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These subspaces increase with n. The 0-skeleton is a discrete space, and the 1-skeleton a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when X has infinite dimension, in the sense that the Xn do not become constant as n → ∞.
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https://en.wikipedia.org/wiki/Skeleton_(topology)
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In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient is zero). Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i.e., parallel to the x-axis). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the xy plane.
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https://en.wikipedia.org/wiki/Stationary_points
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In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in analysis and topology, continuous functions, and so on.
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https://en.wikipedia.org/wiki/Hom_space
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In category theory, morphism is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism.The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions. In category theory, morphisms are sometimes also called arrows.
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https://en.wikipedia.org/wiki/Hom_space
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In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S ( n , k ) {\displaystyle S(n,k)} or { n k } {\displaystyle \textstyle \left\{{n \atop k}\right\}} . Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. They are named after James Stirling.
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https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
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The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the second kind. Identities linking the two kinds appear in the article on Stirling numbers.
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https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
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In mathematics, particularly in combinatorics, given a family of sets, here called a collection C, a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of C (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal: One variation is that there is a bijection f from the transversal to C such that x is an element of f(x) for each x in the transversal.
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https://en.wikipedia.org/wiki/Transversal_(combinatorics)
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In this case, the transversal is also called a system of distinct representatives (SDR). : 29 The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of C. In this situation, the members of the system of representatives are not necessarily distinct. : 692: 322 In computer science, computing transversals is useful in several application domains, with the input family of sets often being described as a hypergraph.
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https://en.wikipedia.org/wiki/Transversal_(combinatorics)
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In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
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https://en.wikipedia.org/wiki/Compact_Riemann_surface
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The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions.
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https://en.wikipedia.org/wiki/Compact_Riemann_surface
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A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and real projective plane do not. Geometrical facts about Riemann surfaces are as "nice" as possible, and they often provide the intuition and motivation for generalizations to other curves, manifolds or varieties. The Riemann–Roch theorem is a prime example of this influence.
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https://en.wikipedia.org/wiki/Compact_Riemann_surface
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In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Sergei A. Abramov in 1989.
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https://en.wikipedia.org/wiki/Abramov's_algorithm
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In mathematics, particularly in differential geometry, a Zoll surface, named after Otto Zoll, is a surface homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length. While the usual unit-sphere metric on S2 obviously has this property, it also has an infinite-dimensional family of geometrically distinct deformations that are still Zoll surfaces. In particular, most Zoll surfaces do not have constant curvature. Zoll, a student of David Hilbert, discovered the first non-trivial examples.
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https://en.wikipedia.org/wiki/Zoll_surface
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In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. The word osculate is from the Latin osculatus which is a past participle of osculari, meaning to kiss. An osculating plane is thus a plane which "kisses" a submanifold. The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors.
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https://en.wikipedia.org/wiki/Osculating_plane
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In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space, R 2 m , {\displaystyle \mathbb {R} ^{2m},} if m > 0. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the real projective spaces of dimension m cannot be embedded into real (2m − 1)-space if m is a power of two (as can be seen from a characteristic class argument, also due to Whitney). The weak Whitney embedding theorem states that any continuous function from an n-dimensional manifold to an m-dimensional manifold may be approximated by a smooth embedding provided m > 2n. Whitney similarly proved that such a map could be approximated by an immersion provided m > 2n − 1. This last result is sometimes called the Whitney immersion theorem.
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https://en.wikipedia.org/wiki/Whitney_embedding_theorem
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In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or other related invariant property thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form geometric shapes that resemble tongues, in which case they are called Arnold tongues.Arnold tongues are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes and cardiac electric waves. Sometimes the frequency of oscillation depends on, or is constrained (i.e., phase-locked or mode-locked, in some contexts) based on some quantity, and it is often of interest to study this relation. For instance, the outset of a tumor triggers in the area a series of substance (mainly proteins) oscillations that interact with each other; simulations show that these interactions cause Arnold tongues to appear, that is, the frequency of some oscillations constrain the others, and this can be used to control tumor growth.Other examples where Arnold tongues can be found include the inharmonicity of musical instruments, orbital resonance and tidal locking of orbiting moons, mode-locking in fiber optics and phase-locked loops and other electronic oscillators, as well as in cardiac rhythms, heart arrhythmias and cell cycle.One of the simplest physical models that exhibits mode-locking consists of two rotating disks connected by a weak spring.
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https://en.wikipedia.org/wiki/Phase_locking
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One disk is allowed to spin freely, and the other is driven by a motor. Mode locking occurs when the freely-spinning disk turns at a frequency that is a rational multiple of that of the driven rotator. The simplest mathematical model that exhibits mode-locking is the circle map, which attempts to capture the motion of the spinning disks at discrete time intervals.
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https://en.wikipedia.org/wiki/Phase_locking
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In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.
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https://en.wikipedia.org/wiki/Bifurcation_diagram
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In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the flow of the system. More precisely, one considers a periodic orbit with initial conditions within a section of the space, which leaves that section afterwards, and observes the point at which this orbit first returns to the section. One then creates a map to send the first point to the second, hence the name first recurrence map. The transversality of the Poincaré section means that periodic orbits starting on the subspace flow through it and not parallel to it.
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https://en.wikipedia.org/wiki/Poincaré_map
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A Poincaré map can be interpreted as a discrete dynamical system with a state space that is one dimension smaller than the original continuous dynamical system. Because it preserves many properties of periodic and quasiperiodic orbits of the original system and has a lower-dimensional state space, it is often used for analyzing the original system in a simpler way. In practice this is not always possible as there is no general method to construct a Poincaré map.
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https://en.wikipedia.org/wiki/Poincaré_map
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A Poincaré map differs from a recurrence plot in that space, not time, determines when to plot a point. For instance, the locus of the Moon when the Earth is at perihelion is a recurrence plot; the locus of the Moon when it passes through the plane perpendicular to the Earth's orbit and passing through the Sun and the Earth at perihelion is a Poincaré map. It was used by Michel Hénon to study the motion of stars in a galaxy, because the path of a star projected onto a plane looks like a tangled mess, while the Poincaré map shows the structure more clearly.
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https://en.wikipedia.org/wiki/Poincaré_map
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In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be used as a placeholder in objects such as polynomials and formal power series. In particular: It does not designate a constant or a parameter of the problem.
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https://en.wikipedia.org/wiki/Indeterminate_(variable)
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It is not an unknown that could be solved for. It is not a variable designating a function argument, or a variable being summed or integrated over. It is not any type of bound variable. It is just a symbol used in an entirely formal way.When used as placeholders, a common operation is to substitute mathematical expressions (of an appropriate type) for the indeterminates. By a common abuse of language, mathematical texts may not clearly distinguish indeterminates from ordinary variables.
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https://en.wikipedia.org/wiki/Indeterminate_(variable)
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In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form ∑ i = 1 ∞ r i x i {\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}} where x 1 , x 2 , … {\displaystyle x_{1},x_{2},\ldots } are all elements of a topological vector space X {\displaystyle X} , and all r 1 , r 2 , … {\displaystyle r_{1},r_{2},\ldots } are non-negative real numbers that sum to 1 {\displaystyle 1} (that is, such that ∑ i = 1 ∞ r i = 1 {\displaystyle \sum _{i=1}^{\infty }r_{i}=1} ).
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https://en.wikipedia.org/wiki/Convex_series
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In mathematics, particularly in functional analysis and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle.
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https://en.wikipedia.org/wiki/Ursescu_theorem
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In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.
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https://en.wikipedia.org/wiki/Approximation_of_the_identity
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In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f: X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.
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https://en.wikipedia.org/wiki/Closed_linear_operator
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In mathematics, particularly in functional analysis and topology, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their graph. In its most elementary form, the closed graph theorem states that a linear function between two Banach spaces is continuous if and only if the graph of that function is closed. The closed graph theorem has extensive application throughout functional analysis, because it can control whether a partially-defined linear operator admits continuous extensions. For this reason, it has been generalized to many circumstances beyond the elementary formulation above.
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https://en.wikipedia.org/wiki/Closed_graph_theorem_(functional_analysis)
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In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space X such that the topology of X coincides with the Mackey topology τ(X,X′), the finest topology which still preserves the continuous dual. They are named after George Mackey.
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https://en.wikipedia.org/wiki/Mackey_space
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In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator. Bornological spaces were first studied by George Mackey. The name was coined by Bourbaki after borné, the French word for "bounded".
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https://en.wikipedia.org/wiki/Mackey_convergence
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In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators.
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https://en.wikipedia.org/wiki/Projective_measurement
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The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements. They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state.
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https://en.wikipedia.org/wiki/Projective_measurement
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In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms.
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https://en.wikipedia.org/wiki/Seminormed_space
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In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.
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https://en.wikipedia.org/wiki/Webbed_space
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In mathematics, particularly in functional analysis, the Krein-Smulian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein and Vitold Shmulyan, who published them in 1940.
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https://en.wikipedia.org/wiki/Krein–Smulian_theorem
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In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number λ {\displaystyle \lambda } is said to be in the spectrum of a bounded linear operator T {\displaystyle T} if T − λ I {\displaystyle T-\lambda I} either has no set-theoretic inverse; or the set-theoretic inverse is either unbounded or defined on a non-dense subset.Here, I {\displaystyle I} is the identity operator. By the closed graph theorem, λ {\displaystyle \lambda } is in the spectrum if and only if the bounded operator T − λ I: V → V {\displaystyle T-\lambda I:V\to V} is non-bijective on V {\displaystyle V} . The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.
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https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
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The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator R on the Hilbert space ℓ2, ( x 1 , x 2 , … ) ↦ ( 0 , x 1 , x 2 , … ) .
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https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
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{\displaystyle (x_{1},x_{2},\dots )\mapsto (0,x_{1},x_{2},\dots ).} This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc. On the other hand, 0 is in the spectrum because although the operator R − 0 (i.e. R itself) is invertible, the inverse is defined on a set which is not dense in ℓ2. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.
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https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
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The notion of spectrum extends to unbounded (i.e. not necessarily bounded) operators. A complex number λ is said to be in the spectrum of an unbounded operator T: X → X {\displaystyle T:\,X\to X} defined on domain D ( T ) ⊆ X {\displaystyle D(T)\subseteq X} if there is no bounded inverse ( T − λ I ) − 1: X → D ( T ) {\displaystyle (T-\lambda I)^{-1}:\,X\to D(T)} defined on the whole of X . {\displaystyle X.}
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https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
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If T is closed (which includes the case when T is bounded), boundedness of ( T − λ I ) − 1 {\displaystyle (T-\lambda I)^{-1}} follows automatically from its existence. The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
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https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
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In mathematics, particularly in group theory, the Frattini subgroup Φ ( G ) {\displaystyle \Phi (G)} of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivial group {e} or a Prüfer group, it is defined by Φ ( G ) = G {\displaystyle \Phi (G)=G} . It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of "small elements" (see the "non-generator" characterization below). It is named after Giovanni Frattini, who defined the concept in a paper published in 1885.
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https://en.wikipedia.org/wiki/Frattini_subgroup
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In mathematics, particularly in homological algebra and algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimension (namely 3 ≤ cd ( G ) ≤ n {\displaystyle 3\leq \operatorname {cd} (G)\leq n} ), one can construct an aspherical CW complex X of dimension n whose fundamental group is G. The theorem is named after Polish mathematician Samuel Eilenberg and Romanian mathematician Tudor Ganea. The theorem was first published in a short paper in 1957 in the Annals of Mathematics.
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https://en.wikipedia.org/wiki/Eilenberg–Ganea_theorem
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In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by Daniel G. Quillen (1967). In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.
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https://en.wikipedia.org/wiki/Pointed_model_category
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In mathematics, particularly in integral calculus, the localization theorem allows, under certain conditions, to infer the nullity of a function given only information about its continuity and the value of its integral. Let F(x) be a real-valued function defined on some open interval Ω of the real line that is continuous in Ω. Let D be an arbitrary subinterval contained in Ω. The theorem states the following implication: A simple proof is as follows: if there were a point x0 within Ω for which F(x0) ≠ 0, then the continuity of F would require the existence of a neighborhood of x0 in which the value of F was nonzero, and in particular of the same sign than in x0. Since such a neighborhood N, which can be taken to be arbitrarily small, must however be of a nonzero width on the real line, the integral of F over N would evaluate to a nonzero value. However, since x0 is part of the open set Ω, all neighborhoods of x0 smaller than the distance of x0 to the frontier of Ω are included within it, and so the integral of F over them must evaluate to zero. Having reached the contradiction that ∫N F(x) dx must be both zero and nonzero, the initial hypothesis must be wrong, and thus there is no x0 in Ω for which F(x0) ≠ 0. The theorem is easily generalized to multivariate functions, replacing intervals with the more general concept of connected open sets, that is, domains, and the original function with some F(x): Rn → R, with the constraints of continuity and nullity of its integral over any subdomain D ⊂ Ω. The proof is completely analogous to the single variable case, and concludes with the impossibility of finding a point x0 ∈ Ω such that F(x0) ≠ 0.
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https://en.wikipedia.org/wiki/Localization_theorem
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In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).
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https://en.wikipedia.org/wiki/Matrix_analysis
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In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing" means each is a proper subspace of the next (see filtration): { 0 } = V 0 ⊂ V 1 ⊂ V 2 ⊂ ⋯ ⊂ V k = V . {\displaystyle \{0\}=V_{0}\subset V_{1}\subset V_{2}\subset \cdots \subset V_{k}=V.} The term flag is motivated by a particular example resembling a flag: the zero point, a line, and a plane correspond to a nail, a staff, and a sheet of fabric.If we write that dimVi = di then we have 0 = d 0 < d 1 < d 2 < ⋯ < d k = n , {\displaystyle 0=d_{0}
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https://en.wikipedia.org/wiki/Flag_(linear_algebra)
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In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition: p. 38 In terms of the entries of the matrix, if a i j {\textstyle a_{ij}} denotes the entry in the i {\textstyle i} -th row and j {\textstyle j} -th column, then the skew-symmetric condition is equivalent to
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https://en.wikipedia.org/wiki/Antisymmetric_matrices
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In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix.
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https://en.wikipedia.org/wiki/Matrix_multiplication
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The product of matrices A and B is denoted as AB.Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational applications of linear algebra.
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https://en.wikipedia.org/wiki/Matrix_multiplication
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In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis: where each index i1, i2, ..., in takes values 1, 2, ..., n. There are nn indexed values of εi1i2...in, which can be arranged into an n-dimensional array. The key defining property of the symbol is total antisymmetry in the indices.
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https://en.wikipedia.org/wiki/Levi-Civita_symbol
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When any two indices are interchanged, equal or not, the symbol is negated: If any two indices are equal, the symbol is zero. When all indices are unequal, we have: where p (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble i1, i2, ..., in into the order 1, 2, ..., n, and the factor (−1)p is called the sign, or signature of the permutation. The value ε1 2 ... n must be defined, else the particular values of the symbol for all permutations are indeterminate.
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https://en.wikipedia.org/wiki/Levi-Civita_symbol
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Most authors choose ε1 2 ... n = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article.
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https://en.wikipedia.org/wiki/Levi-Civita_symbol
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The term "n-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol n matches the dimensionality of the vector space in question, which may be Euclidean or non-Euclidean, for example, R 3 {\displaystyle \mathbb {R} ^{3}} or Minkowski space. The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems; however it can be interpreted as a tensor density. The Levi-Civita symbol allows the determinant of a square matrix, and the cross product of two vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation.
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https://en.wikipedia.org/wiki/Levi-Civita_symbol
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In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix. The result is named after Issai Schur (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in Journal für die reine und angewandte Mathematik.) We remark that the converse of the theorem holds in the following sense. If M {\displaystyle M} is a symmetric matrix and the Hadamard product M ∘ N {\displaystyle M\circ N} is positive definite for all positive definite matrices N {\displaystyle N} , then M {\displaystyle M} itself is positive definite.
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https://en.wikipedia.org/wiki/Schur_product_theorem
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In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded".
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https://en.wikipedia.org/wiki/Bounded_(set_theory)
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In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows (when pre-multiplying, to form PA) or columns (when post-multiplying, to form AP) of the matrix A.
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https://en.wikipedia.org/wiki/Permutation_matrices
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In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences. Unlike Euclid's classical proof, Furstenberg's proof is a proof by contradiction. The proof was published in 1955 in the American Mathematical Monthly while Furstenberg was still an undergraduate student at Yeshiva University.
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https://en.wikipedia.org/wiki/Furstenberg's_proof_of_the_infinitude_of_primes
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In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra.
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https://en.wikipedia.org/wiki/Continuous_functional_calculus
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In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator. In time series analysis, the theorem implies that any stationary discrete-time stochastic process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process.
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https://en.wikipedia.org/wiki/Wold–von_Neumann_decomposition
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In mathematics, particularly in order theory, a pseudocomplement is one generalization of the notion of complement. In a lattice L with bottom element 0, an element x ∈ L is said to have a pseudocomplement if there exists a greatest element x* ∈ L with the property that x ∧ x* = 0. More formally, x* = max{ y ∈ L | x ∧ y = 0 }. The lattice L itself is called a pseudocomplemented lattice if every element of L is pseudocomplemented.
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https://en.wikipedia.org/wiki/Pseudocomplement
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Every pseudocomplemented lattice is necessarily bounded, i.e. it has a 1 as well. Since the pseudocomplement is unique by definition (if it exists), a pseudocomplemented lattice can be endowed with a unary operation * mapping every element to its pseudocomplement; this structure is sometimes called a p-algebra. However this latter term may have other meanings in other areas of mathematics.
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https://en.wikipedia.org/wiki/Pseudocomplement
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In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S.Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S. A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.
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https://en.wikipedia.org/wiki/Upper_and_lower_bounds
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In mathematics, particularly in set theory, Fodor's lemma states the following: If κ {\displaystyle \kappa } is a regular, uncountable cardinal, S {\displaystyle S} is a stationary subset of κ {\displaystyle \kappa } , and f: S → κ {\displaystyle f:S\rightarrow \kappa } is regressive (that is, f ( α ) < α {\displaystyle f(\alpha )<\alpha } for any α ∈ S {\displaystyle \alpha \in S} , α ≠ 0 {\displaystyle \alpha \neq 0} ) then there is some γ {\displaystyle \gamma } and some stationary S 0 ⊆ S {\displaystyle S_{0}\subseteq S} such that f ( α ) = γ {\displaystyle f(\alpha )=\gamma } for any α ∈ S 0 {\displaystyle \alpha \in S_{0}} . In modern parlance, the nonstationary ideal is normal. The lemma was first proved by the Hungarian set theorist, Géza Fodor in 1956. It is sometimes also called "The Pressing Down Lemma".
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https://en.wikipedia.org/wiki/Fodor's_lemma
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In mathematics, particularly in set theory, if κ {\displaystyle \kappa } is a regular uncountable cardinal then club ( κ ) , {\displaystyle \operatorname {club} (\kappa ),} the filter of all sets containing a club subset of κ , {\displaystyle \kappa ,} is a κ {\displaystyle \kappa } -complete filter closed under diagonal intersection called the club filter. To see that this is a filter, note that κ ∈ club ( κ ) {\displaystyle \kappa \in \operatorname {club} (\kappa )} since it is thus both closed and unbounded (see club set). If x ∈ club ( κ ) {\displaystyle x\in \operatorname {club} (\kappa )} then any subset of κ {\displaystyle \kappa } containing x {\displaystyle x} is also in club ( κ ) , {\displaystyle \operatorname {club} (\kappa ),} since x , {\displaystyle x,} and therefore anything containing it, contains a club set. It is a κ {\displaystyle \kappa } -complete filter because the intersection of fewer than κ {\displaystyle \kappa } club sets is a club set.
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https://en.wikipedia.org/wiki/Club_filter
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To see this, suppose ⟨ C i ⟩ i < α {\displaystyle \langle C_{i}\rangle _{i<\alpha }} is a sequence of club sets where α < κ . {\displaystyle \alpha <\kappa .}
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https://en.wikipedia.org/wiki/Club_filter
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Obviously C = ⋂ C i {\displaystyle C=\bigcap C_{i}} is closed, since any sequence which appears in C {\displaystyle C} appears in every C i , {\displaystyle C_{i},} and therefore its limit is also in every C i . {\displaystyle C_{i}.} To show that it is unbounded, take some β < κ .
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https://en.wikipedia.org/wiki/Club_filter
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{\displaystyle \beta <\kappa .} Let ⟨ β 1 , i ⟩ {\displaystyle \langle \beta _{1,i}\rangle } be an increasing sequence with β 1 , 1 > β {\displaystyle \beta _{1,1}>\beta } and β 1 , i ∈ C i {\displaystyle \beta _{1,i}\in C_{i}} for every i < α . {\displaystyle i<\alpha .}
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https://en.wikipedia.org/wiki/Club_filter
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Such a sequence can be constructed, since every C i {\displaystyle C_{i}} is unbounded. Since α < κ {\displaystyle \alpha <\kappa } and κ {\displaystyle \kappa } is regular, the limit of this sequence is less than κ . {\displaystyle \kappa .}
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https://en.wikipedia.org/wiki/Club_filter
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We call it β 2 , {\displaystyle \beta _{2},} and define a new sequence ⟨ β 2 , i ⟩ {\displaystyle \langle \beta _{2,i}\rangle } similar to the previous sequence. We can repeat this process, getting a sequence of sequences ⟨ β j , i ⟩ {\displaystyle \langle \beta _{j,i}\rangle } where each element of a sequence is greater than every member of the previous sequences. Then for each i < α , {\displaystyle i<\alpha ,} ⟨ β j , i ⟩ {\displaystyle \langle \beta _{j,i}\rangle } is an increasing sequence contained in C i , {\displaystyle C_{i},} and all these sequences have the same limit (the limit of ⟨ β j , i ⟩ {\displaystyle \langle \beta _{j,i}\rangle } ).
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https://en.wikipedia.org/wiki/Club_filter
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This limit is then contained in every C i , {\displaystyle C_{i},} and therefore C , {\displaystyle C,} and is greater than β . {\displaystyle \beta .} To see that club ( κ ) {\displaystyle \operatorname {club} (\kappa )} is closed under diagonal intersection, let ⟨ C i ⟩ , {\displaystyle \langle C_{i}\rangle ,} i < κ {\displaystyle i<\kappa } be a sequence of club sets, and let C = Δ i < κ C i .
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https://en.wikipedia.org/wiki/Club_filter
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{\displaystyle C=\Delta _{i<\kappa }C_{i}.} To show C {\displaystyle C} is closed, suppose S ⊆ α < κ {\displaystyle S\subseteq \alpha <\kappa } and ⋃ S = α . {\displaystyle \bigcup S=\alpha .}
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https://en.wikipedia.org/wiki/Club_filter
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Then for each γ ∈ S , {\displaystyle \gamma \in S,} γ ∈ C β {\displaystyle \gamma \in C_{\beta }} for all β < γ . {\displaystyle \beta <\gamma .} Since each C β {\displaystyle C_{\beta }} is closed, α ∈ C β {\displaystyle \alpha \in C_{\beta }} for all β < α , {\displaystyle \beta <\alpha ,} so α ∈ C .
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https://en.wikipedia.org/wiki/Club_filter
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{\displaystyle \alpha \in C.} To show C {\displaystyle C} is unbounded, let α < κ , {\displaystyle \alpha <\kappa ,} and define a sequence ξ i , {\displaystyle \xi _{i},} i < ω {\displaystyle i<\omega } as follows: ξ 0 = α , {\displaystyle \xi _{0}=\alpha ,} and ξ i + 1 {\displaystyle \xi _{i+1}} is the minimal element of ⋂ γ < ξ i C γ {\displaystyle \bigcap _{\gamma <\xi _{i}}C_{\gamma }} such that ξ i + 1 > ξ i . {\displaystyle \xi _{i+1}>\xi _{i}.}
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https://en.wikipedia.org/wiki/Club_filter
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Such an element exists since by the above, the intersection of ξ i {\displaystyle \xi _{i}} club sets is club. Then ξ = ⋃ i < ω ξ i > α {\displaystyle \xi =\bigcup _{i<\omega }\xi _{i}>\alpha } and ξ ∈ C , {\displaystyle \xi \in C,} since it is in each C i {\displaystyle C_{i}} with i < ξ . {\displaystyle i<\xi .}
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https://en.wikipedia.org/wiki/Club_filter
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In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph ( ℵ {\displaystyle \,\aleph \,} ).The cardinality of the natural numbers is ℵ 0 {\displaystyle \,\aleph _{0}\,} (read aleph-nought or aleph-zero; the term aleph-null is also sometimes used), the next larger cardinality of a well-ordered set is aleph-one ℵ 1 , {\displaystyle \,\aleph _{1}\;,} then ℵ 2 {\displaystyle \,\aleph _{2}\,} and so on. Continuing in this manner, it is possible to define a cardinal number ℵ α {\displaystyle \,\aleph _{\alpha }\,} for every ordinal number α , {\displaystyle \,\alpha \;,} as described below. The concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. The aleph numbers differ from the infinity ( ∞ {\displaystyle \,\infty \,} ) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.
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https://en.wikipedia.org/wiki/Aleph_One
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In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written ℶ 0 , ℶ 1 , ℶ 2 , ℶ 3 , … {\displaystyle \beth _{0},\ \beth _{1},\ \beth _{2},\ \beth _{3},\ \dots } , where ℶ {\displaystyle \beth } is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers ( ℵ 0 , ℵ 1 , … {\displaystyle \aleph _{0},\ \aleph _{1},\ \dots } ), but unless the generalized continuum hypothesis is true, there are numbers indexed by ℵ {\displaystyle \aleph } that are not indexed by ℶ {\displaystyle \beth } .
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https://en.wikipedia.org/wiki/Beth_number
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In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group.
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https://en.wikipedia.org/wiki/Fully_characteristic_subgroup
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In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as a x ≡ 1 ( mod m ) , {\displaystyle ax\equiv 1{\pmod {m}},} which is the shorthand way of writing the statement that m divides (evenly) the quantity ax − 1, or, put another way, the remainder after dividing ax by the integer m is 1. If a does have an inverse modulo m, then there are an infinite number of solutions of this congruence, which form a congruence class with respect to this modulus. Furthermore, any integer that is congruent to a (i.e., in a's congruence class) has any element of x's congruence class as a modular multiplicative inverse. Using the notation of w ¯ {\displaystyle {\overline {w}}} to indicate the congruence class containing w, this can be expressed by saying that the modulo multiplicative inverse of the congruence class a ¯ {\displaystyle {\overline {a}}} is the congruence class x ¯ {\displaystyle {\overline {x}}} such that: a ¯ ⋅ m x ¯ = 1 ¯ , {\displaystyle {\overline {a}}\cdot _{m}{\overline {x}}={\overline {1}},} where the symbol ⋅ m {\displaystyle \cdot _{m}} denotes the multiplication of equivalence classes modulo m. Written in this way, the analogy with the usual concept of a multiplicative inverse in the set of rational or real numbers is clearly represented, replacing the numbers by congruence classes and altering the binary operation appropriately.
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https://en.wikipedia.org/wiki/Discrete_inverse
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As with the analogous operation on the real numbers, a fundamental use of this operation is in solving, when possible, linear congruences of the form a x ≡ b ( mod m ) . {\displaystyle ax\equiv b{\pmod {m}}.} Finding modular multiplicative inverses also has practical applications in the field of cryptography, e.g. public-key cryptography and the RSA algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm) that can be used for the calculation of modular multiplicative inverses.
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https://en.wikipedia.org/wiki/Discrete_inverse
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In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces. Let X {\displaystyle X} be a locally compact Hausdorff space. Let M ( X ) {\displaystyle M(X)} be the space of complex Radon measures on X , {\displaystyle X,} and C 0 ( X ) ∗ {\displaystyle C_{0}(X)^{*}} denote the dual of C 0 ( X ) , {\displaystyle C_{0}(X),} the Banach space of complex continuous functions on X {\displaystyle X} vanishing at infinity equipped with the uniform norm. By the Riesz representation theorem M ( X ) {\displaystyle M(X)} is isometric to C 0 ( X ) ∗ .
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https://en.wikipedia.org/wiki/Vague_topology
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{\displaystyle C_{0}(X)^{*}.} The isometry maps a measure μ {\displaystyle \mu } to a linear functional I μ ( f ) := ∫ X f d μ . {\displaystyle I_{\mu }(f):=\int _{X}f\,d\mu .}
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https://en.wikipedia.org/wiki/Vague_topology
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The vague topology is the weak-* topology on C 0 ( X ) ∗ . {\displaystyle C_{0}(X)^{*}.} The corresponding topology on M ( X ) {\displaystyle M(X)} induced by the isometry from C 0 ( X ) ∗ {\displaystyle C_{0}(X)^{*}} is also called the vague topology on M ( X ) .
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https://en.wikipedia.org/wiki/Vague_topology
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{\displaystyle M(X).} Thus in particular, a sequence of measures ( μ n ) n ∈ N {\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }} converges vaguely to a measure μ {\displaystyle \mu } whenever for all test functions f ∈ C 0 ( X ) , {\displaystyle f\in C_{0}(X),} ∫ X f d μ n → ∫ X f d μ . {\displaystyle \int _{X}fd\mu _{n}\to \int _{X}fd\mu .}
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https://en.wikipedia.org/wiki/Vague_topology
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It is also not uncommon to define the vague topology by duality with continuous functions having compact support C c ( X ) , {\displaystyle C_{c}(X),} that is, a sequence of measures ( μ n ) n ∈ N {\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }} converges vaguely to a measure μ {\displaystyle \mu } whenever the above convergence holds for all test functions f ∈ C c ( X ) . {\displaystyle f\in C_{c}(X).} This construction gives rise to a different topology.
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https://en.wikipedia.org/wiki/Vague_topology
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In particular, the topology defined by duality with C c ( X ) {\displaystyle C_{c}(X)} can be metrizable whereas the topology defined by duality with C 0 ( X ) {\displaystyle C_{0}(X)} is not. One application of this is to probability theory: for example, the central limit theorem is essentially a statement that if μ n {\displaystyle \mu _{n}} are the probability measures for certain sums of independent random variables, then μ n {\displaystyle \mu _{n}} converge weakly (and then vaguely) to a normal distribution, that is, the measure μ n {\displaystyle \mu _{n}} is "approximately normal" for large n . {\displaystyle n.}
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https://en.wikipedia.org/wiki/Vague_topology
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In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety Gr ( k , d , n ) {\displaystyle \operatorname {Gr} (k,d,n)} is the fine moduli variety parametrizing all effective algebraic cycles of dimension k − 1 {\displaystyle k-1} and degree d {\displaystyle d} in P n − 1 {\displaystyle \mathbb {P} ^{n-1}} . The Chow variety Gr ( k , d , n ) {\displaystyle \operatorname {Gr} (k,d,n)} may be constructed via a Chow embedding into a sufficiently large projective space.
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https://en.wikipedia.org/wiki/Chow_quotient
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This is a direct generalization of the construction of a Grassmannian variety via the Plücker embedding, as Grassmannians are the d = 1 {\displaystyle d=1} case of Chow varieties. Chow varieties are distinct from Chow groups, which are the abelian group of all algebraic cycles on a variety (not necessarily projective space) up to rational equivalence. Both are named for Wei-Liang Chow(周煒良), a pioneer in the study of algebraic cycles.
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https://en.wikipedia.org/wiki/Chow_quotient
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In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map.
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https://en.wikipedia.org/wiki/Preimage_theorem
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In mathematics, particularly in the fields of nonlinear dynamics and the calculus of variations, the Chaplygin problem is an isoperimetric problem with a differential constraint. Specifically, the problem is to determine what flight path an airplane in a constant wind field should take in order to encircle the maximum possible area in a given amount of time. The airplane is assumed to be constrained to move in a plane, moving at a constant airspeed v, for time T, and the wind is assumed to move in a constant direction with speed w. The solution of the problem is that the airplane should travel in an ellipse whose major axis is perpendicular to w, with eccentricity w/v.
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https://en.wikipedia.org/wiki/Chaplygin_problem
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In mathematics, particularly in the fields of number theory and combinatorics, the rank of a partition of a positive integer is a certain integer associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced by Freeman Dyson in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan. A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of the Durfee square of the partition.
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https://en.wikipedia.org/wiki/Rank_of_a_partition
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In mathematics, particularly in the study of functions of several complex variables, Ushiki's theorem, named after S. Ushiki, states that certain well-behaved functions cannot have certain kinds of well-behaved invariant manifolds.
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https://en.wikipedia.org/wiki/Ushiki's_Theorem
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In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points (for example in Euclidean space) defined in a way broader than for semianalytic sets (roughly speaking, those satisfying conditions requiring certain real power series to be positive there). Subanalytic sets still have a reasonable local description in terms of submanifolds.
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https://en.wikipedia.org/wiki/Semianalytic_set
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In mathematics, particularly in the subfields of set theory and topology, a set C {\displaystyle C} is said to be saturated with respect to a function f: X → Y {\displaystyle f:X\to Y} if C {\displaystyle C} is a subset of f {\displaystyle f} 's domain X {\displaystyle X} and if whenever f {\displaystyle f} sends two points c ∈ C {\displaystyle c\in C} and x ∈ X {\displaystyle x\in X} to the same value then x {\displaystyle x} belongs to C {\displaystyle C} (that is, if f ( x ) = f ( c ) {\displaystyle f(x)=f(c)} then x ∈ C {\displaystyle x\in C} ). Said more succinctly, the set C {\displaystyle C} is called saturated if C = f − 1 ( f ( C ) ) . {\displaystyle C=f^{-1}(f(C)).}
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https://en.wikipedia.org/wiki/Saturated_set
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In topology, a subset of a topological space ( X , τ ) {\displaystyle (X,\tau )} is saturated if it is equal to an intersection of open subsets of X . {\displaystyle X.} In a T1 space every set is saturated.
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https://en.wikipedia.org/wiki/Saturated_set
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In mathematics, particularly in the theory of C*-algebras, a uniformly hyperfinite, or UHF, algebra is a C*-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras.
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https://en.wikipedia.org/wiki/Uniformly_hyperfinite_algebra
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In mathematics, particularly in the theory of spinors, the Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of 2⌊n/2⌋ × 2⌊n/2⌋ matrices. They generalize the Pauli matrices to n dimensions, and are a specific construction of higher-dimensional gamma matrices. They are named for Richard Brauer and Hermann Weyl, and were one of the earliest systematic constructions of spinors from a representation theoretic standpoint. The matrices are formed by taking tensor products of the Pauli matrices, and the space of spinors in n dimensions may then be realized as the column vectors of size 2⌊n/2⌋ on which the Weyl–Brauer matrices act.
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https://en.wikipedia.org/wiki/Weyl-Brauer_matrices
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In mathematics, particularly in universal algebra and category theory, transport of structure refers to the process whereby a mathematical object acquires a new structure and its canonical definitions, as a result of being isomorphic to (or otherwise identified with) another object with a pre-existing structure. Definitions by transport of structure are regarded as canonical. Since mathematical structures are often defined in reference to an underlying space, many examples of transport of structure involve spaces and mappings between them. For example, if V {\displaystyle V} and W {\displaystyle W} are vector spaces with ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} being an inner product on W {\displaystyle W} , such that there is an isomorphism ϕ {\displaystyle \phi } from V {\displaystyle V} to W {\displaystyle W} , then one can define an inner product {\displaystyle } on V {\displaystyle V} by the following rule: = ( ϕ ( v 1 ) , ϕ ( v 2 ) ) {\displaystyle =(\phi (v_{1}),\phi (v_{2}))} Although the equation makes sense even when ϕ {\displaystyle \phi } is not an isomorphism, it only defines an inner product on V {\displaystyle V} when ϕ {\displaystyle \phi } is, since otherwise it will cause {\displaystyle } to be degenerate.
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https://en.wikipedia.org/wiki/Transport_of_structure
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The idea is that ϕ {\displaystyle \phi } allows one to consider V {\displaystyle V} and W {\displaystyle W} as "the same" vector space, and by following this analogy, then one can transport an inner product from one space to the other. A more elaborated example comes from differential topology, in which the notion of smooth manifold is involved: if M {\displaystyle M} is such a manifold, and if X {\displaystyle X} is any topological space which is homeomorphic to M {\displaystyle M} , then one can consider X {\displaystyle X} as a smooth manifold as well. That is, given a homeomorphism ϕ: X → M {\displaystyle \phi \colon X\to M} , one can define coordinate charts on X {\displaystyle X} by "pulling back" coordinate charts on M {\displaystyle M} through ϕ {\displaystyle \phi } .
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https://en.wikipedia.org/wiki/Transport_of_structure
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That is, X {\displaystyle X} is a smooth manifold via transport of structure. This is a special case of transport of structures in general.The second example also illustrates why "transport of structure" is not always desirable. Namely, one can take M {\displaystyle M} to be the plane, and X {\displaystyle X} to be an infinite one-sided cone.
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https://en.wikipedia.org/wiki/Transport_of_structure
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By "flattening" the cone, a homeomorphism of X {\displaystyle X} and M {\displaystyle M} can be obtained, and therefore the structure of a smooth manifold on X {\displaystyle X} , but the cone is not "naturally" a smooth manifold. That is, one can consider X {\displaystyle X} as a subspace of 3-space, in which context it is not smooth at the cone point. A more surprising example is that of exotic spheres, discovered by Milnor, which states that there are exactly 28 smooth manifolds which are homeomorphic (but by definition not diffeomorphic) to S 7 {\displaystyle S^{7}} , the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a canonical isomorphism between the two objects.
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https://en.wikipedia.org/wiki/Transport_of_structure
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