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Stanley's Reciprocity Theorem
In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone (defined below) and the generating function of the cone's interior. Definitions A rational cone is the set of all ''d''-tuples :(''a''1, ..., ''a''''d'') of nonnegative integers satisfying a system of inequalities :M\left begina_1 \\ \vdots \\ a_d\end\right\geq \left begin0 \\ \vdots \\ 0\end\right/math> where ''M'' is a matrix of integers. A ''d''-tuple satisfying the corresponding ''strict'' inequalities, i.e., with ">" rather than "≥", is in the ''interior'' of the cone. The generating function of such a cone is :F(x_1,\dots,x_d)=\sum_ x_1^\cdots x_d^. The generating function ''F''int(''x''1, ..., ''x''''d'') of the interior of the cone is defined in the same way, but one sums over ''d''-tuples in the interior rather than in the whole cone. It can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrhart Polynomial
In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane. These polynomials are named after Eugène Ehrhart who introduced them in the 1960s. Definition Informally, if is a polytope, and is the polytope formed by expanding by a factor of in each dimension, then is the number of integer lattice points in . More formally, consider a lattice \mathcal in Euclidean space \R^n and a -dimensional polytope in \R^n with the property that all vertices of the polytope are points of the lattice. (A common example is \mathcal = \Z^n and a polytope for which all vertices have integer coordinates.) For any positive integer , let be the -fold dilation of (the polytope formed by multiplying each vertex coordinate, in a basis fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrhart Polynomials
In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane. These polynomials are named after Eugène Ehrhart who introduced them in the 1960s. Definition Informally, if is a polytope, and is the polytope formed by expanding by a factor of in each dimension, then is the number of integer lattice points in . More formally, consider a lattice \mathcal in Euclidean space \R^n and a -dimensional polytope in \R^n with the property that all vertices of the polytope are points of the lattice. (A common example is \mathcal = \Z^n and a polytope for which all vertices have integer coordinates.) For any positive integer , let be the -fold dilation of (the polytope formed by multiplying each vertex coordinate, in a basis for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Methods Of Contour Integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. It also has various applications in physics. Contour integration methods include: * direct integration of a complex-valued function along a curve in the complex plane * application of the Cauchy integral formula * application of the residue theorem One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums. Curves in the complex plane In complex analysis, a contour is a type of curve in the complex plane. In contour integration, contours provide a precise definition of the curves on which an integral may be suitab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mike Develin
Michael Lee Develin (born August 27, 1980) is an American mathematician known for his work in combinatorics and discrete geometry. Early life Mike Develin was born in Hobart, Tasmania Hobart ( ) is the List of Australian capital cities, capital and most populous city of the island state of Tasmania, Australia. Located in Tasmania's south-east on the estuary of the River Derwent (Tasmania), River Derwent, it is the southernmo .... He moved to the United States with his Korean mother, living in New York City. He attended Stuyvesant High School, where he was captain of the math team, and entered Harvard University at the age of 16. At 22, he received his PhD from University of California, Berkeley, UC Berkeley, doing his dissertation on Topics in Discrete Geometry. He was awarded the 2003 American Institute of Mathematics five-year fellowship. Mathematics Develin is a 2-time William Lowell Putnam Mathematical Competition, Putnam fellow in 1997 and 1998. He studied under adviso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matthias Beck
Matthias is a name derived from the Greek Ματθαίος, in origin similar to Matthew. Notable people Notable people named Matthias include the following: Religion * Saint Matthias, chosen as an apostle in Acts 1:21–26 to replace Judas Iscariot * Matthias of Trakai (–1453), Lithuanian clergyman, bishop of Samogitia and of Vilnius * Matthias Flacius, Lutheran reformer * Matthias the Prophet, see Robert Matthews (religious impostor) Claimed to be the reincarnation of the original Matthias during the Second Great Awakening * Matthias F. Cowley, Latter-day Saint apostle Arts * Matthias Bamert (born 1942), Swiss composer * Matthias Barr (1831-1911), Scottish poet * Matthias Grünewald, highly regarded painter from the German Renaissance * Matthias Jabs, German guitarist and songwriter * Matthías Jochumsson, Icelandic poet * Matthias Lechner, German film art director * Matthias Menck, German audio engineer, electronic music producer and DJ * Matthias Paul (actor), German a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field . In this case, one speaks of a rational function and a rational fraction ''over ''. The values of the variables may be taken in any field containing . Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is . The set of rational functions over a field is a field, the field of fractions of the ring of the polynomial functions over . Definitions A function f is called a rational function if it can be written in the form : f(x) = \frac where P and Q are polynomial functions of x and Q is not the zero function. The domain of f is the set of all values of x for which the denominator Q(x) is not zero. How ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Inequalities
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by and , respectively the less-than and greater-than signs). Notation There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' or ''a'' ≦ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The nota ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonnegative Integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like jersey numbers on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
