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In mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product ∏ n = 1 ∞ a n = a 1 a 2 a 3 ⋯ {\displaystyle \prod _{n=1}^{\infty }a_{n}=a_{1}a_{2}a_{3}\cdots } is defined to be the limit of the partial products a1a2...an as n increases without bound. The product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums.
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https://en.wikipedia.org/wiki/Infinite_product
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Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence an as n increases without bound must be 1, while the converse is in general not true. The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète (Viète's formula, the first published infinite product in mathematics) and John Wallis (Wallis product): 2 π = 2 2 ⋅ 2 + 2 2 ⋅ 2 + 2 + 2 2 ⋅ ⋯ {\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \;\cdots } π 2 = ( 2 1 ⋅ 2 3 ) ⋅ ( 4 3 ⋅ 4 5 ) ⋅ ( 6 5 ⋅ 6 7 ) ⋅ ( 8 7 ⋅ 8 9 ) ⋅ ⋯ = ∏ n = 1 ∞ ( 4 n 2 4 n 2 − 1 ) . {\displaystyle {\frac {\pi }{2}}=\left({\frac {2}{1}}\cdot {\frac {2}{3}}\right)\cdot \left({\frac {4}{3}}\cdot {\frac {4}{5}}\right)\cdot \left({\frac {6}{5}}\cdot {\frac {6}{7}}\right)\cdot \left({\frac {8}{7}}\cdot {\frac {8}{9}}\right)\cdot \;\cdots =\prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right).}
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https://en.wikipedia.org/wiki/Infinite_product
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In mathematics, for example in the study of statistical properties of graphs, a null model is a type of random object that matches one specific object in some of its features, or more generally satisfies a collection of constraints, but which is otherwise taken to be an unbiasedly random structure. The null model is used as a term of comparison, to verify whether the object in question displays some non-trivial features (properties that wouldn't be expected on the basis of chance alone or as a consequence of the constraints), such as community structure in graphs. An appropriate null model behaves in accordance with a reasonable null hypothesis for the behavior of the system under investigation. One null model of utility in the study of complex networks is that proposed by Newman and Girvan, consisting of a randomized version of an original graph G {\displaystyle G} , produced through edges being rewired at random, under the constraint that the expected degree of each vertex matches the degree of the vertex in the original graph.The null model is the basic concept behind the definition of modularity, a function which evaluates the goodness of partitions of a graph into clusters.
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https://en.wikipedia.org/wiki/Null_model
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In particular, given a graph G {\displaystyle G} and a specific community partition σ: V ( G ) → { 1 , . . . , b } {\displaystyle \sigma :V(G)\rightarrow \{1,...,b\}} (an assignment of a community-index σ ( v ) {\displaystyle \sigma (v)} (here taken as an integer from 1 {\displaystyle 1} to b {\displaystyle b} ) to each vertex v ∈ V ( G ) {\displaystyle v\in V(G)} in the graph), the modularity measures the difference between the number of links from/to each pair of communities, from that expected in a graph that is completely random in all respects other than the set of degrees of each of the vertices (the degree sequence). In other words, the modularity contrasts the exhibited community structure in G {\displaystyle G} with that of a null model, which in this case is the configuration model (the maximally random graph subject to a constraint on the degree of each vertex).
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https://en.wikipedia.org/wiki/Null_model
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In mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that bk = a. In number theory, the more commonly used term is index: we can write x = indr a (mod m) (read "the index of a to the base r modulo m") for rx ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1. Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography, such as ElGamal, base their security on the assumption that the discrete logarithm problem (DLP) over carefully chosen groups has no efficient solution.
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https://en.wikipedia.org/wiki/Discrete_Logarithm
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In mathematics, for positive integers k and s, a vectorial addition chain is a sequence V of k-dimensional vectors of nonnegative integers vi for −k + 1 ≤ i ≤ s together with a sequence w, such that v−k+1 = v−k+2 = ⋮ ⋮ v0 = vi =vj+vr for all 1≤i≤s with -k+1≤j, r≤i-1 vs = w = (w1,...ws), wi=(j,r).For example, a vectorial addition chain for is V=(,,,,,,,,,,,) w=((-2,-1),(1,1),(2,2),(-2,3),(4,4),(1,5),(0,6),(7,7),(0,8))Vectorial addition chains are well suited to perform multi-exponentiation: Input: Elements x0,...,xk-1 of an abelian group G and a vectorial addition chain of dimension k computing Output:The element x0n0...xk-1nr-1for i =-k+1 to 0 do yi → xi+k-1 for i = 1 to s do yi →yj×yr return ys
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https://en.wikipedia.org/wiki/Vectorial_addition_chain
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In mathematics, forcing is a method of constructing new models M of set theory by adding a generic subset G of a poset P to a model M. The poset P used will determine what statements hold in the new universe (the 'extension'); to force a statement of interest thus requires construction of a suitable P. This article lists some of the posets P that have been used in this construction.
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https://en.wikipedia.org/wiki/List_of_forcing_notions
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In mathematics, formal moduli are an aspect of the theory of moduli spaces (of algebraic varieties or vector bundles, for example), closely linked to deformation theory and formal geometry. Roughly speaking, deformation theory can provide the Taylor polynomial level of information about deformations, while formal moduli theory can assemble consistent Taylor polynomials to make a formal power series theory. The step to moduli spaces, properly speaking, is an algebraization question, and has been largely put on a firm basis by Artin's approximation theorem. A formal universal deformation is by definition a formal scheme over a complete local ring, with special fiber the scheme over a field being studied, and with a universal property amongst such set-ups. The local ring in question is then the carrier of the formal moduli.
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https://en.wikipedia.org/wiki/Formal_moduli
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In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abstraction.
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https://en.wikipedia.org/wiki/Function_application
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In mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f: X → Y and g: Y → Z are composed to yield a function that maps x in domain X to g(f(x)) in codomain Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f: X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.The notation g ∘ f is read as "g of f ", "g after f ", "g circle f ", "g round f ", "g about f ", "g composed with f ", "g following f ", "f then g", or "g on f ", or "the composition of g and f ". Intuitively, composing functions is a chaining process in which the output of function f feeds the input of function g. The composition of functions is a special case of the composition of relations, sometimes also denoted by ∘ {\displaystyle \circ } . As a result, all properties of composition of relations are true of composition of functions, such as the property of associativity. Composition of functions is different from multiplication of functions (if defined at all), and has some quite different properties; in particular, composition of functions is not commutative.
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https://en.wikipedia.org/wiki/Function_composition
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In mathematics, functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. A parabola is a specific type of function.
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https://en.wikipedia.org/wiki/List_of_types_of_functions
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In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures; possibility/necessity measures; and probability measures, which are a subset of classical measures.
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https://en.wikipedia.org/wiki/Fuzzy_measure_theory
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In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
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https://en.wikipedia.org/wiki/Fuzzy_set_theory
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At the same time, Salii (1965) defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics (De Cock, Bodenhofer & Kerre 2000), decision-making (Kuzmin 1982), and clustering (Bezdek 1978), are special cases of L-relations when L is the unit interval . In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition—an element either belongs or does not belong to the set.
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https://en.wikipedia.org/wiki/Fuzzy_set_theory
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By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval . Fuzzy sets generalize classical sets, since the indicator functions (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes values 0 or 1. In fuzzy set theory, classical bivalent sets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.
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https://en.wikipedia.org/wiki/Fuzzy_set_theory
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In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size.
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https://en.wikipedia.org/wiki/Point-set_topology
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Connected sets are sets that cannot be divided into two pieces that are far apart.The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are.
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https://en.wikipedia.org/wiki/Point-set_topology
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Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space. Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.
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https://en.wikipedia.org/wiki/Point-set_topology
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In mathematics, generalized Verma modules are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the 1970s. The motivation for their study is that their homomorphisms correspond to invariant differential operators over generalized flag manifolds. The study of these operators is an important part of the theory of parabolic geometries.
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https://en.wikipedia.org/wiki/Generalized_Verma_module
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In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges.
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https://en.wikipedia.org/wiki/Generalized_functions
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They are applied extensively, especially in physics and engineering. A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and more contemporary developments in certain directions are closely related to ideas of Mikio Sato, on what he calls algebraic analysis. Important influences on the subject have been the technical requirements of theories of partial differential equations, and group representation theory.
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https://en.wikipedia.org/wiki/Generalized_functions
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In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).
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https://en.wikipedia.org/wiki/Generalised_mean
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In mathematics, genus (PL: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.
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https://en.wikipedia.org/wiki/Genus_(mathematics)
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In mathematics, geology, and cartography, a surface map is a 2D perspective representation of a 3-dimensional surface. Surface maps usually represent real-world entities such as landforms or the surfaces of objects. They can, however, serve as an abstraction where the third, or even all of the dimensions correspond to non-spatial data. In this capacity they act more as graphs than maps.
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https://en.wikipedia.org/wiki/Surface_map
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In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including vector calculus, differential geometry, and differential forms.
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https://en.wikipedia.org/wiki/Geometric_calculus
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In mathematics, geometric class field theory is an extension of class field theory to higher-dimensional geometrical objects: much the same way as class field theory describes the abelianization of the Galois group of a local or global field, geometric class field theory describes the abelianized fundamental group of higher dimensional schemes in terms of data related to algebraic cycles.
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https://en.wikipedia.org/wiki/Geometric_class_field_theory
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In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper (Hilbert 1893) in classical invariant theory. Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles.
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https://en.wikipedia.org/wiki/Semistable_point
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In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth.
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https://en.wikipedia.org/wiki/Geometric_measure_theory
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In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
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https://en.wikipedia.org/wiki/Geometric_topology
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In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory. Sharp distinctions between geometry and topology can be drawn, however, as discussed below.It is also the title of a journal Geometry & Topology that covers these topics.
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https://en.wikipedia.org/wiki/Geometry_and_topology
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In mathematics, given a G-torsor X → Y and a stack F, the descent along torsors says there is a canonical equivalence between F(Y), the category of Y-points and F(X)G, the category of G-equivariant X-points. It is a basic example of descent, since it says the "equivariant data" (which is an additional data) allows one to "descend" from X to Y. When G is the Galois group of a finite Galois extension L/K, for the G-torsor Spec L → Spec K {\displaystyle \operatorname {Spec} L\to \operatorname {Spec} K} , this generalizes classical Galois descent (cf. field of definition). For example, one can take F to be the stack of quasi-coherent sheaves (in an appropriate topology). Then F(X)G consists of equivariant sheaves on X; thus, the descent in this case says that to give an equivariant sheaf on X is to give a sheaf on the quotient X/G.
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https://en.wikipedia.org/wiki/Galois_descent
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In mathematics, given a category C, a quotient of an object X by an equivalence relation f: R → X × X {\displaystyle f:R\to X\times X} is a coequalizer for the pair of maps R → f X × X → pr i X , i = 1 , 2 , {\displaystyle R\ {\overset {f}{\to }}\ X\times X\ {\overset {\operatorname {pr} _{i}}{\to }}\ X,\ \ i=1,2,} where R is an object in C and "f is an equivalence relation" means that, for any object T in C, the image (which is a set) of f: R ( T ) = Mor ( T , R ) → X ( T ) × X ( T ) {\displaystyle f:R(T)=\operatorname {Mor} (T,R)\to X(T)\times X(T)} is an equivalence relation; that is, a reflexive, symmetric and transitive relation. The basic case in practice is when C is the category of all schemes over some scheme S. But the notion is flexible and one can also take C to be the category of sheaves.
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https://en.wikipedia.org/wiki/Quotient_by_an_equivalence_relation
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In mathematics, given a collection S of subsets of a set X, an exact hitting set X* is a subset of X such that each subset in S contains exactly one element in X*. One says that each subset in S is hit by exactly one element in X*. In computer science, the exact hitting set problem is a decision problem to find an exact hitting set or else determine none exists. The exact hitting set problem is an abstract exact cover problem.
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https://en.wikipedia.org/wiki/Exact_cover
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In the notation above, P is the set X, Q is a collection S of subsets of X, R is the binary relation "is contained in" between elements and subsets, and R−1 restricted to Q × P* is the function "contains" from subsets to selected elements. Whereas an exact cover problem involves selecting subsets and the relation "contains" from subsets to elements, an exact hitting set problem involves selecting elements and the relation "is contained in" from elements to subsets. In a sense, an exact hitting set problem is the inverse of the exact cover problem involving the same set and collection of subsets.
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https://en.wikipedia.org/wiki/Exact_cover
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In mathematics, given a field F {\displaystyle \mathbb {F} } , nonnegative integers m , n {\displaystyle m,n} , and a matrix A ∈ F m × n {\displaystyle A\in \mathbb {F} ^{m\times n}} , a rank decomposition or rank factorization of A is a factorization of A of the form A = CF, where C ∈ F m × r {\displaystyle C\in \mathbb {F} ^{m\times r}} and F ∈ F r × n {\displaystyle F\in \mathbb {F} ^{r\times n}} , where r = rank A {\displaystyle r=\operatorname {rank} A} is the rank of A {\displaystyle A} .
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https://en.wikipedia.org/wiki/Rank_factorization
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In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general G-modules. The term G-module is also used for the more general notion of an R-module on which G acts linearly (i.e. as a group of R-module automorphisms).
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https://en.wikipedia.org/wiki/G-module
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In mathematics, given a locally Lebesgue integrable function f {\displaystyle f} on R k {\displaystyle \mathbb {R} ^{k}} , a point x {\displaystyle x} in the domain of f {\displaystyle f} is a Lebesgue point if lim r → 0 + 1 λ ( B ( x , r ) ) ∫ B ( x , r ) | f ( y ) − f ( x ) | d y = 0. {\displaystyle \lim _{r\rightarrow 0^{+}}{\frac {1}{\lambda (B(x,r))}}\int _{B(x,r)}\!|f(y)-f(x)|\,\mathrm {d} y=0.} Here, B ( x , r ) {\displaystyle B(x,r)} is a ball centered at x {\displaystyle x} with radius r > 0 {\displaystyle r>0} , and λ ( B ( x , r ) ) {\displaystyle \lambda (B(x,r))} is its Lebesgue measure. The Lebesgue points of f {\displaystyle f} are thus points where f {\displaystyle f} does not oscillate too much, in an average sense.The Lebesgue differentiation theorem states that, given any f ∈ L 1 ( R k ) {\displaystyle f\in L^{1}(\mathbb {R} ^{k})} , almost every x {\displaystyle x} is a Lebesgue point of f {\displaystyle f} . == References ==
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https://en.wikipedia.org/wiki/Lebesgue_point
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In mathematics, given a non-empty set of objects of finite extension in d {\displaystyle d} -dimensional space, for example a set of points, a bounding sphere, enclosing sphere or enclosing ball for that set is an d {\displaystyle d} -dimensional solid sphere containing all of these objects. Used in computer graphics and computational geometry, a bounding sphere is a special type of bounding volume. There are several fast and simple bounding sphere construction algorithms with a high practical value in real-time computer graphics applications.In statistics and operations research, the objects are typically points, and generally the sphere of interest is the minimal bounding sphere, that is, the sphere with minimal radius among all bounding spheres.
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https://en.wikipedia.org/wiki/Smallest_bounding_sphere
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It may be proven that such a sphere is unique: If there are two of them, then the objects in question lie within their intersection. But an intersection of two non-coinciding spheres of equal radius is contained in a sphere of smaller radius. The problem of computing the center of a minimal bounding sphere is also known as the "unweighted Euclidean 1-center problem".
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https://en.wikipedia.org/wiki/Smallest_bounding_sphere
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In mathematics, given a partial order ⪯ {\displaystyle \preceq } and ⊑ {\displaystyle \sqsubseteq } on a set A {\displaystyle A} and B {\displaystyle B} , respectively, the product order (also called the coordinatewise order or componentwise order) is a partial ordering ≤ {\displaystyle \leq } on the Cartesian product A × B . {\displaystyle A\times B.} Given two pairs ( a 1 , b 1 ) {\displaystyle \left(a_{1},b_{1}\right)} and ( a 2 , b 2 ) {\displaystyle \left(a_{2},b_{2}\right)} in A × B , {\displaystyle A\times B,} declare that ( a 1 , b 1 ) ≤ ( a 2 , b 2 ) {\displaystyle \left(a_{1},b_{1}\right)\leq \left(a_{2},b_{2}\right)} if a 1 ⪯ a 2 {\displaystyle a_{1}\preceq a_{2}} and b 1 ⊑ b 2 . {\displaystyle b_{1}\sqsubseteq b_{2}.}
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https://en.wikipedia.org/wiki/Product_order
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Another possible ordering on A × B {\displaystyle A\times B} is the lexicographical order, which is a total ordering. However the product order of two total orders is not in general total; for example, the pairs ( 0 , 1 ) {\displaystyle (0,1)} and ( 1 , 0 ) {\displaystyle (1,0)} are incomparable in the product order of the ordering 0 < 1 {\displaystyle 0<1} with itself. The lexicographic combination of two total orders is a linear extension of their product order, and thus the product order is a subrelation of the lexicographic order.The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions.The product order generalizes to arbitrary (possibly infinitary) Cartesian products.
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https://en.wikipedia.org/wiki/Product_order
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Suppose A ≠ ∅ {\displaystyle A\neq \varnothing } is a set and for every a ∈ A , {\displaystyle a\in A,} ( I a , ≤ ) {\displaystyle \left(I_{a},\leq \right)} is a preordered set. Then the product preorder on ∏ a ∈ A I a {\displaystyle \prod _{a\in A}I_{a}} is defined by declaring for any i ∙ = ( i a ) a ∈ A {\displaystyle i_{\bullet }=\left(i_{a}\right)_{a\in A}} and j ∙ = ( j a ) a ∈ A {\displaystyle j_{\bullet }=\left(j_{a}\right)_{a\in A}} in ∏ a ∈ A I a , {\displaystyle \prod _{a\in A}I_{a},} that i ∙ ≤ j ∙ {\displaystyle i_{\bullet }\leq j_{\bullet }} if and only if i a ≤ j a {\displaystyle i_{a}\leq j_{a}} for every a ∈ A . {\displaystyle a\in A.}
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https://en.wikipedia.org/wiki/Product_order
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If every ( I a , ≤ ) {\displaystyle \left(I_{a},\leq \right)} is a partial order then so is the product preorder. Furthermore, given a set A , {\displaystyle A,} the product order over the Cartesian product ∏ a ∈ A { 0 , 1 } {\displaystyle \prod _{a\in A}\{0,1\}} can be identified with the inclusion ordering of subsets of A .
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https://en.wikipedia.org/wiki/Product_order
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{\displaystyle A.} The notion applies equally well to preorders. The product order is also the categorical product in a number of richer categories, including lattices and Boolean algebras.
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https://en.wikipedia.org/wiki/Product_order
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In mathematics, given a quiver Q with set of vertices Q0 and set of arrows Q1, a representation of Q assigns a vector space Vi to each vertex and a linear map V(α): V(s(α)) → V(t(α)) to each arrow α, where s(α), t(α) are, respectively, the starting and the ending vertices of α. Given an element d ∈ N {\displaystyle \mathbb {N} } Q0, the set of representations of Q with dim Vi = d(i) for each i has a vector space structure. It is naturally endowed with an action of the algebraic group Πi∈Q0 GL(d(i)) by simultaneous base change. Such action induces one on the ring of functions. The ones which are invariants up to a character of the group are called semi-invariants. They form a ring whose structure reflects representation-theoretical properties of the quiver.
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https://en.wikipedia.org/wiki/Semi-invariant_of_a_quiver
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In mathematics, given a ring R, the K-theory spectrum of R is an Ω-spectrum K R {\displaystyle K_{R}} whose nth term is given by, writing Σ R {\displaystyle \Sigma R} for the suspension of R, ( K R ) n = K 0 ( Σ n R ) × B G L ( Σ n R ) + {\displaystyle (K_{R})_{n}=K_{0}(\Sigma ^{n}R)\times BGL(\Sigma ^{n}R)^{+}} ,where "+" means the Quillen's + construction. By definition, K i ( R ) = π i ( K R ) {\displaystyle K_{i}(R)=\pi _{i}(K_{R})} . == References ==
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https://en.wikipedia.org/wiki/K-theory_spectrum
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In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector. Similarly, a vector at a point on a surface can be broken down the same way. More generally, given a submanifold N of a manifold M, and a vector in the tangent space to M at a point of N, it can be decomposed into the component tangent to N and the component normal to N.
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https://en.wikipedia.org/wiki/Tangential_and_normal_components
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In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0. In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, X = A + B, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres: The null cone is also the union of the isotropic lines through the origin.
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https://en.wikipedia.org/wiki/Null_cone
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In mathematics, given an action σ: G × S X → X {\displaystyle \sigma :G\times _{S}X\to X} of a group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules together with the isomorphism of O G × S X {\displaystyle {\mathcal {O}}_{G\times _{S}X}} -modules ϕ: σ ∗ F → ≃ p 2 ∗ F {\displaystyle \phi :\sigma ^{*}F\xrightarrow {\simeq } p_{2}^{*}F} that satisfies the cocycle condition: writing m for multiplication, p 23 ∗ ϕ ∘ ( 1 G × σ ) ∗ ϕ = ( m × 1 X ) ∗ ϕ {\displaystyle p_{23}^{*}\phi \circ (1_{G}\times \sigma )^{*}\phi =(m\times 1_{X})^{*}\phi } .
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https://en.wikipedia.org/wiki/Linearized_line_bundle
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In mathematics, given an additive subgroup Γ ⊂ R {\displaystyle \Gamma \subset \mathbb {R} } , the Novikov ring Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} of Γ {\displaystyle \Gamma } is the subring of Z ] {\displaystyle \mathbb {Z} \!]} consisting of formal sums ∑ n γ i t γ i {\displaystyle \sum n_{\gamma _{i}}t^{\gamma _{i}}} such that γ 1 > γ 2 > ⋯ {\displaystyle \gamma _{1}>\gamma _{2}>\cdots } and γ i → − ∞ {\displaystyle \gamma _{i}\to -\infty } . The notion was introduced by Sergei Novikov in the papers that initiated the generalization of Morse theory using a closed one-form instead of a function.
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https://en.wikipedia.org/wiki/Novikov_ring
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The notion is used in quantum cohomology, among the others. The Novikov ring Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} is a principal ideal domain. Let S be the subset of Z {\displaystyle \mathbb {Z} } consisting of those with leading term 1. Since the elements of S are unit elements of Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} , the localization Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} of Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} with respect to S is a subring of Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} called the "rational part" of Nov ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} ; it is also a principal ideal domain.
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https://en.wikipedia.org/wiki/Novikov_ring
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In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that h ( u ∗ v ) = h ( u ) ⋅ h ( v ) {\displaystyle h(u*v)=h(u)\cdot h(v)} where the group operation on the left side of the equation is that of G and on the right side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, h ( e G ) = e H {\displaystyle h(e_{G})=e_{H}} and it also maps inverses to inverses in the sense that h ( u − 1 ) = h ( u ) − 1 . {\displaystyle h\left(u^{-1}\right)=h(u)^{-1}.\,} Hence one can say that h "is compatible with the group structure". Older notations for the homomorphism h(x) may be xh or xh, though this may be confused as an index or a general subscript. In automata theory, sometimes homomorphisms are written to the right of their arguments without parentheses, so that h(x) becomes simply x h {\displaystyle xh} .In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.
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https://en.wikipedia.org/wiki/Group_homomorphisms
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In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for the product measure. Let ( X 1 , Σ 1 ) {\displaystyle (X_{1},\Sigma _{1})} and ( X 2 , Σ 2 ) {\displaystyle (X_{2},\Sigma _{2})} be two measurable spaces, that is, Σ 1 {\displaystyle \Sigma _{1}} and Σ 2 {\displaystyle \Sigma _{2}} are sigma algebras on X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} respectively, and let μ 1 {\displaystyle \mu _{1}} and μ 2 {\displaystyle \mu _{2}} be measures on these spaces. Denote by Σ 1 ⊗ Σ 2 {\displaystyle \Sigma _{1}\otimes \Sigma _{2}} the sigma algebra on the Cartesian product X 1 × X 2 {\displaystyle X_{1}\times X_{2}} generated by subsets of the form B 1 × B 2 {\displaystyle B_{1}\times B_{2}} , where B 1 ∈ Σ 1 {\displaystyle B_{1}\in \Sigma _{1}} and B 2 ∈ Σ 2 .
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https://en.wikipedia.org/wiki/Product_measure
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{\displaystyle B_{2}\in \Sigma _{2}.} This sigma algebra is called the tensor-product σ-algebra on the product space. A product measure μ 1 × μ 2 {\displaystyle \mu _{1}\times \mu _{2}} (also denoted by μ 1 ⊗ μ 2 {\displaystyle \mu _{1}\otimes \mu _{2}} by many authors) is defined to be a measure on the measurable space ( X 1 × X 2 , Σ 1 ⊗ Σ 2 ) {\displaystyle (X_{1}\times X_{2},\Sigma _{1}\otimes \Sigma _{2})} satisfying the property ( μ 1 × μ 2 ) ( B 1 × B 2 ) = μ 1 ( B 1 ) μ 2 ( B 2 ) {\displaystyle (\mu _{1}\times \mu _{2})(B_{1}\times B_{2})=\mu _{1}(B_{1})\mu _{2}(B_{2})} for all B 1 ∈ Σ 1 , B 2 ∈ Σ 2 {\displaystyle B_{1}\in \Sigma _{1},\ B_{2}\in \Sigma _{2}} .
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https://en.wikipedia.org/wiki/Product_measure
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(In multiplying measures, some of which are infinite, we define the product to be zero if any factor is zero.) In fact, when the spaces are σ {\displaystyle \sigma } -finite, the product measure is uniquely defined, and for every measurable set E, ( μ 1 × μ 2 ) ( E ) = ∫ X 2 μ 1 ( E y ) d μ 2 ( y ) = ∫ X 1 μ 2 ( E x ) d μ 1 ( x ) , {\displaystyle (\mu _{1}\times \mu _{2})(E)=\int _{X_{2}}\mu _{1}(E^{y})\,d\mu _{2}(y)=\int _{X_{1}}\mu _{2}(E_{x})\,d\mu _{1}(x),} where E x = { y ∈ X 2 | ( x , y ) ∈ E } {\displaystyle E_{x}=\{y\in X_{2}|(x,y)\in E\}} and E y = { x ∈ X 1 | ( x , y ) ∈ E } {\displaystyle E^{y}=\{x\in X_{1}|(x,y)\in E\}} , which are both measurable sets. The existence of this measure is guaranteed by the Hahn–Kolmogorov theorem.
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https://en.wikipedia.org/wiki/Product_measure
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The uniqueness of product measure is guaranteed only in the case that both ( X 1 , Σ 1 , μ 1 ) {\displaystyle (X_{1},\Sigma _{1},\mu _{1})} and ( X 2 , Σ 2 , μ 2 ) {\displaystyle (X_{2},\Sigma _{2},\mu _{2})} are σ-finite. The Borel measures on the Euclidean space Rn can be obtained as the product of n copies of Borel measures on the real line R. Even if the two factors of the product space are complete measure spaces, the product space may not be. Consequently, the completion procedure is needed to extend the Borel measure into the Lebesgue measure, or to extend the product of two Lebesgue measures to give the Lebesgue measure on the product space. The opposite construction to the formation of the product of two measures is disintegration, which in some sense "splits" a given measure into a family of measures that can be integrated to give the original measure.
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https://en.wikipedia.org/wiki/Product_measure
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In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its image has a supremum in Q, and that supremum is the image of the supremum of D, i.e. ⊔ f = f ( ⊔ D ) {\displaystyle \sqcup f=f(\sqcup D)} , where ⊔ {\displaystyle \sqcup } is the directed join. When Q {\displaystyle Q} is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets.A subset O of a partially ordered set P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have non-empty intersection with O. The Scott-open subsets of a partially ordered set P form a topology on P, the Scott topology. A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology.The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.Scott-continuous functions show up in the study of models for lambda calculi and the denotational semantics of computer programs.
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https://en.wikipedia.org/wiki/Scott_continuous
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In mathematics, given two submanifolds A and B of a manifold X intersecting in two points p and q, a Whitney disc is a mapping from the two-dimensional disc D, with two marked points, to X, such that the two marked points go to p and q, one boundary arc of D goes to A and the other to B.Their existence and embeddedness is crucial in proving the cobordism theorem, where it is used to cancel the intersection points; and its failure in low dimensions corresponds to not being able to embed a Whitney disc. Casson handles are an important technical tool for constructing the embedded Whitney disc relevant to many results on topological four-manifolds. Pseudoholomorphic Whitney discs are counted by the differential in Lagrangian intersection Floer homology. == References ==
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https://en.wikipedia.org/wiki/Whitney_disk
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In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis uses techniques in infinite-dimensional manifold theory and topological spaces of mappings to classify behaviors of differential equations, particularly nonlinear differential equations. These spaces can include singularities and hence catastrophe theory is a part of global analysis. Optimization problems, such as finding geodesics on Riemannian manifolds, can be solved using differential equations, so that the calculus of variations overlaps with global analysis. Global analysis finds application in physics in the study of dynamical systems and topological quantum field theory.
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https://en.wikipedia.org/wiki/Global_analysis
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In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. It is particularly useful in machine learning for minimizing the cost or loss function.
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https://en.wikipedia.org/wiki/Gradient_descent
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Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades.A simple extension of gradient descent, stochastic gradient descent, serves as the most basic algorithm used for training most deep networks today.
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https://en.wikipedia.org/wiki/Gradient_descent
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In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.
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https://en.wikipedia.org/wiki/Algorithmic_graph_theory
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In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Algebraic graph theory has close links with group theory. Algebraic graph theory has been applied to many areas including dynamic systems and complexity.
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https://en.wikipedia.org/wiki/Algorithmic_graph_theory
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In mathematics, helical boundary conditions are a variation on periodic boundary conditions. Helical boundary conditions provide a method for determining the index of a lattice site's neighbours when each lattice site is indexed by just a single coordinate. On a lattice of dimension d where the lattice sites are numbered from 1 to N and L is the width (i.e. number of elements per row) of the lattice in all but the last dimension, the neighbors of site i are: ( i ± 1 ) mod N {\displaystyle (i\pm 1)\mod N} ( i ± L ) mod N {\displaystyle (i\pm L)\mod N} … {\displaystyle \ldots } ( i ± L d − 1 ) mod N {\displaystyle (i\pm L^{d-1})\mod N} where the modulo operator is used. It is not necessary that N = Ld. Helical boundary conditions make it possible to use only one coordinate to describe arbitrary-dimensional lattices.
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https://en.wikipedia.org/wiki/Helical_boundary_conditions
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In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid. In higher category theory, the concept of higher categorical structures, such as (∞-categories), allows for a more robust treatment of homotopy theory, enabling one to capture finer homotopical distinctions, such as differentiating two topological spaces that have the same fundamental group, but differ in their higher homotopy groups. This approach is particularly valuable when dealing with spaces with intricate topological features, such as the Eilenberg-MacLane space.
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https://en.wikipedia.org/wiki/Higher_category_theory
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In mathematics, holes are examined in a number of ways. One of these is in homology, which is a general way of associating certain algebraic objects to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology, and homology was originally a rigorous mathematical method for defining and categorizing holes in a mathematical object called a manifold. The initial motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.
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https://en.wikipedia.org/wiki/Blind_hole
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For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. Because a hole is immaterial, it is not immediately obvious how to define one or distinguish it from others. Another is the notion of homotopy group: these are invariants of a topological space that, when non-trivial (one also says in this case that the space is not k-connected), detect the presence of "holes" in the sense that the space contains a sphere that cannot be contracted to a point. The term of hole is often used informally when discussing these objects.For surfaces a notion closer to the intuitive meaning exists: the genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. In layman's terms, it is exactly the number of "holes" the surface has, when represented as a submanifold in 3-space.
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https://en.wikipedia.org/wiki/Blind_hole
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In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is to construct an operator, f(T), which naturally extends the function f from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of T to the bounded operators. This article will discuss the case where T is a bounded linear operator on some Banach space. In particular, T can be a square matrix with complex entries, a case which will be used to illustrate functional calculus and provide some heuristic insights for the assumptions involved in the general construction.
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https://en.wikipedia.org/wiki/Polynomial_functional_calculus
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In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.
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https://en.wikipedia.org/wiki/Homogeneous_coordinates
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If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane.
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https://en.wikipedia.org/wiki/Homogeneous_coordinates
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In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth. The following list given by Melvin Hochster is considered definitive for this area. In the sequel, A , R {\displaystyle A,R} , and S {\displaystyle S} refer to Noetherian commutative rings; R {\displaystyle R} will be a local ring with maximal ideal m R {\displaystyle m_{R}} , and M {\displaystyle M} and N {\displaystyle N} are finitely generated R {\displaystyle R} -modules.
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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The Zero Divisor Theorem. If M ≠ 0 {\displaystyle M\neq 0} has finite projective dimension and r ∈ R {\displaystyle r\in R} is not a zero divisor on M {\displaystyle M} , then r {\displaystyle r} is not a zero divisor on R {\displaystyle R} . Bass's Question.
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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If M ≠ 0 {\displaystyle M\neq 0} has a finite injective resolution then R {\displaystyle R} is a Cohen–Macaulay ring. The Intersection Theorem. If M ⊗ R N ≠ 0 {\displaystyle M\otimes _{R}N\neq 0} has finite length, then the Krull dimension of N (i.e., the dimension of R modulo the annihilator of N) is at most the projective dimension of M. The New Intersection Theorem.
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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Let 0 → G n → ⋯ → G 0 → 0 {\displaystyle 0\to G_{n}\to \cdots \to G_{0}\to 0} denote a finite complex of free R-modules such that ⨁ i H i ( G ∙ ) {\displaystyle \bigoplus \nolimits _{i}H_{i}(G_{\bullet })} has finite length but is not 0. Then the (Krull dimension) dim R ≤ n {\displaystyle \dim R\leq n} . The Improved New Intersection Conjecture.
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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Let 0 → G n → ⋯ → G 0 → 0 {\displaystyle 0\to G_{n}\to \cdots \to G_{0}\to 0} denote a finite complex of free R-modules such that H i ( G ∙ ) {\displaystyle H_{i}(G_{\bullet })} has finite length for i > 0 {\displaystyle i>0} and H 0 ( G ∙ ) {\displaystyle H_{0}(G_{\bullet })} has a minimal generator that is killed by a power of the maximal ideal of R. Then dim R ≤ n {\displaystyle \dim R\leq n} . The Direct Summand Conjecture. If R ⊆ S {\displaystyle R\subseteq S} is a module-finite ring extension with R regular (here, R need not be local but the problem reduces at once to the local case), then R is a direct summand of S as an R-module.
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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The conjecture was proven by Yves André using a theory of perfectoid spaces. The Canonical Element Conjecture. Let x 1 , … , x d {\displaystyle x_{1},\ldots ,x_{d}} be a system of parameters for R, let F ∙ {\displaystyle F_{\bullet }} be a free R-resolution of the residue field of R with F 0 = R {\displaystyle F_{0}=R} , and let K ∙ {\displaystyle K_{\bullet }} denote the Koszul complex of R with respect to x 1 , … , x d {\displaystyle x_{1},\ldots ,x_{d}} .
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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Lift the identity map R = K 0 → F 0 = R {\displaystyle R=K_{0}\to F_{0}=R} to a map of complexes. Then no matter what the choice of system of parameters or lifting, the last map from R = K d → F d {\displaystyle R=K_{d}\to F_{d}} is not 0. Existence of Balanced Big Cohen–Macaulay Modules Conjecture.
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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There exists a (not necessarily finitely generated) R-module W such that mRW ≠ W and every system of parameters for R is a regular sequence on W. Cohen-Macaulayness of Direct Summands Conjecture. If R is a direct summand of a regular ring S as an R-module, then R is Cohen–Macaulay (R need not be local, but the result reduces at once to the case where R is local). The Vanishing Conjecture for Maps of Tor.
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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Let A ⊆ R → S {\displaystyle A\subseteq R\to S} be homomorphisms where R is not necessarily local (one can reduce to that case however), with A, S regular and R finitely generated as an A-module. Let W be any A-module. Then the map Tor i A ( W , R ) → Tor i A ( W , S ) {\displaystyle \operatorname {Tor} _{i}^{A}(W,R)\to \operatorname {Tor} _{i}^{A}(W,S)} is zero for all i ≥ 1 {\displaystyle i\geq 1} .
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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The Strong Direct Summand Conjecture. Let R ⊆ S {\displaystyle R\subseteq S} be a map of complete local domains, and let Q be a height one prime ideal of S lying over x R {\displaystyle xR} , where R and R / x R {\displaystyle R/xR} are both regular. Then x R {\displaystyle xR} is a direct summand of Q considered as R-modules.
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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Existence of Weakly Functorial Big Cohen-Macaulay Algebras Conjecture. Let R → S {\displaystyle R\to S} be a local homomorphism of complete local domains. Then there exists an R-algebra BR that is a balanced big Cohen–Macaulay algebra for R, an S-algebra B S {\displaystyle B_{S}} that is a balanced big Cohen-Macaulay algebra for S, and a homomorphism BR → BS such that the natural square given by these maps commutes.
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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Serre's Conjecture on Multiplicities. (cf. Serre's multiplicity conjectures.) Suppose that R is regular of dimension d and that M ⊗ R N {\displaystyle M\otimes _{R}N} has finite length. Then χ ( M , N ) {\displaystyle \chi (M,N)} , defined as the alternating sum of the lengths of the modules Tor i R ( M , N ) {\displaystyle \operatorname {Tor} _{i}^{R}(M,N)} is 0 if dim M + dim N < d {\displaystyle \dim M+\dim N
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https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra
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In mathematics, homological stability is any of a number of theorems asserting that the group homology of a series of groups G 1 ⊂ G 2 ⊂ ⋯ {\displaystyle G_{1}\subset G_{2}\subset \cdots } is stable, i.e., H i ( G n ) {\displaystyle H_{i}(G_{n})} is independent of n when n is large enough (depending on i). The smallest n such that the maps H i ( G n ) → H i ( G n + 1 ) {\displaystyle H_{i}(G_{n})\to H_{i}(G_{n+1})} is an isomorphism is referred to as the stable range. The concept of homological stability was pioneered by Daniel Quillen whose proof technique has been adapted in various situations.
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https://en.wikipedia.org/wiki/Homological_stability
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In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.
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https://en.wikipedia.org/wiki/Homology_(mathematics)
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The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes.
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https://en.wikipedia.org/wiki/Homology_(mathematics)
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Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold. Loosely speaking, a cycle is a closed submanifold, a boundary is a cycle which is also the boundary of a submanifold, and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries. A homology class is thus represented by a cycle which is not the boundary of any submanifold: the cycle represents a hole, namely a hypothetical manifold whose boundary would be that cycle, but which is "not there".
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https://en.wikipedia.org/wiki/Homology_(mathematics)
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There are many different homology theories. A particular type of mathematical object, such as a topological space or a group, may have one or more associated homology theories. When the underlying object has a geometric interpretation as topological spaces do, the nth homology group represents behavior in dimension n. Most homology groups or modules may be formulated as derived functors on appropriate abelian categories, measuring the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a derived category.
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https://en.wikipedia.org/wiki/Homology_(mathematics)
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In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the abelian aspects as special cases. The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categories. This subject has received much attention in recent years due to new foundational work of Vladimir Voevodsky, Eric Friedlander, Andrei Suslin, and others resulting in the A1 homotopy theory for quasiprojective varieties over a field. Voevodsky has used this new algebraic homotopy theory to prove the Milnor conjecture (for which he was awarded the Fields Medal) and later, in collaboration with Markus Rost, the full Bloch–Kato conjecture.
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https://en.wikipedia.org/wiki/Homotopic_algebra
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In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.
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https://en.wikipedia.org/wiki/Relative_homotopy_group
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Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the n-th homotopy group, π n ( X ) , {\displaystyle \pi _{n}(X),} of the given space X with base point. Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but topological spaces that are not homeomorphic can have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan.
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https://en.wikipedia.org/wiki/Relative_homotopy_group
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In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
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https://en.wikipedia.org/wiki/Algebraic_Topology
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In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is learned as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories).
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https://en.wikipedia.org/wiki/Abstract_homotopy_theory
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In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions. Such an operation is often also called hyperbolic Dehn filling, as Dehn surgery proper refers to a "drill and fill" operation on a link which consists of drilling out a neighborhood of the link and then filling back in with solid tori. Hyperbolic Dehn surgery actually only involves "filling".
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https://en.wikipedia.org/wiki/Dehn_filling
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We will generally assume that a hyperbolic 3-manifold is complete. Suppose M is a cusped hyperbolic 3-manifold with n cusps. M can be thought of, topologically, as the interior of a compact manifold with toral boundary.
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https://en.wikipedia.org/wiki/Dehn_filling
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Suppose we have chosen a meridian and longitude for each boundary torus, i.e. simple closed curves that are generators for the fundamental group of the torus. Let M ( u 1 , u 2 , … , u n ) {\displaystyle M(u_{1},u_{2},\dots ,u_{n})} denote the manifold obtained from M by filling in the i-th boundary torus with a solid torus using the slope u i = p i / q i {\displaystyle u_{i}=p_{i}/q_{i}} where each pair p i {\displaystyle p_{i}} and q i {\displaystyle q_{i}} are coprime integers. We allow a u i {\displaystyle u_{i}} to be ∞ {\displaystyle \infty } which means we do not fill in that cusp, i.e. do the "empty" Dehn filling.
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https://en.wikipedia.org/wiki/Dehn_filling
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So M = M ( ∞ , … , ∞ ) {\displaystyle M(\infty ,\dots ,\infty )} . We equip the space H of finite volume hyperbolic 3-manifolds with the geometric topology. Thurston's hyperbolic Dehn surgery theorem states: M ( u 1 , u 2 , … , u n ) {\displaystyle M(u_{1},u_{2},\dots ,u_{n})} is hyperbolic as long as a finite set of exceptional slopes E i {\displaystyle E_{i}} is avoided for the i-th cusp for each i. In addition, M ( u 1 , u 2 , … , u n ) {\displaystyle M(u_{1},u_{2},\dots ,u_{n})} converges to M in H as all p i 2 + q i 2 → ∞ {\displaystyle p_{i}^{2}+q_{i}^{2}\rightarrow \infty } for all p i / q i {\displaystyle p_{i}/q_{i}} corresponding to non-empty Dehn fillings u i {\displaystyle u_{i}} .
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https://en.wikipedia.org/wiki/Dehn_filling
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This theorem is due to William Thurston and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in H. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem. Another important result by Thurston is that volume decreases under hyperbolic Dehn filling.
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https://en.wikipedia.org/wiki/Dehn_filling
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In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic. The proof relies on basic properties of the Gromov norm. Jørgensen also showed that the volume function on this space is a continuous, proper function.
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https://en.wikipedia.org/wiki/Dehn_filling
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Thus by the previous results, nontrivial limits in H are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has ordinal type ω ω {\displaystyle \omega ^{\omega }} .
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https://en.wikipedia.org/wiki/Dehn_filling
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This result is known as the Thurston-Jørgensen theorem. Further work characterizing this set was done by Gromov. The figure-eight knot and the (-2, 3, 7) pretzel knot are the only two knots whose complements are known to have more than 6 exceptional surgeries; they have 10 and 7, respectively.
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https://en.wikipedia.org/wiki/Dehn_filling
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