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Vertex algebra Summary Vertex_algebras Physically, the vertex operators arising from holomorphic field insertions at points in two-dimensional conformal field theory admit operator product expansions when insertions collide, and these satisfy precisely the relations specified in the definition of vertex operator algebr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Vexillary involution Summary Vexillary_involution In mathematics, a vexillary permutation is a permutation μ of the positive integers containing no subpermutation isomorphic to the permutation (2143); in other words, there do not exist four numbers i < j < k < l with μ(j) < μ(i) < μ(l) < μ(k). They were introduced by L... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Volume element Summary Differential_volume_element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form d V = ρ ( u 1 , u 2 , u 3 ) d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Volume element Summary Differential_volume_element The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riemannian volume form Summary Volume_form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M {\displaystyle M} of dimension n {\displaystyle n} , a volume form is an n {\displaystyle n} -form. It is an element of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riemannian volume form Summary Volume_form An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riemannian volume form Summary Volume_form In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a twisted volume form or pseudo-volume form. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riemannian volume form Summary Volume_form It also defines a measure, but exists on any differentiable manifold, orientable or not. Kähler manifolds, being complex manifolds, are naturally oriented, and so possess a volume form. More generally, the n {\displaystyle n} th exterior power of the symplectic form on a sympl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Von Neumann algebras Summary Commutative_von_Neumann_algebras In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were original... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Von Neumann algebras Summary Commutative_von_Neumann_algebras Two basic examples of von Neumann algebras are as follows: The ring L ∞ ( R ) {\displaystyle L^{\infty }(\mathbb {R} )} of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication op... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Von Neumann algebras Summary Commutative_von_Neumann_algebras Introductory accounts of von Neumann algebras are given in the online notes of Jones (2003) and Wassermann (1991) and the books by Dixmier (1981), Schwartz (1967), Blackadar (2005) and Sakai (1971). The three volume work by Takesaki (1979) gives an encyclope... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Von Neumann regular element Summary Von_Neumann_regular_ring In mathematics, a von Neumann regular ring is a ring R (associative, with 1, not necessarily commutative) such that for every element a in R there exists an x in R with a = axa. One may think of x as a "weak inverse" of the element a; in general x is not uniq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Von Neumann regular element Summary Von_Neumann_regular_ring Von Neumann regular rings should not be confused with the unrelated regular rings and regular local rings of commutative algebra. An element a of a ring is called a von Neumann regular element if there exists an x such that a = axa. An ideal i {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Wavelet transforms Summary Wavelet_compression In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wave... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weak Hausdorff space Summary Weak_Hausdorff In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. In particular, every Hausdorff space is weak Hausdorff. As a separation property, it is str... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weak Hausdorff space Summary Weak_Hausdorff Specifically, every weak Hausdorff space is a T1 space.The notion was introduced by M. C. McCord to remedy an inconvenience of working with the category of Hausdorff spaces. It is often used in tandem with compactly generated spaces in algebraic topology. For that, see the ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie algebra bundle Summary Lie_algebra_bundle In mathematics, a weak Lie algebra bundle ξ = ( ξ , p , X , θ ) {\displaystyle \xi =(\xi ,p,X,\theta )\,} is a vector bundle ξ {\displaystyle \xi \,} over a base space X together with a morphism θ: ξ ⊗ ξ → ξ {\displaystyle \theta :\xi \otimes \xi \rightarrow \xi } which ind... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie algebra bundle Summary Lie_algebra_bundle Let denote the Lie bracket of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} and deform it by the real parameter as: x = x ⋅ {\displaystyle _{x}=x\cdot } for X , Y ∈ s o ( 3 ) {\displaystyle X,Y\in {\mathfrak {so}}(3)} and x ∈ R {\displaystyle x\in \mathbb {R} } . Lie's t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie algebra bundle Summary Lie_algebra_bundle In general globally the total space might fail to be Hausdorff. But if all fibres of a real Lie algebra bundle over a topological space are mutually isomorphic as Lie algebras, then it is a locally trivial Lie algebra bundle. This result was proved by proving that the real ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie algebra bundle Summary Lie_algebra_bundle Suppose the base space is Hausdorff and fibers of total space are isomorphic as Lie algebras then there exists a Hausdorff Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle. Every semi simple Lie algebra bundle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harmonic Maass form Summary 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 e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weak derivative Summary 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 par... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weak equivalence (homotopy theory) Summary Weak_equivalence_(homotopy_theory) In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A model category is a category wi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized solution Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized solution Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weak trace-class operator Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weakly compact cardinal Summary 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 original... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weakly holomorphic modular form Summary Weakly_holomorphic_modular_form In mathematics, a weakly holomorphic modular form is similar to a holomorphic modular form, except that it is allowed to have poles at cusps. Examples include modular functions and modular forms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weakly symmetric space Summary Weakly_symmetric_space In mathematics, a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg in the 1950s as a generalisation of symmetric space, due to Élie Cartan. Geometrically the spaces are defined as complete Riemannian manifolds such that any t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Web (differential geometry) Summary 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 |
Weighing matrix Summary Weighing_matrices 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 {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weighted Voronoi diagram Summary 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 m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weighted Voronoi diagram Summary Weighted_Voronoi_diagram The idea is that larger weights indicate more important sites, and such sites will get bigger Voronoi cells. In a multiplicatively weighted Voronoi diagram, the distance between a point and a site is divided by the (positive) weight of the site. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weighted Voronoi diagram Summary 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 ha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weighted Voronoi diagram Summary Weighted_Voronoi_diagram Since crystals may grow in empty space only and are continuous objects, a natural variation is the crystal Voronoi diagram, in which the cells are defined somewhat differently. In an additively weighted Voronoi diagram, weights are subtracted from the distances.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-defined expression Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-defined expression Summary 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}}} ,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-ordering property Summary 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-ord... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-ordering property Summary Well-ordering_property Every element s of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see § ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-ordering property Summary 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 theore... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-posed problem (numerical analysis) Summary Well-posed_problem 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 inc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-posed problem (numerical analysis) Summary Well-posed_problem 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 obta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-posed problem (numerical analysis) Summary Well-posed_problem Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers. Problems in nonlinear complex systems (so-called chaotic systems) provide well-known exampl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-posed problem (numerical analysis) Summary Well-posed_problem 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Well-posed problem (numerical analysis) Summary Well-posed_problem 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zero of a function Summary Root_of_a_polynomial 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zero of a function Summary Root_of_a_polynomial 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} -... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zero vector Summary Zero_ideal 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 |
Zero-dimensional space Summary Nildimensional_space In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical illustration of a nildi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zeta function Summary Zeta_functions 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zonal polynomial Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zonal spherical function Summary 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 repr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zonal spherical function Summary 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 contin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zonal spherical function Summary 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 f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gamma-object Summary Gamma-object In mathematics, a Γ-object of a pointed category C is a contravariant functor from Γ to C. The basic example is Segal's so-called Γ-space, which may be thought of as a generalization of simplicial abelian group (or simplicial abelian monoid). More precisely, one can define a Gamma spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Delta set Summary Delta_set In mathematics, a Δ-set S, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Γ-space Summary Γ-space In mathematics, a γ {\displaystyle \gamma } -space is a topological space that satisfies a certain a basic selection principle. An infinite cover of a topological space is an ω {\displaystyle \omega } -cover if every finite subset of this space is contained in some member of the cover, and the w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pi-system Summary Pi-system In mathematics, a π-system (or pi-system) on a set Ω {\displaystyle \Omega } is a collection P {\displaystyle P} of certain subsets of Ω , {\displaystyle \Omega ,} such that P {\displaystyle P} is non-empty. If A , B ∈ P {\displaystyle A,B\in P} then A ∩ B ∈ P . {\displaystyle A\cap B\in P.}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pi-system Summary 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-algebr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pi-system Summary Pi-system π-systems are also useful for checking independence of random variables. This is desirable because in practice, π-systems are often simpler to work with than 𝜎-algebras. For example, it may be awkward to work with 𝜎-algebras generated by infinitely many sets σ ( E 1 , E 2 , … ) . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pi-system Summary 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generalized abstract nonsense Summary General_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 f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abuse of notation Summary Abuse_of_notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors and confusion at the same time)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abuse of notation Summary Abuse_of_notation A related concept is abuse of language or abuse of terminology, where a term — rather than a notation — is misused. Abuse of language is an almost synonymous expression for abuses that are non-notational by nature. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abuse of notation Summary 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 d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left associative operator Nonassociativity of floating point calculation Associative_law > Non-associative operation > Nonassociativity of floating point calculation In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, the addition and multiplication of floating ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Additive K-theory Summary 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 an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Admissible representation Summary Smooth_representation In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Affine geometry Summary 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Affine geometry Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Affine geometry Summary 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 uniq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
L-theory Summary L-theory In mathematics, algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory, also known as "Hermitian K-theory", is important in surgery theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic cobordism Summary Algebraic_cobordism In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi-projective schemes over a field. It was introduced by Marc Levine and Fabien Morel (2001, 2001b). An oriented cohomology theory on the category of smooth quasi-projective schemes Sm o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic cobordism Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lefschetz principle Summary 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 analyti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic space Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pure injective module Summary Algebraically_compact_module In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Linearly disjoint Summary Linearly_disjoint In mathematics, algebras A, B over a field k inside some field extension Ω {\displaystyle \Omega } of k are said to be linearly disjoint over k if the following equivalent conditions are met: (i) The map A ⊗ k B → A B {\displaystyle A\otimes _{k}B\to AB} induced by ( x , y ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Linearly disjoint Summary 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 {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasimodular form Summary 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 almos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Almost ring theory Summary Almost_ring In mathematics, almost modules and almost rings are certain objects interpolating between rings and their fields of fractions. They were introduced by Gerd Faltings (1988) in his study of p-adic Hodge theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Wiener amalgam space Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Wiener amalgam space Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abel equation of the first kind Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Picard stack Summary Abelian_2-group 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, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Adams operation Summary 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 im... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Albert algebra Summary Exceptional_Jordan_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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Albert algebra Summary Exceptional_Jordan_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 grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Albert algebra Summary Exceptional_Jordan_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–... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alexander matrix Summary 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 |
Apollonian gasket Summary Apollonian_gasket In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Appell polynomials Summary Appell_sequence In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence { p n ( x ) } n = 0 , 1 , 2 , … {\displaystyle \{p_{n}(x)\}_{n=0,1,2,\ldots }} satisfying the identity d d x p n ( x ) = n p n − 1 ( x ) , {\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arf ring Summary Arf_ring In mathematics, an Arf ring was defined by Lipman (1971) to be a 1-dimensional commutative semi-local Macaulay ring satisfying some extra conditions studied by Cahit Arf (1948). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artin L-function Summary Artin_root_number 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 parti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Artin–Schreier curve Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Azumaya algebra Summary 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
E n-ring Summary En_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
EP matrix Summary 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 al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eells–Kuiper manifold Summary 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 Nicol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eichler order Summary 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 |
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