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A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes. Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic functions. Algebraic geometry based on formal schemes is called formal algebraic geometry.
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https://en.wikipedia.org/wiki/Formal_algebraic_geometry
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In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces. Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds.
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https://en.wikipedia.org/wiki/Grothendieck-Riemann-Roch_theorem
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The Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves. The theorem has been very influential, not least for the development of the Atiyah–Singer index theorem.
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https://en.wikipedia.org/wiki/Grothendieck-Riemann-Roch_theorem
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Conversely, complex analytic analogues of the Grothendieck–Riemann–Roch theorem can be proved using the index theorem for families. Alexander Grothendieck gave a first proof in a 1957 manuscript, later published. Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck's proof in 1958. Later, Grothendieck and his collaborators simplified and generalized the proof.
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https://en.wikipedia.org/wiki/Grothendieck-Riemann-Roch_theorem
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In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties. Base change is a closely related notion.
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https://en.wikipedia.org/wiki/Fiber_product_of_schemes
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In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f: X → Y {\displaystyle f:X\to Y} , the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
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https://en.wikipedia.org/wiki/Inverse_image_functor
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In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod. One can define a homology theory as a sequence of functors satisfying the Eilenberg–Steenrod axioms. The axiomatic approach, which was developed in 1945, allows one to prove results, such as the Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms.If one omits the dimension axiom (described below), then the remaining axioms define what is called an extraordinary homology theory. Extraordinary cohomology theories first arose in K-theory and cobordism.
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https://en.wikipedia.org/wiki/Dimension_axiom
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In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space X × Y {\displaystyle X\times Y} and those of the spaces X {\displaystyle X} and Y {\displaystyle Y} . The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem.
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https://en.wikipedia.org/wiki/Eilenberg–Zilber_theorem
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In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this. Throughout this article E {\displaystyle E} is an oriented, real vector bundle of rank r {\displaystyle r} over a base space X {\displaystyle X} .
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https://en.wikipedia.org/wiki/Euler_class
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In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944.
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https://en.wikipedia.org/wiki/Cup_product
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In mathematics, specifically in axiomatic set theory, a Hartogs number is an ordinal number associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X. If X cannot be well-ordered then there cannot be an injection from X to α. However, the cardinal number of α is still a minimal cardinal not less than or equal to the cardinality of X. (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of X.) The map taking X to α is sometimes called Hartogs's function. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well-orderable sets. The existence of the Hartogs number was proved by Friedrich Hartogs in 1915, using Zermelo–Fraenkel set theory alone (that is, without using the axiom of choice).
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https://en.wikipedia.org/wiki/Hartogs_number
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In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where f ′ {\displaystyle f'} is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely f ′ , {\displaystyle f',} scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f. This follows directly from the chain rule:
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https://en.wikipedia.org/wiki/Logarithmic_differential
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In mathematics, specifically in category theory and algebraic topology, the Baez–Dolan stabilization hypothesis, proposed in (Baez & Dolan 1995), states that suspension of a weak n-category has no more essential effect after n + 2 times. Precisely, it states that the suspension functor n C a t k → n C a t k + 1 {\displaystyle {\mathsf {nCat}}_{k}\to {\mathsf {nCat}}_{k+1}} is an equivalence for k ≥ n + 2 {\displaystyle k\geq n+2} .
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https://en.wikipedia.org/wiki/Stabilization_hypothesis
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In mathematics, specifically in category theory, Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in 1970 in the general context of enriched functor categories. Day convolution acts as a tensor product for a monoidal category structure on the category of functors {\displaystyle } over some monoidal category V {\displaystyle V} .
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https://en.wikipedia.org/wiki/Day_convolution
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In mathematics, specifically in category theory, F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor F, the signature. F-algebras can also be used to represent data structures used in programming, such as lists and trees. The main related concepts are initial F-algebras which may serve to encapsulate the induction principle, and the dual construction F-coalgebras.
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https://en.wikipedia.org/wiki/F-algebra
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In mathematics, specifically in category theory, a functor F: C → D {\displaystyle F:C\to D} is essentially surjective (or dense) if each object d {\displaystyle d} of D {\displaystyle D} is isomorphic to an object of the form F c {\displaystyle Fc} for some object c {\displaystyle c} of C {\displaystyle C} . Any functor that is part of an equivalence of categories is essentially surjective. As a partial converse, any full and faithful functor that is essentially surjective is part of an equivalence of categories.
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https://en.wikipedia.org/wiki/Essentially_surjective_functor
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In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear); C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts; given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is by definition the kernel of f), as does the coequaliser (this is by definition the cokernel of f).Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2.
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https://en.wikipedia.org/wiki/Preabelian_category
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In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: and where + is the group operation. Some authors have used the term additive category for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see § Special cases below).
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https://en.wikipedia.org/wiki/Preadditive_categories
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In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel. Recall that an idempotent morphism p {\displaystyle p} is an endomorphism of an object with the property that p ∘ p = p {\displaystyle p\circ p=p} . Elementary considerations show that every idempotent then has a cokernel. The pseudo-abelian condition is stronger than preadditivity, but it is weaker than the requirement that every morphism have a kernel and cokernel, as is true for abelian categories. Synonyms in the literature for pseudo-abelian include pseudoabelian and Karoubian.
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https://en.wikipedia.org/wiki/Pseudo-abelian_category
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In mathematics, specifically in category theory, a quasi-abelian category is a pre-abelian category in which the pushout of a kernel along arbitrary morphisms is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel.
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https://en.wikipedia.org/wiki/Quasi-abelian_category
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In mathematics, specifically in category theory, a semi-abelian category is a pre-abelian category in which the induced morphism f ¯: coim f → im f {\displaystyle {\overline {f}}:\operatorname {coim} f\rightarrow \operatorname {im} f} is a bimorphism, i.e., a monomorphism and an epimorphism, for every morphism f {\displaystyle f} .
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https://en.wikipedia.org/wiki/Semi-abelian_category
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In mathematics, specifically in category theory, an F {\displaystyle F} -coalgebra is a structure defined according to a functor F {\displaystyle F} , with specific properties as defined below. For both algebras and coalgebras, a functor is a convenient and general way of organizing a signature. This has applications in computer science: examples of coalgebras include lazy, infinite data structures, such as streams, and also transition systems. F {\displaystyle F} -coalgebras are dual to F {\displaystyle F} -algebras. Just as the class of all algebras for a given signature and equational theory form a variety, so does the class of all F {\displaystyle F} -coalgebras satisfying a given equational theory form a covariety, where the signature is given by F {\displaystyle F} .
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https://en.wikipedia.org/wiki/F-coalgebra
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In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.
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https://en.wikipedia.org/wiki/Additive_category
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In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an exponential law.
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https://en.wikipedia.org/wiki/Exponential_object
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In mathematics, specifically in category theory, an extranatural transformation is a generalization of the notion of natural transformation.
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https://en.wikipedia.org/wiki/Extranatural_transformation
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In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.
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https://en.wikipedia.org/wiki/Canonical_bifunctor
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In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms. The initial object of Cat is the empty category 0, which is the category of no objects and no morphisms.
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https://en.wikipedia.org/wiki/Category_of_small_categories
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The terminal object is the terminal category or trivial category 1 with a single object and morphism.The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”. But it is possible to form a quasicategory (meaning objects and morphisms merely form a conglomerate) of all categories.
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https://en.wikipedia.org/wiki/Category_of_small_categories
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In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an element from every non-empty set.
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https://en.wikipedia.org/wiki/Axiom_of_global_choice
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In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings of toric varieties.
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https://en.wikipedia.org/wiki/Normal_polytope
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In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial P is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree d in n variables for each positive integer d ≤ n, and it is formed by adding together all distinct products of d distinct variables.
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https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial
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In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with integral coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the rationals, but not over the integers.
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https://en.wikipedia.org/wiki/Power_sum_symmetric_polynomial
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In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk.
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https://en.wikipedia.org/wiki/Fatou's_theorem
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In mathematics, specifically in complex geometry, the Kähler quotient of a Kähler manifold X {\displaystyle X} by a Lie group G {\displaystyle G} acting on X {\displaystyle X} by preserving the Kähler structure and with moment map μ: X → g ∗ {\displaystyle \mu :X\to {\mathfrak {g}}^{*}} (with respect to the Kähler form) is the quotient μ − 1 ( 0 ) / G . {\displaystyle \mu ^{-1}(0)/G.} If G {\displaystyle G} acts freely and properly, then μ − 1 ( 0 ) / G {\displaystyle \mu ^{-1}(0)/G} is a new Kähler manifold whose Kähler form is given by the symplectic quotient construction.By the Kempf-Ness theorem, a Kähler quotient by a compact Lie group G {\displaystyle G} is closely related to a geometric invariant theory quotient by the complexification of G {\displaystyle G} .
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https://en.wikipedia.org/wiki/Kähler_quotient
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In mathematics, specifically in computational geometry, geometric nonrobustness is a problem wherein branching decisions in computational geometry algorithms are based on approximate numerical computations, leading to various forms of unreliability including ill-formed output and software failure through crashing or infinite loops. For instance, algorithms for problems like the construction of a convex hull rely on testing whether certain "numerical predicates" have values that are positive, negative, or zero. If an inexact floating-point computation causes a value that is near zero to have a different sign than its exact value, the resulting inconsistencies can propagate through the algorithm causing it to produce output that is far from the correct output, or even to crash. One method for avoiding this problem involves using integers rather than floating point numbers for all coordinates and other quantities represented by the algorithm, and determining the precision required for all calculations to avoid integer overflow conditions.
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https://en.wikipedia.org/wiki/Robust_geometric_computation
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For instance, two-dimensional convex hulls can be computed using predicates that test the sign of quadratic polynomials, and therefore may require twice as many bits of precision within these calculations as the input numbers. When integer arithmetic cannot be used (for instance, when the result of a calculation is an algebraic number rather than an integer or rational number), a second method is to use symbolic algebra to perform all computations with exactly represented algebraic numbers rather than numerical approximations to them. A third method, sometimes called a "floating point filter", is to compute numerical predicates first using an inexact method based on floating-point arithmetic, but to maintain bounds on how accurate the result is, and repeat the calculation using slower symbolic algebra methods or numerically with additional precision when these bounds do not separate the calculated value from zero.
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https://en.wikipedia.org/wiki/Robust_geometric_computation
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In mathematics, specifically in control theory, subspace identification (SID) aims at identifying linear time invariant (LTI) state space models from input-output data. SID does not require that the user parametrizes the system matrices before solving a parametric optimization problem and, as a consequence, SID methods do not suffer from problems related to local minima that often lead to unsatisfactory identification results.
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https://en.wikipedia.org/wiki/Subspace_identification_method
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In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.
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https://en.wikipedia.org/wiki/Equilibrium_points
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In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form g = φ ( d x 1 2 + ⋯ + d x n 2 ) , {\displaystyle g=\varphi (dx_{1}^{2}+\cdots +dx_{n}^{2}),} where φ {\displaystyle \varphi } is a positive smooth function. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.)
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https://en.wikipedia.org/wiki/Isothermal_coordinates
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Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally conformally flat. In dimension 3, a Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes. In dimensions > 3, a metric is locally conformally flat if and only if its Weyl tensor vanishes.
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https://en.wikipedia.org/wiki/Isothermal_coordinates
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In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
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https://en.wikipedia.org/wiki/Morse–Bott_function
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Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics (critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory.
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https://en.wikipedia.org/wiki/Morse–Bott_function
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In mathematics, specifically in differential topology, a Kervaire manifold K 4 n + 2 {\displaystyle K^{4n+2}} is a piecewise-linear manifold of dimension 4 n + 2 {\displaystyle 4n+2} constructed by Michel Kervaire (1960) by plumbing together the tangent bundles of two ( 2 n + 1 ) {\displaystyle (2n+1)} -spheres, and then gluing a ball to the result. In 10 dimensions this gives a piecewise-linear manifold with no smooth structure.
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https://en.wikipedia.org/wiki/Kervaire_manifold
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In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. The method is also occasionally known as the "cross your heart" method because lines resembling a heart outline can be drawn to remember which things to multiply together. Given an equation like a b = c d , {\displaystyle {\frac {a}{b}}={\frac {c}{d}},} where b and d are not zero, one can cross-multiply to get a d = b c or a = b c d . {\displaystyle ad=bc\quad {\text{or}}\quad a={\frac {bc}{d}}.} In Euclidean geometry the same calculation can be achieved by considering the ratios as those of similar triangles.
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https://en.wikipedia.org/wiki/Rule_of_proportion
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In mathematics, specifically in functional analysis and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessarily and sufficient condition for a Hausdorff pre-Hilbert space to be a Hilbert space in terms of the canonical isometry of a pre-Hilbert space into its anti-dual.
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https://en.wikipedia.org/wiki/Fundamental_theorem_of_Hilbert_spaces
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In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C {\displaystyle \mathbb {C} } ) to linear operators valued in topological vector spaces (TVSs).
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https://en.wikipedia.org/wiki/Vector-valued_Hahn–Banach_theorems
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In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X {\displaystyle X} that has a partial order ≤ {\displaystyle \,\leq \,} making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets. Ordered vector lattices have important applications in spectral theory.
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https://en.wikipedia.org/wiki/Topological_vector_lattice
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In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) X that has a partial order ≤ making it into an ordered vector space whose positive cone C := { x ∈ X: x ≥ 0 } {\displaystyle C:=\left\{x\in X:x\geq 0\right\}} is a closed subset of X. Ordered TVS have important applications in spectral theory.
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https://en.wikipedia.org/wiki/Ordered_topological_vector_space
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In mathematics, specifically in functional analysis, a Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are inner (that is of the form a ↦ a . x − x . a {\displaystyle a\mapsto a.x-x.a} for some x {\displaystyle x} in the dual module). An equivalent characterization is that A is amenable if and only if it has a virtual diagonal.
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https://en.wikipedia.org/wiki/Amenable_Banach_algebra
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In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: A is a topologically closed set in the norm topology of operators. A is closed under the operation of taking adjoints of operators.Another important class of non-Hilbert C*-algebras includes the algebra C 0 ( X ) {\displaystyle C_{0}(X)} of complex-valued continuous functions on X that vanish at infinity, where X is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables.
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https://en.wikipedia.org/wiki/C*_algebra
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This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras, which culminated in a series of papers on rings of operators. These papers considered a special class of C*-algebras which are now known as von Neumann algebras.
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https://en.wikipedia.org/wiki/C*_algebra
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Around 1943, the work of Israel Gelfand and Mark Naimark yielded an abstract characterisation of C*-algebras making no reference to operators on a Hilbert space. C*-algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics. Another active area of research is the program to obtain classification, or to determine the extent of which classification is possible, for separable simple nuclear C*-algebras.
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https://en.wikipedia.org/wiki/C*_algebra
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In mathematics, specifically in functional analysis, a family G {\displaystyle {\mathcal {G}}} of subsets a topological vector space (TVS) X {\displaystyle X} is said to be saturated if G {\displaystyle {\mathcal {G}}} contains a non-empty subset of X {\displaystyle X} and if for every G ∈ G , {\displaystyle G\in {\mathcal {G}},} the following conditions all hold: G {\displaystyle {\mathcal {G}}} contains every subset of G {\displaystyle G} ; the union of any finite collection of elements of G {\displaystyle {\mathcal {G}}} is an element of G {\displaystyle {\mathcal {G}}} ; for every scalar a , {\displaystyle a,} G {\displaystyle {\mathcal {G}}} contains a G {\displaystyle aG} ; the closed convex balanced hull of G {\displaystyle G} belongs to G . {\displaystyle {\mathcal {G}}.}
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https://en.wikipedia.org/wiki/Saturated_family
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In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions ƒ: D → C {\displaystyle \mathbb {C} } ,(where D is the open unit disk in the complex plane C {\displaystyle \mathbb {C} } ) that extend to a continuous function on the closure of D. That is, A ( D ) = H ∞ ( D ) ∩ C ( D ¯ ) , {\displaystyle A(\mathbf {D} )=H^{\infty }(\mathbf {D} )\cap C({\overline {\mathbf {D} }}),} where H∞(D) denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space). When endowed with the pointwise addition (ƒ + g)(z) = ƒ(z) + g(z), and pointwise multiplication (ƒg)(z) = ƒ(z)g(z), this set becomes an algebra over C, since if ƒ and g belong to the disk algebra then so do ƒ + g and ƒg. Given the uniform norm, ‖ f ‖ = sup { | f ( z ) | ∣ z ∈ D } = max { | f ( z ) | ∣ z ∈ D ¯ } , {\displaystyle \|f\|=\sup\{|f(z)|\mid z\in \mathbf {D} \}=\max\{|f(z)|\mid z\in {\overline {\mathbf {D} }}\},} by construction it becomes a uniform algebra and a commutative Banach algebra.
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https://en.wikipedia.org/wiki/Disk_algebra
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By construction the disc algebra is a closed subalgebra of the Hardy space H∞. In contrast to the stronger requirement that a continuous extension to the circle exists, it is a lemma of Fatou that a general element of H∞ can be radially extended to the circle almost everywhere. == References ==
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https://en.wikipedia.org/wiki/Disk_algebra
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In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961). Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension n = p + q + 1 {\displaystyle n=p+q+1} , could be described as removing an imbedded sphere of dimension p from M. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary.
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https://en.wikipedia.org/wiki/Algebraic_surgery_theory
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This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood manifold M and perform surgery on it to produce a manifold M′ having some desired property, in such a way that the effects on the homology, homotopy groups, or other invariants of the manifold are known. A relatively easy argument using Morse theory shows that a manifold can be obtained from another one by a sequence of spherical modifications if and only if those two belong to the same cobordism class.The classification of exotic spheres by Michel Kervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.
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https://en.wikipedia.org/wiki/Algebraic_surgery_theory
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In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a mathematical tree called a hydra where, usually, the goal is to cut off the hydra's "heads" while the hydra simultaneously expands itself. Hydra games can be used to generate large numbers or infinite ordinals or prove the strength of certain mathematical theories.Unlike their combinatorial counterparts like TREE and SCG, no search is required to compute these fast-growing function values – one must simply keep applying the transformation rule to the tree until the game says to stop.
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https://en.wikipedia.org/wiki/Hydra_game
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In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in which the common order is p are a particular kind of p-group. A group for which p = 2 (that is, an elementary abelian 2-group) is sometimes called a Boolean group.Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ)n for n a non-negative integer (sometimes called the group's rank).
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https://en.wikipedia.org/wiki/Elementary_abelian
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Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript notation means the n-fold direct product of groups.In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p. (Note that in the finite case the direct product and direct sum coincide, but this is not so in the infinite case.) In the rest of this article, all groups are assumed finite.
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https://en.wikipedia.org/wiki/Elementary_abelian
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In mathematics, specifically in group theory, residue-class-wise affine groups are certain permutation groups acting on Z {\displaystyle \mathbb {Z} } (the integers), whose elements are bijective residue-class-wise affine mappings. A mapping f: Z → Z {\displaystyle f:\mathbb {Z} \rightarrow \mathbb {Z} } is called residue-class-wise affine if there is a nonzero integer m {\displaystyle m} such that the restrictions of f {\displaystyle f} to the residue classes (mod m {\displaystyle m} ) are all affine. This means that for any residue class r ( m ) ∈ Z / m Z {\displaystyle r(m)\in \mathbb {Z} /m\mathbb {Z} } there are coefficients a r ( m ) , b r ( m ) , c r ( m ) ∈ Z {\displaystyle a_{r(m)},b_{r(m)},c_{r(m)}\in \mathbb {Z} } such that the restriction of the mapping f {\displaystyle f} to the set r ( m ) = { r + k m ∣ k ∈ Z } {\displaystyle r(m)=\{r+km\mid k\in \mathbb {Z} \}} is given by f | r ( m ): r ( m ) → Z , n ↦ a r ( m ) ⋅ n + b r ( m ) c r ( m ) {\displaystyle f|_{r(m)}:r(m)\rightarrow \mathbb {Z} ,\ n\mapsto {\frac {a_{r(m)}\cdot n+b_{r(m)}}{c_{r(m)}}}} .Residue-class-wise affine groups are countable, and they are accessible to computational investigations.
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https://en.wikipedia.org/wiki/Residue-class-wise_affine_group
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Many of them act multiply transitively on Z {\displaystyle \mathbb {Z} } or on subsets thereof. A particularly basic type of residue-class-wise affine permutations are the class transpositions: given disjoint residue classes r 1 ( m 1 ) {\displaystyle r_{1}(m_{1})} and r 2 ( m 2 ) {\displaystyle r_{2}(m_{2})} , the corresponding class transposition is the permutation of Z {\displaystyle \mathbb {Z} } which interchanges r 1 + k m 1 {\displaystyle r_{1}+km_{1}} and r 2 + k m 2 {\displaystyle r_{2}+km_{2}} for every k ∈ Z {\displaystyle k\in \mathbb {Z} } and which fixes everything else. Here it is assumed that 0 ≤ r 1 < m 1 {\displaystyle 0\leq r_{1}
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https://en.wikipedia.org/wiki/Residue-class-wise_affine_group
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In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th roots. The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.
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https://en.wikipedia.org/wiki/Quasicyclic_group
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In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. an outer semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation.As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension).
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https://en.wikipedia.org/wiki/Semi-direct_product
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In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G ⊕ H {\displaystyle G\oplus H} . Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.
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https://en.wikipedia.org/wiki/Direct_product_(group_theory)
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In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups. The name "groups of Lie type" is due to the close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers. Dieudonné (1971) and Carter (1989) are standard references for groups of Lie type.
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https://en.wikipedia.org/wiki/Groups_of_Lie_type
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In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer.
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https://en.wikipedia.org/wiki/Commensurability_(group_theory)
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In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
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https://en.wikipedia.org/wiki/Generalized_homology_theory
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From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with functions and pullbacks in geometric situations: given spaces X and Y, and some kind of function F on Y, for any mapping f: X → Y, composition with f gives rise to a function F ∘ f on X. The most important cohomology theories have a product, the cup product, which gives them a ring structure. Because of this feature, cohomology is usually a stronger invariant than homology.
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https://en.wikipedia.org/wiki/Generalized_homology_theory
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In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism". The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., pseudo-functor, pseudoalgebra.
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https://en.wikipedia.org/wiki/Coherence_theorem
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In mathematics, specifically in homotopy theory in the context of a model category M, a fibrant object A of M is an object that has a fibration to the terminal object of the category.
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https://en.wikipedia.org/wiki/Fibrant_object
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In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle E G → B G {\displaystyle EG\to BG} . As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space.
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https://en.wikipedia.org/wiki/Classifying_space
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This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial, that is, BG is an Eilenberg–MacLane space, or a K(G,1).
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https://en.wikipedia.org/wiki/Classifying_space
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In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a complete quadrilateral is a system of four lines, no three of which pass through the same point, and the six points of intersection of these lines. The complete quadrangle was called a tetrastigm by Lachlan (1893), and the complete quadrilateral was called a tetragram; those terms are occasionally still used.
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https://en.wikipedia.org/wiki/Complete_quadrilateral
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In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite was originally proved by Helmut Hasse, and the general result was proved by Cahit Arf.
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https://en.wikipedia.org/wiki/Hasse–Arf_theorem
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In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
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https://en.wikipedia.org/wiki/Borel_probability_measure
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In mathematics, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.
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https://en.wikipedia.org/wiki/Trivial_measure
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In mathematics, specifically in number theory, Newman's conjecture is a conjecture about the behavior of the partition function modulo any integer. Specifically, it states that for any integers m and r such that 0 ≤ r ≤ m − 1 {\displaystyle 0\leq r\leq m-1} , the value of the partition function p ( n ) {\displaystyle p(n)} satisfies the congruence p ( n ) ≡ r ( mod m ) {\displaystyle p(n)\equiv r{\pmod {m}}} for infinitely many non-negative integers n. It was formulated by mathematician Morris Newman in 1960. It is unsolved as of 2020.
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https://en.wikipedia.org/wiki/Newman's_conjecture
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In mathematics, specifically in number theory, a Cunningham number is a certain kind of integer named after English mathematician A. J. C. Cunningham.
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https://en.wikipedia.org/wiki/Cunningham_number
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In mathematics, specifically in number theory, a binomial number is an integer which can be obtained by evaluating a homogeneous polynomial containing two terms. It is a generalization of a Cunningham number.
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https://en.wikipedia.org/wiki/Binomial_number
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In mathematics, specifically in number theory, the extremal orders of an arithmetic function are best possible bounds of the given arithmetic function. Specifically, if f(n) is an arithmetic function and m(n) is a non-decreasing function that is ultimately positive and lim inf n → ∞ f ( n ) m ( n ) = 1 {\displaystyle \liminf _{n\to \infty }{\frac {f(n)}{m(n)}}=1} we say that m is a minimal order for f. Similarly if M(n) is a non-decreasing function that is ultimately positive and lim sup n → ∞ f ( n ) M ( n ) = 1 {\displaystyle \limsup _{n\to \infty }{\frac {f(n)}{M(n)}}=1} we say that M is a maximal order for f.: 80 Here, lim inf n → ∞ {\displaystyle \liminf _{n\to \infty }} and lim sup n → ∞ {\displaystyle \limsup _{n\to \infty }} denote the limit inferior and limit superior, respectively. The subject was first studied systematically by Ramanujan starting in 1915.: 87
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https://en.wikipedia.org/wiki/Extremal_orders_of_an_arithmetic_function
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In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology of the classifying space being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the group's reduced C*-algebra is a purely analytical object. The conjecture, if true, would have some older famous conjectures as consequences.
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https://en.wikipedia.org/wiki/Baum–Connes_conjecture
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For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for discrete torsion-free groups, and the injectivity is closely related to the Novikov conjecture. The conjecture is also closely related to index theory, as the assembly map μ {\displaystyle \mu } is a sort of index, and it plays a major role in Alain Connes' noncommutative geometry program. The origins of the conjecture go back to Fredholm theory, the Atiyah–Singer index theorem and the interplay of geometry with operator K-theory as expressed in the works of Brown, Douglas and Fillmore, among many other motivating subjects.
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https://en.wikipedia.org/wiki/Baum–Connes_conjecture
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In mathematics, specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle A^{*}} on that space according to the rule ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ , {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product on the vector space. The adjoint may also be called the Hermitian conjugate or simply the Hermitian after Charles Hermite. It is often denoted by A† in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators are represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose).
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https://en.wikipedia.org/wiki/Hermitian_conjugate
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The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H {\displaystyle H} . The definition has been further extended to include unbounded densely defined operators whose domain is topologically dense in - but not necessarily equal to - H . {\displaystyle H.}
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https://en.wikipedia.org/wiki/Hermitian_conjugate
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In mathematics, specifically in order theory and functional analysis, a Fréchet lattice is a topological vector lattice that is also a Fréchet space. Fréchet lattices are important in the theory of topological vector lattices.
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https://en.wikipedia.org/wiki/Fréchet_lattice
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In mathematics, specifically in order theory and functional analysis, a band in a vector lattice X {\displaystyle X} is a subspace M {\displaystyle M} of X {\displaystyle X} that is solid and such that for all S ⊆ M {\displaystyle S\subseteq M} such that x = sup S {\displaystyle x=\sup S} exists in X , {\displaystyle X,} we have x ∈ M . {\displaystyle x\in M.} The smallest band containing a subset S {\displaystyle S} of X {\displaystyle X} is called the band generated by S {\displaystyle S} in X . {\displaystyle X.} A band generated by a singleton set is called a principal band.
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https://en.wikipedia.org/wiki/Band_(order_theory)
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In mathematics, specifically in order theory and functional analysis, a filter F {\displaystyle {\mathcal {F}}} in an order complete vector lattice X {\displaystyle X} is order convergent if it contains an order bounded subset (that is, is contained in an interval of the form := { x ∈ X: a ≤ x and x ≤ b } {\displaystyle :=\{x\in X:a\leq x{\text{ and }}x\leq b\}} ) and if F , {\displaystyle {\mathcal {F}},} where OBound ( X ) {\displaystyle \operatorname {OBound} (X)} is the set of all order bounded subsets of X, in which case this common value is called the order limit of F {\displaystyle {\mathcal {F}}} in X . {\displaystyle X.} Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology.
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https://en.wikipedia.org/wiki/Order_convergence
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In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices.
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https://en.wikipedia.org/wiki/Locally_convex_vector_lattice
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In mathematics, specifically in order theory and functional analysis, a normed lattice is a topological vector lattice that is also a normed space whose unit ball is a solid set. Normed lattices are important in the theory of topological vector lattices. They are closely related to Banach vector lattices, which are normed vector lattices that are also Banach spaces.
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https://en.wikipedia.org/wiki/Normed_lattice
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In mathematics, specifically in order theory and functional analysis, a sequence of positive elements ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} in a preordered vector space X {\displaystyle X} (that is, x i ≥ 0 {\displaystyle x_{i}\geq 0} for all i {\displaystyle i} ) is called order summable if sup n = 1 , 2 , … ∑ i = 1 n x i {\displaystyle \sup _{n=1,2,\ldots }\sum _{i=1}^{n}x_{i}} exists in X {\displaystyle X} . For any 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } , we say that a sequence ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} of positive elements of X {\displaystyle X} is of type ℓ p {\displaystyle \ell ^{p}} if there exists some z ∈ X {\displaystyle z\in X} and some sequence ( c i ) i = 1 ∞ {\displaystyle \left(c_{i}\right)_{i=1}^{\infty }} in ℓ p {\displaystyle \ell ^{p}} such that 0 ≤ x i ≤ c i z {\displaystyle 0\leq x_{i}\leq c_{i}z} for all i {\displaystyle i} .The notion of order summable sequences is related to the completeness of the order topology.
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https://en.wikipedia.org/wiki/Order_summable
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In mathematics, specifically in order theory and functional analysis, a subset S {\displaystyle S} of a vector lattice is said to be solid and is called an ideal if for all s ∈ S {\displaystyle s\in S} and x ∈ X , {\displaystyle x\in X,} if | x | ≤ | s | {\displaystyle |x|\leq |s|} then x ∈ S . {\displaystyle x\in S.} An ordered vector space whose order is Archimedean is said to be Archimedean ordered.
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https://en.wikipedia.org/wiki/Solid_set
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If S ⊆ X {\displaystyle S\subseteq X} then the ideal generated by S {\displaystyle S} is the smallest ideal in X {\displaystyle X} containing S . {\displaystyle S.} An ideal generated by a singleton set is called a principal ideal in X . {\displaystyle X.}
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https://en.wikipedia.org/wiki/Solid_set
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In mathematics, specifically in order theory and functional analysis, an abstract L-space, an AL-space, or an abstract Lebesgue space is a Banach lattice ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} whose norm is additive on the positive cone of X.In probability theory, it means the standard probability space.
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https://en.wikipedia.org/wiki/Abstract_L-space
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In mathematics, specifically in order theory and functional analysis, an abstract m-space or an AM-space is a Banach lattice ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} whose norm satisfies ‖ sup { x , y } ‖ = sup { ‖ x ‖ , ‖ y ‖ } {\displaystyle \left\|\sup\{x,y\}\right\|=\sup \left\{\|x\|,\|y\|\right\}} for all x and y in the positive cone of X. We say that an AM-space X is an AM-space with unit if in addition there exists some u ≥ 0 in X such that the interval := { z ∈ X: −u ≤ z and z ≤ u } is equal to the unit ball of X; such an element u is unique and an order unit of X.
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https://en.wikipedia.org/wiki/Abstract_m-space
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In mathematics, specifically in order theory and functional analysis, an element x {\displaystyle x} of a vector lattice X {\displaystyle X} is called a weak order unit in X {\displaystyle X} if x ≥ 0 {\displaystyle x\geq 0} and also for all y ∈ X , {\displaystyle y\in X,} inf { x , | y | } = 0 implies y = 0. {\displaystyle \inf\{x,|y|\}=0{\text{ implies }}y=0.}
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https://en.wikipedia.org/wiki/Weak_order_unit
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In mathematics, specifically in order theory and functional analysis, an element x {\displaystyle x} of an ordered topological vector space X {\displaystyle X} is called a quasi-interior point of the positive cone C {\displaystyle C} of X {\displaystyle X} if x ≥ 0 {\displaystyle x\geq 0} and if the order interval := { z ∈ Z: 0 ≤ z and z ≤ x } {\displaystyle :=\{z\in Z:0\leq z{\text{ and }}z\leq x\}} is a total subset of X {\displaystyle X} ; that is, if the linear span of {\displaystyle } is a dense subset of X . {\displaystyle X.}
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https://en.wikipedia.org/wiki/Quasi-interior_point
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In mathematics, specifically in order theory and functional analysis, if C {\displaystyle C} is a cone at 0 in a vector space X {\displaystyle X} such that 0 ∈ C , {\displaystyle 0\in C,} then a subset S ⊆ X {\displaystyle S\subseteq X} is said to be C {\displaystyle C} -saturated if S = C , {\displaystyle S=_{C},} where C := ( S + C ) ∩ ( S − C ) . {\displaystyle _{C}:=(S+C)\cap (S-C).} Given a subset S ⊆ X , {\displaystyle S\subseteq X,} the C {\displaystyle C} -saturated hull of S {\displaystyle S} is the smallest C {\displaystyle C} -saturated subset of X {\displaystyle X} that contains S .
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https://en.wikipedia.org/wiki/Cone-saturated
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{\displaystyle S.} If F {\displaystyle {\mathcal {F}}} is a collection of subsets of X {\displaystyle X} then C := { C: F ∈ F } . {\displaystyle \left_{C}:=\left\{_{C}:F\in {\mathcal {F}}\right\}.}
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https://en.wikipedia.org/wiki/Cone-saturated
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If T {\displaystyle {\mathcal {T}}} is a collection of subsets of X {\displaystyle X} and if F {\displaystyle {\mathcal {F}}} is a subset of T {\displaystyle {\mathcal {T}}} then F {\displaystyle {\mathcal {F}}} is a fundamental subfamily of T {\displaystyle {\mathcal {T}}} if every T ∈ T {\displaystyle T\in {\mathcal {T}}} is contained as a subset of some element of F . {\displaystyle {\mathcal {F}}.} If G {\displaystyle {\mathcal {G}}} is a family of subsets of a TVS X {\displaystyle X} then a cone C {\displaystyle C} in X {\displaystyle X} is called a G {\displaystyle {\mathcal {G}}} -cone if { C ¯: G ∈ G } {\displaystyle \left\{{\overline {_{C}}}:G\in {\mathcal {G}}\right\}} is a fundamental subfamily of G {\displaystyle {\mathcal {G}}} and C {\displaystyle C} is a strict G {\displaystyle {\mathcal {G}}} -cone if { C: B ∈ B } {\displaystyle \left\{_{C}:B\in {\mathcal {B}}\right\}} is a fundamental subfamily of B . {\displaystyle {\mathcal {B}}.} C {\displaystyle C} -saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.
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https://en.wikipedia.org/wiki/Cone-saturated
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