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wiki_4400_chunk_0 | Z-group | In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups: in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. in the study of infinite groups, a Z-group is a group which possesses a very general ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4401_chunk_0 | Z-group | Such groups are elementarily equivalent to the integers ( Z , + , < ) {\displaystyle (\mathbb {Z} ,+,<)} . Z-groups are an alternative presentation of Presburger arithmetic. occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4402_chunk_0 | Schanuel's lemma | In mathematics, especially in the area of algebra known as module theory, Schanuel's lemma, named after Stephen Schanuel, allows one to compare how far modules depart from being projective. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of dimension shifting. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4403_chunk_0 | Representation ring | In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representations of the group. Elements of the representation ring are sometimes call... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4404_chunk_0 | Ore polynomial | In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore polynomials. Ore extensions appear in several natural contexts, including s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4405_chunk_0 | Ore ring | In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a mult... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4406_chunk_0 | Pure subgroup | In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4407_chunk_0 | Essential subgroup | In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group. The concept was generalized to essential submodules. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4408_chunk_0 | Shapiro's lemma | In mathematics, especially in the areas of abstract algebra dealing with group cohomology or relative homological algebra, Shapiro's lemma, also known as the Eckmann–Shapiro lemma, relates extensions of modules over one ring to extensions over another, especially the group ring of a group and of a subgroup. It thus rel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4409_chunk_0 | Subdirect product | In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product. The notion was introduced by Bir... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4410_chunk_0 | Compact stencil | In mathematics, especially in the areas of numerical analysis called numerical partial differential equations, a compact stencil is a type of stencil that uses only nine nodes for its discretization method in two dimensions. It uses only the center node and the adjacent nodes. For any structured grid utilizing a compac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4411_chunk_0 | Polynomial expression | In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4412_chunk_0 | Polynomial expression | Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have bee... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4413_chunk_0 | Connected ring | In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions: A possesses no non-trivial (that is, not equal to 1 or 0) idempotent elements; the spectrum of A with the Zariski topology is a connected space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4414_chunk_0 | Pure subring | In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those module... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4415_chunk_0 | Schur functor | In mathematics, especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative ring to itself. They generalize the constructions of exterior powers and symmetric powers of a vector space. Schur functors are indexed by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4416_chunk_0 | Irreducible ring | A ring with an irreducible spectrum is a ring whose spectrum is irreducible as a topological space. "Meet-irreducible" rings are referred to as "irreducible rings" in commutative algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4417_chunk_0 | Irreducible ring | This article adopts the term "meet-irreducible" in order to distinguish between the several types being discussed. Meet-irreducible rings play an important part in commutative algebra, and directly irreducible and subdirectly irreducible rings play a role in the general theory of structure for rings. Subdirectly irredu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4418_chunk_0 | Lyndon–Hochschild–Serre spectral sequence | In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the total group G. The spect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4419_chunk_0 | Upper central series | In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4420_chunk_0 | Class function | In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant on the conjugacy classes of G. In other words, it is invariant under the conjugation map on G. Such functions play a basic role in representation theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4421_chunk_0 | Frobenius ring | In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Bra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4422_chunk_0 | Graph algebra | In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by McNulty and Shallon, and has seen many uses in the field of universal algebra since then. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4423_chunk_0 | Projective linear group | In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4424_chunk_0 | Projective linear group | Here SZ is the center of SL, and is naturally identified with the group of nth roots of unity in F (where n is the dimension of V and F is the base field). PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called projective linear transformation, p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4425_chunk_0 | Projective linear group | If V is the n-dimensional vector space over a field F, namely V = Fn, the alternate notations PGL(n, F) and PSL(n, F) are also used. Note that PGL(n, F) and PSL(n, F) are isomorphic if and only if every element of F has an nth root in F. As an example, note that PGL(2, C) = PSL(2, C), but that PGL(2, R) > PSL(2, R); th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4426_chunk_0 | Stable manifold theorem | In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly states that the existence of a local diffeomorphism near a fixed point implies the e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4427_chunk_0 | Limit set | In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. A sys... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4428_chunk_0 | Kuranishi theory | In mathematics, especially in topology, a Kuranishi structure is a smooth analogue of scheme structure. If a topological space is endowed with a Kuranishi structure, then locally it can be identified with the zero set of a smooth map ( f 1 , … , f k ): R n + k → R k {\displaystyle (f_{1},\ldots ,f_{k})\colon \mathbb {R... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4429_chunk_0 | Quasipositive matrix | In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are non-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrix A which satisfies A = ( a i j ) ; a i j ≥ 0 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4430_chunk_0 | M-matrix | In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4431_chunk_0 | Exchange matrix | In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrices, where the 1 elements reside on the antidiagonal and all other elements are zero. In other words, they are 'row-reversed' or... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4432_chunk_0 | Set function | In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line R ∪ { ± ∞ } , {\displaystyle \mathbb {R} \cup \{\pm \infty \},} which consists of the real numbers R {\displaystyle \mathbb {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4433_chunk_0 | Paranormal operator | In mathematics, especially operator theory, a paranormal operator is a generalization of a normal operator. More precisely, a bounded linear operator T on a complex Hilbert space H is said to be paranormal if: ‖ T 2 x ‖ ≥ ‖ T x ‖ 2 {\displaystyle \|T^{2}x\|\geq \|Tx\|^{2}} for every unit vector x in H. The class of par... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4434_chunk_0 | Weak order | In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4435_chunk_0 | Harmonic measure | In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space R n {\displaystyle R^{n}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4436_chunk_0 | Flat function | In mathematics, especially real analysis, a flat function is a smooth function f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } all of whose derivatives vanish at a given point x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} } . The flat functions are, in some sense, the antitheses of the analytic functions. An anal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4437_chunk_0 | Flat function | {\displaystyle f(x)\sim \lim _{n\to \infty }\sum _{k=0}^{n}{\frac {f^{(k)}(x_{0})}{k!}}(x-x_{0})^{k}.} In the case of a flat function, all derivatives vanish at x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} } , i.e. f ( k ) ( x 0 ) = 0 {\displaystyle f^{(k)}(x_{0})=0} for all k ∈ N {\displaystyle k\in \mathbb {N} } . Thi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4438_chunk_0 | Flat function | In the language of Taylor's theorem, the non-constant part of the function always lies in the remainder R n ( x ) {\displaystyle R_{n}(x)} for all n ∈ N {\displaystyle n\in \mathbb {N} } . The function need not be flat at just one point. Trivially, constant functions on R {\displaystyle \mathbb {R} } are flat everywher... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4439_chunk_0 | Frobenius characteristic map | In mathematics, especially representation theory and combinatorics, a Frobenius characteristic map is an isometric isomorphism between the ring of characters of symmetric groups and the ring of symmetric functions. It builds a bridge between representation theory of the symmetric groups and algebraic combinatorics. Thi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4440_chunk_0 | Regular ideal | In mathematics, especially ring theory, a regular ideal can refer to multiple concepts. In operator theory, a right ideal i {\displaystyle {\mathfrak {i}}} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that e x − x ∈ i {\displaystyle ex-x\in {\mathfrak {i}}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4441_chunk_0 | Behnke–Stein theorem on Stein manifolds | In mathematics, especially several complex variables, the Behnke–Stein theorem states that a connected, non-compact (open) Riemann surface is a Stein manifold. In other words, it states that there is a nonconstant single-valued holomorphic function (univalent function) on such a Riemann surface. It is a generalization ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4442_chunk_0 | Behnke–Stein theorem | In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence G k ⊂ C n {\displaystyle G_{k}\subset \mathbb {C} ^{n}} (i.e., G k ⊂ G k + 1 {\displaystyle G_{k}\subset G_{k+1}} ) of domains of holomorphy is again a domain of holomorphy. It was proved by Hein... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4443_chunk_0 | Weyl's law | In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the d = 2 , 3 {\displaystyle d=2,3} case) by Hermann Weyl for eigenvalues for the Laplace–Beltrami operator acting on functions that vanis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4444_chunk_0 | Parker vector | In mathematics, especially the field of group theory, the Parker vector is an integer vector that describes a permutation group in terms of the cycle structure of its elements. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4445_chunk_0 | Oka–Weil theorem | In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4446_chunk_0 | Einstein summation notation | In mathematics, especially the usage of linear algebra in mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4447_chunk_0 | Perfect map | In mathematics, especially topology, a perfect map is a particular kind of continuous function between topological spaces. Perfect maps are weaker than homeomorphisms, but strong enough to preserve some topological properties such as local compactness that are not always preserved by continuous maps. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4448_chunk_0 | Closed differential form | In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4449_chunk_0 | Essential dimension | In mathematics, essential dimension is an invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by J. Buhler and Z. Reichstein and in its most generality defined by A. Merkurjev.Basically, essential dimension measures the complexity of algebraic structures vi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4450_chunk_0 | Even ordinal | In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite induction proofs. The literature contains a few equivalent definitions of the parity of an ordinal α: Every limit ordinal (including 0) is even. The successor of an eve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4451_chunk_0 | Even ordinal | Let α = λ + n, where λ is a limit ordinal and n is a natural number. The parity of α is the parity of n. Let n be the finite term of the Cantor normal form of α. The parity of α is the parity of n. Let α = ωβ + n, where n is a natural number. The parity of α is the parity of n. If α = 2β, then α is even. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4452_chunk_0 | Even ordinal | Otherwise α = 2β + 1 and α is odd.Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number (ω + 3)2 = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4453_chunk_0 | Even ordinal | Given an infinite cardinal κ, or generally any limit ordinal κ, κ is order-isomorphic to both its subset of even ordinals and its subset of odd ordinals. Hence one has the cardinal sum κ + κ = κ. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4454_chunk_0 | Even function | In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4455_chunk_0 | Analytic function of a matrix | In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4456_chunk_0 | Exponential polynomial | In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4457_chunk_0 | Laws of exponents | In mathematics, exponentiation is an operation involving two numbers, the base and the exponent or power. Exponentiation is written as bn, where b is the base and n is the power; this is pronounced as "b (raised) to the (power of) n". When n is a positive integer, exponentiation corresponds to repeated multiplication o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4458_chunk_0 | Laws of exponents | Starting from the basic fact stated above that, for any positive integer n {\displaystyle n} , b n {\displaystyle b^{n}} is n {\displaystyle n} occurrences of b {\displaystyle b} all multiplied by each other, several other properties of exponentiation directly follow. In particular: In other words, when multiplying a b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4459_chunk_0 | Laws of exponents | Consider the question of what b − 1 {\displaystyle b^{-1}} should mean. In order to respect the "exponents add" rule, it must be the case that b − 1 × b 1 = b − 1 + 1 = b 0 = 1 {\displaystyle b^{-1}\times b^{1}=b^{-1+1}=b^{0}=1} . Dividing both sides by b 1 {\displaystyle b^{1}} gives b − 1 = 1 / b 1 {\displaystyle b^{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4460_chunk_0 | Laws of exponents | Therefore, r = 1 2 {\displaystyle r={\frac {1}{2}}} , so b = b 1 / 2 {\displaystyle {\sqrt {b}}=b^{1/2}} . The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices. Expon... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4461_chunk_0 | Extrapolation method | In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4462_chunk_0 | Factorization | In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4463_chunk_0 | Factorization | Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4464_chunk_0 | Factorization | In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4465_chunk_0 | Factorization | In this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see factoriza... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4466_chunk_0 | Factorization | Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, ev... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4467_chunk_0 | Field Arithmetic | In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4468_chunk_0 | Finite field arithmetic | In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinitely many different finite fields. Their number of elements is necessarily o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4469_chunk_0 | Finite-dimensional distribution | In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or finite collection of times). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4470_chunk_0 | Flat convergence | In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Federer and Fleming in 1960. It forms a fundamental part of the field of geometric measure theory. The notion was applied to find... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4471_chunk_0 | Hermitian matrix | In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient R ( M , x ) , {\displaystyle R(M,\mathbf {x} ),} is defined as:: p. 234 For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x H {\displaystyle \m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4472_chunk_0 | Hermitian matrix | {\displaystyle c.} Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value λ min {\displaystyle \lambda _{\min }} (the smallest eigenvalue of M) when x {\displaystyle \mathbf {x} } is v min {\displaystyle \m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4473_chunk_0 | Hermitian matrix | Similarly, R ( M , x ) ≤ λ max {\displaystyle R(M,\mathbf {x} )\leq \lambda _{\max }} and R ( M , v max ) = λ max . {\displaystyle R(M,\mathbf {v} _{\max })=\lambda _{\max }.} The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4474_chunk_0 | Hermitian matrix | It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration. The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional anal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4475_chunk_0 | Hermitian matrix | When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, λ max {\displaystyle \lambda _{\max }} is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh quotient R(M, x) for a fixed... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4476_chunk_0 | Infinite product | In mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product ∏ n = 1 ∞ a n = a 1 a 2 a 3 ⋯ {\displaystyle \prod _{n=1}^{\infty }a_{n}=a_{1}a_{2}a_{3}\cdots } is defined to be the limit of the partial products a1a2...an as n increases without bound. The product is said to converge when the limi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4477_chunk_0 | Infinite product | Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence an as n increases without bound must be 1, while the converse is in general ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4478_chunk_0 | Null model | In mathematics, for example in the study of statistical properties of graphs, a null model is a type of random object that matches one specific object in some of its features, or more generally satisfies a collection of constraints, but which is otherwise taken to be an unbiasedly random structure. The null model is us... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4479_chunk_0 | Discrete Logarithm | In mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that bk = a. In number theory, the more commonly used term is index: we can write x = indr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4480_chunk_0 | Formal moduli | In mathematics, formal moduli are an aspect of the theory of moduli spaces (of algebraic varieties or vector bundles, for example), closely linked to deformation theory and formal geometry. Roughly speaking, deformation theory can provide the Taylor polynomial level of information about deformations, while formal modul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4481_chunk_0 | Function application | In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abstraction. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4482_chunk_0 | Function composition | In mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f: X → Y and g: Y → Z are composed to yield a function t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4483_chunk_0 | Fuzzy set theory | In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4484_chunk_0 | Fuzzy set theory | At the same time, Salii (1965) defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics (De Cock, Bodenhofer & Kerre 2000), decision-making (Kuzmin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4485_chunk_0 | Fuzzy set theory | By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval . Fuzzy sets generalize classical sets, since the indicator functions (aka characteristic functions) of classical sets are special... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4486_chunk_0 | Generalised mean | In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4487_chunk_0 | Geometric calculus | In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including vector calculus, differential geometry, and differential forms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4488_chunk_0 | Semistable point | In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper (Hilbert 1893) in classical invariant theory. Geometric invariant theory studies an ac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4489_chunk_0 | Geometric measure theory | In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4490_chunk_0 | Geometry and topology | In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Wei... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4491_chunk_0 | Exact cover | In the notation above, P is the set X, Q is a collection S of subsets of X, R is the binary relation "is contained in" between elements and subsets, and R−1 restricted to Q × P* is the function "contains" from subsets to selected elements. Whereas an exact cover problem involves selecting subsets and the relation "cont... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4492_chunk_0 | Rank factorization | In mathematics, given a field F {\displaystyle \mathbb {F} } , nonnegative integers m , n {\displaystyle m,n} , and a matrix A ∈ F m × n {\displaystyle A\in \mathbb {F} ^{m\times n}} , a rank decomposition or rank factorization of A is a factorization of A of the form A = CF, where C ∈ F m × r {\displaystyle C\in \math... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4493_chunk_0 | Lebesgue point | In mathematics, given a locally Lebesgue integrable function f {\displaystyle f} on R k {\displaystyle \mathbb {R} ^{k}} , a point x {\displaystyle x} in the domain of f {\displaystyle f} is a Lebesgue point if lim r → 0 + 1 λ ( B ( x , r ) ) ∫ B ( x , r ) | f ( y ) − f ( x ) | d y = 0. {\displaystyle \lim _{r\rightarr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4494_chunk_0 | Smallest bounding sphere | In mathematics, given a non-empty set of objects of finite extension in d {\displaystyle d} -dimensional space, for example a set of points, a bounding sphere, enclosing sphere or enclosing ball for that set is an d {\displaystyle d} -dimensional solid sphere containing all of these objects. Used in computer graphics a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4495_chunk_0 | Semi-invariant of a quiver | In mathematics, given a quiver Q with set of vertices Q0 and set of arrows Q1, a representation of Q assigns a vector space Vi to each vertex and a linear map V(α): V(s(α)) → V(t(α)) to each arrow α, where s(α), t(α) are, respectively, the starting and the ending vertices of α. Given an element d ∈ N {\displaystyle \ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4496_chunk_0 | Tangential and normal components | In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector. Similarly, a vector at a point on a surfac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4497_chunk_0 | Null cone | In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0. In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4498_chunk_0 | Linearized line bundle | In mathematics, given an action σ: G × S X → X {\displaystyle \sigma :G\times _{S}X\to X} of a group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules together with the isomorphism of O G × S X {\displaystyle {\mathcal {O}}_{G\times _{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4499_chunk_0 | Novikov ring | In mathematics, given an additive subgroup Γ ⊂ R {\displaystyle \Gamma \subset \mathbb {R} } , the Novikov ring Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} of Γ {\displaystyle \Gamma } is the subring of Z ] {\displaystyle \mathbb {Z} \!]} consisting of formal sums ∑ n γ i t γ i {\displaystyle \sum n_{\ga... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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