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wiki_3600_chunk_0 | Harmonic Maass form | In mathematics, a weak Maass form is a smooth function f {\displaystyle f} on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps. If the eigenvalu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3601_chunk_0 | Weak derivative | In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space L 1 ( ) {\displaystyle L^{1}()} . The method of integration by parts holds that for differentiable functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3602_chunk_0 | Generalized solution | In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3603_chunk_0 | Generalized solution | Somewhat surprisingly, a differential equation may have solutions which are not differentiable; and the weak formulation allows one to find such solutions. Weak solutions are important because many differential equations encountered in modelling real-world phenomena do not admit of sufficiently smooth solutions, and th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3604_chunk_0 | Weak trace-class operator | In mathematics, a weak trace class operator is a compact operator on a separable Hilbert space H with singular values the same order as the harmonic sequence. When the dimension of H is infinite, the ideal of weak trace-class operators is strictly larger than the ideal of trace class operators, and has fundamentally di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3605_chunk_0 | Weakly compact cardinal | In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.) Form... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3606_chunk_0 | Web (differential geometry) | In mathematics, a web permits an intrinsic characterization in terms of Riemannian geometry of the additive separation of variables in the Hamilton–Jacobi equation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3607_chunk_0 | Weighing matrix | In mathematics, a weighing matrix of order n {\displaystyle n} and weight w {\displaystyle w} is a matrix W {\displaystyle W} with entries from the set { 0 , 1 , − 1 } {\displaystyle \{0,1,-1\}} such that: W W T = w I n {\displaystyle WW^{\mathsf {T}}=wI_{n}} Where W T {\displaystyle W^{\mathsf {T}}} is the transpose o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3608_chunk_0 | Weighted Voronoi diagram | In mathematics, a weighted Voronoi diagram in n dimensions is a generalization of a Voronoi diagram. The Voronoi cells in a weighted Voronoi diagram are defined in terms of a distance function. The distance function may specify the usual Euclidean distance, or may be some other, special distance function. In weighted V... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3609_chunk_0 | Weighted Voronoi diagram | In the plane under the ordinary Euclidean distance, the multiplicatively weighted Voronoi diagram is also called circular Dirichlet tessellation and its edges are circular arcs and straight line segments. A Voronoi cell may be non-convex, disconnected and may have holes. This diagram arises, e.g., as a model of crystal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3610_chunk_0 | Well-defined expression | In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill defined or ambiguous. A function is well defined if it gives the same result when the representation of the inp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3611_chunk_0 | Well-defined expression | The term well defined can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well defined is not the same as a function that is undefined. For example, if f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} , then even though f ( 0 ) {\displaystyle f(0)} is undefi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3612_chunk_0 | Well-ordering property | In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set. In some academic articles and textbooks thes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3613_chunk_0 | Well-ordering property | The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible. Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3614_chunk_0 | Well-posed problem (numerical analysis) | In mathematics, a well-posed problem is one for which the following properties hold: The problem has a solution The solution is unique The solution's behavior changes continuously with the initial conditionsExamples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3615_chunk_0 | Well-posed problem (numerical analysis) | For example, the inverse heat equation, deducing a previous distribution of temperature from final data, is not well-posed in that the solution is highly sensitive to changes in the final data. Continuum models must often be discretized in order to obtain a numerical solution. While solutions may be continuous with res... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3616_chunk_0 | Well-posed problem (numerical analysis) | If the problem is well-posed, then it stands a good chance of solution on a computer using a stable algorithm. If it is not well-posed, it needs to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3617_chunk_0 | Well-posed problem (numerical analysis) | This process is known as regularization. Tikhonov regularization is one of the most commonly used for regularization of linear ill-posed problems. The definition of a well-posed problem comes from the work of Jacques Hadamard on mathematical modeling of physical phenomena. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3618_chunk_0 | Zero of a function | In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle x} of the domain of f {\displaystyle f} such that f ( x ) {\displaystyle f(x)} vanishes at x {\displaystyle x} ; that is, the function f {\displaystyle f} at... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3619_chunk_0 | Zero of a function | For example, the polynomial f {\displaystyle f} of degree two, defined by f ( x ) = x 2 − 5 x + 6 {\displaystyle f(x)=x^{2}-5x+6} has the two roots (or zeros) that are 2 and 3. If the function maps real numbers to real numbers, then its zeros are the x {\displaystyle x} -coordinates of the points where its graph meets ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3620_chunk_0 | Zero vector | In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3621_chunk_0 | Zeta function | In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.} Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa–Kaneko zeta fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3622_chunk_0 | Zonal polynomial | In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials. They appear as zonal spherical functions of the Gelfand pairs ( S 2 n , H n ) {\displaystyle (S_{2n},H_{n})} (here, H n {\displaystyle H_{n}} is the hyperoctah... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3623_chunk_0 | Zonal spherical function | In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3624_chunk_0 | Zonal spherical function | The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series. Zonal spherical ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3625_chunk_0 | Zonal spherical function | For complex groups, the theory simplifies significantly, because G is the complexification of K, and the formulas are related to analytic continuations of the Weyl character formula on K. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement. Apart from their group t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3626_chunk_0 | Pi-system | The importance of π-systems arises from the fact that if two probability measures agree on a π-system, then they agree on the 𝜎-algebra generated by that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they hold for the generated 𝜎-algebra as well. This is the case ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3627_chunk_0 | Pi-system | {\displaystyle \sigma (E_{1},E_{2},\ldots ).} So instead we may examine the union of all 𝜎-algebras generated by finitely many sets ⋃ n σ ( E 1 , … , E n ) . {\textstyle \bigcup _{n}\sigma (E_{1},\ldots ,E_{n}).} This forms a π-system that generates the desired 𝜎-algebra. Another example is the collection of all inte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3628_chunk_0 | Generalized abstract nonsense | In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are nonderogatory terms used by mathematicians to describe long, theoretical parts of a proof they skip over when readers are expected to be familiar with them. These terms are mainly used for abstract meth... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3629_chunk_0 | Abuse of notation | For example, while the word representation properly designates a group homomorphism from a group G to GL(V), where V is a vector space, it is common to call V "a representation of G". Another common abuse of language consists in identifying two mathematical objects that are different, but canonically isomorphic. Other ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3630_chunk_0 | Left associative operator | In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, the addition and multiplication of floating point numbers is not associative, as rounding errors are introduced when dissimilar-sized values are joined together.To illustrate this, consider a floating point rep... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3631_chunk_0 | Additive K-theory | In mathematics, additive K-theory means some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group GL has everywhere been replaced by its Lie algebra gl. It is not, therefore, one theory but a way of creating additive or infinitesimal analogues of multiplicative theories. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3632_chunk_0 | Affine geometry | In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of paral... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3633_chunk_0 | Affine geometry | Comparisons of figures in affine geometry are made with affine transformations, which are mappings that preserve alignment of points and parallelism of lines. Affine geometry can be developed in two ways that are essentially equivalent.In synthetic geometry, an affine space is a set of points to which is associated a s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3634_chunk_0 | Affine geometry | In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3635_chunk_0 | Algebraic cobordism | In particular they are "oriented", which means roughly that they behave well on vector bundles; this is closely related to the condition that a generalized cohomology theory has a complex orientation. Over a field of characteristic 0, algebraic cobordism is the universal oriented cohomology theory for smooth varieties.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3636_chunk_0 | Lefschetz principle | In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3637_chunk_0 | Algebraic space | In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3638_chunk_0 | Linearly disjoint | However, there are examples where A ⊗ k B {\displaystyle A\otimes _{k}B} is a domain but A and B are not linearly disjoint: for example, A = B = k(t), the field of rational functions over k. One also has: A, B are linearly disjoint over k if and only if subfields of Ω {\displaystyle \Omega } generated by A , B {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3639_chunk_0 | Quasimodular form | In mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms that are polynomials in 1/Im(τ) with coefficients that are holomorphic functions of τ. A quasimodular form is the holomorphic part of an almost holomorphic modular form. An almost holomo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3640_chunk_0 | Wiener amalgam space | In mathematics, amalgam spaces categorize functions with regard to their local and global behavior. While the concept of function spaces treating local and global behavior separately was already known earlier, Wiener amalgams, as the term is used today, were introduced by Hans Georg Feichtinger in 1980. The concept is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3641_chunk_0 | Wiener amalgam space | Then the Wiener amalgam space with local component X {\displaystyle X} and global component L m p {\displaystyle L_{m}^{p}} , a weighted L p {\displaystyle L^{p}} space with non-negative weight m {\displaystyle m} , is defined by W ( X , L p ) = { f: ( ∫ R d ‖ f ( ⋅ ) g ¯ ( ⋅ − x ) ‖ X p m ( x ) p d x ) 1 / p < ∞ } , {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3642_chunk_0 | Abel equation of the first kind | In mathematics, an Abel equation of the first kind, named after Niels Henrik Abel, is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the form y ′ = f 3 ( x ) y 3 + f 2 ( x ) y 2 + f 1 ( x ) y + f 0 ( x ) {\displaystyle y'=f_{3}(x)y^{3}+f_{2}(x)y^{2}+f_{1}(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3643_chunk_0 | Picard stack | In mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups. More concretely, they are given by groupoids A {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3644_chunk_0 | Adams operation | In mathematics, an Adams operation, denoted ψk for natural numbers k, is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. The basic idea is to implement some fundamental identities in s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3645_chunk_0 | Albert algebra | In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism. One of them, which was ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3646_chunk_0 | Albert algebra | Another is defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution. Over any algebraically closed field, there is just one Albert algebra, and its automorphism group G is the simple split group of type F4. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3647_chunk_0 | Albert algebra | (For example, the complexifications of the three Albert algebras over the real numbers are isomorphic Albert algebras over the complex numbers.) Because of this, for a general field F, the Albert algebras are classified by the Galois cohomology group H1(F,G).The Kantor–Koecher–Tits construction applied to an Albert alg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3648_chunk_0 | Alexander matrix | In mathematics, an Alexander matrix is a presentation matrix for the Alexander invariant of a knot. The determinant of an Alexander matrix is the Alexander polynomial for the knot. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3649_chunk_0 | Artin L-function | In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3650_chunk_0 | Artin–Schreier curve | In mathematics, an Artin–Schreier curve is a plane curve defined over an algebraically closed field of characteristic p {\displaystyle p} by an equation y p − y = f ( x ) {\displaystyle y^{p}-y=f(x)} for some rational function f {\displaystyle f} over that field. One of the most important examples of such curves is hyp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3651_chunk_0 | Azumaya algebra | In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3652_chunk_0 | E n-ring | In mathematics, an E n {\displaystyle {\mathcal {E}}_{n}} -algebra in a symmetric monoidal infinity category C consists of the following data: An object A ( U ) {\displaystyle A(U)} for any open subset U of Rn homeomorphic to an n-disk. A multiplication map: μ: A ( U 1 ) ⊗ ⋯ ⊗ A ( U m ) → A ( V ) {\displaystyle \mu :A(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3653_chunk_0 | EP matrix | In mathematics, an EP matrix (or range-Hermitian matrix or RPN matrix) is a square matrix A whose range is equal to the range of its conjugate transpose A*. Another equivalent characterization of EP matrices is that the range of A is orthogonal to the nullspace of A. Thus, EP matrices are also known as RPN (Range Perpe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3654_chunk_0 | Eells–Kuiper manifold | In mathematics, an Eells–Kuiper manifold is a compactification of R n {\displaystyle \mathbb {R} ^{n}} by a sphere of dimension n / 2 {\displaystyle n/2} , where n = 2 , 4 , 8 {\displaystyle n=2,4,8} , or 16 {\displaystyle 16} . It is named after James Eells and Nicolaas Kuiper. If n = 2 {\displaystyle n=2} , the Eells... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3655_chunk_0 | Eichler order | In mathematics, an Eichler order, named after Martin Eichler, is an order of a quaternion algebra that is the intersection of two maximal orders. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3656_chunk_0 | Engel subalgebra | In mathematics, an Engel subalgebra of a Lie algebra with respect to some element x is the subalgebra of elements annihilated by some power of ad x. Engel subalgebras are named after Friedrich Engel. For finite-dimensional Lie algebras over infinite fields the minimal Engel subalgebras are the Cartan subalgebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3657_chunk_0 | Erdös cardinal | In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958). A cardinal κ is called α-Erdős if for every function f: κ< ω → {0, 1}, there is a set of order type α that is homogeneous for f . In the notation of the parti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3658_chunk_0 | Euler systems | In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by Kolyvagin (1990) in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper Kolyvagin (1988) and the work of Thaine (1988). Euler systems ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3659_chunk_0 | Cauchy–Euler equation | In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3660_chunk_0 | H-matrix (iterative method) | In mathematics, an H-matrix is a matrix whose comparison matrix is an M-matrix. It is useful in iterative methods. Definition: Let A = (aij) be a n × n complex matrix. Then comparison matrix M(A) of complex matrix A is defined as M(A) = αij where αij = −|Aij| for all i ≠ j, 1 ≤ i,j ≤ n and αij = |Aij| for all i = j, 1 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3661_chunk_0 | I-bundle | In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. An I-bundle is said to be twisted if it is not trivial. Two simple examples of I-bundles are the annulus ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3662_chunk_0 | I-bundle | The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product S 1 × I {\displaystyle S^{1}\times I} , and the Möbius band is a non-trivial or twisted bundle. Both bundles are 2-manifolds, but the annulus is an orientable manifold while the Möbius band is a non-orientable manifold. Curious... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3663_chunk_0 | I-bundle | These observations are simple well known facts on elementary 3-manifolds. Line bundles are both I-bundles and vector bundles of rank one. When considering I-bundles, one is interested mostly in their topological properties and not their possible vector properties, as one might be for line bundles. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3664_chunk_0 | Igusa zeta-function | In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3665_chunk_0 | L function | In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. The Riemann zeta function is an example... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3666_chunk_0 | L function | The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3667_chunk_0 | LB-space | In mathematics, an LB-space, also written (LB)-space, is a topological vector space X {\displaystyle X} that is a locally convex inductive limit of a countable inductive system ( X n , i n m ) {\displaystyle (X_{n},i_{nm})} of Banach spaces. This means that X {\displaystyle X} is a direct limit of a direct system ( X n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3668_chunk_0 | LB-space | This means that the topology induced on X n {\displaystyle X_{n}} by X n + 1 {\displaystyle X_{n+1}} is identical to the original topology on X n . {\displaystyle X_{n}.} Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space," so when reading mathematical literature, its recommended to always... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3669_chunk_0 | LF-space | In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system ( X n , i n m ) {\displaystyle (X_{n},i_{nm})} of Fréchet spaces. This means that X is a direct limit of a direct system ( X n , i n m ) {\displaystyle (X_... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3670_chunk_0 | LF-space | If each of the bonding maps i n m {\displaystyle i_{nm}} is an embedding of TVSs then the LF-space is called a strict LF-space. This means that the subspace topology induced on Xn by Xn+1 is identical to the original topology on Xn. Some authors (e.g. Schaefer) define the term "LF-space" to mean "strict LF-space," so w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3671_chunk_0 | O*-algebra | In mathematics, an O*-algebra is an algebra of possibly unbounded operators defined on a dense subspace of a Hilbert space. The original examples were described by Borchers (1962) and Uhlmann (1962), who studied some examples of O*-algebras, called Borchers algebras, arising from the Wightman axioms of quantum field th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3672_chunk_0 | Ockham algebras | In mathematics, an Ockham algebra is a bounded distributive lattice with a dual endomorphism, that is, an operation ~ satisfying ~(x ∧ y) = ~x ∨ ~y, ~(x ∨ y) = ~x ∧ ~y, ~0 = 1, ~1 = 0. They were introduced by Berman (1977), and were named after William of Ockham by Urquhart (1979). Ockham algebras form a variety. Examp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3673_chunk_0 | Oper (mathematics) | In mathematics, an Oper is a principal connection, or in more elementary terms a type of differential operator. They were first defined and used by Vladimir Drinfeld and Vladimir Sokolov to study how the KdV equation and related integrable PDEs correspond to algebraic structures known as Kac–Moody algebras. Their moder... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3674_chunk_0 | Rvachev function | In mathematics, an R-function, or Rvachev function, is a real-valued function whose sign does not change if none of the signs of its arguments change; that is, its sign is determined solely by the signs of its arguments.Interpreting positive values as true and negative values as false, an R-function is transformed into... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3675_chunk_0 | Homological algebra | In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative attempt to unify sever... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3676_chunk_0 | Homological algebra | The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3677_chunk_0 | Homological algebra | Abelian categories are named after Niels Henrik Abel. More concretely, a category is abelian if it has a zero object, it has all binary products and binary coproducts, and it has all kernels and cokernels. all monomorphisms and epimorphisms are normal. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3678_chunk_0 | Abelian category | Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an ab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3679_chunk_0 | Fundamental theorem of finite abelian groups | In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3680_chunk_0 | Abelian integral | In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form ∫ z 0 z R ( x , w ) d x , {\displaystyle \int _{z_{0}}^{z}R(x,w)\,dx,} where R ( x , w ) {\displaystyle R(x,w)} is an arbitrary rational function of the two variables x {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3681_chunk_0 | Abelian surface | In mathematics, an abelian surface is a 2-dimensional abelian variety. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bilinear relations. Essentially, these are conditions on the parameter space ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3682_chunk_0 | Abelian variety of CM-type | In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(A). The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century. One of the major achievements i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3683_chunk_0 | Abelian variety of CM-type | The formal definition is that End Q ( A ) {\displaystyle \operatorname {End} _{\mathbb {Q} }(A)} the tensor product of End(A) with the rational number field Q, should contain a commutative subring of dimension 2d over Q. When d = 1 this can only be a quadratic field, and one recovers the cases where End(A) is an orde... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3684_chunk_0 | Abelian variety of CM-type | To construct such varieties in the same style as for elliptic curves, starting with a lattice Λ in Cd, one must take into account the Riemann relations of abelian variety theory. The CM-type is a description of the action of a (maximal) commutative subring L of EndQ(A) on the holomorphic tangent space of A at the ident... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3685_chunk_0 | Abelian variety of CM-type | It is known that all such possible CM-types can be realised. Basic results of Goro Shimura and Yutaka Taniyama compute the Hasse–Weil L-function of A, in terms of the CM-type and a Hecke L-function with Hecke character, having infinity-type derived from it. These generalise the results of Max Deuring for the elliptic c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3686_chunk_0 | Absolute presentation of a group | In mathematics, an absolute presentation is one method of defining a group.Recall that to define a group G {\displaystyle G} by means of a presentation, one specifies a set S {\displaystyle S} of generators so that every element of the group can be written as a product of some of these generators, and a set R {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3687_chunk_0 | Absolutely integrable | In mathematics, an absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite. For a real-valued function, since where both ∫ f + ( x ) d x {\textstyle \int f^{+}(x)\,dx} and ∫ f − ( x ) d x {\textstyle \int f^{-}(x)\,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3688_chunk_0 | Absolutely integrable | Let us define where ℜ f ( x ) {\displaystyle \Re f(x)} and ℑ f ( x ) {\displaystyle \Im f(x)} are the real and imaginary parts of f ( x ) {\displaystyle f(x)} . Then so This shows that the sum of the four integrals (in the middle) is finite if and only if the integral of the absolute value is finite, and the function i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3689_chunk_0 | Absorbing set (random dynamical systems) | In mathematics, an absorbing set for a random dynamical system is a subset of the phase space. A dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. The absorbing set eventually contains the image of any bounded set under the cocycle ("flow") of the random d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3690_chunk_0 | Abstract cell complexes | In mathematics, an abstract cell complex is an abstract set with Alexandrov topology in which a non-negative integer number called dimension is assigned to each point. The complex is called “abstract” since its points, which are called “cells”, are not subsets of a Hausdorff space as is the case in Euclidean and CW com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3691_chunk_0 | Abstract differential equation | In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic abstract space (a Hilbert space, a Banach space, etc.). Equations of this kind arise e.g. in the study of partial differential equations: if to one of the variables ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3692_chunk_0 | Abstract differential equation | An exhaustive treatment of the homogeneous ( f = 0 {\displaystyle f=0} ) case with a constant operator is given by the theory of C0-semigroups. Very often, the study of other abstract differential equations amounts (by e.g. reduction to a set of equations of the first order) to the study of this equation. The theory of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3693_chunk_0 | Acyclic space | In mathematics, an acyclic space is a nonempty topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of X are isomorphic to the corresponding homology groups of a point. In other words, using the idea of reduced homology,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3694_chunk_0 | Acyclic space | The condition of acyclicity on a space X implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of X to the circle or to the higher spheres is null-homotopic. If a space X is contractible, then it is also acyclic, by the homotopy invariance of homology. The converse is not true, in gene... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3695_chunk_0 | Addition theorem | In mathematics, an addition theorem is a formula such as that for the exponential function: ex + y = ex · ey,that expresses, for a particular function f, f(x + y) in terms of f(x) and f(y). Slightly more generally, as is the case with the trigonometric functions sin and cos, several functions may be involved; this is m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3696_chunk_0 | Addition theorem | An algebraic addition theorem is one in which G can be taken to be a vector of polynomials, in some set of variables. The conclusion of the mathematicians of the time was that the theory of abelian functions essentially exhausted the interesting possibilities: considered as a functional equation to be solved with polyn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3697_chunk_0 | Addition theorem | The connected, projective variety examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an abelian variety by rather weak conditions on its group law. The so-called quasi-abelian functions are all known to come from extensions of abelian varieties by commutative affine gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3698_chunk_0 | Sigma additive | In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, μ ( A ∪ B ) = μ ( A ) + μ ( B ) . {\textstyle \mu (A\cup B)=\mu (A)+\mu (B).} If this additivity property holds for any two... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3699_chunk_0 | Sigma additive | However, a finitely additive set function might not have the additivity property for a union of an infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, μ ( ⋃ n = 1 ∞ A n ) = ∑ n = 1 ∞ μ ( A n ) . {\textstyle \mu \left(\bigcup _... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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