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Noncommutative harmonic analysis Summary Non-commutative_harmonic_analysis (For Pontryagin duality the Plancherel measure is some Haar measure on the dual group to G, the only issue therefore being its normalization.) For general locally compact groups, or even countable discrete groups, the von Neumann group algebra n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Noncommutative projective geometry Summary Non-commutative_projective_geometry In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Noncommutative residue Summary Noncommutative_residue In mathematics, noncommutative residue, defined independently by M. Wodzicki (1984) and Guillemin (1985), is a certain trace on the algebra of pseudodifferential operators on a compact differentiable manifold that is expressed via a local density. In the case of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Noncommutative topology Summary Non-commutative_topology In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts. The term has its origins in the Gelfand–Naimark theorem, which implies the duality of the category of locally compact Hausdorff spaces and t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nonlinear model Summary Nonlinear_model In mathematics, nonlinear modelling is empirical or semi-empirical modelling which takes at least some nonlinearities into account. Nonlinear modelling in practice therefore means modelling of phenomena in which independent variables affecting the system can show complex and syne... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nonlinear model Summary Nonlinear_model Thus the nonlinear modelling can utilize production data or experimental results while taking into account complex nonlinear behaviours of modelled phenomena which are in most cases practically impossible to be modelled by means of traditional mathematical approaches, such as phe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Non-linear programming Summary Non-Linear_Optimization In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nonstandard calculus Summary Nonstandard_calculus In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic. Non-r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nonstandard calculus Summary Nonstandard_calculus (See history of calculus.) For almost one hundred years thereafter, mathematicians such as Richard Courant viewed infinitesimals as being naive and vague or meaningless.Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nuclear map Summary Nuclear_operator In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nuclear operators between Banach spaces Summary Nuclear_operators_between_Banach_spaces In mathematics, nuclear operators between Banach spaces are a linear operators between Banach spaces in infinite dimensions that share some of the properties of their counter-part in finite dimension. In Hilbert spaces such operator... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nuclear spaces Summary Nuclear_space In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nuclear spaces Summary Nuclear_space Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold. All finite-dimensional vector spaces are nuclear. There are no Banach spaces that are nuclear, except for the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Domain decomposition Summary Domain_decomposition_method In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Domain decomposition Summary Domain_decomposition_method In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the Schwarz alternating method and the additive Schwarz method. Many domain decomposition methods can be written and a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Domain decomposition Summary Domain_decomposition_method In non-overlapping methods, the subdomains intersect only on their interface. In primal methods, such as Balancing domain decomposition and BDDC, the continuity of the solution across subdomain interface is enforced by representing the value of the solution on al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Domain decomposition Summary Domain_decomposition_method The FETI-DP method is hybrid between a dual and a primal method. Non-overlapping domain decomposition methods are also called iterative substructuring methods. Mortar methods are discretization methods for partial differential equations, which use separate discre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Domain decomposition Summary Domain_decomposition_method The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution. In the engineering practice in the finite element method, continuity of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Obstruction theory Summary Obstruction_theory In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants. In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of li... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scale invariance Scale-invariant curves and self-similarity Scaling_invariance > Scale-invariant curves and self-similarity In mathematics, one can consider the scaling properties of a function or curve f (x) under rescalings of the variable x. That is, one is interested in the shape of f (λx) for some scale factor λ, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Scale invariance Scale-invariant curves and self-similarity Scaling_invariance > Scale-invariant curves and self-similarity In polar coordinates (r, θ), the spiral can be written as θ = 1 b ln ( r / a ) . {\displaystyle \theta ={\frac {1}{b}}\ln(r/a)~.} Allowing for rotations of the curve, it is invariant under all r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Product of group subsets Summary Product_of_group_subsets In mathematics, one can define a product of group subsets in a natural way. If S and T are subsets of a group G, then their product is the subset of G defined by S T = { s t: s ∈ S and t ∈ T } . {\displaystyle ST=\{st:s\in S{\text{ and }}t\in T\}.} The subsets S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Product of group subsets Summary Product_of_group_subsets The associativity of this product follows from that of the group product. The product of group subsets therefore defines a natural monoid structure on the power set of G. A lot more can be said in the case where S and T are subgroups. The product of two subgroup... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Directly indecomposable Summary Internal_direct_product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Continuous embedding Summary Continuous_embedding In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Operator K-theory Summary Operator_K-theory In mathematics, operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Operator Theory Summary Operator_Theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and conside... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Omega and agemo subgroup Summary Omega_and_agemo_subgroup In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group. They were introduced in (Hall 1933) where they were used to describe a class of finite p-groups whose structure was su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riesz projector Summary Riesz_projector In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ehresmann's lemma Summary Ehresmann's_fibration_theorem In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that if a smooth mapping f: M → N {\displaystyle f\colon M\rightarrow N} , where M {\displaystyle M} and N {\displaystyle N} are smooth manifolds, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orbit capacity Summary Orbit_capacity In mathematics, orbit capacity of a subset of a topological dynamical system may be thought of heuristically as a “topological dynamical probability measure” of the subset. More precisely, its value for a set is a tight upper bound for the normalized number of visits of orbits in t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordinal logic Summary Ordinal_logic In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics. The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems.While Gödel sh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orientation reversing Summary Orientable_surface In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is orientable if such a consiste... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orientation reversing Summary Orientable_surface A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orientation reversing Summary Orientable_surface Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more struct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal coordinate system Summary Orthogonal_coordinate In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots ,q^{d})} in which the coordinate hypersurfaces all meet at right angles (note that superscripts are indices, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal functions Summary Orthogonal_system In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: ⟨ f , g ⟩ = ∫ f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal functions Summary Orthogonal_system As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-produc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal polynomials on the unit circle Summary Orthogonal_polynomials_on_the_unit_circle In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. Th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonality (mathematics) Summary Orthogonality_(mathematics) In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal subspace Summary Orthogonality_(quantum_mechanics) In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity. Orthogonality is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gell-Mann matrices Trace orthonormality Gell-Mann_matrices > Properties > Trace orthonormality In mathematics, orthonormality typically implies a norm which has a value of unity (1). Gell-Mann matrices, however, are normalized to a value of 2. Thus, the trace of the pairwise product results in the ortho-normalization c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gell-Mann matrices Trace orthonormality Gell-Mann_matrices > Properties > Trace orthonormality In this three-dimensional matrix representation, the Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices λ 3 {\displaystyle \lambda _{3}} and λ 8 {\displaystyle \lambda _{8}} , whi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gell-Mann matrices Trace orthonormality Gell-Mann_matrices > Properties > Trace orthonormality The SU(2) Casimirs of these subalgebras mutually commute. However, any unitary similarity transformation of these subalgebras will yield SU(2) subalgebras. There is an uncountable number of such transformations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Overconvergent modular form Summary Overconvergent_modular_form In mathematics, overconvergent modular forms are special p-adic modular forms that are elements of certain p-adic Banach spaces (usually infinite dimensional) containing classical spaces of modular forms as subspaces. They were introduced by Nicholas M. Ka... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-adic Hodge theory Summary P-adic_Hodge_theory In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as Qp). The theory has its beginnings in Jean-Pierre Serre and John Tate's study o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-adic Teichmuller theory Summary P-adic_Teichmüller_theory In mathematics, p-adic Teichmüller theory describes the "uniformization" of p-adic curves and their moduli, generalizing the usual Teichmüller theory that describes the uniformization of Riemann surfaces and their moduli. It was introduced and developed by Shi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-adic analysis Summary P-adic_analysis In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers. The theory of complex-valued numerical functions on the p-adic numbers is part of the theory of locally compact groups. The usual meaning taken f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-adic analysis Summary P-adic_analysis Some applications have required the development of p-adic functional analysis and spectral theory. In many ways p-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of p-adic numbers is mu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-adic cohomology Summary P-adic_cohomology In mathematics, p-adic cohomology means a cohomology theory for varieties of characteristic p whose values are modules over a ring of p-adic integers. Examples (in roughly historical order) include: Serre's Witt vector cohomology Monsky–Washnitzer cohomology Infinitesimal coh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parabolic cylindrical coordinates Summary Parabolic_cylindrical_coordinates In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular z {\displaystyle z} -direction. Hence, the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Philosophy of cusp forms Summary Philosophy_of_cusp_forms In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups. If G is a reductive algebraic group and P = M A N {\displaystyle P=MAN} is the Langlands decomposition of a para... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Even number Summary Odd_numbers In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not. For example, −4, 0, 82 are even because By contrast, −3, 5, 7, 21 are odd numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Even number Summary Odd_numbers The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Even number Summary Odd_numbers In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Even number Summary Odd_numbers That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8. The same idea will work using any even base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1; and it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stable range condition Summary Stable_range_condition In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring R {\displaystyle R} is the smallest integer n {\displaystyle n} such that whenever v 0 , v 1 , . . . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stable range condition Summary Stable_range_condition , v n {\displaystyle v_{0},v_{1},...,v_{n}} in R {\displaystyle R} generate the unit ideal (they form a unimodular row), there exist some t 1 , . . . , t n {\displaystyle t_{1},...,t_{n}} in R {\displaystyle R} such that the elements v i − v 0 t i {\displaystyle v_{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Flexible identity Summary Flexible_identity In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: a ∙ ( b ∙ a ) = ( a ∙ b ) ∙ a {\displaystyle a\bullet \left(b\bullet a\right)=\left(a\bullet b\right)\bullet a} for any two elements a and b of the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Separably closed field Summary Algebraic_closure In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma or the weaker ultrafilter lemma, it can be shown that every field h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mayer–Vietoris sequence Summary Mayer-Vietoris_sequence In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mayer–Vietoris sequence Summary Mayer-Vietoris_sequence It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces. The Mayer–Vietoris sequence holds for a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mayer–Vietoris sequence Summary Mayer-Vietoris_sequence Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the Mayer–Vietoris sequence in the hope of obtaining partial information. Many spaces encountered in topology are constructed by piecing together ver... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cohomotopy group Summary Cohomotopy_set In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and basepoint-preserving continuous maps to the category of sets and functions. They are dual to the homotopy groups, but less st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kan-Thurston theorem Summary Kan-Thurston_theorem In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group G {\displaystyle G} to every path-connected topological space X {\displaystyle X} in such a way that the group cohomology of G {\displaystyle G} is the same as the coho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Vertical tangent Summary Vertical_tangent In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universal element Summary Universal_element In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invertible module Summary Invertible_module In mathematics, particularly commutative algebra, an invertible module is intuitively a module that has an inverse with respect to the tensor product. Invertible modules form the foundation for the definition of invertible sheaves in algebraic geometry. Formally, a finitely g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Berlekamp's algorithm Summary Berlekamp's_algorithm In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was inven... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Finsler geometry Summary Finsler_manifold In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski functional F(x, −) is provided on each tangent space TxM, that enables one to define the length of any smooth curve γ: → M as L ( γ ) =... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double tangent bundle Summary Double_tangent_bundle In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM,M) of a smooth manifold M . A note on notation: in this article, w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Secondary vector bundle structure Summary Secondary_vector_bundle_structure In mathematics, particularly differential topology, the secondary vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle (E, p, M), induced by the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
James' theorem Summary James's_theorem In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space X {\displaystyle X} is reflexive if and only if every continuous linear functional's norm on X {\displaystyle X} attains its supremum on the closed unit ball in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Topology of uniform convergence Summary Space_of_linear_maps In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The article operator... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dunford–Schwartz theorem Summary Dunford–Schwartz_theorem In mathematics, particularly functional analysis, the Dunford–Schwartz theorem, named after Nelson Dunford and Jacob T. Schwartz, states that the averages of powers of certain norm-bounded operators on L1 converge in a suitable sense. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unit distance graph Summary Unit_distance_graph In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graphs from a broader definitio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unit distance graph Summary Unit_distance_graph The generalized Petersen graphs are non-strict unit distance graphs. An unsolved problem of Paul Erdős asks how many edges a unit distance graph on n {\displaystyle n} vertices can have. The best known lower bound is slightly above linear in n {\displaystyle n} —far from ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unit distance graph Summary Unit_distance_graph The number of colors required to color unit distance graphs is also unknown (the Hadwiger–Nelson problem): some unit distance graphs require five colors, and every unit distance graph can be colored with seven colors. For every algebraic number there is a unit distance gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unit distance graph Summary Unit_distance_graph It is possible to construct a unit distance graph efficiently, given its points. Finding all unit distances has applications in pattern matching, where it can be a first step in finding congruent copies of larger patterns. However, determining whether a given graph can be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Directed Acyclic Graph Summary Directed_acyclic_graphs In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called arcs), with each edge directed from one vertex to another, such that... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left exact functor Summary Left_exact_functor In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Zig-zag lemma Summary Zig-zag_lemma In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stably finite ring Summary Stably_finite_ring In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1. This is a stronger property for a ring than having the invariant basis nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free semigroup with involution Summary Monoid_with_involution In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary opera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free semigroup with involution Summary Monoid_with_involution An example from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free semigroup with involution Summary Monoid_with_involution Another example, coming from formal language theory, is the free semigroup generated by a nonempty set (an alphabet), with string concatenation as the binary operation, and the involution being the map which reverses the linear order of the letters in a stri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Adjunction (field theory) Summary Field_extension In mathematics, particularly in algebra, a field extension is a pair of fields K ⊆ L , {\displaystyle K\subseteq L,} such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indeterminate equations Summary Indeterminate_equation In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation a x + b y = c {\displaystyle ax+by=c} is a simple indeterminate equation, as is x 2 = 1 {\displaystyle x^{2}=1} . ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indeterminate equations Summary Indeterminate_equation In fact, in some cases it might even have infinitely many solutions. Some of the prominent examples of indeterminate equations include: Univariate polynomial equation: a n x n + a n − 1 x n − 1 + ⋯ + a 2 x 2 + a 1 x + a 0 = 0 , {\displaystyle a_{n}x^{n}+a_{n-1}x^{n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indeterminate equations Summary Indeterminate_equation Pell's equation: x 2 − P y 2 = 1 , {\displaystyle \ x^{2}-Py^{2}=1,} where P {\displaystyle P} is a given integer that is not a square number, and in which the variables x {\displaystyle x} and y {\displaystyle y} are required to be integers. The equation of Pythag... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indeterminate system Summary Indeterminate_system In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions). In the case of a linear system, the system may be said to be underspec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Locally free module Summary Projective_module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Injective envelope Summary Injective_hull In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in (Eckmann & Schopf 1953). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal polarization Summary Abelian_Variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal polarization Summary Abelian_Variety Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal polarization Summary Abelian_Variety Localization techniques lead naturally from abelian varieties defined over number fields to ones defined over finite fields and various local fields. Since a number field is the fraction field of a Dedekind domain, for any nonzero prime of your Dedekind domain, there is a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal polarization Summary Abelian_Variety Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field. Abelian varieties appear naturally as ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Alexander–Spanier cohomology Summary Alexander-Spanier_cohomology In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tautness (topology) Summary Tautness_(topology) In mathematics, particularly in algebraic topology, a taut pair is a topological pair whose direct limit of cohomology module of open neighborhood of that pair which is directed downward by inclusion is isomorphic to the cohomology module of original pair. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Skeleton (topology) Summary N-skeleton In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions m ≤ n. In other words, given an induc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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