Cayley's theorem Representation of groups by permutations whose elements are the permutations of the underlying set of G. Explicitly, for each , the left-multiplication-by-g map sending each element x to gx is a permutation of G, andthe map sending each element g to is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of . The homomorphism When G is finite, is finite too. . for some ; for instance, the order 6 group is not only isomorphic to a subgroup of , but also (trivially) isomorphic to a subgroup of .[3] The problem of finding the minimal-order symmetric group into which a given group G embeds is rather difficult.[4][5] Alperin and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups".[6] When G is infinite, is infinite, but Cayley's theorem still applies. History[edit] Background[edit] .[13] In particular, taking A to be the underlying set of a group G produces a symmetric group denoted
Category:Integer sequences From Wikipedia, the free encyclopedia This category includes not only articles about certain types of integer sequences, but also articles about theorems and conjectures pertaining to, and properties of, integer sequences. Subcategories This category has the following 10 subcategories, out of 10 total. Pages in category "Integer sequences" The following 200 pages are in this category, out of approximately 219 total. (previous page) (next page)(previous page) (next page)
Statistical physics Branch of physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Scope[edit] Statistical mechanics[edit] Statistical mechanics provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life, therefore explaining thermodynamics as a natural result of statistics, classical mechanics, and quantum mechanics at the microscopic level. One of the most important equations in statistical mechanics (akin to , which is essentially a weighted sum of all possible states available to a system. where is the Boltzmann constant, is temperature and is energy of state . See also[edit]
Zero-sum problem From Wikipedia, the free encyclopedia Mathematical problem In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0. The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv.[1] They proved that for the group of integers modulo n, Explicitly this says that any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n, but that the same is not true of multisets of size 2n − 2. More general results than this theorem exist, such as Olson's theorem, Kemnitz's conjecture (proved by Christian Reiher in 2003[3]), and the weighted EGZ theorem (proved by David J. See also[edit] References[edit] External links[edit] Further reading[edit]
Probability theory Branch of mathematics concerning probability As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data.[1] Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. [2] History of probability[edit] The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the "problem of points").[3] Christiaan Huygens published a book on the subject in 1657.[4] In the 19th century, what is considered the classical definition of probability was completed by Pierre Laplace.[5] Treatment[edit] . .
Wilf–Zeilberger pair Definition[edit] Together, these conditions ensure that because the function G telescopes: Therefore, that is The constant does not depend on n. If F and G form a WZ pair, then they satisfy the relation where is a rational function of n and k and is called the WZ proof certificate. Example[edit] A Wilf–Zeilberger pair can be used to verify the identity Divide the identity by its right-hand side: Use the proof certificate to verify that the left-hand side does not depend on n, where Now F and G form a Wilf–Zeilberger pair. To prove that the constant in the right-hand side of the identity is 1, substitute n = 0, for instance. References[edit] See also[edit] External links[edit] Gosper's algorithm gives a method for generating WZ pairs when they exist.Generatingfunctionology provides details on the WZ method of identity certification.
Symmetric group Type of group in abstract algebra defined over a finite set of symbols consists of the permutations that can be performed on the symbols.[1] Since there are is Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. is isomorphic to a subgroup of the symmetric group on (the underlying set of) Definition and first properties[edit] The symmetric group on a finite set is the group whose elements are all bijective functions from to and whose group operation is that of function composition.[1] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. , and .[1] If is the set .
Weighing matrix Mathematical weight device In mathematics, a weighing matrix of order and weight with entries from the set such that: Where is the transpose of and is the identity matrix of order . is also called the degree of the matrix. is often denoted by Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. Properties[edit] Some properties are immediate from the definition. is a , then: The rows of are pairwise orthogonal. A weighing matrix is a generalization of Hadamard matrix, which does not allow zero entries.[3] As two special cases, a is a Hadamard matrix[3] and a is equivalent to a conference matrix. Applications[edit] Experiment design[edit] Weighing matrices take their name from the problem of measuring the weight of multiple objects. , then measuring the weights of objects and subtracting the (equally imprecise) tare weight will result in a final measurement with a variance of An order matrix can be used to represent the placement of trials. .
Probability interpretations Philosophical interpretation of the axioms of probability The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory. There are two broad categories[1][2] of probability interpretations which can be called "physical" and "evidential" probabilities. Physical probabilities, which are also called objective or frequency probabilities, are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. Some interpretations of probability are associated with approaches to statistical inference, including theories of estimation and hypothesis testing. Philosophy[edit] Classical definition[edit] Frequentism[edit] in