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wiki_5100_chunk_0 | Tapering (mathematics) | In mathematics, physics, and theoretical computer graphics, tapering is a kind of shape deformation. Just as an affine transformation, such as scaling or shearing, is a first-order model of shape deformation, tapering is a higher order deformation just as twisting and bending. Tapering can be thought of as non-constant... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5101_chunk_0 | Tapering (mathematics) | To create a nonlinear taper, instead of scaling in x and y for all z with constants as in: q = p , {\displaystyle q={\begin{bmatrix}a&0&0\\0&b&0\\0&0&1\end{bmatrix}}p,} let a and b be functions of z so that: q = p . {\displaystyle q={\begin{bmatrix}a(p_{z})&0&0\\0&b(p_{z})&0\\0&0&1\end{bmatrix}}p.} An example of a li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5102_chunk_0 | Frequency space | Although it is common to refer to the magnitude portion as the frequency response of a signal, the phase portion is required to uniquely define the signal. A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5103_chunk_0 | Frequency space | The "spectrum" of frequency components is the frequency-domain representation of the signal. The inverse Fourier transform converts the frequency-domain function back to the time-domain function. A spectrum analyzer is a tool commonly used to visualize electronic signals in the frequency domain. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5104_chunk_0 | Frequency space | A frequency-domain representation may describe either a static function or a particular time period of a dynamic function (signal or system). The frequency transform of a dynamic function is performed over a finite time period of that function and assumes the function repeats infinitely outside of that time period. Som... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5105_chunk_0 | Planar algebra | In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5106_chunk_0 | Plurisubharmonic function | In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Ri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5107_chunk_0 | Point-free geometry | In mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as connection theory. Point-free geometry was first formulated in Whitehead (1919, 1920), not as a th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5108_chunk_0 | Topology of pointwise convergence | In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5109_chunk_0 | Poly-Bernoulli number | {\displaystyle B_{k}^{(-k)}\sim (k! )^{2}{\sqrt {\frac {1}{k\pi (1-\log 2)}}}\left({\frac {1}{\log 2}}\right)^{2k+1},\quad {\text{as }}k\rightarrow \infty .} For a positive integer n and a prime number p, the poly-Bernoulli numbers satisfy B n ( − p ) ≡ 2 n ( mod p ) , {\displaystyle B_{n}^{(-p)}\equiv 2^{n}{\pmod {p}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5110_chunk_0 | Polynomial identity testing | In mathematics, polynomial identity testing (PIT) is the problem of efficiently determining whether two multivariate polynomials are identical. More formally, a PIT algorithm is given an arithmetic circuit that computes a polynomial p in a field, and decides whether p is the zero polynomial. Determining the computation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5111_chunk_0 | Positive definite | In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: Positive-definite bilinear form Positive-definite function Positive-definite function on a group Positive-definite functional Pos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5112_chunk_0 | Potential flow around a circular cylinder | In mathematics, potential flow around a circular cylinder is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5113_chunk_0 | Power method | In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A {\displaystyle A} , the algorithm will produce a number λ {\displaystyle \lambda } , which is the greatest (in absolute value) eigenvalue of A {\displaystyle A} , and a nonzero vector v {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5114_chunk_0 | Pluriharmonic function | In mathematics, precisely in the theory of functions of several complex variables, a pluriharmonic function is a real valued function which is locally the real part of a holomorphic function of several complex variables. Sometimes such a function is referred to as n-harmonic function, where n ≥ 2 is the dimension of th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5115_chunk_0 | Primitive recursive ordinal function | In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They were introduced by Jensen & Karp (1971). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5116_chunk_0 | Probabilistic metric space | In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions.Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mappin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5117_chunk_0 | Projections onto convex sets | In mathematics, projections onto convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets. It is a very simple algorithm and has been rediscovered many times. The simplest case, when the sets are affine spaces, was analyzed by Jo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5118_chunk_0 | Projections onto convex sets | For general closed convex sets, the limit point need not be the projection. Classical work on the case of two closed convex sets shows that the rate of convergence of the iterates is linear. There are now extensions that consider cases when there are more than two sets, or when the sets are not convex, or that give fas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5119_chunk_0 | Projections onto convex sets | Analysis of POCS and related methods attempt to show that the algorithm converges (and if so, find the rate of convergence), and whether it converges to the projection of the original point. These questions are largely known for simple cases, but a topic of active research for the extensions. There are also variants of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5120_chunk_0 | Projective differential geometry | In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifolds, that are invariant under transformations of the projective group. This is a mixture of the approaches from Riemannian... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5121_chunk_0 | Projective differential geometry | Élie Cartan formulated the idea of a general projective connection, as part of his method of moving frames; abstractly speaking, this is the level of generality at which the Erlangen program can be reconciled with differential geometry, while it also develops the oldest part of the theory (for the projective line), nam... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5122_chunk_0 | Axioms of projective geometry | In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5123_chunk_0 | Axioms of projective geometry | The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5124_chunk_0 | Axioms of projective geometry | Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5125_chunk_0 | Axioms of projective geometry | See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5126_chunk_0 | Axioms of projective geometry | Several major types of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme resulting in the study of the classical groups) were motivated by projective geometry. It was also a subject with many practitioners for its own sake, as synthetic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5127_chunk_0 | Projectivization | In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space P ( V ) {\displaystyle {\mathbb {P} }(V)} , whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalar multiplication defines a subset of P ( V ) {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5128_chunk_0 | Generic property | In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5129_chunk_0 | Generic property | There are many different notions of "generic" (what is meant by "almost all") in mathematics, with corresponding dual notions of "almost none" (negligible set); the two main classes are: In measure theory, a generic property is one that holds almost everywhere, with the dual concept being null set, meaning "with probab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5130_chunk_0 | Pseudoanalytic function | In mathematics, pseudoanalytic functions are functions introduced by Lipman Bers (1950, 1951, 1953, 1956) that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5131_chunk_0 | Quadratic Jordan algebra | In mathematics, quadratic Jordan algebras are a generalization of Jordan algebras introduced by Kevin McCrimmon (1966). The fundamental identities of the quadratic representation of a linear Jordan algebra are used as axioms to define a quadratic Jordan algebra over a field of arbitrary characteristic. There is a unifo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5132_chunk_0 | Quadrature (mathematics) | In mathematics, quadrature is a historical term for the process of determining area. This term is still used in the context of differential equations, where "solving an equation by quadrature" or "reduction to quadrature" means expressing its solution in terms of integrals. Quadrature problems served as one of the main... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5133_chunk_0 | Quasi-bialgebra | In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible element Φ {\displaystyle \Phi } which controls the non-coassociativity. On... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5134_chunk_0 | Quaternionic analysis | In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. As with complex and real analysis, it is possible to study the concepts of a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5135_chunk_0 | Random graphs | In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5136_chunk_0 | Real algebraic variety | In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings). Semialgebraic geometry is the study of semialgebraic sets, i.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5137_chunk_0 | Reduced homology | In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5138_chunk_0 | Reduction (mathematics) | In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator a whole number) is called "reducing a fraction". Rewriting a radical (or "root") expression w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5139_chunk_0 | Scientific reductionism | In mathematics, reductionism can be interpreted as the philosophy that all mathematics can (or ought to) be based on a common foundation, which for modern mathematics is usually axiomatic set theory. Ernst Zermelo was one of the major advocates of such an opinion; he also developed much of axiomatic set theory. It has ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5140_chunk_0 | Scientific reductionism | Yet Gödel proved that, for any consistent recursively enumerable axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are (model-theoretically) true propositions about the natural numbers that cannot be proved from the axioms. Such propositions are known as formally undecidable prop... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5141_chunk_0 | Rigid cohomology | In mathematics, rigid cohomology is a p-adic cohomology theory introduced by Berthelot (1986). It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky–Washnitzer cohomology to non-affine varieties. For a scheme X of finite type over a perfect field k, there are rigid cohomolog... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5142_chunk_0 | Unital ring | In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5143_chunk_0 | Unital ring | (Some authors use the term "rng" with a missing "i" to refer to the more general structure that omits this last requirement; see § Notes on the definition.) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has profound implications on its behavior. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5144_chunk_0 | Unital ring | Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry. The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields. Examples of commutative rings include the set of integers with their standard addition and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5145_chunk_0 | Unital ring | Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology. The conceptualization of rings spanned the 1870s to the 1920s, with key contribut... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5146_chunk_0 | Scalar multiplication | In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector—without chan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5147_chunk_0 | Scattering process | In mathematics, scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a differential equation is known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role of time. One then asks what... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5148_chunk_0 | Scattering process | As a result, the solutions often have a spectrum that can be identified with a Hilbert space, and scattering is described by a certain map, the S matrix, on Hilbert spaces. Spaces with a discrete spectrum correspond to bound states in quantum mechanics, while a continuous spectrum is associated with scattering states. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5149_chunk_0 | Secondary calculus | In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation. It is a sophisticated theory at the level of jet spaces and employing algebraic methods. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5150_chunk_0 | Semi-simple category | In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5151_chunk_0 | Semi-simple category | Now Maschke's theorem says that any finite-dimensional representation of a finite group is a direct sum of simple representations (provided the characteristic of the base field does not divide the order of the group). So in the case of finite groups with this condition, every finite-dimensional representation is semi-s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5152_chunk_0 | Semi-simple category | For example, Weyl's theorem on complete reducibility says a finite-dimensional representation of a semisimple compact Lie group is semisimple. A square matrix (in other words a linear operator T: V → V {\displaystyle T:V\to V} with V finite dimensional vector space) is said to be simple if its only invariant subspaces ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5153_chunk_0 | Separation of variables | In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5154_chunk_0 | Series acceleration | In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5155_chunk_0 | Subset inclusion | In mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5156_chunk_0 | Set inversion | In mathematics, set inversion is the problem of characterizing the preimage X of a set Y by a function f, i.e., X = f −1(Y ) = {x ∈ Rn | f(x) ∈ Y }. It can also be viewed as the problem of describing the solution set of the quantified constraint "Y(f (x))", where Y( y) is a constraint, e.g. an inequality, describing th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5157_chunk_0 | Set-theoretic topology | In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5158_chunk_0 | Sheaf cohomology | In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5159_chunk_0 | Sheaf cohomology | It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to suc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5160_chunk_0 | Sieved orthogonal polynomials | In mathematics, sieved orthogonal polynomials are orthogonal polynomials whose recurrence relations are formed by sieving the recurrence relations of another family; in other words, some of the recurrence relations are replaced by simpler ones. The first examples were the sieved ultraspherical polynomials introduced by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5161_chunk_0 | Pole–zero plot | In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as: Stability Causal system / anticausal system Region of convergence (ROC) Minimum phase / non minimum ph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5162_chunk_0 | Negative frequency | In mathematics, signed frequency (negative and positive frequency) expands upon the concept of frequency, from just an absolute value representing how often some repeating event occurs, to also have a positive or negative sign representing one of two opposing orientations for occurrences of those events. The following ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5163_chunk_0 | Signed measure | In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to aqcuire sign. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5164_chunk_0 | Sine and cosine | More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic phenomena such as sound and lig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5165_chunk_0 | Singular integral operators of convolution type | In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5166_chunk_0 | Singular integral operators of convolution type | Continuity on Lp spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5167_chunk_0 | Singular integral operators on closed curves | In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5168_chunk_0 | Singular integral operators on closed curves | The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve. The theory for the original curve can be deduced from that of the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution type. The Hilbert transfo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5169_chunk_0 | Singular integral operator | In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator T ( f ) ( x ) = ∫ K ( x , y ) f ( y ) d y , {\displaystyle T(f)(x)=\int K(x,y)f(y)\,dy,} whose kernel function K... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5170_chunk_0 | Non-analytic smooth function | In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below. One of the most impo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5171_chunk_0 | Non-analytic smooth function | The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic cas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5172_chunk_0 | Stochastic processes and boundary value problems | In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5173_chunk_0 | Non-positive curvature | In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero. The notion of curvature ex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5174_chunk_0 | Synge theorem | In mathematics, specifically Riemannian geometry, Synge's theorem is a classical result relating the curvature of a Riemannian manifold to its topology. It is named for John Lighton Synge, who proved it in 1936. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5175_chunk_0 | Totally ordered abelian group | In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant. This may have different meanings. We say that (G, ≤) is a: left-ordered group if ≤ is left-invariant, that is a ≤ b implies ca ≤ cb for all a, b, c in G,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5176_chunk_0 | Square class | In mathematics, specifically abstract algebra, a square class of a field F {\displaystyle F} is an element of the square class group, the quotient group F × / F × 2 {\displaystyle F^{\times }/F^{\times 2}} of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5177_chunk_0 | Square class | The reason is that if V {\displaystyle V} is an F {\displaystyle F} -vector space and q: V → F {\displaystyle q:V\to F} is a quadratic form and v {\displaystyle v} is an element of V {\displaystyle V} such that q ( v ) = a ∈ F × {\displaystyle q(v)=a\in F^{\times }} , then for all u ∈ F × {\displaystyle u\in F^{\times ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5178_chunk_0 | Artinian module | In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5179_chunk_0 | Artinian ring | In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5180_chunk_0 | Artinian ring | For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Wedderburn–Artin theorem characterizes every simple Artinian ring as a ring of matrices over a division ring. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5181_chunk_0 | Artinian ring | Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem is that a left (resp. right) Artinian ring is automatically a left (resp. right) Noetherian ring. This is not true for general modules; that is, an Artinian module need not be a Noetherian module. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5182_chunk_0 | Associate (ring theory) | In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5183_chunk_0 | Associate (ring theory) | "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral dom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5184_chunk_0 | Norm (abelian group) | In mathematics, specifically abstract algebra, if ( G , + ) {\displaystyle (G,+)} is an (abelian) group with identity element e {\displaystyle e} then ν: G → R {\displaystyle \nu \colon G\to \mathbb {R} } is said to be a norm on ( G , + ) {\displaystyle (G,+)} if: Positive definiteness: ν ( g ) > 0 for all g ≠ e and ν ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5185_chunk_0 | First ring isomorphism theorem | In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5186_chunk_0 | Opposite algebra | In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring (R, +, ⋅) is the ring (R, +, ∗) whose multiplication ∗ is defined by a ∗ b = b ⋅ a for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5187_chunk_0 | Romanov's theorem | In mathematics, specifically additive number theory, Romanov's theorem is a mathematical theorem proved by Nikolai Pavlovich Romanov. It states that given a fixed base b, the set of numbers that are the sum of a prime and a positive integer power of b has a positive lower asymptotic density. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5188_chunk_0 | Donaldson-Thomas theory | In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5189_chunk_0 | Exceptional curve | In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map f: X → Y {\displaystyle f:X\rightarrow Y} of varieties is a kind of 'large' subvariety of X {\displaystyle X} which is 'crushed' by f {\displaystyle f} , in a certain definite sense. More strictly, f has an associated exceptional ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5190_chunk_0 | Exceptional curve | A codimension-1 subvariety Z ⊂ X {\displaystyle Z\subset X} is said to be exceptional if f ( Z ) {\displaystyle f(Z)} has codimension at least 2 as a subvariety of Y {\displaystyle Y} . One may then define the exceptional divisor of f {\displaystyle f} to be ∑ i Z i ∈ D i v ( X ) , {\displaystyle \sum _{i}Z_{i}\in Div(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5191_chunk_0 | Valuative criterion | In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide whether a morphism of algebraic varieties, or more generally schemes, is universally closed, separated, or proper. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5192_chunk_0 | Mapping cylinder | In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f {\displaystyle f} between topological spaces X {\displaystyle X} and Y {\displaystyle Y} is the quotient M f = ( ( × X ) ⨿ Y ) / ∼ {\displaystyle M_{f}=((\times X)\amalg Y)\,/\,\sim } where the ⨿ {\displaystyle \amalg } de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5193_chunk_0 | Covariant functor | Thus, functors are important in all areas within mathematics to which category theory is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in a linguistic context; see function word. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5194_chunk_0 | Posetal category | For example, under this assumption, a poset may be defined as a small posetal category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category, and a Boolean algebra as a small posetal finitely cocomplete *-autonomous category.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5195_chunk_0 | Left adjoint | In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5196_chunk_0 | Hypergeometric function identities | In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. WZ pairs are named after Herbert S. Wilf and Doron Zeilberger, and are instrumental in the evaluation of many sums involving binomial coefficients, factori... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5197_chunk_0 | Hilbert's basis theorem | In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5198_chunk_0 | Divided power structure | In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form x n / n ! {\displaystyle x^{n}/n!} meaningful even when it is not possible to actually divide by n ! {\displaystyle n!} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5199_chunk_0 | Primary ideal | In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n > 0. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number. The notion of primary ideals is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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