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Projective differential geometry Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective differential geometry Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axioms of projective geometry Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axioms of projective geometry Summary 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axioms of projective geometry Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axioms of projective geometry Summary 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 coordinate... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Axioms of projective geometry Summary 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projectivization Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generic property Summary 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,"... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generic property Summary 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 co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudoanalytic function Summary 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 |
Quadratic Jordan algebra Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadratic variation Summary Quadratic_variation In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quadrature (mathematics) Summary 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quantale Summary Quantale In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras). Quantales are sometimes referred to as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-bialgebra Summary 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 } wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quaternionic analysis Summary Quaternion_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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quaternionic projective line Summary Quaternionic_projective_space In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H . {\displaystyle \mathbb {H} .} Quaternionic projective spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
History of quaternions Summary History_of_quaternions In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, but independently discovered by Irish mathema... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Racks and quandles Summary Racks_and_quandles In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams. While mainly used to obtain invariants of knots, they can be viewed as algebraic constructions in their own right. In p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Random graphs Summary 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 the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Random graphs Summary Random_graphs From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Random group Summary Random_group In mathematics, random groups are certain groups obtained by a probabilistic construction. They were introduced by Misha Gromov to answer questions such as "What does a typical group look like?" It so happens that, once a precise definition is given, random groups satisfy some properti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rational reconstruction (mathematics) Summary Rational_reconstruction_(mathematics) In mathematics, rational reconstruction is a method that allows one to recover a rational number from its value modulo a sufficiently large integer. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real algebraic variety Summary Real_algebraic_curve 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). Semialgeb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real projective space Summary Real_projective_space In mathematics, real projective space, denoted R P n {\displaystyle \mathbb {RP} ^{n}} or P n ( R ) , {\displaystyle \mathbb {P} _{n}(\mathbb {R} ),} is the topological space of lines passing through the origin 0 in the real space R n + 1 . {\displaystyle \mathbb {R} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Real tree Summary Real_tree In mathematics, real trees (also called R {\displaystyle \mathbb {R} } -trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and probability theory. They are also the simplest examples of G... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reduced homology Summary Reduced_cohomology 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 Ale... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reduction (mathematics) Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scientific reductionism In mathematics Scientific_reductionism > Definitions > In mathematics 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scientific reductionism In mathematics Scientific_reductionism > Definitions > In mathematics 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reflection symmetry Summary Reflection_symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Relative root system Summary Relative_root_system In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a symmetric space and its d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weil descent Summary Weil_descent In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fiel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Rigid cohomology Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Suslin rigidity Summary Rigidity_(K-theory) In mathematics, rigidity of K-theory encompasses results relating algebraic K-theory of different rings. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unital ring Summary Unit_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 multiplicat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unital ring Summary Unit_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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unital ring Summary Unit_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 wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unital ring Summary Unit_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scalar multiplication Summary Scalar_multiple 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 multi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scattering process Mathematical framework Scattering_process > Theory > Mathematical framework 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scattering process Mathematical framework Scattering_process > Theory > Mathematical framework Solutions to differential equations are often posed on manifolds. Frequently, the means to the solution requires the study of the spectrum of an operator on the manifold. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scattering process Mathematical framework Scattering_process > Theory > Mathematical framework 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Secondary calculus Summary 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 |
Self similarity Self-affinity Self-similarity > Self-affinity In mathematics, self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x- and y-directions. This means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic affine ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semi-infinite Summary Semi-infinite In mathematics, semi-infinite objects are objects which are infinite or unbounded in some but not all possible ways. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semi-simple category Summary 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semi-simple category Summary 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semi-simple category Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Separation of variables Summary 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 sid... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Series acceleration Summary Accelerated_convergence 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subset inclusion Summary Strict_subset 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 inclusi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Set inversion Summary 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 constrain... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Set-theoretic topology Summary 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 |
Sheaf cohomology Summary Cohomology_of_sheaves 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.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sheaf cohomology Summary Cohomology_of_sheaves From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp. Leray's definitions were simplified and clarified in the 1950s. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sheaf cohomology Summary Cohomology_of_sheaves 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sieved Jacobi polynomials Summary Sieved_Jacobi_polynomials In mathematics, sieved Jacobi polynomials are a family of sieved orthogonal polynomials, introduced by Askey (1984). Their recurrence relations are a modified (or "sieved") version of the recurrence relations for Jacobi polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sieved Pollaczek polynomials Summary Sieved_Pollaczek_polynomials In mathematics, sieved Pollaczek polynomials are a family of sieved orthogonal polynomials, introduced by Ismail (1985). Their recurrence relations are a modified (or "sieved") version of the recurrence relations for Pollaczek polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sieved orthogonal polynomials Summary 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 fir... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pole–zero plot Summary 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 convergenc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Negative frequency Summary 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 f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Signed measure Summary 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 |
Simple homotopy theory Summary Simple_homotopy_theory In mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space. It was originated by Whitehead in his 1950 paper "Simple homotopy type". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sine and cosine Summary Cosine_function In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sine and cosine Summary Cosine_function 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Singular integral operators of convolution type Summary Beurling_transform 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 translation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Singular integral operators of convolution type Summary Beurling_transform 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Singular integral operators on closed curves Summary 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 Cauc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Singular integral operators on closed curves Summary 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Singular integral operator Summary Singular_integrals 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Singularity theory Summary Singularity_theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Singularity theory Summary Singularity_theory The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string. Perhaps the string will also touch itself without crossing, like an underlined "U". This is another kind of singularity... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Singularity theory Summary Singularity_theory Unlike the double point, it is not stable, in the sense that a small push will lift the bottom of the "U" away from the "underline". Vladimir Arnold defines the main goal of singularity theory as describing how objects depend on parameters, particularly in cases where the p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Size theory Summary Size_theory In mathematics, size theory studies the properties of topological spaces endowed with R k {\displaystyle \mathbb {R} ^{k}} -valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance between size p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Non-analytic smooth function Summary 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, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Non-analytic smooth function Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sociable number Summary Sociable_number In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sociable number Summary Sociable_number For the sequence to be sociable, the sequence must be cyclic and return to its starting point. The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle. If the period of the sequence is 1, the number is a sociable number of order... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Solid partition Summary Solid_partition In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon. A solid partition of n {\displaystyle n} is a three-dimensional array of non-negative integers n i , j , k {\displaystyle n_{i,j,k}} (with indices ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Solid partition Summary Solid_partition Let p 3 ( n ) {\displaystyle p_{3}(n)} denote the number of solid partitions of n {\displaystyle n} . As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called three-dimensional partitions in notation where plane partitions are two-d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stochastic processes and boundary value problems Summary Characteristic_operator 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 mo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of mathematical functions Summary List_of_mathematical_functions In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed ou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Non-positive curvature Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double complex Summary Double_complex That is d h ∘ d v + d v ∘ d h = 0. {\displaystyle d_{h}\circ d_{v}+d_{v}\circ d_{h}=0.} This eases the definition of Total Complexes. By setting f p , q = ( − 1 ) p d p , q v: C p , q → C p , q − 1 {\displaystyle f_{p,q}=(-1)^{p}d_{p,q}^{v}\colon C_{p,q}\to C_{p,q-1}} , we can swit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Synge theorem Summary 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 |
Totally ordered abelian group Summary Totally_ordered_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-in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Square class Summary 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 e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Square class Summary 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 ∈ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artinian module Summary 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 itse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artinian module Summary Artinian_module Like Noetherian modules, Artinian modules enjoy the following heredity property: If M is an Artinian R-module, then so is any submodule and any quotient of M.The converse also holds: If M is any R-module and N any Artinian submodule such that M/N is Artinian, then M is Artinian.A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artinian ring Summary Simple_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 fir... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artinian ring Summary Simple_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 |
Artinian ring Summary Simple_Artinian_ring This implies that a simple ring is left Artinian if and only if it is right Artinian. The same definition and terminology can be applied to modules, with ideals replaced by submodules. Although the descending chain condition appears dual to the ascending chain condition, in ri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artinian ring Summary Simple_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 |
Associate (ring theory) Summary Integral_domains 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 divisibili... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Associate (ring theory) Summary Integral_domains "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 mul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Norm (abelian group) Summary 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: Posit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
First ring isomorphism theorem Summary Noether_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 group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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