|
Ira Gessel
Ira Martin Gessel (born 9 April 1951 in Philadelphia, Pennsylvania) is an American mathematician, known for his work in combinatorics. He is a long-time faculty member at Brandeis University and resides in Arlington, Massachusetts. Education and career Gessel studied at Harvard University graduating ''magna cum laude'' in 1973. There, he became a Putnam Fellow in 1972, alongside Arthur Rubin and David Vogan. He received his Ph.D. at MIT and was the first student of Richard P. Stanley. He was then a postdoctoral fellow at the Thomas J. Watson Research Center, IBM Watson Research Center and MIT. He then joined Brandeis University faculty in 1984. He was promoted to Professor of Mathematics and Computer Science in 1990, became a chair in 1996–98, and Professor Emeritus in 2015. Gessel is a prolific contributor to enumerative combinatorics, enumerative and algebraic combinatorics. He is credited with the invention of quasisymmetric functions in 1984 and foundational work on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philadelphia
Philadelphia, often called Philly, is the largest city in the Commonwealth of Pennsylvania, the sixth-largest city in the U.S., the second-largest city in both the Northeast megalopolis and Mid-Atlantic regions after New York City. Since 1854, the city has been coextensive with Philadelphia County, the most populous county in Pennsylvania and the urban core of the Delaware Valley, the nation's seventh-largest and one of world's largest metropolitan regions, with 6.245 million residents . The city's population at the 2020 census was 1,603,797, and over 56 million people live within of Philadelphia. Philadelphia was founded in 1682 by William Penn, an English Quaker. The city served as capital of the Pennsylvania Colony during the British colonial era and went on to play a historic and vital role as the central meeting place for the nation's founding fathers whose plans and actions in Philadelphia ultimately inspired the American Revolution and the nation's inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Inversion Theorem
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. Statement Suppose is defined as a function of by an equation of the form :z = f(w) where is analytic at a point and f'(a)\neq 0. Then it is possible to ''invert'' or ''solve'' the equation for , expressing it in the form w=g(z) given by a power series : g(z) = a + \sum_^ g_n \frac, where : g_n = \lim_ \frac \left left( \frac \right)^n \right The theorem further states that this series has a non-zero radius of convergence, i.e., g(z) represents an analytic function of in a neighbourhood of z= f(a). This is also called reversion of series. If the assertions about analyticity are omitted, the formula is also valid for formal power series and can be generalized in various ways: It can be formulated for functions of several variables; it can be extended to provide a ready formula for for an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dixon's Identity
In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem, and can now be routinely proved by computer algorithms . Statements The original identity, from , is :\sum_^(-1)^^3 =\frac. A generalization, also sometimes called Dixon's identity, is :\sum_(-1)^k = \frac where ''a'', ''b'', and ''c'' are non-negative integers . The sum on the left can be written as the terminating well-poised hypergeometric series :_3F_2(-2a,-a-b,-a-c;1+b-a,1+c-a;1) and the identity follows as a limiting case (as ''a'' tends to an integer) of Dixon's theorem evaluating a well-poised 3''F''2 generalized hypergeometric series at 1, from : :\;_3F_2 (a,b,c;1+a-b,1+a-c;1)= \frac . This holds for Re(1 + ''a'' − ''b'' &minus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Permutation
In combinatorial mathematics, a Stirling permutation of order ''k'' is a permutation of the multiset 1, 1, 2, 2, ..., ''k'', ''k'' (with two copies of each value from 1 to ''k'') with the additional property that, for each value ''i'' appearing in the permutation, the values between the two copies of ''i'' are larger than ''i''. For instance, the 15 Stirling permutations of order three are :1,1,2,2,3,3; 1,2,2,1,3,3; 2,2,1,1,3,3; :1,1,2,3,3,2; 1,2,2,3,3,1; 2,2,1,3,3,1; :1,1,3,3,2,2; 1,2,3,3,2,1; 2,2,3,3,1,1; :1,3,3,1,2,2; 1,3,3,2,2,1; 2,3,3,2,1,1; :3,3,1,1,2,2; 3,3,1,2,2,1; 3,3,2,2,1,1. The number of Stirling permutations of order ''k'' is given by the double factorial (2''k'' − 1)!!. Stirling permutations were introduced by in order to show that certain numbers (the numbers of Stirling permutations with a fixed number of descents) are non-negative. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyson Conjecture
In mathematics, the Dyson conjecture is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Kenneth G. Wilson, Wilson and Gunson. George Andrews (mathematician), Andrews generalized it to the q-Dyson conjecture, proved by Doron Zeilberger, Zeilberger and David Bressoud, Bressoud and sometimes called the Zeilberger–Bressoud theorem. Ian G. Macdonald, Macdonald generalized it further to more general root systems with the Macdonald constant term conjecture, proved by Ivan Cherednik, Cherednik. Dyson conjecture The Dyson conjecture states that the Laurent polynomial :\prod _(1-t_i/t_j)^ has constant term :\frac. The conjecture was first proved independently by and . later found a short proof, by observing that the Laurent polynomials, and therefore their constant terms, satisfy the recursion relations :F(a_1,\dots,a_n) = \sum_^nF(a_1,\dots,a_i-1,\dots,a_n). The case ''n'' = 3 of Dyson's conjecture follows from the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindström–Gessel–Viennot Lemma
In Mathematics, the Lindström–Gessel–Viennot lemma provides a way to count the number of tuples of non-intersecting lattice paths, or, more generally, paths on a directed graph. It was proved by Gessel–Viennot in 1985, based on previous work of Lindström published in 1973. Statement Let ''G'' be a locally finite directed acyclic graph. This means that each vertex has finite degree, and that ''G'' contains no directed cycles. Consider base vertices A = \ and destination vertices B = \, and also assign a weight \omega_ to each directed edge ''e''. These edge weights are assumed to belong to some commutative ring. For each directed path ''P'' between two vertices, let \omega(P) be the product of the weights of the edges of the path. For any two vertices ''a'' and ''b'', write ''e''(''a'',''b'') for the sum e(a,b) = \sum_ \omega(P) over all paths from ''a'' to ''b''. This is well-defined if between any two points there are only finitely many paths; but even in the gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Committee To End Pay Toilets In America
The Committee to End Pay Toilets in America, or CEPTIA, was a 1970s grass-roots political organization which was one of the main forces behind the elimination of pay toilets in many American cities and states. History Founded in 1970 by then-nineteen-year-old Ira Gessel, the Committee's purpose was to "eliminate pay toilets in the U.S. through legislation and public pressure." Starting a national crusade to cast away coin-operated commodes, Gessel told newsmen, "You can have a fifty-dollar bill, but if you don't have a dime, that metal box is between you and relief." Membership in the organization cost only $0.25, and members received the Committee's newsletter, the ''Free Toilet Paper''. Headquartered in Dayton, Ohio, USA, the group had as many as 1,500 members, in seven chapters. The group also sponsored the Thomas Crapper Memorial Award, which was given to "the person who has made an outstanding contribution to the cause of CEPTIA and free toilets." In 1973, Chicago bec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassroots
A grassroots movement is one that uses the people in a given district, region or community as the basis for a political or economic movement. Grassroots movements and organizations use collective action from the local level to effect change at the local, regional, national or international level. Grassroots movements are associated with bottom-up, rather than top-down decision making, and are sometimes considered more natural or spontaneous than more traditional power structures. Grassroots movements, using self-organization, encourage community members to contribute by taking responsibility and action for their community. Grassroots movements utilize a variety of strategies from fundraising and registering voters, to simply encouraging political conversation. Goals of specific movements vary and change, but the movements are consistent in their focus on increasing mass participation in politics. These political movements may begin as small and at the local level, but grassroots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David P
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". was, according to the Hebrew Bible, the third king of the United Kingdom of Israel. In the Books of Samuel, he is described as a young shepherd and harpist who gains fame by slaying Goliath, a champion of the Philistines, in southern Canaan. David becomes a favourite of Saul, the first king of Israel; he also forges a notably close friendship with Jonathan, a son of Saul. However, under the paranoia that David is seeking to usurp the throne, Saul attempts to kill David, forcing the latter to go into hiding and effectively operate as a fugitive for several years. After Saul and Jonathan are both killed in battle against the Philistines, a 30-year-old David is anointed king over all of Israel and Judah. Following his rise to power, David ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doron Zeilberger
Doron Zeilberger (דורון ציילברגר, born 2 July 1950 in Haifa, Israel) is an Israeli mathematician, known for his work in combinatorics. Education and career He received his doctorate from the Weizmann Institute of Science in 1976, under the direction of Harry Dym, with the thesis "New Approaches and Results in the Theory of Discrete Analytic Functions." He is a Board of Governors Professor of Mathematics at Rutgers University. Contributions Zeilberger has made contributions to combinatorics, hypergeometric identities, and q-series. Zeilberger gave the first proof of the alternating sign matrix conjecture, noteworthy not only for its mathematical content, but also for the fact that Zeilberger recruited nearly a hundred volunteer checkers to "pre-referee" the paper. In 2011, together with Manuel Kauers and Christoph Koutschan, Zeilberger proved the ''q''-TSPP conjecture, which was independently stated in 1983 by George Andrews and David P. Robbins. Zeilberger is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christoph Koutschan
Christoph Koutschan is a German mathematician and computer scientist. He is currently with the Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences. Education Christoph Koutschan (born 12 December 1978 in Dillingen an der Donau, Germany) is a German mathematician and computer scientist. He studied computer science at the University of Erlangen-Nuremberg in Germany from 1999 to 2005 and then moved to the Research Institute for Symbolic Computation (RISC) in Linz, Austria, where he completed his PhD in symbolic computation in 2009 under the supervision of Peter Paule. Career Koutschan is working on computer algebra, particularly on holonomic functions, with applications to combinatorics, special functions, knot theory, and physics. Together with Doron Zeilberger and Manuel Kauers, Koutschan proved two famous open conjectures in combinatorics using large scale computer algebra calculations. Both proofs appeared in the Pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
