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wiki_6600_chunk_0 | Complete Fermi–Dirac integral | In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by F j ( x ) = 1 Γ ( j + 1 ) ∫ 0 ∞ t j e t − x + 1 d t , ( j > − 1 ) {\displaystyle F_{j}(x)={\frac {1}{\Gamma (j+1)}}\int _{0}^{\infty }{\frac {t^{j}}{e^{t-x}+1}}\,dt,\qquad (j>-1)} This equals − Li j ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6601_chunk_0 | Witt algebra | In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6602_chunk_0 | Conjugate pair | This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a 2 + b 2 {\displaystyle a^{2}+b^{2}} (or r 2 {\displaystyle r^{2}} in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6603_chunk_0 | Complex conjugate of a vector space | In mathematics, the complex conjugate of a complex vector space V {\displaystyle V\,} is a complex vector space V ¯ {\displaystyle {\overline {V}}} , which has the same elements and additive group structure as V , {\displaystyle V,} but whose scalar multiplication involves conjugation of the scalars. In other words, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6604_chunk_0 | Complex conjugate of a vector space | The letter v {\displaystyle v} stands for a vector in V , {\displaystyle V,} α {\displaystyle \alpha } is a complex number, and α ¯ {\displaystyle {\overline {\alpha }}} denotes the complex conjugate of α . {\displaystyle \alpha .} More concretely, the complex conjugate vector space is the same underlying real vector s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6605_chunk_0 | Complex conjugate root theorem | In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P.It follows from this (and the fundamental theorem of algebra) that, if the degree of a re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6606_chunk_0 | Gauss plane | The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6607_chunk_0 | Complex projective plane | In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates ( Z 1 , Z 2 , Z 3 ) ∈ C 3 , ( Z 1 , Z 2 , Z 3 ) ≠ ( 0 , 0 , 0 ) {\displaystyle (Z_{1},Z_{2},Z_{3})\in \mathbf {C} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6608_chunk_0 | Complexification | In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for V (a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6609_chunk_0 | Complexification (Lie group) | In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6610_chunk_0 | Complexification (Lie group) | They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear. For compact Lie groups, the complexification, sometimes called the Chevalley complexification after Claude Chevalley, can be defined as the group of complex characters of the Hopf algebra of representative functions,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6611_chunk_0 | Composition operator | In mathematics, the composition operator C ϕ {\displaystyle C_{\phi }} with symbol ϕ {\displaystyle \phi } is a linear operator defined by the rule where f ∘ ϕ {\displaystyle f\circ \phi } denotes function composition. The study of composition operators is covered by AMS category 47B33. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6612_chunk_0 | Compound of three octahedra | In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6613_chunk_0 | Generalised metric | In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field. In general, when we define metric space the distance function is taken to be a real-valued function. The real numbers form an ordered field w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6614_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6615_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6616_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6617_chunk_0 | Residuated mapping | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6618_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6619_chunk_0 | Equations defining abelian varieties | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6620_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6621_chunk_0 | Groupoid algebra | In mathematics, the concept of groupoid algebra generalizes the notion of group algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6622_chunk_0 | Irreducible (mathematics) | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6623_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6624_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6625_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6626_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6627_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6628_chunk_0 | Ess sup | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6629_chunk_0 | Conductor of an elliptic curve | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6630_chunk_0 | Conductor-discriminant formula | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6631_chunk_0 | Cone of curves | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6632_chunk_0 | Conformal group of spacetime | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6633_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6634_chunk_0 | Conformal radius | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6635_chunk_0 | Huhn's theorem | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6636_chunk_0 | Preconditioned conjugate gradient method | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6637_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6638_chunk_0 | Conjugate (square roots) | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6639_chunk_0 | Adjoint matrix | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6640_chunk_0 | Connective constant | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6641_chunk_0 | Continuous wavelet transform | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6642_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6643_chunk_0 | Continuum function | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6644_chunk_0 | Courant–Friedrichs–Lewy condition | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6645_chunk_0 | Converse (logic) | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6646_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6647_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6648_chunk_0 | Convolution power | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6649_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6650_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6651_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6652_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6653_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6654_chunk_0 | Convolution theorem | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6655_chunk_0 | Corona theorem | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6656_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6657_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6658_chunk_0 | Correlation immunity | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6659_chunk_0 | Coset construction | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6660_chunk_0 | Coshc function | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6661_chunk_0 | Cotangent complex | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6662_chunk_0 | Covariant differential | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6663_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6664_chunk_0 | Crenel function | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6665_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6666_chunk_0 | Xyzzy (mnemonic) | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6667_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6668_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6669_chunk_0 | Curvature of a measure | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6670_chunk_0 | Curve complex | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6671_chunk_0 | Curve-shortening flow | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6672_chunk_0 | Curve-shortening flow | It loses area at a constant rate, and its perimeter decreases as quickly as possible for any continuous curve evolution. If the curve is non-convex, its total absolute curvature decreases monotonically, until it becomes convex. Once convex, the isoperimetric ratio of the curve decreases as the curve converges to a circ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6673_chunk_0 | Curve-shortening flow | An approximation to the curve-shortening flow can be computed numerically, by approximating the curve as a polygon and using the finite difference method to calculate the motion of each polygon vertex. Alternative methods include computing a convolution of polygon vertices and then resampling vertices on the resulting ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6674_chunk_0 | Curve-shortening flow | Later, it was applied in image analysis to give a multi-scale representation of shapes. It can also model reaction–diffusion systems, and the behavior of cellular automata. The curve-shortening flow can be used to find closed geodesics on Riemannian manifolds, and as a model for the behavior of higher-dimensional flows... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6675_chunk_0 | Cyclotomic identity | In mathematics, the cyclotomic identity states that 1 1 − α z = ∏ j = 1 ∞ ( 1 1 − z j ) M ( α , j ) {\displaystyle {1 \over 1-\alpha z}=\prod _{j=1}^{\infty }\left({1 \over 1-z^{j}}\right)^{M(\alpha ,j)}} where M is Moreau's necklace-counting function, M ( α , n ) = 1 n ∑ d | n μ ( n d ) α d , {\displaystyle M(\alpha ,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6676_chunk_0 | Cylinder set | In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6677_chunk_0 | Cylindrical harmonics | In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, ∇ 2 V = 0 {\displaystyle \nabla ^{2}V=0} , expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function Vn(k) is the product of thre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6678_chunk_0 | De Franchis theorem | In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism group of X is finite (see though Hurwitz's automorphisms theorem). More generall... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6679_chunk_0 | Degree (algebraic geometry) | In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety with n hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6680_chunk_0 | Degree (algebraic geometry) | This is a generalization of Bézout's theorem (For a proof, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem). The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space. The degree of a hype... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6681_chunk_0 | Degree (algebraic geometry) | A generalization of Bézout's theorem asserts that, if an intersection of n projective hypersurfaces has codimension n, then the degree of the intersection is the product of the degrees of the hypersurfaces. The degree of a projective variety is the evaluation at 1 of the numerator of the Hilbert series of its coordinat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6682_chunk_0 | Generalized derivative | In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6683_chunk_0 | First-order derivative expression | In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the obj... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6684_chunk_0 | First-order derivative expression | The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6685_chunk_0 | First-order derivative expression | In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the ch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6686_chunk_0 | First-order derivative expression | For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6687_chunk_0 | Derived category | In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6688_chunk_0 | Derived category | The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkable strides. The basic theory of Verdier was written down in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6689_chunk_0 | Derived category | The original impulse to develop the "derived" formalism came from the need to find a suitable formulation of Grothendieck's coherent duality theory. Derived categories have since become indispensable also outside of algebraic geometry, for example in the formulation of the theory of D-modules and microlocal analysis. R... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6690_chunk_0 | Determinant (mathematics) | In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6691_chunk_0 | Determinant (mathematics) | The determinant of a product of matrices is the product of their determinants (which follows directly from the above properties). The determinant of a 2 × 2 matrix is | a b c d | = a d − b c , {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc,} and the determinant of a 3 × 3 matrix is | a b c d e f g h i | = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6692_chunk_0 | Determinant (mathematics) | The determinant of an n × n matrix can be defined in several equivalent ways, the most common being Leibniz formula, which expresses the determinant as a sum of n ! {\displaystyle n!} (the factorial of n) signed products of matrix entries. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6693_chunk_0 | Determinant (mathematics) | It can be computed by the Laplace expansion, which expresses the determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which expresses the determinant as the product of the diagonal entries of a diagonal matrix that is obtained by a succession of elementary row operations. De... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6694_chunk_0 | Determinant (mathematics) | For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a matr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6695_chunk_0 | Determinant method | In mathematics, the determinant method is any of a family of techniques in analytic number theory. The name was coined by Roger Heath-Brown and comes from the fact that the center piece of the method is estimating a certain determinant. Its main application is to give an upper bound for the number of rational points of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6696_chunk_0 | Difference of two squares | In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity a 2 − b 2 = ( a + b ) ( a − b ) {\displaystyle a^{2}-b^{2}=(a+b)(a-b)} in elementary algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6697_chunk_0 | Differentiable surface | In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6698_chunk_0 | Differentiable surface | An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_6699_chunk_0 | Gauss's digamma theorem | In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z ) . {\displaystyle \psi (z)={\frac {\mathrm {d} }{\mathrm {d} z}}\ln \Gamma (z)={\frac {\Gamma '(z)}{\Gamma (z)}}.} It is the first of the polygamma functions. This functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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