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In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one poi... | https://en.wikipedia.org/wiki/Projective_spaces |
There are two classes of definitions. In synthetic geometry, point and line are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defi... | https://en.wikipedia.org/wiki/Projective_spaces |
Using linear algebra, a projective space of dimension n is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space V of dimension n + 1. Equivalently, it is the quotient set of V \ {0} by the equivalence relation "being on the same vector line". As a vector line intersects ... | https://en.wikipedia.org/wiki/Projective_spaces |
A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane. Projective spaces are widely used in geometry, as allowing simpler statements and simpler proofs. For example, in affine geometry, two distinct lines in a plane intersect in at most one point, while, in ... | https://en.wikipedia.org/wiki/Projective_spaces |
In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume. | https://en.wikipedia.org/wiki/Relatively_hyperbolic_group |
In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function. If A, B are posets, a function f: A → B is defined to be monotone if it is order-preserving: that is, if x ≤ y implies f(x) ≤ f(y). This is equivalent to the condition that ... | https://en.wikipedia.org/wiki/Residuated_mapping |
In general the preimage under f of a principal down-set need not be a principal down-set. If all of them are, f is called residuated. | https://en.wikipedia.org/wiki/Residuated_mapping |
The notion of residuated map can be generalized to a binary operator (or any higher arity) via component-wise residuation. This approach gives rise to notions of left and right division in a partially ordered magma, additionally endowing it with a quasigroup structure. (One speaks only of residuated algebra for higher ... | https://en.wikipedia.org/wiki/Residuated_mapping |
In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss such equations. T... | https://en.wikipedia.org/wiki/Equations_defining_abelian_varieties |
In mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers. Given an operation denoted here ∗, and an identity element denoted e, if x ∗ y = e, one says that x is a left inverse of y, and that y is a right inverse of x. (An identity element is an element ... | https://en.wikipedia.org/wiki/Left_inverse_element |
Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an invertible element, also called a unit, is an element that is invertible und... | https://en.wikipedia.org/wiki/Left_inverse_element |
Inverses are commonly used in groups—where every element is invertible, and rings—where invertible elements are also called units. They are also commonly used for operations that are not defined for all possible operands, such as inverse matrices and inverse functions. | https://en.wikipedia.org/wiki/Left_inverse_element |
This has been generalized to category theory, where, by definition, an isomorphism is an invertible morphism. The word 'inverse' is derived from Latin: inversus that means 'turned upside down', 'overturned'. This may take its origin from the case of fractions, where the (multiplicative) inverse is obtained by exchangin... | https://en.wikipedia.org/wiki/Left_inverse_element |
In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of GDSs is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result. T... | https://en.wikipedia.org/wiki/Graph_dynamical_system |
As such, the research typically involves techniques from, e.g., graph theory, combinatorics, algebra, and dynamical systems rather than differential geometry. In principle, one could define and study GDSs over an infinite graph (e.g. cellular automata or probabilistic cellular automata over Z k {\displaystyle \mathbb {... | https://en.wikipedia.org/wiki/Graph_dynamical_system |
In mathematics, the concept of groupoid algebra generalizes the notion of group algebra. | https://en.wikipedia.org/wiki/Groupoid_algebra |
In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial. In rep... | https://en.wikipedia.org/wiki/Irreducible_(mathematics) |
Similarly, an irreducible module is another name for a simple module. Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where ... | https://en.wikipedia.org/wiki/Irreducible_(mathematics) |
In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space. A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix (that has more than one block of positive size). (Replacing non... | https://en.wikipedia.org/wiki/Irreducible_(mathematics) |
A detailed definition is given here. Also, a Markov chain is irreducible if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state. In the theory of manifolds, an n-manifold is irreducible if any embedded (n − 1)-sphere bounds an embedded n-ball. | https://en.wikipedia.org/wiki/Irreducible_(mathematics) |
Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds. The notions of irreducibility in algebra and manifold theory are related. An n-manifold is called prime, if it cannot be written as a connected sum of two n-mani... | https://en.wikipedia.org/wiki/Irreducible_(mathematics) |
An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime b... | https://en.wikipedia.org/wiki/Irreducible_(mathematics) |
See, for example, Prime decomposition (3-manifold). A topological space is irreducible if it is not the union of two proper closed subsets. This notion is used in algebraic geometry, where spaces are equipped with the Zariski topology; it is not of much significance for Hausdorff spaces. | https://en.wikipedia.org/wiki/Irreducible_(mathematics) |
See also irreducible component, algebraic variety. In universal algebra, irreducible can refer to the inability to represent an algebraic structure as a composition of simpler structures using a product construction; for example subdirectly irreducible. A 3-manifold is P²-irreducible if it is irreducible and contains n... | https://en.wikipedia.org/wiki/Irreducible_(mathematics) |
In mathematics, the concept of quantity is an ancient one extending back to the time of Aristotle and earlier. Aristotle regarded quantity as a fundamental ontological and scientific category. In Aristotle's ontology, quantity or quantum was classified into two different types, which he characterized as follows: Quantu... | https://en.wikipedia.org/wiki/Mathematical_quantity |
Plurality means that which is divisible potentially into non-continuous parts, magnitude that which is divisible into continuous parts; of magnitude, that which is continuous in one dimension is length; in two breadth, in three depth. Of these, limited plurality is number, limited length is a line, breadth a surface, d... | https://en.wikipedia.org/wiki/Mathematical_quantity |
For Aristotle and Euclid, relations were conceived as whole numbers (Michell, 1993). John Wallis later conceived of ratios of magnitudes as real numbers: When a comparison in terms of ratio is made, the resultant ratio often leaves the genus of quantities compared, and passes into the numerical genus, whatever the gen... | https://en.wikipedia.org/wiki/Mathematical_quantity |
In mathematics, the concept of symmetry is studied with the notion of a mathematical group. Every polyhedron has an associated symmetry group, which is the set of all transformations (Euclidean isometries) which leave the polyhedron invariant. The order of the symmetry group is the number of symmetries of the polyhedro... | https://en.wikipedia.org/wiki/Regular_solid |
The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, ... | https://en.wikipedia.org/wiki/Regular_solid |
One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is regular if and only if it is vertex-uniform, edge-uniform, and face-uniform. There are only three symmetry groups associated with the Platonic s... | https://en.wikipedia.org/wiki/Regular_solid |
This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice versa. The three polyhedral groups are: the tetrahedral group T, the octahedral group O (which is also the symmetry group of the cube), and the icosahedral group I (which is als... | https://en.wikipedia.org/wiki/Regular_solid |
The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. | https://en.wikipedia.org/wiki/Regular_solid |
All Platonic solids except the tetrahedron are centrally symmetric, meaning they are preserved under reflection through the origin. The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parenthesis (likewise ... | https://en.wikipedia.org/wiki/Regular_solid |
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, that is, except on a set o... | https://en.wikipedia.org/wiki/Ess_sup |
In mathematics, the conductor of an elliptic curve over the field of rational numbers, or more generally a local or global field, is an integral ideal analogous to the Artin conductor of a Galois representation. It is given as a product of prime ideals, together with associated exponents, which encode the ramification ... | https://en.wikipedia.org/wiki/Conductor_of_an_elliptic_curve |
In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by Hasse (1926, 1930) for abelian extensions and by Artin (1931) for Galois extensions, is a formula calculating the relative discriminant of a finite Galois extension L / K {\displaystyle L/K} of local or global fields ... | https://en.wikipedia.org/wiki/Conductor-discriminant_formula |
In mathematics, the cone condition is a property which may be satisfied by a subset of a Euclidean space. Informally, it requires that for each point in the subset a cone with vertex in that point must be contained in the subset itself, and so the subset is "non-flat". | https://en.wikipedia.org/wiki/Cone_condition |
In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X {\displaystyle X} is a combinatorial invariant of importance to the birational geometry of X {\displaystyle X} . | https://en.wikipedia.org/wiki/Cone_of_curves |
In mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the conformal gauge of X, that is, the class of all metric spaces quasisymmetric to X. | https://en.wikipedia.org/wiki/Conformal_dimension |
In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important: The conformal or... | https://en.wikipedia.org/wiki/Conformal_group_of_spacetime |
If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V Q ( T x ) = λ 2 Q ( x ) {\displaystyle Q(Tx)=\lambda ^{2}Q(x)} For a definite quadratic form, the conformal orthogonal gro... | https://en.wikipedia.org/wiki/Conformal_group_of_spacetime |
In mathematics, the conformal radius is a way to measure the size of a simply connected planar domain D viewed from a point z in it. As opposed to notions using Euclidean distance (say, the radius of the largest inscribed disk with center z), this notion is well-suited to use in complex analysis, in particular in confo... | https://en.wikipedia.org/wiki/Conformal_radius |
In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep i... | https://en.wikipedia.org/wiki/Huhn's_theorem |
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled ... | https://en.wikipedia.org/wiki/Preconditioned_conjugate_gradient_method |
The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched it.The biconjugate gradient method provides a generalization to non-symmetric ... | https://en.wikipedia.org/wiki/Preconditioned_conjugate_gradient_method |
In mathematics, the conjugate of an expression of the form a + b d {\displaystyle a+b{\sqrt {d}}} is a − b d , {\displaystyle a-b{\sqrt {d}},} provided that d {\displaystyle {\sqrt {d}}} does not appear in a and b. One says also that the two expressions are conjugate. In particular, the two solutions of a quadratic equ... | https://en.wikipedia.org/wiki/Conjugate_(square_roots) |
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m × n {\displaystyle m\times n} complex matrix A {\displaystyle {\boldsymbol {A}}} is an n × m {\displaystyle n\times m} matrix obtained by transposing A {\displaystyle {\boldsymbol {A}}} and applying complex conjugate on each entry (... | https://en.wikipedia.org/wiki/Adjoint_matrix |
In mathematics, the connective constant is a numerical quantity associated with self-avoiding walks on a lattice. It is studied in connection with the notion of universality in two-dimensional statistical physics models. While the connective constant depends on the choice of lattice so itself is not universal (similarl... | https://en.wikipedia.org/wiki/Connective_constant |
In mathematics, the constant problem is the problem of deciding whether a given expression is equal to zero. | https://en.wikipedia.org/wiki/Constant_problem |
In mathematics, the constant sheaf on a topological space X {\displaystyle X} associated to a set A {\displaystyle A} is a sheaf of sets on X {\displaystyle X} whose stalks are all equal to A {\displaystyle A} . It is denoted by A _ {\displaystyle {\underline {A}}} or A X {\displaystyle A_{X}} . The constant presheaf w... | https://en.wikipedia.org/wiki/Constant_sheaf |
The constant sheaf associated to A {\displaystyle A} is the sheafification of the constant presheaf associated to A {\displaystyle A} . This sheaf identifies with the sheaf of locally constant A {\displaystyle A} -valued functions on X {\displaystyle X} .In certain cases, the set A {\displaystyle A} may be replaced wit... | https://en.wikipedia.org/wiki/Constant_sheaf |
In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by p n ( x ; a , b , c , d ) = i n ( a + c ) n ( a + d ) n n ! 3 F 2 ( − n , n + a + b + c + d − 1 , a + ... | https://en.wikipedia.org/wiki/Continuous_Hahn_polynomials |
}}{}_{3}F_{2}\left({\begin{array}{c}-n,n+a+b+c+d-1,a+ix\\a+c,a+d\end{array}};1\right)} Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. Closely related polynomials include the dual Hahn polynomials Rn(x;γ,δ,N), the Hahn polynomials Qn(x;a,b,c), and the continuou... | https://en.wikipedia.org/wiki/Continuous_Hahn_polynomials |
In mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://en.wikipedia.org/wiki/Continuous_big_q-Hermite_polynomials |
In mathematics, the continuous dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by S n ( x 2 ; a , b , c ) = 3 F 2 ( − n , a + i x , a − i x ; a + b , a + c ; 1 ) . {\displaystyle ... | https://en.wikipedia.org/wiki/Continuous_dual_Hahn_polynomials |
In mathematics, the continuous dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://en.wikipedia.org/wiki/Continuous_dual_q-Hahn_polynomials |
In mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://en.wikipedia.org/wiki/Continuous_q-Hahn_polynomials |
In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://en.wikipedia.org/wiki/Continuous_q-Hermite_polynomials |
In mathematics, the continuous q-Jacobi polynomials P(α,β)n(x|q), introduced by Askey & Wilson (1985), are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://en.wikipedia.org/wiki/Continuous_q-Jacobi_polynomials |
In mathematics, the continuous q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://en.wikipedia.org/wiki/Continuous_q-Laguerre_polynomials |
In mathematics, the continuous q-Legendre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme.Koekoek, Lesky & Swarttouw (2010) give a detailed list of their properties. | https://en.wikipedia.org/wiki/Continuous_q-Legendre_polynomials |
In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. The continuous wavelet transform of a function x ( t ) {\displaystyle x(t)} at a scale ... | https://en.wikipedia.org/wiki/Continuous_wavelet_transform |
x ( t ) = C ψ − 1 ∫ 0 ∞ ∫ − ∞ ∞ X w ( a , b ) 1 | a | 1 / 2 ψ ~ ( t − b a ) d b d a a 2 {\displaystyle x(t)=C_{\psi }^{-1}\int _{0}^{\infty }\int _{-\infty }^{\infty }X_{w}(a,b){\frac {1}{|a|^{1/2}}}{\tilde {\psi }}\left({\frac {t-b}{a}}\right)\,db\ {\frac {da}{a^{2}}}} ψ ~ ( t ) {\displaystyle {\tilde {\psi }}(t)} is ... | https://en.wikipedia.org/wiki/Continuous_wavelet_transform |
In mathematics, the continuum function is κ ↦ 2 κ {\displaystyle \kappa \mapsto 2^{\kappa }} , i.e. raising 2 to the power of κ using cardinal exponentiation. Given a cardinal number, it is the cardinality of the power set of a set of the given cardinality. | https://en.wikipedia.org/wiki/Continuum_function |
In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical soluti... | https://en.wikipedia.org/wiki/Courant–Friedrichs–Lewy_condition |
In mathematics, the converse of a theorem of the form P → Q will be Q → P. The converse may or may not be true, and even if true, the proof may be difficult. For example, the Four-vertex theorem was proved in 1912, but its converse was proved only in 1997.In practice, when determining the converse of a mathematical the... | https://en.wikipedia.org/wiki/Converse_(logic) |
In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if X {\displaystyle X} and Y {\displaystyle Y} are sets and... | https://en.wikipedia.org/wiki/Converse_relation |
In set-builder notation, L T = { ( y , x ) ∈ Y × X: ( x , y ) ∈ L } . {\displaystyle L^{\operatorname {T} }=\{(y,x)\in Y\times X:(x,y)\in L\}.} The notation is analogous with that for an inverse function. | https://en.wikipedia.org/wiki/Converse_relation |
Although many functions do not have an inverse, every relation does have a unique converse. The unary operation that maps a relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on th... | https://en.wikipedia.org/wiki/Converse_relation |
Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation is also called the transpose relation. It has also been called the opposite or dual of the original relation, or the inverse of the original relation, or the r... | https://en.wikipedia.org/wiki/Converse_relation |
In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if x {\displaystyle x} is a function on Euclidean space Rd and n {\displaystyle n} is a natural number, then the convolution power is defined by x ∗ n = x ∗ x ∗ x ∗ ⋯ ∗ x ∗ x ⏟ n , x ∗ 0 = δ 0 {\displaystyle x^{*n}=\under... | https://en.wikipedia.org/wiki/Convolution_power |
Equivalently, x ∗ n / σ n {\displaystyle x^{*n}/\sigma {\sqrt {n}}} tends weakly to the standard normal distribution. In some cases, it is possible to define powers x*t for arbitrary real t > 0. If μ is a probability measure, then μ is infinitely divisible provided there exists, for each positive integer n, a probabili... | https://en.wikipedia.org/wiki/Convolution_power |
{\displaystyle \mu _{1/n}^{*n}=\mu .} That is, a measure is infinitely divisible if it is possible to define all nth roots. Not every probability measure is infinitely divisible, and a characterization of infinitely divisible measures is of central importance in the abstract theory of stochastic processes. | https://en.wikipedia.org/wiki/Convolution_power |
Intuitively, a measure should be infinitely divisible provided it has a well-defined "convolution logarithm." The natural candidate for measures having such a logarithm are those of (generalized) Poisson type, given in the form π α , μ = e − α ∑ n = 0 ∞ α n n ! | https://en.wikipedia.org/wiki/Convolution_power |
μ ∗ n . {\displaystyle \pi _{\alpha ,\mu }=e^{-\alpha }\sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{n! }}\mu ^{*n}.} | https://en.wikipedia.org/wiki/Convolution_power |
In fact, the Lévy–Khinchin theorem states that a necessary and sufficient condition for a measure to be infinitely divisible is that it must lie in the closure, with respect to the vague topology, of the class of Poisson measures (Stroock 1993, §3.2). Many applications of the convolution power rely on being able to def... | https://en.wikipedia.org/wiki/Convolution_power |
{\displaystyle F^{*}(x)=a_{0}\delta _{0}+\sum _{n=1}^{\infty }a_{n}x^{*n}.} If x ∈ L1(Rd) or more generally is a finite Borel measure on Rd, then the latter series converges absolutely in norm provided that the norm of x is less than the radius of convergence of the original series defining F(z). In particular, it is p... | https://en.wikipedia.org/wiki/Convolution_power |
. {\displaystyle \exp ^{*}(x)=\delta _{0}+\sum _{n=1}^{\infty }{\frac {x^{*n}}{n!}}.} It is not generally possible to extend this definition to arbitrary distributions, although a class of distributions on which this series still converges in an appropriate weak sense is identified by Ben Chrouda, El Oued & Ouerdiane (... | https://en.wikipedia.org/wiki/Convolution_power |
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g... | https://en.wikipedia.org/wiki/Convolution_theorem |
In mathematics, the corona or corona set of a topological space X is the complement βX\X of the space in its Stone–Čech compactification βX. A topological space is said to be σ-compact if it is the union of countably many compact subspaces, and locally compact if every point has a neighbourhood with compact closure. Th... | https://en.wikipedia.org/wiki/Corona_set |
In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by Kakutani (1941) and proved by Lennart Carleson (1962). The commutative Banach algebra and Hardy space H∞ consists of the bounded holomorphic functions on the open unit disc D. Its... | https://en.wikipedia.org/wiki/Corona_theorem |
In 1979 Thomas Wolff gave a simplified (but unpublished) proof of the corona theorem, described in (Koosis 1980) and (Gamelin 1980). Cole later showed that this result cannot be extended to all open Riemann surfaces (Gamelin 1978). | https://en.wikipedia.org/wiki/Corona_theorem |
As a by-product, of Carleson's work, the Carleson measure was invented which itself is a very useful tool in modern function theory. It remains an open question whether there are versions of the corona theorem for every planar domain or for higher-dimensional domains. Note that if one assumes the continuity up to the b... | https://en.wikipedia.org/wiki/Corona_theorem |
In mathematics, the correlation immunity of a Boolean function is a measure of the degree to which its outputs are uncorrelated with some subset of its inputs. Specifically, a Boolean function is said to be correlation-immune of order m if every subset of m or fewer variables in x 1 , x 2 , … , x n {\displaystyle x_{1}... | https://en.wikipedia.org/wiki/Correlation_immunity |
In mathematics, the coset construction (or GKO construction) is a method of constructing unitary highest weight representations of the Virasoro algebra, introduced by Peter Goddard, Adrian Kent and David Olive (1986). The construction produces the complete discrete series of highest weight representations of the Viraso... | https://en.wikipedia.org/wiki/Coset_construction |
In mathematics, the coshc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For z ≠ 0 {\displaystyle z\neq 0} , it is defined as It is a solution of the following differential equation: | https://en.wikipedia.org/wiki/Coshc_function |
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X → Y {\displaystyle f:X\to Y} is a morphism of geometric or algebraic objects, the corresponding cotangent complex L X / Y ∙... | https://en.wikipedia.org/wiki/Cotangent_complex |
Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cot... | https://en.wikipedia.org/wiki/Cotangent_complex |
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal conne... | https://en.wikipedia.org/wiki/Covariant_differential |
The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation.This article presents an introduction to the covariant derivative of a vector field with... | https://en.wikipedia.org/wiki/Covariant_differential |
In mathematics, the crank conjecture was a conjecture about the existence of the crank of a partition that separates partitions of a number congruent to 6 mod 11 into 11 equal classes. The conjecture was introduced by Dyson (1944) and proved by Andrews and Garvan (1987). | https://en.wikipedia.org/wiki/Crank_conjecture |
In mathematics, the crenel function is a periodic discontinuous function P(x) defined as 1 for x belonging to a given interval and 0 outside of it. It can be presented as a difference between two Heaviside step functions of amplitude 1. It is used in crystallography to account for irregularities in the occupation of at... | https://en.wikipedia.org/wiki/Crenel_function |
{\displaystyle P_{k}(\Delta ,x)={\frac {\exp(2\pi i\,kx)\sin(\pi k\Delta )}{\pi k}}=\Delta \cdot \mathrm {sinc} (\pi k\Delta )\cdot \mathrm {e} ^{2\pi i\,kx}.} with the Sinc function. == References == | https://en.wikipedia.org/wiki/Crenel_function |
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E {\displaystyle E} ), and is denoted by the symbol × {\displaystyle \times } . Given ... | https://en.wikipedia.org/wiki/Xyzzy_(mnemonic) |
It should not be confused with the dot product (projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are not linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product ... | https://en.wikipedia.org/wiki/Xyzzy_(mnemonic) |
The cross product is anticommutative (that is, a × b = − b × a) and is distributive over addition (that is, a × (b + c) = a × b + a × c). The space E {\displaystyle E} together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross p... | https://en.wikipedia.org/wiki/Xyzzy_(mnemonic) |
In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space. The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimens... | https://en.wikipedia.org/wiki/Xyzzy_(mnemonic) |
In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of curvature in geometry. In the form presented below, the concept was introduced in 1995 by the mathematician Mark S. Melnikov; accordi... | https://en.wikipedia.org/wiki/Curvature_of_a_measure |
In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed curves on S. The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups.... | https://en.wikipedia.org/wiki/Curve_complex |
In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. ... | https://en.wikipedia.org/wiki/Curve-shortening_flow |
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