# Combinatorics

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Cayley's theorem Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems that are true for subgroups of permutation groups are true for groups in general. Nevertheless, 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".[4] The regular action used in the standard proof of Cayley's theorem does not produce the representation of G in a minimal-order permutation group. , itself already a symmetric group of order 6, would be represented by the regular action as a subgroup of (a group of order 720).[5] The problem of finding an embedding of a group in a minimal-order symmetric group is rather more difficult.[6][7] History The theorem was later published by Walther Dyck in 1882[12] and is attributed to Dyck in the first edition of Burnside's book.[13] . Firstly, suppose with .

Algebraic combinatorics The Fano matroid, derived from the Fano plane. Matroids are one of many areas studied in algebraic combinatorics. Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. However, within the last decade or so, algebraic combinatorics came to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. See also References Bannai, Eiichi; Ito, Tatsuro (1984).

Measure-preserving dynamical system In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Definition A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. with the following structure: is a set, is a σ-algebra over , is a probability measure, so that μ(X) = 1, and μ(∅) = 0, is a measurable transformation which preserves the measure , i.e., . , the identity function on X;, whenever all the terms are well-defined;, whenever all the terms are well-defined. The earlier, simpler case fits into this framework by definingTs = Ts for s ∈ N. The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem. Examples Examples include: Homomorphisms The concept of a homomorphism and an isomorphism may be defined. Consider two dynamical systems and . The system is then called a factor of is defined as T.

Statistical physics Statistical physics is a branch of physics that 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. Its applications include many problems in the fields of physics, biology, chemistry, neurology, and even some social sciences, such as sociology. Statistical mechanics 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 (analogous to , which is essentially a weighted sum of all possible states available to a system. where . See also

Enumerative combinatorics Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The simplest such functions are closed formulas, which can be expressed as a composition of elementary functions such as factorials, powers, and so on. Finally, f(n) may be expressed by a formal power series, called its generating function, which is most commonly either the ordinary generating function or the exponential generating function is an asymptotic approximation to if as . .

Maximal ergodic theorem The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics. Suppose that is a probability space, that is a (possibly noninvertible) measure-preserving transformation, and that . Define by Then the maximal ergodic theorem states that for any λ ∈ R. This theorem is used to prove the point-wise ergodic theorem. Keane, Michael; Petersen, Karl (2006), "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem", Institute of Mathematical Statistics Lecture Notes - Monograph Series, Institute of Mathematical Statistics Lecture Notes - Monograph Series 48: 248–251, doi:10.1214/074921706000000266, ISBN 0-940600-64-1 .

Probability theory Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem. 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. History of probability Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. . . . If is

Combinatorial principles In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. Rule of sum The rule of sum is an intuitive principle stating that if there are a possible outcomes for an event (or ways to do something) and b possible outcomes for another event (or ways to do another thing), and the two events cannot both occur (or the two things can't both be done), then there are a + b total possible outcomes for the events (or total possible ways to do one of the things). More formally, the sum of the sizes of two disjoint sets is equal to the size of their union. Rule of product The rule of product is another intuitive principle stating that if there are a ways to do something and b ways to do another thing, then there are a · b ways to do both things. Inclusion-exclusion principle Inclusion–exclusion illustrated for three sets Generally, according to this principle, if A1, ..., An are finite sets, then References

Ergodic theory Ergodic theory (Greek: έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics. The problem of metric classification of systems is another important part of the abstract ergodic theory. An outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of entropy for dynamical systems. Ergodic transformations Ergodic theory is often concerned with ergodic transformations. Let T : X → X be a measure-preserving transformation on a measure space (X, Σ, μ), with μ(X) = 1. Examples Evolution of an ensemble of classical systems in phase space (top). Ergodic theorems Let T: X → X be a measure-preserving transformation on a measure space (X, Σ, μ) and suppose ƒ is a μ-integrable function, i.e. ƒ ∈ L1(μ). In general the time average and space average may be different. where

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