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Opposite algebra Summary Opposite_algebra In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring (R, +, ⋅) is the ring (R, +, ∗) whose multipl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Romanov's theorem Summary Romanov's_theorem In mathematics, specifically additive number theory, Romanov's theorem is a mathematical theorem proved by Nikolai Pavlovich Romanov. It states that given a fixed base b, the set of numbers that are the sum of a prime and a positive integer power of b has a positive lower asy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Donaldson-Thomas theory Summary Donaldson–Thomas_theory In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Donaldson-Thomas theory Summary Donaldson–Thomas_theory Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas. Donaldson–Thomas theory is physically motivated by certain BPS states that occur in string... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Exceptional curve Summary Exceptional_curve In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map f: X → Y {\displaystyle f:X\rightarrow Y} of varieties is a kind of 'large' subvariety of X {\displaystyle X} which is 'crushed' by f {\displaystyle f} , in a certain definite sense. Mor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Exceptional curve Summary Exceptional_curve A codimension-1 subvariety Z ⊂ X {\displaystyle Z\subset X} is said to be exceptional if f ( Z ) {\displaystyle f(Z)} has codimension at least 2 as a subvariety of Y {\displaystyle Y} . One may then define the exceptional divisor of f {\displaystyle f} to be ∑ i Z i ∈ D i v (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Valuative criterion Summary Valuative_criterion In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide whether a morphism of algebraic varieties, or more generally schemes, is universally closed, separated, or proper. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eilenberg–MacLane spectra Summary Eilenberg–MacLane_space In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer. A connected topological space X is called an Eilenberg–MacLane space of type K ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eilenberg–MacLane spectra Summary Eilenberg–MacLane_space Thus, one may consider K ( G , n ) {\displaystyle K(G,n)} as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a K ( G , n ) {\displaystyle K(G,n)} " or as "a model of K ( G , n ) {\displaystyle K(G,n)} ". ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eilenberg–MacLane spectra Summary Eilenberg–MacLane_space The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s. As such, an Eilenberg–MacLane space is a special kind of topological space that in homotopy theory can be regarded as a building block for CW-complexes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semilocally simply connected Summary Semi-locally_simply_connected In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space X is semi-locally simply connected if there is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cohomology ring Summary Cohomology_ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cohomology ring Summary Cohomology_ring {\displaystyle H^{k}(X;R)\times H^{\ell }(X;R)\to H^{k+\ell }(X;R).} The cup product gives a multiplication on the direct sum of the cohomology groups H ∙ ( X ; R ) = ⨁ k ∈ N H k ( X ; R ) . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cohomology ring Summary Cohomology_ring {\displaystyle H^{\bullet }(X;R)=\bigoplus _{k\in \mathbb {N} }H^{k}(X;R).} This multiplication turns H•(X;R) into a ring. In fact, it is naturally an N-graded ring with the nonnegative integer k serving as the degree. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cohomology ring Summary Cohomology_ring The cup product respects this grading. The cohomology ring is graded-commutative in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree k and ℓ; we have ( α k ⌣ β ℓ ) = ( − 1 ) k ℓ ( β ℓ ⌣ α k ) . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cohomology ring Summary Cohomology_ring {\displaystyle (\alpha ^{k}\smile \beta ^{\ell })=(-1)^{k\ell }(\beta ^{\ell }\smile \alpha ^{k}).} A numerical invariant derived from the cohomology ring is the cup-length, which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero resul... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mapping cylinder Summary Mapping_telescope In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f {\displaystyle f} between topological spaces X {\displaystyle X} and Y {\displaystyle Y} is the quotient M f = ( ( × X ) ⨿ Y ) / ∼ {\displaystyle M_{f}=((\times X)\amalg Y)\,/\,\s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mapping cylinder Summary Mapping_telescope It is common to write M f {\displaystyle Mf} for M f {\displaystyle M_{f}} , and to use the notation ⊔ f {\displaystyle \sqcup _{f}} or ∪ f {\displaystyle \cup _{f}} for the mapping cylinder construction. That is, one writes M f = ( × X ) ∪ f Y {\displaystyle Mf=(\times X)\cu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Adams resolution Summary Adams_resolution In mathematics, specifically algebraic topology, there is a resolution analogous to free resolutions of spectra yielding a tool for constructing the Adams spectral sequence. Essentially, the idea is to take a connective spectrum of finite type X {\displaystyle X} and iterativel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cocycle condition Summary Čech_cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
First Feigenbaum constant Summary First_Feigenbaum_constant In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Generator (category theory) Summary Generator_(category_theory) In mathematics, specifically category theory, a family of generators (or family of separators) of a category C {\displaystyle {\mathcal {C}}} is a collection G ⊆ O b ( C ) {\displaystyle {\mathcal {G}}\subseteq Ob({\mathcal {C}})} of objects in C {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Covariant functor Summary Covariance_and_contravariance_of_functors In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps betwee... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Covariant functor Summary Covariance_and_contravariance_of_functors Thus, functors are important in all areas within mathematics to which category theory is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Posetal category Summary Posetal_category In mathematics, specifically category theory, a posetal category, or thin category, is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class (or a preordered set, if its objects form a set). As suggested by the nam... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Posetal category Summary Posetal_category When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a codiscrete posetal category corresponds to an inconsistent theory understood as one satisfying the axiom x = y at all types. Viewing a 2-category as an enrich... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Posetal category Summary Posetal_category For example, under this assumption, a poset may be defined as a small posetal category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category, and a Boolean algebra as a small posetal ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasitopos Summary Quasitopos In mathematics, specifically category theory, a quasitopos is a generalization of a topos. A topos has a subobject classifier classifying all subobjects, but in a quasitopos, only strong subobjects are classified. Quasitoposes are also required to be finitely cocomplete and locally cartesi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subcategory Summary Full_embedding In mathematics, specifically category theory, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left adjoint Summary Right_adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left adjoint Summary Right_adjoint Naturality here means that there are natural isomorphisms between the pair of functors C ( F − , X ): D → S e t {\displaystyle {\mathcal {C}}(F-,X):{\mathcal {D}}\to \mathrm {Set} } and D ( − , G X ): D → S e t {\displaystyle {\mathcal {D}}(-,GX):{\mathcal {D}}\to \mathrm {Set} } for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Left adjoint Summary Right_adjoint We write F ⊣ G {\displaystyle F\dashv G} . An adjunction between categories C {\displaystyle {\mathcal {C}}} and D {\displaystyle {\mathcal {D}}} is somewhat akin to a "weak form" of an equivalence between C {\displaystyle {\mathcal {C}}} and D {\displaystyle {\mathcal {D}}} , and ind... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Essential monomorphism Summary Essential_monomorphism In mathematics, specifically category theory, an essential monomorphism is a monomorphism f in a category C such that for a morphism g in C, the morphism g ∘ f {\displaystyle g\circ f} is a monomorphism only when g is a monomorphism. Essential monomorphisms in a cat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Overcategory Summary Overcategory In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track of data surrounding a fixed object X {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hypergeometric function identities Summary WZ_theory In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. WZ pairs are named after Herbert S. Wilf and Doron Zeilberger, and are instrumental in the evaluation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert's basis theorem Summary Hilbert's_basis_theorem In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Divided power structure Summary Divided_power_structure In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form x n / n ! {\displaystyle x^{n}/n!} meaningful even when it is not possible to actually divide by n ! {\displaystyle n!} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Primary ideal Summary Primary_ideal In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n > 0. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime num... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Primary ideal Summary Primary_ideal Consequently, an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal value Summary Principal_value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. A simple case arises in taking the square root of a positive real number. For example, 4 has two squ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Computable ordinal Summary Constructive_ordinal In mathematics, specifically computability and set theory, an ordinal α {\displaystyle \alpha } is said to be computable or recursive if there is a computable well-ordering of a computable subset of the natural numbers having the order type α {\displaystyle \alpha } . It ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Computable ordinal Summary Constructive_ordinal The Church–Kleene ordinal is a limit ordinal. An ordinal is computable if and only if it is smaller than ω 1 C K {\displaystyle \omega _{1}^{CK}} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Computable ordinal Summary Constructive_ordinal Since there are only countably many computable relations, there are also only countably many computable ordinals. Thus, ω 1 C K {\displaystyle \omega _{1}^{CK}} is countable. The computable ordinals are exactly the ordinals that have an ordinal notation in Kleene's O {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Normal fan Summary Normal_fan In mathematics, specifically convex geometry, the normal fan of a convex polytope P is a polyhedral fan that is dual to P. Normal fans have applications to polyhedral combinatorics, linear programming, tropical geometry and other areas of mathematics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Milnor's sphere Summary Milnor's_sphere In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnorpg 14 was trying to understand the structure of ( n − 1 ) {\displaystyle (n-1)} -connected manifolds of dimension 2 n {\displaystyle 2n} (since n {\displaystyle n} -connected 2 n {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inverse function theorem Summary Inverse_function_theorem In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riemann curvature Summary Curvature_of_Riemannian_manifolds In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to defi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Virtual fundamental class Summary Virtual_fundamental_class In mathematics, specifically enumerative geometry, the virtual fundamental class E ∙ vir {\displaystyle _{E^{\bullet }}^{\text{vir}}} of a space X {\displaystyle X} is a replacement of the classical fundamental class ∈ A ∗ ( X ) {\displaystyle \in A^{*}(X)} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Virtual fundamental class Summary Virtual_fundamental_class One such example is in the moduli space M ¯ 1 , n ( P 2 , 1 ) {\displaystyle {\overline {\mathcal {M}}}_{1,n}(\mathbb {P} ^{2},1)} for H {\displaystyle H} the class of a line in P 2 {\displaystyle \mathbb {P} ^{2}} . The non-compact "smooth" component is empt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mercer's theorem Summary Mercer's_theorem In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in (Mercer 1909), is one of the most notable results of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Approximation property Summary Approximation_property In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true. Every Hilbert space has this property. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Approximation property Summary Approximation_property There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955). Later many other counterexamples were found. The space of bounded operators on ℓ 2 {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Schatten class operator Summary Schatten_class_operator In mathematics, specifically functional analysis, a pth Schatten-class operator is a bounded linear operator on a Hilbert space with finite pth Schatten norm. The space of pth Schatten-class operators is a Banach space with respect to the Schatten norm. Via polar ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Schatten class operator Summary Schatten_class_operator {\displaystyle \|ST\|_{S_{1}}\leq \|S\|_{S_{p}}\|T\|_{S_{q}}\ {\mbox{if}}\ S\in S_{p},\ T\in S_{q}{\mbox{ and }}1/p+1/q=1.} If we denote by S ∞ {\displaystyle S_{\infty }} the Banach space of compact operators on H with respect to the operator norm, the above Höld... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Schatten class operator Summary Schatten_class_operator From this it follows that ϕ: S p → S q ′ {\displaystyle \phi :S_{p}\rightarrow S_{q}'} , T ↦ t r ( T ⋅ ) {\displaystyle T\mapsto \mathrm {tr} (T\cdot )} is a well-defined contraction. (Here the prime denotes (topological) dual.) Observe that the 2nd Schatten class... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unconditional convergence Summary Unconditional_convergence In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trace class operator Summary Trace_class_operator In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators gener... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trace class operator Summary Trace_class_operator In quantum mechanics, mixed states are described by density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Schatten norm Summary Schatten_norm In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Barrier cone Summary Barrier_cone In mathematics, specifically functional analysis, the barrier cone is a cone associated to any non-empty subset of a Banach space. It is closely related to the notions of support functions and polar sets. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Von Neumann bicommutant theorem Summary Von_Neumann_double_commutant_theorem In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection bet... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Von Neumann bicommutant theorem Summary Von_Neumann_double_commutant_theorem Then the closures of M in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant M′′ of M.This algebra is called the von Neumann algebra generated by M. There are several other topologie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Von Neumann bicommutant theorem Summary Von_Neumann_double_commutant_theorem One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact space Summary Compact_metric_space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact space Summary Compact_metric_space However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact space Summary Compact_metric_space One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.The Bolzano–Weierstrass theorem states that a subset of Euclidean space... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact space Summary Compact_metric_space Since neither 0 nor 1 are members of the open unit interval (0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact space Summary Compact_metric_space Compactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions. The Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compact space Summary Compact_metric_space In general topological spaces, however, these notions of compactness are not necessarily equivalent. The most useful notion — and the standard definition of the unqualified term compactness — is phrased in terms of the existence of finite families of open sets that "cover" the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Geometric group action Summary Geometric_group_action In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Borel conjecture Summary Borel_conjecture In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equival... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Classification of manifolds Summary Classification_of_manifolds In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cauchy's theorem (group theory) Summary Cauchy's_theorem_(group_theory) In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sylow system Summary Hall_subgroup In mathematics, specifically group theory, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced by the group theorist Philip Hall (1928). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Descendant tree (group theory) Summary Descendant_tree_(group_theory) In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order p n {\displaystyle p^{n}} , for a fixed prime numbe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Descendant tree (group theory) Summary Descendant_tree_(group_theory) It turned out that descendant trees of a particular kind, the so-called pruned coclass trees whose infinitely many vertices share a common coclass r {\displaystyle r} , reveal a repeating finite pattern. These two crucial properties of finiteness and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Descendant tree (group theory) Summary Descendant_tree_(group_theory) Consequently, descendant trees play a fundamental role in the classification of finite p-groups. By means of kernels and targets of Artin transfer homomorphisms, descendant trees can be endowed with additional structure. An important question is how ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nilpotent Lie group Summary Normalizer_condition In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}. Intuitively, a nilpotent group is a gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nilpotent Lie group Summary Normalizer_condition It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Right coset Summary Left_coset In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as doe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Right coset Summary Left_coset The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Noetherian group Summary Noetherian_group In mathematics, specifically group theory, a subgroup series of a group G {\displaystyle G} is a chain of subgroups: 1 = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G {\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G} where 1 {\displaystyle 1} is the trivial subgroup. Subgroup series can simp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Abnormal subgroup Summary Abnormal_subgroup In mathematics, specifically group theory, an abnormal subgroup is a subgroup H of a group G such that for all x in G, x lies in the subgroup generated by H and H x, where H x denotes the conjugate subgroup xHx−1. Here are some facts relating abnormality to other subgroup pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-group generation algorithm Summary P-group_generation_algorithm In mathematics, specifically group theory, finite groups of prime power order p n {\displaystyle p^{n}} , for a fixed prime number p {\displaystyle p} and varying integer exponents n ≥ 0 {\displaystyle n\geq 0} , are briefly called finite p-groups. The p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-primary group Summary P-primary_group In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isoclinism of groups Summary Isoclinism_of_groups In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by Hall (1940) to help classify and understand p-groups, although it is applicable to all groups. Isoclinism also has cons... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isoclinism of groups Summary Isoclinism_of_groups 6.7). The word "isoclinism" comes from the Greek ισοκλινης meaning equal slope. Some textbooks discussing isoclinism include Berkovich (2008, §29) and Blackburn, Neumann & Venkataraman (2007, §21.2) and Suzuki (1986, pp. 92–95). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free product Summary Amalgamated_product In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free product Summary Amalgamated_product Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group. Therefore, the free product is not the coproduct in the category of abelian groups. The free product is important in algebraic topology because of van Kampen's theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Free product Summary Amalgamated_product In particular, the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is, under certain conditions given in the Seifert van-Kampen theorem, the free product of the fundamental groups of the spaces. Free pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Component group Summary Connected_component_of_the_identity In mathematics, specifically group theory, the identity component of a group G refers to several closely related notions of the largest connected subgroup of G containing the identity element. In point set topology, the identity component of a topological grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Finite index Summary Index_of_a_subgroup In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted | G: H | {\displaystyle |G:H|} or {\displaystyle } or ( G: H ) {\displaystyle (G... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Finite index Summary Index_of_a_subgroup More generally, | Z: n Z | = n {\displaystyle |\mathbb {Z} :n\mathbb {Z} |=n} for any positive integer n. When G is finite, the formula may be written as | G: H | = | G | / | H | {\displaystyle |G:H|=|G|/|H|} , and it implies Lagrange's theorem that | H | {\displaystyle |H|} div... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
H-object Summary H-object In mathematics, specifically homotopical algebra, an H-object is a categorical generalization of an H-space, which can be defined in any category C {\displaystyle {\mathcal {C}}} with a product × {\displaystyle \times } and an initial object ∗ {\displaystyle *} . These are useful constructions... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime element Summary Prime_element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Torsion-free abelian group Summary Torsion-free_abelian_groups_of_rank_1 In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Power associative Summary Power-associative_algebra In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete homogeneous symmetric polynomial Summary Complete_homogeneous_symmetric_polynomial In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polyn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weak Lefschetz theorem Summary Lefschetz_theorem In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Formal algebraic geometry Summary Formal_scheme In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this rea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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