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Combinatorics

Combinatorics
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Tesseract A generalization of the cube to dimensions greater than three is called a "hypercube", "n-cube" or "measure polytope".[1] The tesseract is the four-dimensional hypercube, or 4-cube. According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek τέσσερεις ακτίνες ("four rays"), referring to the four lines from each vertex to other vertices.[2] In this publication, as well as some of Hinton's later work, the word was occasionally spelled "tessaract." Some people[citation needed] have called the same figure a tetracube, and also simply a hypercube (although a tetracube can also mean a polycube made of four cubes, and the term hypercube is also used with dimensions greater than 4). Geometry[edit] Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. A tesseract is bounded by eight hyperplanes (xi = ±1). See also[edit]

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

Dimensions: A Walk Through Mathematics A film for a wide audience! Nine chapters, two hours of maths, that take you gradually up to the fourth dimension. Mathematical vertigo guaranteed! Dimension Two - Hipparchus shows us how to describe the position of any point on Earth with two numbers... and explains the stereographic projection: how to draw a map of the world. Dimension Three - M.C. The Fourth Dimension - Mathematician Ludwig Schläfli talks about objects that live in the fourth dimension... and shows a parade of four-dimensional polytopes, strange objects with 24, 120 and even 600 faces! Complex Numbers - Mathematician Adrien Douady explains complex numbers. Fibration - Mathematician Heinz Hopf explains his "fibration". Proof - Mathematician Bernhard Riemann explains the importance of proofs in mathematics. Watch the full documentary now (playlist - )

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]

Probability theory Branch of mathematics concerning probability 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 or sequential estimation. History of probability[edit] Treatment[edit] Motivation[edit] . . The function

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 .

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. Classical definition [edit] If we denote by in trials, then if

Bijection One-to-one correspondence A function is bijective if and only if it is invertible; that is, a function is bijective if and only if there is a function the inverse of f, such that each of the two ways for composing the two functions produces an identity function: for each in and For example, the multiplication by two defines a bijection from the integers to the even numbers, which has the division by two as its inverse function. A function is bijective if and only if it is both injective (or one-to-one)—meaning that each element in the codomain is mapped to from at most one element of the domain—and surjective (or onto)—meaning that each element of the codomain is mapped to from at least one element of the domain. The elementary operation of counting establishes a bijection from some finite set to the first natural numbers (1, 2, 3, ...), up to the number of elements in the counted set. Definition[edit] Examples[edit] Batting line-up of a baseball or cricket team[edit] Inverses[edit] is Notes[edit]

Game of chance Roulette is a game of chance, no strategy can give players advantages, the outcome is determined by pure chance A game of chance is a game whose outcome is strongly influenced by some randomizing device, and upon which contestants may choose to wager money or anything of monetary value. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from a container. A game of chance may have some skill element to it, however, chance generally plays a greater role in determining the outcome than skill. A game of skill, on the other hand, also may have elements of chance, but with skill playing a greater role in determining the outcome. Any game of chance that involves anything of monetary value is gambling. Gambling is known in nearly all human societies, even though many have passed laws restricting it. Addiction[edit] He must play regularly: the issue here is to know from when the subject performs "too much." See also[edit] References[edit]

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