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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 di... | https://en.wikipedia.org/wiki/Skeleton_(topology) |
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 seve... | https://en.wikipedia.org/wiki/Stationary_points |
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... | https://en.wikipedia.org/wiki/Hom_space |
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 opera... | https://en.wikipedia.org/wiki/Hom_space |
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\}} . Stirli... | https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind |
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. | https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind |
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... | https://en.wikipedia.org/wiki/Transversal_(combinatorics) |
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. : 6... | https://en.wikipedia.org/wiki/Transversal_(combinatorics) |
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... | https://en.wikipedia.org/wiki/Compact_Riemann_surface |
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. Ev... | https://en.wikipedia.org/wiki/Compact_Riemann_surface |
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 ... | https://en.wikipedia.org/wiki/Compact_Riemann_surface |
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. | https://en.wikipedia.org/wiki/Abramov's_algorithm |
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-... | https://en.wikipedia.org/wiki/Zoll_surface |
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 t... | https://en.wikipedia.org/wiki/Osculating_plane |
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 ... | https://en.wikipedia.org/wiki/Whitney_embedding_theorem |
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 r... | https://en.wikipedia.org/wiki/Phase_locking |
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 ... | https://en.wikipedia.org/wiki/Phase_locking |
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 ... | https://en.wikipedia.org/wiki/Bifurcation_diagram |
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... | https://en.wikipedia.org/wiki/Poincaré_map |
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 ... | https://en.wikipedia.org/wiki/Poincaré_map |
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 ... | https://en.wikipedia.org/wiki/Poincaré_map |
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 pr... | https://en.wikipedia.org/wiki/Indeterminate_(variable) |
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 express... | https://en.wikipedia.org/wiki/Indeterminate_(variable) |
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 ... | https://en.wikipedia.org/wiki/Convex_series |
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. | https://en.wikipedia.org/wiki/Ursescu_theorem |
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. | https://en.wikipedia.org/wiki/Approximation_of_the_identity |
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 theore... | https://en.wikipedia.org/wiki/Closed_linear_operator |
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 contin... | https://en.wikipedia.org/wiki/Closed_graph_theorem_(functional_analysis) |
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. | https://en.wikipedia.org/wiki/Mackey_space |
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 ad... | https://en.wikipedia.org/wiki/Mackey_convergence |
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 sel... | https://en.wikipedia.org/wiki/Projective_measurement |
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 g... | https://en.wikipedia.org/wiki/Projective_measurement |
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 top... | https://en.wikipedia.org/wiki/Seminormed_space |
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... | https://en.wikipedia.org/wiki/Webbed_space |
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. | https://en.wikipedia.org/wiki/Krein–Smulian_theorem |
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 operato... | https://en.wikipedia.org/wiki/Spectrum_(functional_analysis) |
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 , … ... | https://en.wikipedia.org/wiki/Spectrum_(functional_analysis) |
{\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 n... | https://en.wikipedia.org/wiki/Spectrum_(functional_analysis) |
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 ) {\... | https://en.wikipedia.org/wiki/Spectrum_(functional_analysis) |
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 m... | https://en.wikipedia.org/wiki/Spectrum_(functional_analysis) |
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} ... | https://en.wikipedia.org/wiki/Frattini_subgroup |
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 c... | https://en.wikipedia.org/wiki/Eilenberg–Ganea_theorem |
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 ca... | https://en.wikipedia.org/wiki/Pointed_model_category |
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 continuo... | https://en.wikipedia.org/wiki/Localization_theorem |
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 (... | https://en.wikipedia.org/wiki/Matrix_analysis |
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... | https://en.wikipedia.org/wiki/Flag_(linear_algebra) |
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 ro... | https://en.wikipedia.org/wiki/Antisymmetric_matrices |
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 ... | https://en.wikipedia.org/wiki/Matrix_multiplication |
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 numerou... | https://en.wikipedia.org/wiki/Matrix_multiplication |
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 mathematici... | https://en.wikipedia.org/wiki/Levi-Civita_symbol |
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, ... | https://en.wikipedia.org/wiki/Levi-Civita_symbol |
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. | https://en.wikipedia.org/wiki/Levi-Civita_symbol |
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 in... | https://en.wikipedia.org/wiki/Levi-Civita_symbol |
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 angewan... | https://en.wikipedia.org/wiki/Schur_product_theorem |
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". | https://en.wikipedia.org/wiki/Bounded_(set_theory) |
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 (whe... | https://en.wikipedia.org/wiki/Permutation_matrices |
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. Unl... | https://en.wikipedia.org/wiki/Furstenberg's_proof_of_the_infinitude_of_primes |
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. | https://en.wikipedia.org/wiki/Continuous_functional_calculus |
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 ... | https://en.wikipedia.org/wiki/Wold–von_Neumann_decomposition |
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 ∧ ... | https://en.wikipedia.org/wiki/Pseudocomplement |
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... | https://en.wikipedia.org/wiki/Pseudocomplement |
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... | https://en.wikipedia.org/wiki/Upper_and_lower_bounds |
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 {\displaystyl... | https://en.wikipedia.org/wiki/Fodor's_lemma |
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 diag... | https://en.wikipedia.org/wiki/Club_filter |
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 .} | https://en.wikipedia.org/wiki/Club_filter |
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 . | https://en.wikipedia.org/wiki/Club_filter |
{\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 .} | https://en.wikipedia.org/wiki/Club_filter |
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 .} | https://en.wikipedia.org/wiki/Club_filter |
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 great... | https://en.wikipedia.org/wiki/Club_filter |
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 < ... | https://en.wikipedia.org/wiki/Club_filter |
{\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 .} | https://en.wikipedia.org/wiki/Club_filter |
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 . | https://en.wikipedia.org/wiki/Club_filter |
{\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... | https://en.wikipedia.org/wiki/Club_filter |
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 .} | https://en.wikipedia.org/wiki/Club_filter |
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 {\displ... | https://en.wikipedia.org/wiki/Aleph_One |
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... | https://en.wikipedia.org/wiki/Beth_number |
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 guar... | https://en.wikipedia.org/wiki/Fully_characteristic_subgroup |
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{\p... | https://en.wikipedia.org/wiki/Discrete_inverse |
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. publ... | https://en.wikipedia.org/wiki/Discrete_inverse |
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)}... | https://en.wikipedia.org/wiki/Vague_topology |
{\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 .} | https://en.wikipedia.org/wiki/Vague_topology |
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 ) . | https://en.wikipedia.org/wiki/Vague_topology |
{\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}... | https://en.wikipedia.org/wiki/Vague_topology |
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 th... | https://en.wikipedia.org/wiki/Vague_topology |
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... | https://en.wikipedia.org/wiki/Vague_topology |
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 f... | https://en.wikipedia.org/wiki/Chow_quotient |
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 s... | https://en.wikipedia.org/wiki/Chow_quotient |
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. | https://en.wikipedia.org/wiki/Preimage_theorem |
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 ... | https://en.wikipedia.org/wiki/Chaplygin_problem |
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... | https://en.wikipedia.org/wiki/Rank_of_a_partition |
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. | https://en.wikipedia.org/wiki/Ushiki's_Theorem |
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 sti... | https://en.wikipedia.org/wiki/Semianalytic_set |
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 p... | https://en.wikipedia.org/wiki/Saturated_set |
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. | https://en.wikipedia.org/wiki/Saturated_set |
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. | https://en.wikipedia.org/wiki/Uniformly_hyperfinite_algebra |
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 ... | https://en.wikipedia.org/wiki/Weyl-Brauer_matrices |
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. Defini... | https://en.wikipedia.org/wiki/Transport_of_structure |
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 s... | https://en.wikipedia.org/wiki/Transport_of_structure |
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 ... | https://en.wikipedia.org/wiki/Transport_of_structure |
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 ... | https://en.wikipedia.org/wiki/Transport_of_structure |
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