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.
Topological vector space
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In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure with the concept of a vector space. Hilbert spaces and Banach spaces are well-known examples, unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers C or the real numbers R. Some authors require the topology on X to be T1, it follows that the space is Hausdorff. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the vector spaces over K and the morphisms are the continuous K-linear maps from one object to another. Every normed vector space has a topological structure, the norm induces a metric. This is a vector space because, The vector addition +, V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm, the scalar multiplication ·, K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the inequality and homogeneity of the norm. Therefore, all Banach spaces and Hilbert spaces are examples of vector spaces. There are topological spaces whose topology is not induced by a norm. These are all examples of Montel spaces, an infinite-dimensional Montel space is never normable. A topological field is a vector space over each of its subfields. A cartesian product of a family of vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set X of all functions f, R → R, with this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following, if is a sequence of elements in X, then fn has limit f in X if and only if fn has limit f for every real number x. This space is complete, but not normable, indeed, every neighborhood of 0 in the topology contains lines
3.
Banach space
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In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn, Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces, the vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. All norms on a vector space are equivalent. Every finite-dimensional normed space over R or C is a Banach space, if X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T, X → Y is denoted by B. In infinite-dimensional spaces, not all maps are continuous. For Y a Banach space, the space B is a Banach space with respect to this norm, if X is a Banach space, the space B = B forms a unital Banach algebra, the multiplication operation is given by the composition of linear maps. If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T, X → Y such that T, if one of the two spaces X or Y is complete then so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, the Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ. Every normed space X can be embedded in a Banach space. More precisely, there is a Banach space Y and an isometric mapping T, X → Y such that T is dense in Y. If Z is another Banach space such that there is an isomorphism from X onto a dense subset of Z. This Banach space Y is the completion of the normed space X, the underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y. The completion of X is often denoted by X ^, the cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ∥ ∥1 = ∥ x ∥ + ∥ y ∥, ∥ ∥ ∞ = max. In this sense, the product X × Y is complete if and only if the two factors are complete. If M is a linear subspace of a normed space X, there is a natural norm on the quotient space X / M
4.
Linear map
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In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE
5.
Homeomorphism
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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος = similar and μορφή = shape, roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. A function f, X → Y between two spaces and is called a homeomorphism if it has the following properties, f is a bijection, f is continuous. A function with three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic, a self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form a relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes, the open interval is homeomorphic to the real numbers R for any a < b. The unit 2-disc D2 and the square in R2 are homeomorphic. An example of a mapping from the square to the disc is, in polar coordinates. The graph of a function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve, a chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the sphere in R3 with a single point removed and the set of all points in R2. If G is a group, its inversion map x ↦ x −1 is a homeomorphism. Also, for any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, rm and Rn are not homeomorphic for m ≠ n. The Euclidean real line is not homeomorphic to the circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2. The third requirement, that f −1 be continuous, is essential, consider for instance the function f, [0, 2π) → S1 defined by f =
6.
Approximation property
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In mathematics, a Banach space is said to have the approximation property, if every compact operator is a limit of finite-rank operators. Every Hilbert space has this property, there are, however, Banach spaces which do not, Per Enflo published the first counterexample in a 1973 article. However, a lot of work in area was done by Grothendieck. Later many other counterexamples were found, the space of bounded operators on ℓ2 does not have the approximation property. The spaces ℓ p for p ≠2 and c 0 have closed subspaces that do not have the approximation property, some other flavours of the AP are studied, Let X be a Banach space and let 1 ≤ λ < ∞. A Banach space is said to have bounded approximation property, if it has the λ -AP for some λ, a Banach space is said to have metric approximation property, if it is 1-AP. A Banach space is said to have compact approximation property, if in the definition of AP an operator of rank is replaced with a compact operator. Every projective limit of Hilbert spaces, as well as any subspace of such a projective limit, every subspace of an arbitrary product of Hilbert spaces possesses the approximation property. Every separable Frechet space that contains a Schauder basis possesses the approximation property, every space with a Schauder basis has the AP, thus a lot of spaces with the AP can be found. For example, the ℓ p spaces, or the symmetric Tsirelson space, a counterexample to the approximation property in Banach spaces. Grothendieck, A. Produits tensoriels topologiques et espaces nucleaires, paul R. Halmos, Has progress in mathematics slowed down. MR1066321 William B. Johnson Complementably universal separable Banach spaces in Robert G. Bartle,1980 Studies in functional analysis, on Enflos example of a Banach space without the approximation property. Séminaire Goulaouic–Schwartz 1972—1973, Équations aux dérivées partielles et analyse fonctionnelle, mR407569 Lindenstrauss, J. Tzafriri, L. Classical Banach Spaces I, Sequence spaces,1977. Nedevski, P. Trojanski, S. P. Enflo solved in the negative Banachs problem on the existence of a basis for every separable Banach space, history of Banach spaces and linear operators. Boston, MA, Birkhäuser Boston, Inc. pp. xxiv+855 pp. ISBN 978-0-8176-4367-6, karen Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics,2002 Springer-Verlag, New York. Schaefer, Helmuth H. Wolff, M. P. Topological Vector Spaces, editura Academiei Republicii Socialiste România, Bucharest, Springer-Verlag, Berlin-New York,1981
7.
Monoidal category
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The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, monoidal categories can be seen as a generalization of these and other examples. In category theory, monoidal categories can be used to define the concept of a monoid object and they are also used in the definition of an enriched category. Monoidal categories have applications outside of category theory proper. They are used to define models for the fragment of intuitionistic linear logic. They also form the foundation for the topological order in condensed matter. Braided monoidal categories have applications in information, quantum field theory. A monoidal category is a category C equipped with a monoidal structure, every monoidal category is monoidally equivalent to a strict monoidal category. Any category with products can be regarded as monoidal with the product as the monoidal product. Such a category is called a cartesian monoidal category. For example, Set, the category of sets with the Cartesian product, Cat, the bicategory of small categories with the product category, where the category with one object and only its identity map is the unit. Dually, any category with finite coproducts is monoidal with the coproduct as the monoidal product, as special cases one has, K-Vect, the category of vector spaces over a field K, with the one-dimensional vector space K serving as the unit. Ab, the category of groups, with the group of integers Z serving as the unit. For any commutative ring R, the category of R-algebras is monoidal with the product of algebras as the product. The category of pointed spaces is monoidal with the smash product serving as the product, the category of all endofunctors on a category C is a strict monoidal category with the composition of functors as the product and the identity functor as the unit. In the case E=Cat, we get the endofunctors example above, bounded-above meet semilattices are strict symmetric monoidal categories, the product is meet and the identity is the top element. As a special case of monoidal categories, we consider monoidal preorders and this sort of structure comes up in the theory of string rewriting systems, but it is plentiful in pure mathematics as well. We now present the general case and its well-known that a preorder can be considered as a category C, such that for every two objects c, c ′ ∈ O b, there exists at most one morphism c → c ′ in C
8.
Kernel (category theory)
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Intuitively, the kernel of the morphism f, X → Y is the most general morphism k, K → X that yields zero when composed with f. Note that kernel pairs and difference kernels sometimes go by the kernel, while related. In order to define a kernel in the general category-theoretical sense, in that case, if f, X → Y is an arbitrary morphism in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y. In symbols, ker = eq To be more explicit, the universal property can be used. Note that in many contexts, one would refer to the object K as the kernel. In any case, one can show that k is always a monomorphism, one may prefer to think of the kernel as the pair rather than as simply K or k alone. Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of modules over a fixed ring. Note that in the category of monoids, category-theoretic kernels exist just as for groups, therefore, the notion of kernel studied in monoid theory is slightly different. In the category of rings, there are no kernels in the sense, indeed. Nevertheless, there is still a notion of kernel studied in ring theory that corresponds to kernels in the category of pseudo-rings. In the category of pointed spaces, if f, X → Y is a continuous pointed map, then the preimage of the distinguished point. The inclusion map of K into X is the kernel of f. The dual concept to that of kernel is that of cokernel and that is, the kernel of a morphism is its cokernel in the opposite category, and vice versa. As mentioned above, a kernel is a type of binary equaliser, conversely, in a preadditive category, every binary equaliser can be constructed as a kernel. To be specific, the equaliser of the f and g is the kernel of the difference g − f. It is because of this fact that binary equalisers are called difference kernels, every kernel, like any other equaliser, is a monomorphism. Conversely, a monomorphism is called if it is the kernel of some morphism. A category is called normal if every monomorphism is normal, abelian categories, in particular, are always normal
9.
Direct limit
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In mathematics, a direct limit is a colimit of a directed family of objects. We will first give the definition for algebraic structures like groups and modules, and then the general definition, in this section objects are understood to be sets with a given algebraic structure such as groups, rings, modules, algebras, etc. With this in mind, homomorphisms are understood in the corresponding setting, let ⟨ I, ≤ ⟩ be a directed set. Then the pair ⟨ A i, f i j ⟩ is called a system over I. Here, if x i ∈ A i and x j ∈ A j, x i ∼ x j if there is some k ∈ I such that f i k = f j k. Heuristically, two elements in the disjoint union are equivalent if and only if they become equal in the direct system. An equivalent formulation that highlights the duality to the limit is that an element is equivalent to all its images under the maps of the directed system. One naturally obtains from this definition canonical functions ϕ i, A i → A sending each element to its equivalence class, the algebraic operations on A are defined such that these maps become morphisms. An important property is that taking direct limits in the category of modules is an exact functor, the direct limit can be defined in an arbitrary category C by means of a universal property. Let ⟨ X i, f i j ⟩ be a system of objects. A target is a pair ⟨ X, ϕ i ⟩ where X is an object in C and ϕ i, X i → X are morphisms such that ϕ i = ϕ j ∘ f i j. A direct limit is a universally repelling target in the sense that for each target ⟨ Y, ψ i ⟩, the direct limit of ⟨ X i, f i j ⟩ is often denoted lim → X i = X. Unlike for algebraic objects, the limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense and we note that a direct system in a category C admits an alternative description in terms of functors. Any directed poset ⟨ I, ≤ ⟩ can be considered as a small category I where the morphisms consist of arrows i → j if, a direct system is then just a covariant functor I → C. In this case a direct limit is a colimit, a collection of subsets M i of a set M can be partially ordered by inclusion. If the collection is directed, its limit is the union ⋃ M i. Let I be any directed set with a greatest element m, the direct limit of any corresponding direct system is isomorphic to Xm and the canonical morphism φm, Xm → X is an isomorphism
10.
Inverse limit
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In mathematics, the inverse limit is a construction that allows one to glue together several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category and we start with the definition of an inverse system of groups and homomorphisms. Then the pair is called a system of groups and morphisms over I. We define the limit of the inverse system as a particular subgroup of the direct product of the Ais. The inverse limit, A, comes equipped with natural projections πi, the inverse limit and the natural projections satisfy a universal property described in the next section. This same construction may be carried out if the Ais are sets, semigroups, topological spaces, rings, modules, algebras, etc. the inverse limit will also belong to that category. The inverse limit can be defined abstractly in a category by means of a universal property. Let be a system of objects and morphisms in a category C. The inverse limit of this system is an object X in C together with morphisms πi, the inverse limit is often denoted X = lim ← X i with the inverse system being understood. In some categories, the limit does not exist. If it does, however, it is unique in a strong sense and we note that an inverse system in a category C admits an alternative description in terms of functors. Any partially ordered set I can be considered as a category where the morphisms consist of arrows i → j if. An inverse system is then just a contravariant functor I → C, and the inverse limit functor lim ←, C I o p → C is a covariant functor. The ring of integers is the inverse limit of the rings Z/pnZ with the index set being the natural numbers with the usual order. That is, one considers sequences of such that each element of the sequence projects down to the previous ones, namely. The natural topology on the integers is the one implied here. Pro-finite groups are defined as limits of finite groups. Let the index set I of a system have a greatest element m
11.
Monoid (category theory)
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In the above notations, I is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the dual category Cop, suppose that the monoidal category C has a symmetry γ. A monoid M in C is commutative when μ o γ = μ, a monoid object in Set, the category of sets, is a monoid in the usual sense. A monoid object in Top, the category of spaces, is a topological monoid. A monoid object in the category of monoids is just a commutative monoid and this follows easily from the Eckmann–Hilton theorem. A monoid object in the category of complete join-semilattices Sup is a unital quantale, a monoid object in, the category of abelian groups, is a ring. For a commutative ring R, an object in, the category of modules over R, is an R-algebra. The category of graded modules is a graded R-algebra, the category of chain complexes is a dg-algebra. A monoid object in K-Vect, the category of spaces, is a K-algebra. For any category C, the category of its endofunctors has a monoidal structure induced by the composition, a monoid object in is a monad on C. Given two monoids and in a monoidal category C, a morphism f, M → M is a morphism of monoids when f o μ = μ o, f o η = η, in other words, the following diagrams, commute. The category of monoids in C and their morphisms is written MonC. Act-S, the category of monoids acting on sets Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov, Monoids, Acts and Categories, Walter de Gruyter, Berlin ISBN 3-11-015248-7
12.
Enriched category
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In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. In an enriched category, the set of associated with every pair of objects is replaced by an opaque object in some fixed monoidal category of hom-objects. Enriched category theory thus encompasses within the same framework a wide variety of structures including ordinary categories where the hom-set carries additional structure beyond being a set. An enriched category with hom-objects from monoidal category M is said to be an enriched category over M or a category in M. Due to Mac Lanes preference for the letter V in referring to the monoidal category, the first diagram expresses the associativity of composition, That is, the associativity requirement is now taken over by the associator of the hom-category M. In this case, each leading to C in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms from a → b → c → d from C, C and C. Commutativity of the diagram is then merely the statement that both orders of composition give the result, exactly as required for ordinary categories. The identity morphism 1C, C → C that M has for each of its objects by virtue of it being an ordinary category, ordinary categories are categories enriched over, the category of sets with Cartesian product as the monoidal operation, as noted above. 2-Categories are categories enriched over Cat, the category of small categories, in this case the 2-cells between morphisms a → b and the vertical-composition rule that relates them correspond to the morphisms of the ordinary category C and its own composition rule. Locally small categories are categories enriched over, the category of sets with Cartesian product as the monoidal operation. Locally finite categories, by analogy, are categories enriched over, the hom-objects 2 then simply deny or affirm a particular binary relation on the given pair of objects, for the sake of having more familiar notation we can write this relation as a ≤ b. And since all diagrams in 2 commute, this is the content of the enriched category axioms for categories enriched over 2. Preadditive categories are categories enriched over, the category of groups with tensor product as the monoidal operation. If there is a functor from a monoidal category M to a monoidal category N. Every monoidal category M has a monoidal functor M to the category of sets, in many examples this functor is faithful, so a category enriched over M can be described as an ordinary category with certain additional structure or properties. An enriched functor is the generalization of the notion of a functor to enriched categories. Enriched functors are then maps between enriched categories which respect the enriched structure, because the hom-objects need not be sets in an enriched category, one cannot speak of a particular morphism. There is no longer any notion of an identity morphism, nor of a composition of two morphisms
13.
Hopf algebra
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Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other. They have diverse applications ranging from Condensed-matter physics and quantum theory to string theory. Theorem Let A be a finite-dimensional, graded commutative, graded cocommutative Hopf algebra over a field of characteristic 0, then A is a free exterior algebra with generators of odd degree. In the sumless Sweedler notation, this property can also be expressed as S c = c S = ϵ1 for all c ∈ H, as for algebras, one can replace the underlying field K with a commutative ring R in the above definition. The definition of Hopf algebra is self-dual, so if one can define a dual of H, in general, S is an antihomomorphism, so S2 is a homomorphism, which is therefore an automorphism if S was invertible. If S2 = idH, then the Hopf algebra is said to be involutive, if H is finite-dimensional semisimple over a field of characteristic zero, commutative, or cocommutative, then it is involutive. If a bialgebra B admits an antipode S, then S is unique, the antipode is an analog to the inversion map on a group that sends g to g−1. A subalgebra A of a Hopf algebra H is a Hopf subalgebra if it is a subcoalgebra of H, in other words, a Hopf subalgebra A is a Hopf algebra in its own right when the multiplication, comultiplication, counit and antipode of H is restricted to A. The Nichols–Zoeller freeness theorem established that the natural A-module H is free of finite rank if H is finite-dimensional, as a corollary of this and integral theory, a Hopf subalgebra of a semisimple finite-dimensional Hopf algebra is automatically semisimple. Similarly, a Hopf subalgebra A is left normal in H if it is stable under the left adjoint mapping defined by adl = haS. The two conditions of normality are equivalent if the antipode S is bijective, in which case A is said to be a normal Hopf subalgebra, a normal Hopf subalgebra A in H satisfies the condition, HA+ = A+H where A+ denotes the kernel of the counit on K. This normality condition implies that HA+ is a Hopf ideal of H, as a consequence one has a quotient Hopf algebra H/HA+ and epimorphism H → H/A+H, a theory analogous to that of normal subgroups and quotient groups in group theory. A group-like element is an element x such that Δ = x⊗x. The group-like elements form a group with inverse given by the antipode, a primitive element x satisfies Δ = x⊗1 + 1⊗x. Let A be a Hopf algebra, and let M and N be A-modules, then, M ⊗ N is also an A-module, with a, = Δ = = for m ∈ M, n ∈ N and Δ =. The relationship between Δ, ε, and S ensure that certain natural homomorphisms of vector spaces are indeed homomorphisms of A-modules, for instance, the natural isomorphisms of vector spaces M → M ⊗ K and M → K ⊗ M are also isomorphisms of A-modules. Also, the map of vector spaces M* ⊗ M → K with f ⊗ m → f is also a homomorphism of A-modules, however, the map M ⊗ M* → K is not necessarily a homomorphism of A-modules. This observation was actually a source of the notion of Hopf algebra, using this structure, Hopf proved a structure theorem for the cohomology algebra of Lie groups
14.
Duality (mathematics)
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Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues theorem is self-dual in this sense under the standard duality in projective geometry, many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. From a category theory viewpoint, duality can also be seen as a functor and this functor assigns to each space its dual space, and the pullback construction assigns to each arrow f, V → W its dual f∗, W∗ → V∗. In the words of Michael Atiyah, Duality in mathematics is not a theorem, the following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case. A simple, maybe the most simple, duality arises from considering subsets of a fixed set S, to any subset A ⊆ S, the complement Ac consists of all those elements in S which are not contained in A. It is again a subset of S, taking the complement has the following properties, Applying it twice gives back the original set, i. e. c = A. This is referred to by saying that the operation of taking the complement is an involution, an inclusion of sets A ⊆ B is turned into an inclusion in the opposite direction Bc ⊆ Ac. Given two subsets A and B of S, A is contained in Bc if and only if B is contained in Ac. This duality appears in topology as a duality between open and closed subsets of some fixed topological space X, a subset U of X is closed if, because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of sets is open, so dually. The interior of a set is the largest open set contained in it, because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U. A duality in geometry is provided by the cone construction. Given a set C of points in the plane R2, unlike for the complement of sets mentioned above, it is not in general true that applying the dual cone construction twice gives back the original set C. Instead, C ∗ ∗ is the smallest cone containing C which may be bigger than C. Therefore this duality is weaker than the one above, in that Applying the operation twice gives back a possibly bigger set, the other two properties carry over without change, It is still true that an inclusion C ⊆ D is turned into an inclusion in the opposite direction. Given two subsets C and D of the plane, C is contained in D ∗ if, a very important example of a duality arises in linear algebra by associating to any vector space V its dual vector space V*. Its elements are the k-linear maps φ, V → k, the three properties of the dual cone carry over to this type of duality by replacing subsets of R2 by vector space and inclusions of such subsets by linear maps. That is, Applying the operation of taking the dual vector space twice gives another vector space V**, there is always a map V → V**
15.
Pontryagin duality
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The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual. The subject is named after Lev Semenovich Pontryagin who laid down the foundations for the theory of compact abelian groups. Pontryagins treatment relied on the group being second-countable and either compact or discrete and this was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940. Moreover, any function on a group can be recovered from its discrete Fourier transform. Similarly, a group G and its dual group G ^ are not in general isomorphic, more categorically, this is not just an isomorphism of endomorphism algebras, but an isomorphism of categories – see categorical considerations. A topological group is called compact if the underlying topological space is locally compact. Its called abelian if the group is abelian. Examples of locally compact abelian groups are, R n for n a positive integer, the positive real numbers R + with multiplication as operation. This group is isomorphic to by the exponential map, any finite abelian group, with the discrete topology. By the structure theorem for finite groups, all such groups are products of cyclic groups. The integers Z under addition, again with the discrete topology, the circle group, denoted T for torus. This is the group of numbers of modulus 1. T is isomorphic as a group to the quotient group R / Z. The field Q p of p-adic numbers under addition, with the usual p-adic topology, if G is a locally compact abelian group, a character of G is a continuous group homomorphism from G with values in the circle group T. The set of all characters on G can be made into a compact abelian group, called the dual group of G. This topology in general is not metrizable, however, if the group G is a separable locally compact abelian group, then the dual group is metrizable. This is analogous to the space in linear algebra, just as for a vector space V over a field K. More abstractly, these are examples of representable functors, being represented respectively by K and T
16.
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
17.
Cambridge University Press
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Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the worlds oldest publishing house and it also holds letters patent as the Queens Printer. The Presss mission is To further the Universitys mission by disseminating knowledge in the pursuit of education, learning, Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global presence, publishing hubs, and offices in more than 40 countries. Its publishing includes journals, monographs, reference works, textbooks. Cambridge University Press is an enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press and it originated from Letters Patent granted to the University of Cambridge by Henry VIII in 1534, and has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses, authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, and Stephen Hawking. In 1591, Thomass successor, John Legate, printed the first Cambridge Bible, the London Stationers objected strenuously, claiming that they had the monopoly on Bible printing. The universitys response was to point out the provision in its charter to print all manner of books. In July 1697 the Duke of Somerset made a loan of £200 to the university towards the house and presse and James Halman, Registrary of the University. It was in Bentleys time, in 1698, that a body of scholars was appointed to be responsible to the university for the Presss affairs. The Press Syndicates publishing committee still meets regularly, and its role still includes the review, John Baskerville became University Printer in the mid-eighteenth century. Baskervilles concern was the production of the finest possible books using his own type-design, a technological breakthrough was badly needed, and it came when Lord Stanhope perfected the making of stereotype plates. This involved making a mould of the surface of a page of type. The Press was the first to use this technique, and in 1805 produced the technically successful, under the stewardship of C. J. Clay, who was University Printer from 1854 to 1882, the Press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks, during Clays administration, the Press also undertook a sizable co-publishing venture with Oxford, the Revised Version of the Bible, which was begun in 1870 and completed in 1885. It was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories, the Cambridge Modern History was published between 1902 and 1912
18.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, the work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially, until January 2009 JSTOR operated as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR
19.
ArXiv
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In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14,1991, arXiv. org passed the half-million article milestone on October 3,2008, by 2014 the submission rate had grown to more than 8,000 per month. The arXiv was made possible by the low-bandwidth TeX file format, around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Additional modes of access were added, FTP in 1991, Gopher in 1992. The term e-print was quickly adopted to describe the articles and its original domain name was xxx. lanl. gov. Due to LANLs lack of interest in the rapidly expanding technology, in 1999 Ginsparg changed institutions to Cornell University and it is now hosted principally by Cornell, with 8 mirrors around the world. Its existence was one of the factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists regularly upload their papers to arXiv. org for worldwide access, Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv. The annual budget for arXiv is approximately $826,000 for 2013 to 2017, funded jointly by Cornell University Library, annual donations were envisaged to vary in size between $2,300 to $4,000, based on each institution’s usage. As of 14 January 2014,174 institutions have pledged support for the period 2013–2017 on this basis, in September 2011, Cornell University Library took overall administrative and financial responsibility for arXivs operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it was supposed to be a three-hour tour, however, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. The lists of moderators for many sections of the arXiv are publicly available, additionally, an endorsement system was introduced in 2004 as part of an effort to ensure content that is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, new authors from recognized academic institutions generally receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for allegedly restricting scientific inquiry, perelman appears content to forgo the traditional peer-reviewed journal process, stating, If anybody is interested in my way of solving the problem, its all there – let them go and read about it. The arXiv generally re-classifies these works, e. g. in General mathematics, papers can be submitted in any of several formats, including LaTeX, and PDF printed from a word processor other than TeX or LaTeX. The submission is rejected by the software if generating the final PDF file fails, if any image file is too large. ArXiv now allows one to store and modify an incomplete submission, the time stamp on the article is set when the submission is finalized
20.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
21.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of algebra and calculus from the two-dimensional Euclidean plane. A Hilbert space is a vector space possessing the structure of an inner product that allows length. Furthermore, Hilbert spaces are complete, there are limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces, the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis —and ergodic theory, john von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis, geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space, at a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be specified by its coordinates with respect to a set of coordinate axes. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of sequences that are square-summable. The latter space is often in the literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of vectors, denoted by ℝ3. The dot product takes two vectors x and y, and produces a real number x·y, If x and y are represented in Cartesian coordinates, then the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3. The dot product satisfies the properties, It is symmetric in x and y, x · y = y · x. It is linear in its first argument, · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y. It is positive definite, for all x, x · x ≥0, with equality if. An operation on pairs of vectors that, like the dot product, a vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space, multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist