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In materials science, work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context. This strengthening occurs because of dislocation movements and dislocation generation withi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-differential equations... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points. It is named in France after Henri Lebesgue, who studied it in 1904, and named in th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Also, in probability, σ-algebras are pivotal in the definition of conditional expectation. In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic, particularly when the statistic is a function or a random process and the notion of conditional density is not applicabl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real value... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Boundary is a distinct concept: for example, a circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice versa. For example, a subset S of a 2-dimensional real space R2 constrained by two parabolic curves x2 + 1 and x2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
{\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{L^{2}(\Omega )}^{1/4}\|u\|_{H^{2}(\Omega )}^{3/4}.} In 2D, the first inequality still holds, but not the second: let u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) {\displaystyle u\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )} where Ω ⊂ R 2 {\displaystyle \Omega \subset \m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Cesàro summation (also known as the Cesàro mean) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series. This spec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous.
In mathematical analysis, Ehrenpreis's fundamental principle, introduced by Leon Ehrenpreis, states: Every solution of a system (in general, ove... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases. A Fourier integral operator T {\displaystyle T} is given b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains. A related theorem is often called Fubini's theorem for infinite series, which states that if { a m , n } m = 1 , n = 1 ∞ {\textstyle \{a_{m,n}\}_{m=1,n=1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Glaeser's continuity theorem is a characterization of the continuity of the derivative of the square roots of functions of class C 2 {\displaystyle C^{2}} . It was introduced in 1963 by Georges Glaeser, and was later simplified by Jean Dieudonné.The theorem states: Let f: U → R 0 + {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces. The numbers p and q above are said to be Hölder conjugates of each other. The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Krein's condition provides a necessary and sufficient condition for exponential sums { ∑ k = 1 n a k exp ( i λ k x ) , a k ∈ C , λ k ≥ 0 } , {\displaystyle \left\{\sum _{k=1}^{n}a_{k}\exp(i\lambda _{k}x),\quad a_{k}\in \mathbb {C} ,\,\lambda _{k}\geq 0\right\},} to be dense in a weighted L2 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Littlewood's 4/3 inequality, named after John Edensor Littlewood, is an inequality that holds for every complex-valued bilinear form defined on c 0 {\displaystyle c_{0}} , the Banach space of scalar sequences that converge to zero. Precisely, let B: c 0 × c 0 → C {\displaystyle B:c_{0}\times c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s, are generalisations of the more familiar L p {\displaystyle L^{p}} spaces. The Lorentz spaces are denoted by L p , q {\displaystyle L^{p,q}} . Like the L p {\displaystyle L^{p}} spaces, they are characterized by a norm (technically ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Netto's theorem states that continuous bijections of smooth manifolds preserve dimension. That is, there does not exist a continuous bijection between two smooth manifolds of different dimension. It is named after Eugen Netto.The case for maps from a higher-dimensional manifold to a one-dimens... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
They form Jordan curves in the plane. However, by Netto's theorem, they cannot cover the entire plane, unit square, or any other two-dimensional region.
If one relaxes the requirement of continuity, then all smooth manifolds of bounded dimension have equal cardinality, the cardinality of the continuum. Therefore, ther... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which can have an uncountable infinity of basis vectors). Informally, the ide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Strichartz estimates are a family of inequalities for linear dispersive partial differential equations. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces. They were first noted by Robert Strichartz and arose out of connections to the Fourier restriction pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The inequality is a limiting case of Sobolev imbedding and can be stated as the following theorem: Let Ω {\displaystyle \Omega } be a bounded domain in R n {\displaystyle \mathbb {R} ^{n}} satisfying the cone condition. Let m p = n {\displaystyle mp=n} and p > 1 {\displaystyle p>1} . Set A ( t ) = exp ( t n / ( n − m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L 1 {\displaystyle L^{1}} or L 2 {\displaystyle L^{2}} can be approximated by linear combinations of translations of a gi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
As a consequence of the above properties, a real-valued Banach limit also satisfies: lim inf n → ∞ x n ≤ ϕ ( x ) ≤ lim sup n → ∞ x n . {\displaystyle \liminf _{n\to \infty }x_{n}\leq \phi (x)\leq \limsup _{n\to \infty }x_{n}.} The existence of Banach limits is usually proved using the Hahn–Banach theorem (analyst's app... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let G denote the subcollection of F consisting of all balls from the cN disjoint families A1,...,AcN. The less precise following statement is clearly true: every point x ∈ RN belongs to at most cN different balls from the subcollection G, and G remains a cover for E (every point y ∈ E belongs to at least one ball from ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limit. Young measures have applications in the calculus of variations, especia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact spa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In the case of several variables, a function f defined on an open subset Ω of R n {\displaystyle \mathbb {R} ^{n}} is said to have bounded variation if its distributional derivative is a vector-valued finite Radon measure. One of the most important aspects of functions of bounded variation is that they form an algebra ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, a modulus of continuity is a function ω: → used to measure quantitatively the uniform continuity of functions. So, a function f: I → R admits ω as a modulus of continuity if and only if | f ( x ) − f ( y ) | ≤ ω ( | x − y | ) , {\displaystyle |f(x)-f(y)|\leq \omega (|x-y|),} for all x and y ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A special role is played by concave moduli of continuity, especially in connection with extension properties, and with approximation of uniformly continuous functions. For a function between metric spaces, it is equivalent to admit a modulus of continuity that is either concave, or subadditive, or uniformly continuous,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, a positively (or positive) invariant set is a set with the following properties: Suppose x ˙ = f ( x ) {\displaystyle {\dot {x}}=f(x)} is a dynamical system, x ( t , x 0 ) {\displaystyle x(t,x_{0})} is a trajectory, and x 0 {\displaystyle x_{0}} is the initial point. Let O := { x ∈ R n ∣ φ ( x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, a strong measure zero set is a subset A of the real line with the following property: for every sequence (εn) of positive reals there exists a sequence (In) of intervals such that |In| < εn for all n and A is contained in the union of the In. (Here |In| denotes the length of the interval In.) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The following characterization of strong measure zero sets was proved in 1973: A set A ⊆ R has strong measure zero if and only if A + M ≠ R for every meagre set M ⊆ R.This result establishes a connection to the notion of strongly meagre set, defined as follows: A set M ⊆ R is strongly meagre if and only if A + M ≠ R fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over which the in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Such an integral is sometimes described as being of the "first" type or kind if the integrand otherwise satisfies the assumptions of integration. Integrals in the fourth form that are improper because f ( x ) {\displaystyle f(x)} has a vertical asymptote somewhere on the interval {\displaystyle } may be described as b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
If f ( x ) {\displaystyle f(x)} is only continuous on ( a , ∞ ) {\displaystyle (a,\infty )} and not at a {\displaystyle a} itself, then typically this is rewritten as ∫ a ∞ f ( x ) d x = lim t → a + ∫ t c f ( x ) d x + lim b → ∞ ∫ c b f ( x ) d x , {\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{t\to a^{+}}\int _{t}^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alternatively, an iterated limit could be used or a single limit based on the Cauchy principal value. If f ( x ) {\displaystyle f(x)} is continuous on [ a , d ) {\displaystyle [a,d)} and ( d , ∞ ) {\displaystyle (d,\infty )} , with a discontinuity of any kind at d {\displaystyle d} , then ∫ a ∞ f ( x ) d x = lim t → d ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers R {\displaystyle \mathbb {R} } , or a subset of R {\displaystyle \mathbb {R} } that contains an interval of positive length. Most r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the real or complex numbers. This space, denoted by C ( X ) , {\displaystyle {\mathcal {C}}(X),} is a vector space with respect ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
This is often written symbolically as f (n) ~ n2, which is read as "f(n) is asymptotic to n2". An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x) is the number of prime numbers that are less ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, idempotent analysis is the study of idempotent semirings, such as the tropical semiring. The lack of an additive inverse in the semiring is compensated somewhat by the idempotent rule A ⊕ A = A {\displaystyle A\oplus A=A} . == References ==
In mathematical analysis, in particular the subfield... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Extended real-valued function for which the minimization problem is not solved by any one of these three trivial cases are exactly those that are called proper. Many (although not all) results whose hypotheses require that the function be proper add this requirement specifically to exclude these trivial cases. If the p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmander around 1970.
In mathematical analy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p ≥ 1 {\displaystyle p\geq 1} . p-variation is a measure of the regularity or smoothness of a function. Specifically, if f: I → ( M , d ) {\displaystyle f:I\to (M,d)} , where (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively, lower) semicontinuous at a point x 0 {\displaystyle x_{0}} if, roughly speaking, the function values ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions. If π ( x ) = 1 Π ( x ) = 1 Γ ( x + 1 ) {\displaystyle \pi (x)={\frac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for x > 0 by Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t {\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,\mathrm {d} t} as the o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb {R} ^{2}} be the exterior or the interior of a bounded domain with regular boundary, or R 2 {\displaystyle \mathbb {R} ^{2}} itself. Then the Brezis–Gallouët inequality states that there exists a real C {\displaystyle C} only depending on Ω {\displaystyle \Omega } such ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as where n is any nonnegative integer. The kernel functions are periodic with period 2 π {\displaystyle 2\pi } . The importance of the Dirichlet kernel com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f {\displaystyle f} and g {\displaystyle g} are nonnegative measurable real functions vanishing at infinity that are defined on n {\displaystyle n} -dimensional Euclidean space R n {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems.
In mathematical... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.
In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S {\displaystyle S} be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the Rademacher–Menchov theorem, introduced by Rademacher (1922) and Menchoff (1923), gives a sufficient condition for a series of orthogonal functions on an interval to converge almost everywhere.
In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
{\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy.} If there exist real functions p ( x ) > 0 {\displaystyle \,p(x)>0} and q ( y ) > 0 {\displaystyle \,q(y)>0} and numbers α , β > 0 {\displaystyle \,\alpha ,\beta >0} such that ( 1 ) ∫ Y K ( x , y ) q ( y ) d y ≤ α p ( x ) {\displaystyle (1)\qquad \int _{Y}K(x,y)q(y)\,dy\leq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
More precisely: Whitney Covering Lemma (Grafakos 2008, Appendix J) Let Ω {\displaystyle \Omega } be an open non-empty proper subset of R n {\displaystyle \mathbb {R} ^{n}} . Then there exists a family of closed cubes { Q j } j {\displaystyle \{Q_{j}\}_{j}} such that ∪ j Q j = Ω {\displaystyle \cup _{j}Q_{j}=\Omega } an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz cri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity. Mathematically, if f ( t ) {\displaystyle f(t)} in continuous time has (unilateral) Laplace transform F ( s ) {\displaystyle F(s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.Let F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t {\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt} be the (one-sided) Laplace transform of ƒ(t). If f {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum.
In mathematical analysis, the mean v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If U = R n {\displaystyle U=\mathbb {R} ^{n}} then the use of Schwartz functions as test functions gives rise to a certain subspace of D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} whose elem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
It shows that, for curves under uniform convergence, the length of a curve is not a continuous function of the curve.For any smooth curve, polygonal chains with segment lengths decreasing to zero, connecting consecutive vertices along the curve, always converge to the arc length. The failure of the staircase curves to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The name "uniform norm" derives from the fact that a sequence of functions { f n } {\displaystyle \left\{f_{n}\right\}} converges to f {\displaystyle f} under the metric derived from the uniform norm if and only if f n {\displaystyle f_{n}} converges to f {\displaystyle f} uniformly.If f {\displaystyle f} is a continuo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical area of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product with ( a ; q ) 0 = 1. {\displaystyle (a;q)_{0}=1.} It is a q-analog of the Pochhammer symbol ( x ) n = x ( x + 1 ) … ( x + n − 1 ) {\displaystyle (x)_{n}=x(x+1)\dots (x+n-1)} , in the sense that The q-Poc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
By the Hartman–Grobman theorem the non-linear system is topologically equivalent to a linearization of the system about an equilibrium point (x*, y*), which has the form = A , {\displaystyle {\begin{bmatrix}{\frac {du}{dt}}\\{\frac {dv}{dt}}\end{bmatrix}}=\mathbf {A} {\begin{bmatrix}u\\v\end{bmatrix}},} where u = x −... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical complex analysis, Radó's theorem, proved by Tibor Radó (1925), states that every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f: D → D′ be an orientation-preserving homeomorphism betwee... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
In mathematical economics, Topkis's theorem is a result that is useful for establishing comparative statics. The theorem allows researchers to understand how the optimal ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical economics, applied general equilibrium (AGE) models were pioneered by Herbert Scarf at Yale University in 1967, in two papers, and a follow-up book with Terje Hansen in 1973, with the aim of empirically estimating the Arrow–Debreu model of general equilibrium theory with empirical data, to provide "“a g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scarf's fixed-point method was a break-through in the mathematics of computation generally, and specifically in optimization and computational economics. Later researchers continued to develop iterative methods for computing fixed-points, both for topological models like Scarf's and for models described by functions wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical field of combinatorial geometry, the Littlewood–Offord problem is the problem of determining the number of subsums of a set of vectors that fall in a given convex set. More formally, if V is a vector space of dimension d, the problem is to determine, given a finite subset of vectors S and a convex subse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
By subtracting 1 2 ∑ i = 1 n v i {\displaystyle {\frac {1}{2}}\sum _{i=1}^{n}v_{i}} from each possible subsum (that is, by changing the origin and then scaling by a factor of 2), the Littlewood–Offord problem is equivalent to the problem of determining the number of sums of the form ∑ i = 1 n ε i v i {\displaystyle \su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
From this point of view, quaternionic representation of a group G is a group homomorphism φ: G → GL(V, H), the group of invertible quaternion-linear transformations of V. In particular, a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the ide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form ω: V × V → F {\displaystyle \omega \colon V\times V\to \mathbb {F}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. The first application to option pricing was by Phelim Boyle in 1977 (for European options). In 1996, M. Broadie and P. Glasserman showed how... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives du... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finance, the stochastic volatility jump (SVJ) model is suggested by Bates. This model fits the observed implied volatility surface well. The model is a Heston process for stochastic volatility with an added Merton log-normal jump. It assumes the following correlated processes: d S = μ S d t + ν S d Z 1 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finite group theory, a Thompson factorization, introduced by Thompson (1966), is an expression of some finite groups as a product of two subgroups, usually normalizers or centralizers of p-subgroups for some prime p.
In mathematical finite group theory, a block, sometimes called Aschbacher block, is a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
The groups of each of these 8 types were classified by various authors. They consist mainly of groups of Lie type with all roots of the same length over the field with 2 elements, but also include many exceptional cases, including the majority of the sporadic simple groups. Smith (1980) gave a survey of this work. Smit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was started by Higman (1964, 1971). Several of the sporadic simple groups were discovered as rank 3 permutation groups.
In mathematical finite group the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finite group theory, the Baer–Suzuki theorem, proved by Baer (1957) and Suzuki (1965), states that if any two elements of a conjugacy class C of a finite group generate a nilpotent subgroup, then all elements of the conjugacy class C are contained in a nilpotent subgroup. Alperin & Lyons (1971) gave a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension 2 5 . G L 5 ( F 2 ) {\displaystyle 2^{5\,. }\mathrm {GL} _{5}(\mathbb {F} _{2})} of G L 5 ( F 2 ) {\displaystyle \mathrm {GL} _{5}(\mathbb {F} _{2})} by its natural module... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
G L 4 ( F 2 ) {\displaystyle 2^{4\,. }\mathrm {GL} _{4}(\mathbb {F} _{2})} is a maximal subgroup of the sporadic Conway group Co3. The nonsplit extension 2 5 . G L 5 ( F 2 ) {\displaystyle 2^{5\,. }\mathrm {GL} _{5}(\mathbb {F} _{2})} is a maximal subgroup of the Thompson sporadic group Th.
In mathematical finite grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finite group theory, the Puig subgroup, introduced by Puig (1976), is a characteristic subgroup of a p-group analogous to the Thompson subgroup.
In mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson (Held 1969, p.279), gives a formula for the order of a fin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized by A. It originated in the proof of the odd order theorem by Feit and Thompson (1963), where it was used to prove the Thompson un... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem. The uniqueness case covers groups G of characteristic 2 type with e(G) ≥ 3 that have an almost strongly p-embedded maximal 2-local subgroup for all primes p who... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is an important special case. Specifically, if (M, g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weight... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical group theory, a C-group is a group such that the centralizer of any involution has a normal Sylow 2-subgroup. They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups have trivial intersection. The simple C-groups were deter... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical group theory, a subgroup of a group is termed a special abelian subgroup or SA-subgroup if the centralizer of any nonidentity element in the subgroup is precisely the subgroup(Curtis & Reiner 1981, p.354). Equivalently, an SA subgroup is a centrally closed abelian subgroup. Any SA subgroup is a maximal ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H 2 ( G , Z ) {\displaystyle H_{2}(G,\mathbb {Z} )} of a group G. It was introduced by Issai Schur (1904) in his work on projective representations.
In mathematical group theory, the automorphism group of a free grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970.
In mathematical heat conduction, the Green's function number ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Elliott (1907) describes the early history of perpetuants and gives an annotated bibliography. MacMahon conjectured and Stroh proved that the dimension of the space of perpetuants of degree n>2 and weight w is the coefficient of xw of x 2 n − 1 − 1 ( 1 − x 2 ) ( 1 − x 3 ) ⋯ ( 1 − x n ) {\displaystyle {\frac {x^{2^{n-1}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables x and y that remains invariant under the special linear group acting on the variables x and y.
In mathematical invariant theory, the canonizant or canonisant is a covariant of forms rel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical invariant theory, the osculant or tacinvariant or tact invariant is an invariant of a hypersurface that vanishes if the hypersurface touches itself, or an invariant of several hypersurfaces that osculate, meaning that they have a common point where they meet to unusually high order.
In mathematical kno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial refe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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