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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 multiplication ∗ is defined by a ∗ b = b ⋅ a for all a, b in R. The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see § Properties). Monoids, groups, rings, and algebras can all be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc.
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https://en.wikipedia.org/wiki/Opposite_algebra
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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 asymptotic density.
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https://en.wikipedia.org/wiki/Romanov's_theorem
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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 integral of the cohomology class 1 against the virtual fundamental class. The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by Simon Donaldson and Richard Thomas (1998).
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https://en.wikipedia.org/wiki/Donaldson-Thomas_theory
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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 and gauge theorypg 5. This is due to the fact the invariants depend on a stability condition on the derived category D b ( M ) {\displaystyle D^{b}({\mathcal {M}})} of the moduli spaces being studied. Essentially, these stability conditions correspond to points in the Kahler moduli space of a Calabi-Yau manifold, as considered in mirror symmetry, and the resulting subcategory P ⊂ D b ( M ) {\displaystyle {\mathcal {P}}\subset D^{b}({\mathcal {M}})} is the category of BPS states for the corresponding SCFT.
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https://en.wikipedia.org/wiki/Donaldson-Thomas_theory
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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. More strictly, f has an associated exceptional locus which describes how it identifies nearby points in codimension one, and the exceptional divisor is an appropriate algebraic construction whose support is the exceptional locus. The same ideas can be found in the theory of holomorphic mappings of complex manifolds. More precisely, suppose that f: X → Y {\displaystyle f:X\rightarrow Y} is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of X {\displaystyle X} and Y {\displaystyle Y} ).
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https://en.wikipedia.org/wiki/Exceptional_curve
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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 ( X ) , {\displaystyle \sum _{i}Z_{i}\in Div(X),} where the sum is over all exceptional subvarieties of f {\displaystyle f} , and is an element of the group of Weil divisors on X {\displaystyle X} . Consideration of exceptional divisors is crucial in birational geometry: an elementary result (see for instance Shafarevich, II.4.4) shows (under suitable assumptions) that any birational regular map that is not an isomorphism has an exceptional divisor. A particularly important example is the blowup σ: X ~ → X {\displaystyle \sigma :{\tilde {X}}\rightarrow X} of a subvariety W ⊂ X {\displaystyle W\subset X} :in this case the exceptional divisor is exactly the preimage of W {\displaystyle W} .
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https://en.wikipedia.org/wiki/Exceptional_curve
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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.
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https://en.wikipedia.org/wiki/Valuative_criterion
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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 ( G , n ) {\displaystyle K(G,n)} , if it has n-th homotopy group π n ( X ) {\displaystyle \pi _{n}(X)} isomorphic to G and all other homotopy groups trivial. Assuming that G is abelian in the case that n > 1 {\displaystyle n>1} , Eilenberg–MacLane spaces of type K ( G , n ) {\displaystyle K(G,n)} always exist, and are all weak homotopy equivalent.
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https://en.wikipedia.org/wiki/Eilenberg–MacLane_spectra
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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)} ". Moreover, it is common to assume that this space is a CW-complex (which is always possible via CW approximation).
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https://en.wikipedia.org/wiki/Eilenberg–MacLane_spectra
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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 via fibrations in a Postnikov system. These spaces are important in many contexts in algebraic topology, including computations of homotopy groups of spheres, definition of cohomology operations, and for having a strong connection to singular cohomology. A generalised Eilenberg–Maclane space is a space which has the homotopy type of a product of Eilenberg–Maclane spaces ∏ m K ( G m , m ) {\displaystyle \prod _{m}K(G_{m},m)} .
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https://en.wikipedia.org/wiki/Eilenberg–MacLane_spectra
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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 a lower bound on the sizes of the “holes” in X. This condition is necessary for most of the theory of covering spaces, including the existence of a universal cover and the Galois correspondence between covering spaces and subgroups of the fundamental group. Most “nice” spaces such as manifolds and CW complexes are semi-locally simply connected, and topological spaces that do not satisfy this condition are considered somewhat pathological. The standard example of a non-semi-locally simply connected space is the Hawaiian earring.
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https://en.wikipedia.org/wiki/Semilocally_simply_connected
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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 ring structure is also present in other theories such as de Rham cohomology. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant. Specifically, given a sequence of cohomology groups Hk(X;R) on X with coefficients in a commutative ring R (typically R is Zn, Z, Q, R, or C) one can define the cup product, which takes the form H k ( X ; R ) × H ℓ ( X ; R ) → H k + ℓ ( X ; R ) .
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https://en.wikipedia.org/wiki/Cohomology_ring
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{\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 ) .
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https://en.wikipedia.org/wiki/Cohomology_ring
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{\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.
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https://en.wikipedia.org/wiki/Cohomology_ring
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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 ) .
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https://en.wikipedia.org/wiki/Cohomology_ring
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{\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 result. For example a complex projective space has cup-length equal to its complex dimension.
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https://en.wikipedia.org/wiki/Cohomology_ring
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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)\,/\,\sim } where the ⨿ {\displaystyle \amalg } denotes the disjoint union, and ∼ is the equivalence relation generated by ( 0 , x ) ∼ f ( x ) for each x ∈ X . {\displaystyle (0,x)\sim f(x)\quad {\text{for each }}x\in X.} That is, the mapping cylinder M f {\displaystyle M_{f}} is obtained by gluing one end of X × {\displaystyle X\times } to Y {\displaystyle Y} via the map f {\displaystyle f} . Notice that the "top" of the cylinder { 1 } × X {\displaystyle \{1\}\times X} is homeomorphic to X {\displaystyle X} , while the "bottom" is the space f ( X ) ⊂ Y {\displaystyle f(X)\subset Y} .
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https://en.wikipedia.org/wiki/Mapping_cylinder
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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)\cup _{f}Y} with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone C f {\displaystyle Cf} , obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations.
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https://en.wikipedia.org/wiki/Mapping_cylinder
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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 iteratively resolve with other spectra that are in the homotopy kernel of a map resolving the cohomology classes in H ∗ ( X ; Z / p ) {\displaystyle H^{*}(X;\mathbb {Z} /p)} using Eilenberg–MacLane spectra. This construction can be generalized using a spectrum E {\displaystyle E} , such as the Brown–Peterson spectrum B P {\displaystyle BP} , or the complex cobordism spectrum M U {\displaystyle MU} , and is used in the construction of the Adams–Novikov spectral sequencepg 49.
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https://en.wikipedia.org/wiki/Adams_resolution
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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.
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https://en.wikipedia.org/wiki/Cocycle_condition
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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.
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https://en.wikipedia.org/wiki/First_Feigenbaum_constant
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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 {\displaystyle {\mathcal {C}}} , such that for any two distinct morphisms f , g: X → Y {\displaystyle f,g:X\to Y} in C {\displaystyle {\mathcal {C}}} , that is with f ≠ g {\displaystyle f\neq g} , there is some G {\displaystyle G} in G {\displaystyle {\mathcal {G}}} and some morphism h: G → X {\displaystyle h:G\to X} such that f ∘ h ≠ g ∘ h . {\displaystyle f\circ h\neq g\circ h.} If the collection consists of a single object G {\displaystyle G} , we say it is a generator (or separator). Generators are central to the definition of Grothendieck categories. The dual concept is called a cogenerator or coseparator.
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https://en.wikipedia.org/wiki/Generator_(category_theory)
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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 between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories.
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https://en.wikipedia.org/wiki/Covariant_functor
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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 a linguistic context; see function word.
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https://en.wikipedia.org/wiki/Covariant_functor
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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 name, the further requirement that the category be skeletal is often assumed for the definition of "posetal"; in the case of a category that is posetal, being skeletal is equivalent to the requirement that the only isomorphisms are the identity morphisms, equivalently that the preordered class satisfies antisymmetry and hence, if a set, is a poset. All diagrams commute in a posetal category.
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https://en.wikipedia.org/wiki/Posetal_category
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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 enriched category whose hom-objects are categories, the hom-objects of any extension of a posetal category to a 2-category having the same 1-cells are monoids. Some lattice-theoretic structures are definable as posetal categories of a certain kind, usually with the stronger assumption of being skeletal.
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https://en.wikipedia.org/wiki/Posetal_category
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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 finitely cocomplete *-autonomous category. Conversely, categories, distributive categories, finitely cocomplete cartesian closed categories, and finitely cocomplete *-autonomous categories can be considered the respective categorifications of posets, distributive lattices, Heyting algebras, and Boolean algebras. == References ==
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https://en.wikipedia.org/wiki/Posetal_category
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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 cartesian closed. A solid quasitopos is one for which 0 is a strong subobject of 1.
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https://en.wikipedia.org/wiki/Quasitopos
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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 "removing" some of its objects and arrows.
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https://en.wikipedia.org/wiki/Subcategory
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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 left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories C {\displaystyle {\mathcal {C}}} and D {\displaystyle {\mathcal {D}}} is a pair of functors (assumed to be covariant) F: D → C {\displaystyle F:{\mathcal {D}}\rightarrow {\mathcal {C}}} and G: C → D {\displaystyle G:{\mathcal {C}}\rightarrow {\mathcal {D}}} and, for all objects X {\displaystyle X} in C {\displaystyle {\mathcal {C}}} and Y {\displaystyle Y} in D {\displaystyle {\mathcal {D}}} , a bijection between the respective morphism sets h o m C ( F Y , X ) ≅ h o m D ( Y , G X ) {\displaystyle \mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)} such that this family of bijections is natural in X {\displaystyle X} and Y {\displaystyle Y} .
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https://en.wikipedia.org/wiki/Left_adjoint
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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 a fixed X {\displaystyle X} in C {\displaystyle {\mathcal {C}}} , and also the pair of functors C ( F Y , − ): C → S e t {\displaystyle {\mathcal {C}}(FY,-):{\mathcal {C}}\to \mathrm {Set} } and D ( Y , G − ): C → S e t {\displaystyle {\mathcal {D}}(Y,G-):{\mathcal {C}}\to \mathrm {Set} } for a fixed Y {\displaystyle Y} in D {\displaystyle {\mathcal {D}}} . The functor F {\displaystyle F} is called a left adjoint functor or left adjoint to G {\displaystyle G} , while G {\displaystyle G} is called a right adjoint functor or right adjoint to F {\displaystyle F} .
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https://en.wikipedia.org/wiki/Left_adjoint
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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 indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.
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https://en.wikipedia.org/wiki/Left_adjoint
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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 category of modules are those whose image is an essential submodule of the codomain. An injective hull of an object X is an essential monomorphism from X to an injective object. == References ==
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https://en.wikipedia.org/wiki/Essential_monomorphism
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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 {\displaystyle X} in some category C {\displaystyle {\mathcal {C}}} . There is a dual notion of undercategory, which is defined similarly.
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https://en.wikipedia.org/wiki/Overcategory
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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 of many sums involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent and much simpler sum. Although finding WZ pairs by hand is impractical in most cases, Gosper's algorithm provides a sure method to find a function's WZ counterpart, and can be implemented in a symbolic manipulation program.
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https://en.wikipedia.org/wiki/Hypergeometric_function_identities
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In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.
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https://en.wikipedia.org/wiki/Hilbert's_basis_theorem
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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!} .
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https://en.wikipedia.org/wiki/Divided_power_structure
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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 number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem.
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https://en.wikipedia.org/wiki/Primary_ideal
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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 identity.
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https://en.wikipedia.org/wiki/Primary_ideal
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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 square roots: 2 and −2; of these the positive root, 2, is considered the principal root and is denoted as 4 . {\displaystyle {\sqrt {4}}.}
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https://en.wikipedia.org/wiki/Principal_value
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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 is easy to check that ω {\displaystyle \omega } is computable. The successor of a computable ordinal is computable, and the set of all computable ordinals is closed downwards. The supremum of all computable ordinals is called the Church–Kleene ordinal, the first nonrecursive ordinal, and denoted by ω 1 C K {\displaystyle \omega _{1}^{CK}} .
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https://en.wikipedia.org/wiki/Computable_ordinal
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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}} .
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https://en.wikipedia.org/wiki/Computable_ordinal
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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 {\displaystyle {\mathcal {O}}} .
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https://en.wikipedia.org/wiki/Computable_ordinal
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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.
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https://en.wikipedia.org/wiki/Normal_fan
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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 {\displaystyle 2n} -manifolds are homeomorphic to spheres, this is the first non-trivial case after) and found an example of a space which is homotopy equivalent to a sphere, but was not explicitly diffeomorphic. He did this through looking at real vector bundles V → S n {\displaystyle V\to S^{n}} over a sphere and studied the properties of the associated disk bundle. It turns out, the boundary of this bundle is homotopically equivalent to a sphere S 2 n − 1 {\displaystyle S^{2n-1}} , but in certain cases it is not diffeomorphic. This lack of diffeomorphism comes from studying a hypothetical cobordism between this boundary and a sphere, and showing this hypothetical cobordism invalidates certain properties of the Hirzebruch signature theorem.
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https://en.wikipedia.org/wiki/Milnor's_sphere
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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 theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth. The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem.
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https://en.wikipedia.org/wiki/Inverse_function_theorem
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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 define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
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https://en.wikipedia.org/wiki/Riemann_curvature
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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)} in its chow ring which has better behavior with respect to the enumerative problems being considered. In this way, there exists a cycle with can be used for answering specific enumerative problems, such as the number of degree d {\displaystyle d} rational curves on a quintic threefold. For example, in Gromov–Witten theory, the Kontsevich moduli spaces M ¯ g , n ( X , β ) {\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,\beta )} for X {\displaystyle X} a scheme and β {\displaystyle \beta } a class in A 1 ( X ) {\displaystyle A_{1}(X)} , their behavior can be wild at the boundary, such aspg 503 having higher-dimensional components at the boundary than on the main space.
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https://en.wikipedia.org/wiki/Virtual_fundamental_class
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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 empty, but the boundary contains maps of curves f: C → P 2 {\displaystyle f:C\to \mathbb {P} ^{2}} whose components consist of one degree 3 curve which contracts to a point. There is a virtual fundamental class which can then be used to count the number of curves in this family.
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https://en.wikipedia.org/wiki/Virtual_fundamental_class
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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 the work of James Mercer (1883–1932). It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive-definite kernel.
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https://en.wikipedia.org/wiki/Mercer's_theorem
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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.
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https://en.wikipedia.org/wiki/Approximation_property
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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 {\displaystyle \ell ^{2}} does not have the approximation property. The spaces ℓ p {\displaystyle \ell ^{p}} for p ≠ 2 {\displaystyle p\neq 2} and c 0 {\displaystyle c_{0}} (see Sequence space) have closed subspaces that do not have the approximation property.
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https://en.wikipedia.org/wiki/Approximation_property
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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 decomposition, one can prove that the space of pth Schatten class operators is an ideal in B(H). Furthermore, the Schatten norm satisfies a type of Hölder inequality: ‖ S T ‖ S 1 ≤ ‖ S ‖ S p ‖ T ‖ S q if S ∈ S p , T ∈ S q and 1 / p + 1 / q = 1.
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https://en.wikipedia.org/wiki/Schatten_class_operator
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{\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 Hölder-type inequality even holds for p ∈ {\displaystyle p\in } .
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https://en.wikipedia.org/wiki/Schatten_class_operator
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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 is in fact the Hilbert space of Hilbert–Schmidt operators. Moreover, the 1st Schatten class is the space of trace class operators.
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https://en.wikipedia.org/wiki/Schatten_class_operator
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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 to that same value. Unconditional convergence is equivalent to absolute convergence in finite-dimensional vector spaces, but is a weaker property in infinite dimensions.
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https://en.wikipedia.org/wiki/Unconditional_convergence
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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 generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators.
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https://en.wikipedia.org/wiki/Trace_class_operator
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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 nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Note that the trace operator studied in partial differential equations is an unrelated concept.
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https://en.wikipedia.org/wiki/Trace_class_operator
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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.
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https://en.wikipedia.org/wiki/Schatten_norm
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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.
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https://en.wikipedia.org/wiki/Barrier_cone
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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 between the algebraic and topological sides of operator theory. The formal statement of the theorem is as follows: Von Neumann bicommutant theorem. Let M be an algebra consisting of bounded operators on a Hilbert space H, containing the identity operator, and closed under taking adjoints.
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https://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem
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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 topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If M is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra.
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https://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem
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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, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong topologies. It is related to the Jacobson density theorem.
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https://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem
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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 points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval would be compact. Similarly, the space of rational numbers Q {\displaystyle \mathbb {Q} } is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers R {\displaystyle \mathbb {R} } is not compact either, because it excludes the two limiting values + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } .
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https://en.wikipedia.org/wiki/Compact_space
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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.
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https://en.wikipedia.org/wiki/Compact_space
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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 is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval , some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, ... accumulate to 0 (while others accumulate to 1).
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https://en.wikipedia.org/wiki/Compact_space
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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 compact, since it is not bounded. For example, considering R 1 {\displaystyle \mathbb {R} ^{1}} (the real number line), the sequence of points 0, 1, 2, 3, ... has no subsequence that converges to any real number.
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https://en.wikipedia.org/wiki/Compact_space
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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 to classical analysis. Following its initial introduction, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces.
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https://en.wikipedia.org/wiki/Compact_space
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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 space in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally — that is, in a neighborhood of each point — into corresponding statements that hold throughout the space, and many theorems are of this character. The term compact set is sometimes used as a synonym for compact space, but also often refers to a compact subspace of a topological space.
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https://en.wikipedia.org/wiki/Compact_space
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In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.
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https://en.wikipedia.org/wiki/Geometric_group_action
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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 equivalence (namely, homotopy equivalence) should imply a stronger, topological notion (namely, homeomorphism).
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https://en.wikipedia.org/wiki/Borel_conjecture
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In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain.
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https://en.wikipedia.org/wiki/Classification_of_manifolds
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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 p is the smallest positive integer with xp = e, where e is the identity element of G. It is named after Augustin-Louis Cauchy, who discovered it in 1845.The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite group G divides the order of G. Cauchy's theorem implies that for any prime divisor p of the order of G, there is a subgroup of G whose order is p—the cyclic group generated by the element in Cauchy's theorem. Cauchy's theorem is generalized by Sylow's first theorem, which implies that if pn is the maximal power of p dividing the order of G, then G has a subgroup of order pn (and using the fact that a p-group is solvable, one can show that G has subgroups of order pr for any r less than or equal to n).
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https://en.wikipedia.org/wiki/Cauchy's_theorem_(group_theory)
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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).
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https://en.wikipedia.org/wiki/Sylow_system
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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 number p {\displaystyle p} and varying integer exponents n ≥ 0 {\displaystyle n\geq 0} . Such groups are briefly called finite p-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups. Additionally to their order p n {\displaystyle p^{n}} , finite p-groups have two further related invariants, the nilpotency class c {\displaystyle c} and the coclass r = n − c {\displaystyle r=n-c} .
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https://en.wikipedia.org/wiki/Descendant_tree_(group_theory)
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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 periodicity admit a characterization of all members of the tree by finitely many parametrized presentations.
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https://en.wikipedia.org/wiki/Descendant_tree_(group_theory)
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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 the descendant tree T ( R ) {\displaystyle {\mathcal {T}}(R)} can actually be constructed for an assigned starting group which is taken as the root R {\displaystyle R} of the tree. The p-group generation algorithm is a recursive process for constructing the descendant tree of a given finite p-group playing the role of the tree root. This algorithm is implemented in the computational algebra systems GAP and Magma.
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https://en.wikipedia.org/wiki/Descendant_tree_(group_theory)
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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 group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute.
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https://en.wikipedia.org/wiki/Nilpotent_Lie_group
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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 classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series.
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https://en.wikipedia.org/wiki/Nilpotent_Lie_group
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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 does H. Furthermore, H itself is both a left coset and a right coset.
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https://en.wikipedia.org/wiki/Right_coset
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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 for any finite group G, the number of elements of every subgroup H of G divides the number of elements of G. Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes.
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https://en.wikipedia.org/wiki/Right_coset
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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 simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups. A subgroup series is used in the subgroup method. Subgroup series are a special example of the use of filtrations in abstract algebra.
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https://en.wikipedia.org/wiki/Noetherian_group
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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 properties: Every abnormal subgroup is a self-normalizing subgroup, as well as a contranormal subgroup. The only normal subgroup that is also abnormal is the whole group. Every abnormal subgroup is a weakly abnormal subgroup, and every weakly abnormal subgroup is a self-normalizing subgroup. Every abnormal subgroup is a pronormal subgroup, and hence a weakly pronormal subgroup, a paranormal subgroup, and a polynormal subgroup.
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https://en.wikipedia.org/wiki/Abnormal_subgroup
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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-group generation algorithm by M. F. Newman and E. A. O'Brien is a recursive process for constructing the descendant tree of an assigned finite p-group which is taken as the root of the tree.
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https://en.wikipedia.org/wiki/P-group_generation_algorithm
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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 equal to the identity element. The orders of different elements may be different powers of p. Abelian p-groups are also called p-primary or simply primary. A finite group is a p-group if and only if its order (the number of its elements) is a power of p. Given a finite group G, the Sylow theorems guarantee the existence of a subgroup of G of order pn for every prime power pn that divides the order of G. Every finite p-group is nilpotent. The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see Prüfer group, and for an example of an infinite simple p-group, see Tarski monster group.
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https://en.wikipedia.org/wiki/P-primary_group
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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 consequences for the Schur multiplier and the associated aspects of character theory, as described in Suzuki (1982, p. 256) and Conway et al. (1985, p. xxiii, Ch.
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https://en.wikipedia.org/wiki/Isoclinism_of_groups
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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).
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https://en.wikipedia.org/wiki/Isoclinism_of_groups
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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 properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from G ∗ H to K. Unless one of the groups G and H is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group (the universal group with a given set of generators). The free product is the coproduct in the category of groups. That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory.
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https://en.wikipedia.org/wiki/Free_product
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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 theorem, which states that the fundamental group of the union of two path-connected topological spaces whose intersection is also path-connected is always an amalgamated free product of the fundamental groups of the spaces.
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https://en.wikipedia.org/wiki/Free_product
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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 products are also important in Bass–Serre theory, the study of groups acting by automorphisms on trees. Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and HNN extensions. Using the action of the modular group on a certain tessellation of the hyperbolic plane, it follows from this theory that the modular group is isomorphic to the free product of cyclic groups of orders 4 and 6 amalgamated over a cyclic group of order 2.
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https://en.wikipedia.org/wiki/Free_product
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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 group G is the connected component G0 of G that contains the identity element of the group. The identity path component of a topological group G is the path component of G that contains the identity element of the group. In algebraic geometry, the identity component of an algebraic group G over a field k is the identity component of the underlying topological space. The identity component of a group scheme G over a base scheme S is, roughly speaking, the group scheme G0 whose fiber over the point s of S is the connected component (Gs)0 of the fiber Gs, an algebraic group.
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https://en.wikipedia.org/wiki/Component_group
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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:H)} . Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula | G | = | G: H | | H | {\displaystyle |G|=|G:H||H|} (interpret the quantities as cardinal numbers if some of them are infinite). Thus the index | G: H | {\displaystyle |G:H|} measures the "relative sizes" of G and H. For example, let G = Z {\displaystyle G=\mathbb {Z} } be the group of integers under addition, and let H = 2 Z {\displaystyle H=2\mathbb {Z} } be the subgroup consisting of the even integers. Then 2 Z {\displaystyle 2\mathbb {Z} } has two cosets in Z {\displaystyle \mathbb {Z} } , namely the set of even integers and the set of odd integers, so the index | Z: 2 Z | {\displaystyle |\mathbb {Z} :2\mathbb {Z} |} is 2.
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https://en.wikipedia.org/wiki/Finite_index
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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|} divides | G | {\displaystyle |G|} . When G is infinite, | G: H | {\displaystyle |G:H|} is a nonzero cardinal number that may be finite or infinite. For example, | Z: 2 Z | = 2 {\displaystyle |\mathbb {Z} :2\mathbb {Z} |=2} , but | R: Z | {\displaystyle |\mathbb {R} :\mathbb {Z} |} is infinite. If N is a normal subgroup of G, then | G: N | {\displaystyle |G:N|} is equal to the order of the quotient group G / N {\displaystyle G/N} , since the underlying set of G / N {\displaystyle G/N} is the set of cosets of N in G.
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https://en.wikipedia.org/wiki/Finite_index
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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 because they help export some of the ideas from algebraic topology and homotopy theory into other domains, such as in commutative algebra and algebraic geometry.
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https://en.wikipedia.org/wiki/H-object
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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, a concept which is the same in UFDs but not the same in general.
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https://en.wikipedia.org/wiki/Prime_element
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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 with finite order. While finitely generated abelian groups are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case.
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https://en.wikipedia.org/wiki/Torsion-free_abelian_group
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In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
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https://en.wikipedia.org/wiki/Power_associative
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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 polynomial expression in complete homogeneous symmetric polynomials.
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https://en.wikipedia.org/wiki/Complete_homogeneous_symmetric_polynomial
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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 a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories. A far-reaching generalization of the hard Lefschetz theorem is given by the decomposition theorem.
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https://en.wikipedia.org/wiki/Weak_Lefschetz_theorem
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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 reason, formal schemes frequently appear in topics such as deformation theory. But the concept is also used to prove a theorem such as the theorem on formal functions, which is used to deduce theorems of interest for usual schemes.
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https://en.wikipedia.org/wiki/Formal_algebraic_geometry
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