1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Representation of a Lie group
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In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a tool in their study being the use of the corresponding infinitesimal representations of Lie algebras. The physics literature sometimes passes over the distinction between Lie groups and Lie algebras, let us first discuss representations acting on finite-dimensional complex vector spaces. A representation of a Lie group G on a complex vector space V is a smooth group homomorphism Ψ. For n-dimensional V, the group of V is identified with a subset of the complex square matrices of order n. The automorphism group of V is given the structure of a manifold using this identification. The condition that Ψ is smooth, in the definition above, if a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into general linear group GL. This is known as a matrix representation, a representation of a Lie group G on a vector space V is a smooth group homomorphism G→Aut from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into general linear group GL and this is known as a matrix representation. Two representations of G on vector spaces V, W are equivalent if they have the same matrix representations with respect to some choices of bases for V and W. On the Lie algebra level, there is a linear mapping from the Lie algebra of G to End preserving the Lie bracket. See representation of Lie algebras for the Lie algebra theory, if the homomorphism is in fact a monomorphism, the representation is said to be faithful. A unitary representation is defined in the way, except that G maps to unitary matrices. If G is a compact Lie group, every representation is equivalent to a unitary one. This definition can handle representations on infinite-dimensional Hilbert spaces, such representations can be found in e. g. quantum mechanics, but also in Fourier analysis as shown in the following example. Let G=R, and let the complex Hilbert space V be L2 and we define the representation Ψ, R → B by Ψ → f. See also Wigners classification for representations of the Poincaré group, if G is a semisimple group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are indexed by highest weight, the allowable highest weights satisfy a suitable positivity condition, the characters of the irreducible representations are given by the Weyl character formula

3.
Lie algebra representation
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The notion is closely related to that of a representation of a Lie group. In the study of representations of a Lie algebra, a ring, called the universal enveloping algebra. The universality of this says that the category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra. Explicitly, this means that ρ is a map that satisfies ρ = = ρ x ρ y − ρ y ρ x for all x, y in g. The vector space V, together with the representation ρ, is called a g -module, the representation ρ is said to be faithful if it is injective. One can equivalently define a g -module as a vector space V together with a map g × V → V such that ⋅ v = x ⋅ − y ⋅ for all x, y in g and v in V. This is related to the definition by setting x ⋅ v = ρx. The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra g on itself, ad, g → g l, x ↦ ad x, indeed, by virtue of the Jacobi identity, ad is a Lie algebra homomorphism. A Lie algebra representation also arises in nature. e, for example, let c g = g x g −1. Then the differential of c g, G → G at the identity is an element of G L. Denoting it by Ad one obtains a representation Ad of G on the vector space g and this is the adjoint representation of G. Applying the preceding, one gets the Lie algebra representation d Ad and it can be shown that d e Ad = ad, the adjoint representation of g. Then a linear map f, V → W is a homomorphism of g -modules if it is g -equivariant, if f is bijective, V, W are said to be equivalent. Similarly, many other constructions from module theory in abstract algebra carry over to this setting, submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc. Then V is said to be semisimple or completely reducible if it satisfies the following equivalent conditions, V is the sum of its simple submodules. Every submodule of V is a direct summand, for every submodule W of V, if g is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and V is finite-dimensional, then V is semisimple. A Lie algebra is said to be if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive, an element v of V is said to be g -invariant if x v =0 for all x ∈ g. The set of all invariant elements is denoted by V g, V ↦ V g is a left-exact functor

4.
American Mathematical Society
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The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. It was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, john Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, the result was the Bulletin of the New York Mathematical Society, with Fiske as editor-in-chief. The de facto journal, as intended, was influential in increasing membership, the popularity of the Bulletin soon led to Transactions of the American Mathematical Society and Proceedings of the American Mathematical Society, which were also de facto journals. In 1891 Charlotte Scott became the first woman to join the society, the society reorganized under its present name and became a national society in 1894, and that year Scott served as the first woman on the first Council of the American Mathematical Society. In 1951, the headquarters moved from New York City to Providence. The society later added an office in Ann Arbor, Michigan in 1984, in 1954 the society called for the creation of a new teaching degree, a Doctor of Arts in Mathematics, similar to a PhD but without a research thesis. Mary W. Gray challenged that situation by sitting in on the Council meeting in Atlantic City, when she was told she had to leave, she refused saying she would wait until the police came. After that time, Council meetings were open to observers and the process of democratization of the Society had begun, julia Robinson was the first female president of the American Mathematical Society but was unable to complete her term as she was suffering from leukemia. In 1988 the Journal of the American Mathematical Society was created, the 2013 Joint Mathematics Meeting in San Diego drew over 6,600 attendees. Each of the four sections of the AMS hold meetings in the spring. The society also co-sponsors meetings with other mathematical societies. The AMS selects a class of Fellows who have made outstanding contributions to the advancement of mathematics. The AMS publishes Mathematical Reviews, a database of reviews of mathematical publications, various journals, in 1997 the AMS acquired the Chelsea Publishing Company, which it continues to use as an imprint. Blogs, Blog on Blogs e-Mentoring Network in the Mathematical Sciences AMS Graduate Student Blog PhD + Epsilon On the Market Some prizes are awarded jointly with other mathematical organizations. The AMS is led by the President, who is elected for a two-year term, morrey, Jr. Oscar Zariski Nathan Jacobson Saunders Mac Lane Lipman Bers R. H. Andrews Eric M. Friedlander David Vogan Robert L

5.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques, Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars

6.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

7.
Topological space
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Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion, the branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. The utility of the notion of a topology is shown by the fact there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application, the most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so this is given first. Note, A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy and this axiomatization is due to Felix Hausdorff. Let X be a set, the elements of X are usually called points, let N be a function assigning to each x in X a non-empty collection N of subsets of X. The elements of N will be called neighbourhoods of x with respect to N, the function N is called a neighbourhood topology if the axioms below are satisfied, and then X with N is called a topological space. If N is a neighbourhood of x, then x ∈ N, in other words, each point belongs to every one of its neighbourhoods. If N is a subset of X and includes a neighbourhood of x, I. e. every superset of a neighbourhood of a point x in X is again a neighbourhood of x. The intersection of two neighbourhoods of x is a neighbourhood of x, any neighbourhood N of x includes a neighbourhood M of x such that N is a neighbourhood of each point of M. The first three axioms for neighbourhoods have a clear meaning, the fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of X. A standard example of such a system of neighbourhoods is for the real line R, given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. A topological space is a pair, where X is a set and τ is a collection of subsets of X, satisfying the following axioms, The empty set. Any union of members of τ still belongs to τ, the intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X, given X =, the collection τ = of only the two subsets of X required by the axioms forms a topology of X, the trivial topology. Given X =, the collection τ = of six subsets of X forms another topology of X, given X = and the collection τ = P, is a topological space. τ is called the discrete topology, using de Morgans laws, the above axioms defining open sets become axioms defining closed sets, The empty set and X are closed. The intersection of any collection of closed sets is also closed, the union of any finite number of closed sets is also closed

8.
General linear group
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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two matrices is again invertible, and the inverse of an invertible matrix is invertible. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix, for example, the general linear group over R is the group of n×n invertible matrices of real numbers, and is denoted by GLn or GL. More generally, the linear group of degree n over any field F, or a ring R, is the set of n×n invertible matrices with entries from F. Typical notation is GLn or GL, or simply GL if the field is understood, more generally still, the general linear group of a vector space GL is the abstract automorphism group, not necessarily written as matrices. The special linear group, written SL or SLn, is the subgroup of GL consisting of matrices with a determinant of 1, the group GL and its subgroups are often called linear groups or matrix groups. These groups are important in the theory of representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general. The modular group may be realised as a quotient of the linear group SL. If n ≥2, then the group GL is not abelian, if V has finite dimension n, then GL and GL are isomorphic. The isomorphism is not canonical, it depends on a choice of basis in V, in a similar way, for a commutative ring R the group GL may be interpreted as the group of automorphisms of a free R-module M of rank n. One can also define GL for any R-module, but in general this is not isomorphic to GL, over a field F, a matrix is invertible if and only if its determinant is nonzero. Therefore, a definition of GL is as the group of matrices with nonzero determinant. Over a non-commutative ring R, determinants are not at all well behaved, in this case, GL may be defined as the unit group of the matrix ring M. The general linear group GL over the field of numbers is a real Lie group of dimension n2. To see this, note that the set of all n×n real matrices, Mn, the subset GL consists of those matrices whose determinant is non-zero. The determinant is a map, and hence GL is an open affine subvariety of Mn. The Lie algebra of GL, denoted g l n, consists of all n×n real matrices with the serving as the Lie bracket. As a manifold, GL is not connected but rather has two connected components, the matrices with positive determinant and the ones with negative determinant, the identity component, denoted by GL+, consists of the real n×n matrices with positive determinant