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wiki_5300_chunk_0 | Normed lattice | In mathematics, specifically in order theory and functional analysis, a normed lattice is a topological vector lattice that is also a normed space whose unit ball is a solid set. Normed lattices are important in the theory of topological vector lattices. They are closely related to Banach vector lattices, which are nor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5301_chunk_0 | Order summable | In mathematics, specifically in order theory and functional analysis, a sequence of positive elements ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} in a preordered vector space X {\displaystyle X} (that is, x i ≥ 0 {\displaystyle x_{i}\geq 0} for all i {\displaystyle i} ) is called order summable i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5302_chunk_0 | Solid set | In mathematics, specifically in order theory and functional analysis, a subset S {\displaystyle S} of a vector lattice is said to be solid and is called an ideal if for all s ∈ S {\displaystyle s\in S} and x ∈ X , {\displaystyle x\in X,} if | x | ≤ | s | {\displaystyle |x|\leq |s|} then x ∈ S . {\displaystyle x\in S.} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5303_chunk_0 | Weak order unit | In mathematics, specifically in order theory and functional analysis, an element x {\displaystyle x} of a vector lattice X {\displaystyle X} is called a weak order unit in X {\displaystyle X} if x ≥ 0 {\displaystyle x\geq 0} and also for all y ∈ X , {\displaystyle y\in X,} inf { x , | y | } = 0 implies y = 0. {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5304_chunk_0 | Quasi-interior point | In mathematics, specifically in order theory and functional analysis, an element x {\displaystyle x} of an ordered topological vector space X {\displaystyle X} is called a quasi-interior point of the positive cone C {\displaystyle C} of X {\displaystyle X} if x ≥ 0 {\displaystyle x\geq 0} and if the order interval := ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5305_chunk_0 | Cone-saturated | In mathematics, specifically in order theory and functional analysis, if C {\displaystyle C} is a cone at 0 in a vector space X {\displaystyle X} such that 0 ∈ C , {\displaystyle 0\in C,} then a subset S ⊆ X {\displaystyle S\subseteq X} is said to be C {\displaystyle C} -saturated if S = C , {\displaystyle S=_{C},} wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5306_chunk_0 | Cone-saturated | If T {\displaystyle {\mathcal {T}}} is a collection of subsets of X {\displaystyle X} and if F {\displaystyle {\mathcal {F}}} is a subset of T {\displaystyle {\mathcal {T}}} then F {\displaystyle {\mathcal {F}}} is a fundamental subfamily of T {\displaystyle {\mathcal {T}}} if every T ∈ T {\displaystyle T\in {\mathcal ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5307_chunk_0 | Normal cone (functional analysis) | In mathematics, specifically in order theory and functional analysis, if C {\displaystyle C} is a cone at the origin in a topological vector space X {\displaystyle X} such that 0 ∈ C {\displaystyle 0\in C} and if U {\displaystyle {\mathcal {U}}} is the neighborhood filter at the origin, then C {\displaystyle C} is call... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5308_chunk_0 | Order bound dual | In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space X {\displaystyle X} is the set of all linear functionals on X {\displaystyle X} that map order intervals, which are sets of the form := { x ∈ X: a ≤ x and x ≤ b } , {\displaystyle :=\{x\in X:a\leq x{\t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5309_chunk_0 | Order dual (functional analysis) | In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space X {\displaystyle X} is the set Pos ( X ∗ ) − Pos ( X ∗ ) {\displaystyle \operatorname {Pos} \left(X^{*}\right)-\operatorname {Pos} \left(X^{*}\right)} where Pos ( X ∗ ) {\displaystyle \operatorname {Pos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5310_chunk_0 | Order topology (functional analysis) | In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} is the finest locally convex topological vector space (TVS) topology on X {\displaystyle X} for which every order interval is bounded, where an order interval in X {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5311_chunk_0 | Order topology (functional analysis) | This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of ( X , ≤ ) . {\displaystyle (X,\leq ).} For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5312_chunk_0 | Lattice disjoint | In mathematics, specifically in order theory and functional analysis, two elements x and y of a vector lattice X are lattice disjoint or simply disjoint if inf { | x | , | y | } = 0 {\displaystyle \inf \left\{|x|,|y|\right\}=0} , in which case we write x ⊥ y {\displaystyle x\perp y} , where the absolute value of x is d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5313_chunk_0 | Archimedean ordered vector space | In mathematics, specifically in order theory, a binary relation ≤ {\displaystyle \,\leq \,} on a vector space X {\displaystyle X} over the real or complex numbers is called Archimedean if for all x ∈ X , {\displaystyle x\in X,} whenever there exists some y ∈ X {\displaystyle y\in X} such that n x ≤ y {\displaystyle nx\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5314_chunk_0 | Bolzano-Weierstrass theorem | In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The theorem states that each infinite bounded sequence in R n {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5315_chunk_0 | Borel subalgebra | In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra g {\displaystyle {\mathfrak {g}}} is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra g {\displaystyle {\mathfrak {g}}} is the Lie algebra of a complex Lie group, then a Borel subalgebra i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5316_chunk_0 | Isotypic decomposition | In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematica... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5317_chunk_0 | Frobenius formula | In mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations of the symmetric group Sn. Among the other applications, the formula can be used to derive the hook length formula. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5318_chunk_0 | Nilpotent algebra | In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5319_chunk_0 | Hamiltonian action | In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5320_chunk_0 | Symplectic cut | In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues two manifolds together into one. The symplectic cut can also be viewed a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5321_chunk_0 | Symplectic sum | In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a submanifold, often called a fiber sum. The symplectic sum is the inverse of the sy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5322_chunk_0 | Gromov-Witten theory | In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5323_chunk_0 | Gromov-Witten theory | They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5324_chunk_0 | Quantum cohomology ring | In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5325_chunk_0 | Quantum cohomology ring | Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has important implications for enumerative geometry. It also connects to many ideas in mathematical physics and mirror symmetry. In particular, it is ring-isomorphic to symplectic Floer homology. Throughout this article, X is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5326_chunk_0 | Stable map | In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5327_chunk_0 | Hilbert's arithmetic of ends | In mathematics, specifically in the area of hyperbolic geometry, Hilbert's arithmetic of ends is a method for endowing a geometric set, the set of ideal points or "ends" of a hyperbolic plane, with an algebraic structure as a field. It was introduced by German mathematician David Hilbert. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5328_chunk_0 | Fundamental lemma of calculus of variations | In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5329_chunk_0 | Cartesian monoidal category | In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any car... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5330_chunk_0 | Sylow p-subgroup | The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G {\displaystyle G} the order (number of elements) of every subgroup of G {\displaystyle G} divides the order of G {\displaystyle G} . The Sylow theorems state that for every prime factor p {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5331_chunk_0 | Kummer's transformation of series | In mathematics, specifically in the field of numerical analysis, Kummer's transformation of series is a method used to accelerate the convergence of an infinite series. The method was first suggested by Ernst Kummer in 1837. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5332_chunk_0 | Positive semi-orbit | In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system ev... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5333_chunk_0 | Cauchy–Peano theorem | In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5334_chunk_0 | Nakano vanishing theorem | In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira vanishing theorem. Given a compact complex manifold M with a holomorphic line bundle F over M, the Nakano vanishing theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5335_chunk_0 | Chapman–Kolmogorov equation | In mathematics, specifically in the theory of Markovian stochastic processes in probability theory, the Chapman–Kolmogorov equation(CKE) is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. The equation was derived independently by both the British mathem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5336_chunk_0 | Limit of a distribution | In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance, suitably quantified, to the limiting distribution can be made arbitrarily small by selecting a distribution sufficiently far along the sequence. This ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5337_chunk_0 | Pseudoholomorphic curve | In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5338_chunk_0 | Siegel's lemma | In mathematics, specifically in transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials was proven by Axel Thue; Thue's proof used Dirichlet's box principle. C... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5339_chunk_0 | Relative dimension | In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient map V → Q {\displaystyle V\to Q} , the difference dim V − dim Q is the relative dimension; this equals the dimension of the kernel. In fiber bundles, the relative dimension... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5340_chunk_0 | Nondegenerate bilinear form | In mathematics, specifically linear algebra, a degenerate bilinear form f (x, y ) on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V ) given by v ↦ (x ↦ f (x, v )) is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5341_chunk_0 | Copositive matrix | In mathematics, specifically linear algebra, a real matrix A is copositive if x T A x ≥ 0 {\displaystyle x^{T}Ax\geq 0} for every nonnegative vector x ≥ 0 {\displaystyle x\geq 0} . The collection of all copositive matrices is a proper cone; it includes as a subset the collection of real positive-definite matrices. Copo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5342_chunk_0 | Cauchy-Binet formula | In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5343_chunk_0 | Counting measure | In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ∞ {\displaystyle \infty } if the subset is infinite.The counting measure c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5344_chunk_0 | Dyadics | In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5345_chunk_0 | Dyadics | The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A dyadic can be used to contain physical or geometric information, although in general there is no direct way of geometrically interpreting it. The dyadic product is distributive over vector addition, and associat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5346_chunk_0 | Dyadics | The formalism of dyadic algebra is an extension of vector algebra to include the dyadic product of vectors. The dyadic product is also associative with the dot and cross products with other vectors, which allows the dot, cross, and dyadic products to be combined to obtain other scalars, vectors, or dyadics. It also has... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5347_chunk_0 | Dyadics | Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5348_chunk_0 | Dyadics | The notation and terminology are relatively obsolete today. Its uses in physics include continuum mechanics and electromagnetism. In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5349_chunk_0 | Granville number | In mathematics, specifically number theory, Granville numbers, also known as S {\displaystyle {\mathcal {S}}} -perfect numbers, are an extension of the perfect numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5350_chunk_0 | Geometric configuration | In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.Although certain specific configurations had been stud... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5351_chunk_0 | Kirszbraun theorem | In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H1, and H2 is another Hilbert space, and f: U → H 2 {\displaystyle f:U\rightarrow H_{2}} is a Lipschitz-continuous map, then there is a Lipschitz-continuous map F: H 1 → H 2 {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5352_chunk_0 | Kirszbraun theorem | It is for instance possible to construct counterexamples where the domain is a subset of R n {\displaystyle \mathbb {R} ^{n}} with the maximum norm and R m {\displaystyle \mathbb {R} ^{m}} carries the Euclidean norm. More generally, the theorem fails for R m {\displaystyle \mathbb {R} ^{m}} equipped with any ℓ p {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5353_chunk_0 | Coxeter functor | In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra. Tilting theory was motivate... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5354_chunk_0 | Stationary set | In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5355_chunk_0 | Cumulative hierarchy | In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W α {\displaystyle W_{\alpha }} indexed by ordinals α {\displaystyle \alpha } such that W α ⊆ W α + 1 {\displaystyle W_{\alpha }\subseteq W_{\alpha +1}} If λ {\displaystyle \lambda } is a limit ordinal, then W λ = ⋃ α < λ W α {\textstyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5356_chunk_0 | Dimensional operator | In mathematics, specifically set theory, a dimensional operator on a set E is a function from the subsets of E to the subsets of E. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5357_chunk_0 | Cylinder (algebra) | In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. In terms of set-builder notation, that is A × B = { ( a , b ) ∣ a ∈ A and b ∈ B } . {\displaystyle A\times B=\{(a,b)\mid a\in A\ {\mbox{ and}}\ b\in B\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5358_chunk_0 | Cylinder (algebra) | If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5359_chunk_0 | Aleph hypothesis | In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that there is no set whose cardinality is strictly between that of the integers and the real numbers, or equivalently, that any subset of the real numbers is finite, is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5360_chunk_0 | Bregman distance | In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted as probability distributions – notably as... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5361_chunk_0 | Bregman distance | However, they satisfy a generalization of the Pythagorean theorem, and in information geometry the corresponding statistical manifold is interpreted as a (dually) flat manifold. This allows many techniques of optimization theory to be generalized to Bregman divergences, geometrically as generalizations of least squares... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5362_chunk_0 | Normal basis | In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5363_chunk_0 | Davenport–Schmidt theorem | In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irra... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5364_chunk_0 | Cyclic cubic field | In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5365_chunk_0 | Bergman's diamond lemma | In mathematics, specifically the field of abstract algebra, Bergman's Diamond Lemma (after George Bergman) is a method for confirming whether a given set of monomials of an algebra forms a k {\displaystyle k} -basis. It is an extension of Gröbner bases to non-commutative rings. The proof of the lemma gives rise to an a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5366_chunk_0 | Sklyanin algebra | In mathematics, specifically the field of algebra, Sklyanin algebras are a class of noncommutative algebra named after Evgeny Sklyanin. This class of algebras was first studied in the classification of Artin-Schelter regular algebras of global dimension 3 in the 1980s. Sklyanin algebras can be grouped into two differen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5367_chunk_0 | Minkowski space (number field) | In mathematics, specifically the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field.If K is a number field of degree d then there are d distinct embeddings of K into C. We let KC be the image of K in the product Cd, considered as equipped with the usual He... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5368_chunk_0 | Four exponentials conjecture | In mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials. The conjecture, along with two related, stronger conjectures, is at the top... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5369_chunk_0 | Cauchy-Lipschitz theorem | In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5370_chunk_0 | Lie's theorem | In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if π: g → g l ( V ) {\displaystyle \pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)} is a finite-dimensional representation of a solvable Lie algebra, then there's a flag V = V 0 ⊃ V... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5371_chunk_0 | Schanuel's conjecture | In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5372_chunk_0 | Six exponentials theorem | In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a set of exponentials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5373_chunk_0 | 2-functor | In mathematics, specifically, in category theory, a 2-functor is a morphism between 2-categories. They may be defined formally using enrichment by saying that a 2-category is exactly a Cat-enriched category and a 2-functor is a Cat-functor.Explicitly, if C and D are 2-categories then a 2-functor F: C → D {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5374_chunk_0 | Graph spectrum | In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5375_chunk_0 | Spectral theory | In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5376_chunk_0 | Spin geometry | In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics. An important generalisation is the theory of symplectic D... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5377_chunk_0 | Stability theory | In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small va... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5378_chunk_0 | Stable homotopy | In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space X... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5379_chunk_0 | Stable homotopy | The first example is a standard corollary of the Hurewicz theorem, that π n ( S n ) ≅ Z {\displaystyle \pi _{n}(S^{n})\cong \mathbb {Z} } . In the second example the Hopf map, η {\displaystyle \eta } , is mapped to its suspension Σ η {\displaystyle \Sigma \eta } , which generates π 4 ( S 3 ) ≅ Z / 2 {\displaystyle \pi ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5380_chunk_0 | Stable homotopy | According to Freudenthal's theorem, in the stable range the homotopy groups of spheres depend not on the specific dimensions of the spheres in the domain and target, but on the difference in those dimensions. With this in mind the k-th stable stem is π k s := lim n π n + k ( S n ) {\displaystyle \pi _{k}^{s}:=\lim _{n}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5381_chunk_0 | Stable homotopy | A theorem of Goro Nishida states that all elements of positive grading in this ring are nilpotent. Thus the only prime ideals are the primes in π 0 s ≅ Z {\displaystyle \pi _{0}^{s}\cong \mathbb {Z} } . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5382_chunk_0 | Grey box model | In mathematics, statistics, and computational modelling, a grey box model combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information on the smoothness of results, to models that need only parameter values from data or existing literature. Thus, almost a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5383_chunk_0 | Grey box model | These have special analysis methods. In particular linear regression techniques are much more efficient than most non-linear techniques. The model can be deterministic or stochastic (i.e. containing random components) depending on its planned use. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5384_chunk_0 | Regularization (machine learning) | In mathematics, statistics, finance, computer science, particularly in machine learning and inverse problems, regularization is a process that changes the result answer to be "simpler". It is often used to obtain results for ill-posed problems or to prevent overfitting.Although regularization procedures can be divided ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5385_chunk_0 | Regularization (machine learning) | The regularization term, or penalty, imposes a cost on the optimization function to make the optimal solution unique. Implicit regularization is all other forms of regularization. This includes, for example, early stopping, using a robust loss function, and discarding outliers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5386_chunk_0 | Regularization (machine learning) | Implicit regularization is essentially ubiquitous in modern machine learning approaches, including stochastic gradient descent for training deep neural networks, and ensemble methods (such as random forests and gradient boosted trees).In explicit regularization, independent of the problem or model, there is always a da... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5387_chunk_0 | Regularization (machine learning) | There is a whole research branch dealing with all possible regularizations. In practice, one usually tries a specific regularization and then figures out the probability density that corresponds to that regularization to justify the choice. It can also be physically motivated by common sense or intuition. In machine le... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5388_chunk_0 | Stochastic analysis on manifolds | In mathematics, stochastic analysis on manifolds or stochastic differential geometry is the study of stochastic analysis over smooth manifolds. It is therefore a synthesis of stochastic analysis and differential geometry. The connection between analysis and stochastic processes stems from the fundamental relation that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5389_chunk_0 | Stochastic analysis on manifolds | However, its stochastic behaviour determines global aspects of the topology and geometry of the manifold. Brownian motion is defined to be the diffusion process generated by the Laplace-Beltrami operator 1 2 Δ M {\displaystyle {\tfrac {1}{2}}\Delta _{M}} with respect to a manifold M {\displaystyle M} and can be constru... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5390_chunk_0 | Stochastic analysis on manifolds | However, this obstacle can be overcome if the manifold is equipped with a connection: We can then introduce the stochastic horizontal lift of a semimartingale and the stochastic development by the so-called Eells-Elworthy-Malliavin construction.The latter is a generalisation of a horizontal lift of smooth curves to hor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5391_chunk_0 | Stochastic analysis on manifolds | For example, one can apply Brownian motion to the Dirichlet problem at infinity for Cartan-Hadamard manifolds or give a probabilistic proof of the Atiyah-Singer index theorem. Stochastic differential geometry also applies in other areas of mathematics (e.g. mathematical finance). For example, we can convert classical a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5392_chunk_0 | Stochastic geometry | In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5393_chunk_0 | Structural stability | Variants of this notion apply to systems of ordinary differential equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms. Structurally stable systems were introduced by Aleksandr Andronov and Lev Pontryagin in 1937 under the name "systèmes grossiers", or rough systems. They announ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5394_chunk_0 | Structural stability | In this case, structurally stable systems are typical, they form an open dense set in the space of all systems endowed with appropriate topology. In higher dimensions, this is no longer true, indicating that typical dynamics can be very complex (cf. strange attractor). An important class of structurally stable systems ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5395_chunk_0 | Subadditive function | In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5396_chunk_0 | Subharmonic function | In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a lin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5397_chunk_0 | Sigma notation | In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5398_chunk_0 | Sigma notation | Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very often, the elements of a sequence are defined, through a regular pattern, as a function of their place in the sequence. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5399_chunk_0 | Sigma notation | For example, the sum of the first n natural numbers can be denoted as ∑ i = 1 n i . {\textstyle \sum _{i=1}^{n}i.} For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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