
2)
3)
If we don't use the abbreviation, we get the axiom schemes in the following form:
1)
2)
3)
Also, modus ponens becomes:
Sheffer's stroke
Because Sheffer's stroke (also known as NAND operator) is functionally complete, it can be used to create an entire formulation of propositional calculus. NAND formulations use a rule of inference called Nicod's modus ponens:
Nicod's axiom system:
Łukasiewicz's axiom systems:
First:
Second:
Wajsberg's axiom system:
Argonne axiom systems:
First:
Second:
Computer analysis by Argonne has revealed over 60 additional single axiom systems that can be used to formulate NAND propositional calculus.
Implicational propositional calculus
The implicational propositional calculus is the fragment of the classical |
https://en.wikipedia.org/wiki/Baker%27s%20theorem | In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier.
Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1.
History
To simplify notation, let be the set of logarithms to the base e of nonzero algebraic numbers, that is
where denotes the set of complex numbers and denotes the algebraic numbers (the algebraic completion of the rational numbers ). Using this notation, several results in transcendental number theory become much easier to state. For example the Hermite–Lindemann theorem becomes the statement that any nonzero element of is transcendental.
In 1934, Alexander Gelfond and Theodor Schneider independently proved the Gelfond–Schneider theorem. This result is usually stated as: if is algebraic and not equal to 0 or 1, and if is algebraic and irrational, then is transcendental. The exponential function is multi-valued for complex exponents, and this applies to all of its values, which in most cases constitute infinitely many numbers. Equivalently, though, it says that if are linearly independent over the rational numbers, then they are linearly independent over the algebraic numbers. So if and is not zero, then the quotient is either a rational number or transcendental. It cannot be an algebraic irrational number like .
Although proving this result of "rational linear independence implies algebraic linear independence" for two elements of was sufficient for his and Schneider's result, Gelfond felt that it was crucial to extend this result to arbitrarily many elements of Indeed, from :
This problem was solved fourteen years later by Alan Baker and has since had numerous applications not only to transcendence theory but in algebraic number theory and the study of Diophantine equations as well. Baker received the Fields medal in 1970 for both this work and his applications of it to Diophantine equations.
Statement
With the above notation, Baker's theorem is a nonhomogeneous generalization of the Gelfond–Schneider theorem. Specifically it states:
Just as the Gelfond–Schneider theorem is equivalent to the statement about the transcendence of numbers of the form ab, so too Baker's theorem implies the transcendence of numbers of the form
where the bi are all algebraic, irrational, and 1, b1, …, bn are linearly independent over the rationals, and the ai are all algebraic and not 0 or 1.
also gave several versions with explicit constants. For example, if has height at most and all the numbers have height at most then the linear form
is either 0 or satisfies
wher |
https://en.wikipedia.org/wiki/Supporting%20line | In geometry, a supporting line L of a curve C in the plane is a line that contains a point of C, but does not separate any two points of C. In other words, C lies completely in one of the two closed half-planes defined by L and has at least one point on L.
Properties
There can be many supporting lines for a curve at a given point. When a tangent exists at a given point, then it is the unique supporting line at this point, if it does not separate the curve.
Generalizations
The notion of supporting line is also discussed for planar shapes. In this case a supporting line may be defined as a line which has common points with the boundary of the shape, but not with its interior.
The notion of a supporting line to a planar curve or convex shape can be generalized to n dimension as a supporting hyperplane.
Critical support lines
If two bounded connected planar shapes have disjoint convex hulls that are separated by a positive distance, then they necessarily have exactly four common lines of support, the bitangents of the two convex hulls. Two of these lines of support separate the two shapes, and are called critical support lines. Without the assumption of convexity, there may be more or fewer than four lines of support, even if the shapes themselves are disjoint. For instance, if one shape is an annulus that contains the other, then there are no common lines of support, while if each of two shapes consists of a pair of small disks at opposite corners of a square then there may be as many as 16 common lines of support.
References
Geometry |
https://en.wikipedia.org/wiki/Shimura%20correspondence | In number theory, the Shimura correspondence is a correspondence between modular forms F of half integral weight k+1/2, and modular forms f of even weight 2k, discovered by . It has the property that the eigenvalue of a Hecke operator Tn2 on F is equal to the eigenvalue of Tn on f.
Let be a holomorphic cusp form with weight and character . For any prime number p, let
where 's are the eigenvalues of the Hecke operators determined by p.
Using the functional equation of L-function, Shimura showed that
is a holomorphic modular function with weight 2k and character .
Shimura's proof uses the Rankin-Selberg convolution of with the theta series for various Dirichlet characters then applies Weil's converse theorem.
See also
Theta correspondence
References
Modular forms
Langlands program |
https://en.wikipedia.org/wiki/John%20Lodge%20Cowley | John Lodge Cowley (1719, United Kingdom – buried 1797, United Kingdom) was an English cartographer, geologist and mathematician.
John Cowley was a professor of mathematics at the Royal Military Academy, Woolwich, London, for a number of years between 1761 and 1773. He was elected a Fellow of the Royal Society in April, 1768.
His mathematical methods were famous, but he was also an important geographer as well as Cartographer Royal to King George II. He specialised in maps that depicted the counties of the United Kingdom from which arose his most famous work, Counties of England. Cowley published several maps, many of minute detail, and on them often appeared his name although on others appeared the name of Emanuel Bowen acting as engraver. Another one of his works of some note, but less famous than The Counties, was A new and easy introduction to the study of geography which was published by Thomas Cox and James Hodges. The work was structured as questions and answers with decorative maps added later. His maps contained longer titles than those usually found on the standard miniature maps. John Cowley collaborated with Robert Dodsley for several years in the creation of his maps which explains why the maps were ascribed to Dodsley/Cowley. Among his works are remembered the superb engravings representing the constellation drawn on glass globes created by Thomas Heath.
He also published a number of introductory works on solid geometry incorporating fold-up figures: Geometry Made Easy (1752), An Appendix to Euclid's Elements (1758) and The Theory of Perspective Demonstrated (1765).
He taught geometry to subscribers of the St. Martin's Lane Academy, a drawing school established by William Hogarth and John Ellys.
Cowley died in Walworth, Surrey. He had a daughter called Mrs Johnstone who inherited her father's passion for science and over the years instructed many members of the British nobility in the use of globes and maps.
See also
List of cartographers
History of cartography
Willem Blaeu
Joan Blaeu
Glass celestial sphere engraved by Cowley in 1739, in the Science Museum's collection, London
Gallery
References
1719 births
1787 deaths
English cartographers
Fellows of the Royal Society
18th-century English mathematicians
18th-century cartographers
18th-century English people
English geologists |
https://en.wikipedia.org/wiki/Electoral%20wards%20of%20Belfast | The electoral wards of Belfast are subdivisions of the city, used primarily for statistics and elections. Belfast had 51 wards from May 1973, which were revised in May 1985 and again in May 1993. The number of wards was increased to 60 with the 2014 changes in local government. Wards are the smallest administrative unit in Northern Ireland and are set by the Local Government Boundaries Commissioner and reviewed every 8–12 years.
Wards are used to create constituencies for local government authorities, the Northern Ireland Assembly and the House of Commons of the United Kingdom. In elections to Belfast City Council, the 60 wards are split into ten District Electoral Areas, each of which contains between five and seven wards, with the number of councillors it elects equal to the number of wards it contains. The constituencies for elections to the House of Commons and the Assembly are coterminous and are created by amalgamating wards into larger areas, with the city's wards split between the four 'Belfast' constituencies, although these also contain wards from bordering local authorities.
The use of wards for statistical purposes by the Northern Ireland Statistics and Research Agency (NISRA) has changed since the creation of 'Census Output Areas' (5022 in total) and 'Super Output Areas' (890 in total), which were created to address the variance in size of the 582 wards across Northern Ireland. Each ward contains several Super Output Area, which in turn are made up of a number of Census Output Areas.
Current wards
The wards were redrawn for the 2014 elections.
History
From 1928 until May 1973, Belfast was divided into 15 wards, each represented by a total of four aldermen or councillors. The 15 wards were Duncairn, Dock, Clifton, Shankill, Court, Woodvale, Smithfield, Falls, Saint Anne's, Saint George's, Windsor, Cromac, Ormeau, Pottinger and Victoria.
By the early 1970s, population shifts had resulted in significant differences in the electorates of the wards. Although the wards continued to have four representatives on Belfast City Council, electorates varied from less than 5,000 in the smallest ward, Smithfield, to almost 40,000 in the largest ward, Victoria. The review of local government which took place in the early 1970s expanded Belfast to take in some of the newer housing estates on the fringes of the city and attempted to equalise electorates. Initially, Belfast was to have been divided into 52 wards. Following a public review, one ward, Tullycarnet, was excluded from Belfast and became instead the Castlereagh wards of Tullycarnet and Gilnahirk. The remaining 51 wards were intended to elect one member each using the first past the post electoral system. The reintroduction of the single transferable vote method of election meant that a different system was required. The 51 wards were therefore grouped into 8 electoral areas, distinguished by letters, with each electoral area returning either 6 or 7 councillors.
By the early 1980s, pop |
https://en.wikipedia.org/wiki/P-adic%20exponential%20function | In mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the p-adic logarithm.
Definition
The usual exponential function on C is defined by the infinite series
Entirely analogously, one defines the exponential function on Cp, the completion of the algebraic closure of Qp, by
However, unlike exp which converges on all of C, expp only converges on the disc
This is because p-adic series converge if and only if the summands tend to zero, and since the n! in the denominator of each summand tends to make them large p-adically, a small value of z is needed in the numerator. It follows from Legendre's formula that if then tends to , p-adically.
Although the p-adic exponential is sometimes denoted ex, the number e itself has no p-adic analogue. This is because the power series expp(x) does not converge at . It is possible to choose a number e to be a p-th root of expp(p) for , but there are multiple such roots and there is no canonical choice among them.
p-adic logarithm function
The power series
converges for x in Cp satisfying |x|p < 1 and so defines the p-adic logarithm function logp(z) for |z − 1|p < 1 satisfying the usual property logp(zw) = logpz + logpw. The function logp can be extended to all of (the set of nonzero elements of Cp) by imposing that it continues to satisfy this last property and setting logp(p) = 0. Specifically, every element w of can be written as w = pr·ζ·z with r a rational number, ζ a root of unity, and |z − 1|p < 1, in which case logp(w) = logp(z). This function on is sometimes called the Iwasawa logarithm to emphasize the choice of logp(p) = 0. In fact, there is an extension of the logarithm from |z − 1|p < 1 to all of for each choice of logp(p) in Cp.
Properties
If z and w are both in the radius of convergence for expp, then their sum is too and we have the usual addition formula: expp(z + w) = expp(z)expp(w).
Similarly if z and w are nonzero elements of Cp then logp(zw) = logpz + logpw.
For z in the domain of expp, we have expp(logp(1+z)) = 1+z and logp(expp(z)) = z.
The roots of the Iwasawa logarithm logp(z) are exactly the elements of Cp of the form pr·ζ where r is a rational number and ζ is a root of unity.
Note that there is no analogue in Cp of Euler's identity, e2πi = 1. This is a corollary of Strassmann's theorem.
Another major difference to the situation in C is that the domain of convergence of expp is much smaller than that of logp. A modified exponential function — the Artin–Hasse exponential — can be used instead which converges on |z|p < 1.
Notes
References
Chapter 12 of
External links
p-adic exponential and p-adic logarithm
Exponentials
p-adic numbers |
https://en.wikipedia.org/wiki/National%20Institute%20of%20Statistics%20and%20Applied%20Economics | The National Institute of Statistics and Applied Economics (INSEA) () is one of the oldest engineering schools in Morocco and remains to this day one of the most prestigious Moroccan Grandes écoles in engineering. Located in Rabat and created in 1961, its latest naming has changed by Royal Decree from the appellation The Training Centre of Engineers in Statistics in 1967 with the support of the Economic Commission for Africa (ECA).
Introduction
INSEA provides training that gives equal weight to statistics and economic analysis, and offers a specialization in the fields of computing, finance, actuarial and demographic sciences, as well as operations research. It is the first engineering school that has offered training in computers at the national level, and has introduced the first computer into the kingdom in 1974.
INSEA graduates have different sought-after skills that allow them to perform statistical analysis, economic forecasting, as well as engineering of information systems. They are also capable of developing models that can contribute to market analysis, or measure and evaluate different types of risk in a variety of areas.
Furthermore, the training INSEA is not limited to technical aspects of engineering, it also includes management, communication and quality courses related to the socio-economic and political environment of Morocco.
Graduates of INSEA are expected to play a strategic role at various stages of the process of economic and social development, pursuing careers in the public sector, local communities and the private sector: bank companies' Insurance and large companies and national multinationals.
The institute is under the supervision of the High Commissioner for Planning, National Board of Economic Planning of the country.
Specializations and careers
Statistics
Statistics is the study of observations, its application areas are numerous and diverse, in fact almost all areas that lend themselves to numerical observations may be privileged fields of application.
An INSEAist statistician engineer may have different tasks: it is frequently entrusted with the planning of a full statistical study with constraints and objectives to meet. It must determine the number of observations necessary for the conclusions can be expressed with a level of certainty. It must, with experts in the field of study, discuss the variables mesurer.Il must explain the results of statistical analysis and specify, if necessary, restrictions on the conclusions given the methods and level of uncertainty present.
The engineering statistician can play the role of associate and / or consultant, he is also invited, on occasion, to play the role of expert. His expertise is often ad hoc, but still very important for the decision maker.
It may possibly be a member of a team of industrial production, collaboration is then as a specialist in quality assurance and reliability. The experimental research centers routinely use the statisticians.
Applied E |
https://en.wikipedia.org/wiki/Mengenlehreuhr | The Mengenlehreuhr (German for "Set Theory Clock") or Berlin-Uhr ("Berlin Clock") is the first public clock in the world that tells the time by means of illuminated, coloured fields, for which it entered the Guinness Book of Records upon its installation on 17 June 1975. Commissioned by the Senate of Berlin and designed by Dieter Binninger, the original full-sized Mengenlehreuhr was originally located at the Kurfürstendamm on the corner with Uhlandstraße. After the Senate decommissioned it in 1995, the clock was relocated to a site in Budapester Straße in front of Europa-Center, where it stands today.
Time encoding
The Mengenlehreuhr consists of 24 lights which are divided into one circular blinking yellow light on top to denote the seconds, two top rows denoting the hours and two bottom rows denoting the minutes.
The clock is read from the top row to the bottom. The top row of four red fields denote five full hours each, alongside the second row, also of four red fields, which denote one full hour each, displaying the hour value in 24-hour format. The third row consists of eleven yellow-and-red fields, which denote five full minutes each (the red ones also denoting 15, 30 and 45 minutes past), and the bottom row has another four yellow fields, which mark one full minute each. The round yellow light on top blinks to denote odd- (when lit) or even-numbered (when unlit) seconds.
Given the photo of the clock at the top of the article as an example, two fields are lit in the first row (five hours multiplied by two, i.e. ten hours), but no fields are lit in the second row; therefore the hour value is 10. Six fields are lit in the third row (five minutes multiplied by six, i.e. thirty minutes), while the bottom row has one field on (plus one minute). Hence, the lights of the clock altogether tell the time as 10:31.
Kryptos
This clock may be the key to the unsolved section of Kryptos, a sculpture at the CIA headquarters. After revealing that part of the deciphered text of the sculpture, in positions 64-69, reads "BERLIN", the sculptor, Jim Sanborn, gave The New York Times another clue in November 2014, that letters 70–74 in part 4 of the sculpture's code, which read "MZFPK", will become "CLOCK" when decoded, a direct reference to the Berlin Clock. Sanborn further stated that in order to solve section 4, "You'd better delve into that particular clock".
However, Sanborn also said that, "There are several really interesting clocks in Berlin."
References
External links
Europa Center – Set Theory Clock
Aqua Phoenix: Set Theory Clock
Your local time in Berlin-Clock
The actual time in the Berlin-Clock (Flash version)
JS and CSS realization of multiple Berlin clock time zones
German inventions
Set theory
Individual clocks
Clocks in Germany
Products introduced in 1975
1975 establishments in West Germany
Monuments and memorials in Berlin
1970s in West Berlin |
https://en.wikipedia.org/wiki/Uniformization%20%28probability%20theory%29 | In probability theory, uniformization method, (also known as Jensen's method or the randomization method) is a method to compute transient solutions of finite state continuous-time Markov chains, by approximating the process by a discrete-time Markov chain. The original chain is scaled by the fastest transition rate γ, so that transitions occur at the same rate in every state, hence the name. The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero). The method was first introduced by Winfried Grassmann in 1977.
Method description
For a continuous-time Markov chain with transition rate matrix Q, the uniformized discrete-time Markov chain has probability transition matrix , which is defined by
with γ, the uniform rate parameter, chosen such that
In matrix notation:
For a starting distribution (0), the distribution at time t, (t) is computed by
This representation shows that a continuous-time Markov chain can be described by a discrete Markov chain with transition matrix P as defined above where jumps occur according to a Poisson process with intensity γt.
In practice this series is terminated after finitely many terms.
Implementation
Pseudocode for the algorithm is included in Appendix A of Reibman and Trivedi's 1988 paper. Using a parallel version of the algorithm, chains with state spaces of larger than 107 have been analysed.
Limitations
Reibman and Trivedi state that "uniformization is the method of choice for typical problems," though they note that for stiff problems some tailored algorithms are likely to perform better.
External links
Matlab implementation
Notes
Queueing theory
Markov processes |
https://en.wikipedia.org/wiki/Arnaud%20Beauville | Arnaud Beauville (born 10 May 1947) is a French mathematician, whose research interest is algebraic geometry.
Beauville earned his doctorate from Paris Diderot University in 1977, with a thesis regarding Prym varieties and the Schottky problem, under supervision of Jean-Louis Verdier.
He has been a professor at the Université Paris-Sud, then Director of the Mathematics Department at the École Normale Supérieure. He is currently Professor emeritus at the Université de Nice Sophia-Antipolis.
Beauville was a visiting scholar at the Institute for Advanced Study in the summer of 1982. He was an invited speaker at the International Congress of Mathematicians in 1986 at Berkeley. He was a member of Bourbaki.
He has had 25 Ph.D. students, among them Claire Voisin, Olivier Debarre, Yves Laszlo.
In 2012 he became a fellow of the American Mathematical Society.
See also
Beauville surface
References
External links
Personal website
1947 births
Living people
École Normale Supérieure alumni
20th-century French mathematicians
21st-century French mathematicians
Algebraic geometers
Paris Diderot University alumni
Academic staff of the University of Angers
Academic staff of Paris-Sud University
Institute for Advanced Study visiting scholars
Fellows of the American Mathematical Society
Nicolas Bourbaki |
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