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c_pi08vc4tro1s | 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 limi... | Infinite product |
c_x0lxh9zt50v0 | 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 ... | Infinite product |
c_knr1p6sohbs6 | 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 us... | Null model |
c_m66f95h772hy | 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 eac... | Null model |
c_dhhb6cpdj52r | 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... | Discrete Logarithm |
c_8fa4ycsgxhhd | 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 vecto... | Vectorial addition chain |
c_h296a4c0fbx0 | 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 lis... | List of forcing notions |
c_h7x210bybrai | 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 modul... | Formal moduli |
c_s6susbm0hmz2 | 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. | Function application |
c_x9dzzm1bsicm | 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 t... | Function composition |
c_ac8lt0mb2rqs | 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. | List of types of functions |
c_rqtnw14ib8h9 | 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 19... | Fuzzy measure theory |
c_m46x3uitgd3b | 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. | Fuzzy set theory |
c_rbx4fs4n2ozs | 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... | Fuzzy set theory |
c_2i5f3305jl33 | 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... | Fuzzy set theory |
c_1ilk7gas5k39 | 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... | Point-set topology |
c_pzsl2j2vt296 | 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. | Point-set topology |
c_54br6qdwg6ik | 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, ... | Point-set topology |
c_hmcc3gmhop0n | 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 ov... | Generalized Verma module |
c_qifx2zgntsyk | 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 suc... | Generalized functions |
c_bmjzsjogfdq5 | 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 cl... | Generalized functions |
c_om7jpi5dtacq | 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). | Generalised mean |
c_nxqii7amizqg | 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. | Genus (mathematics) |
c_i8ng73jxh0kx | 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 ... | Surface map |
c_f8lf7af5y802 | 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. | Geometric calculus |
c_eelgpw4nt3ad | 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 h... | Geometric class field theory |
c_6dzp9o161vcf | 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 ac... | Semistable point |
c_8k9ff7awe0dk | 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. | Geometric measure theory |
c_xvwgefghhkfh | In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. | Geometric topology |
c_rjp5n6j9g2cx | 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–Wei... | Geometry and topology |
c_znv7mr51skz9 | 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 ... | Galois descent |
c_b51pypgomcyl | 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,} whe... | Quotient by an equivalence relation |
c_4kuag4psrxa5 | 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 exac... | Exact cover |
c_3g6ekga6mcy2 | 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 "cont... | Exact cover |
c_4q8y070vl6vu | 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 \math... | Rank factorization |
c_ygtdg598yfqq | 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 use... | G-module |
c_19qoyefsn3zt | 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\rightarr... | Lebesgue point |
c_jklh02r4t4rh | 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 a... | Smallest bounding sphere |
c_4b34x9hige3x | 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 know... | Smallest bounding sphere |
c_y81c4ytetmlu | 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 ... | Product order |
c_acxkflq55abs | 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... | Product order |
c_3zmnehmulr0v | 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 {\displaysty... | Product order |
c_oj985lxyftck | 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... | Product order |
c_943ejs5q3kt3 | {\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. | Product order |
c_m43w8ypbgz37 | 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 \ma... | Semi-invariant of a quiver |
c_yvd9vjiza00q | 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... | K-theory spectrum |
c_xzkbomnn9a7p | 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 surfac... | Tangential and normal components |
c_hqc4dpj0tgms | 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 fo... | Null cone |
c_ii9mqlxaeec3 | 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 _{... | Linearized line bundle |
c_5a6iij90hahs | 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_{\ga... | Novikov ring |
c_2eccz1grr1v6 | 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 ) {\d... | Novikov ring |
c_qs8vcji0lq8c | 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 rig... | Group homomorphisms |
c_mu2tmpsabkqv | 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 t... | Product measure |
c_0ipnc3k8v6t5 | {\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 spac... | Product measure |
c_1ct5im5hairh | (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 ) ... | Product measure |
c_p3oi4dkvwuqk | 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 ... | Product measure |
c_pcge6gjut7a8 | 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 t... | Scott continuous |
c_8c34h2zw6x8p | 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... | Whitney disk |
c_lq3s3gmgzzno | 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 differe... | Global analysis |
c_nic8i2mj5iu4 | 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, b... | Gradient descent |
c_tbgb73ggit8t | 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 ... | Gradient descent |
c_eyvko44ghk0i | 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... | Algorithmic graph theory |
c_td0kzv1sztng | 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. | Algorithmic graph theory |
c_9zgs99atudn7 | 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 number... | Helical boundary conditions |
c_29vpgkzf62oz | 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 t... | Higher category theory |
c_k0nvh2pg8zvw | 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 mathematica... | Blind hole |
c_ma35ia4atbue | 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 on... | Blind hole |
c_bttde1qln6k8 | 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 ... | Polynomial functional calculus |
c_tbpqda3yfmvj | 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 poin... | Homogeneous coordinates |
c_lw0vl9n2yrr3 | 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 c... | Homogeneous coordinates |
c_wox8gtykdbzz | 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 ... | Homological conjectures in commutative algebra |
c_yaj249if8fw0 | 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. | Homological conjectures in commutative algebra |
c_4ivmtgl60w7p | 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 proje... | Homological conjectures in commutative algebra |
c_nsg3qkts6ow6 | 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... | Homological conjectures in commutative algebra |
c_i3nezk9ebbzc | 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 ... | Homological conjectures in commutative algebra |
c_5bbljv5fad4p | 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}... | Homological conjectures in commutative algebra |
c_buyrabbmwdbm | 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. | Homological conjectures in commutative algebra |
c_9j4xt98y7q7f | 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 r... | Homological conjectures in commutative algebra |
c_fnt85li9iun1 | 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,... | Homological conjectures in commutative algebra |
c_wwluz7ffv936 | 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-... | Homological conjectures in commutative algebra |
c_pn1iismxlr4o | 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 a... | Homological conjectures in commutative algebra |
c_5i0vpcpjlj6i | 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 ) {\displ... | Homological conjectures in commutative algebra |
c_6j61drcea4z2 | 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... | Homological stability |
c_mjwey4sddtwf | 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,... | Homology (mathematics) |
c_oyhwd449igj1 | 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-dimensi... | Homology (mathematics) |
c_p79inchxd9lt | 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 boundari... | Homology (mathematics) |
c_ty1z5x1tu8rx | 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. ... | Homology (mathematics) |
c_r2sm1rtjpd0r | 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 a... | Homotopic algebra |
c_t1uiz32baouy | 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 basi... | Relative homotopy group |
c_jgwm4seixy2t | 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 to... | Relative homotopy group |
c_kku8tdn9htke | 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. | Algebraic Topology |
c_kry93k8e91mn | 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 al... | Abstract homotopy theory |
c_a25v9jgsvdeo | 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... | Dehn filling |
c_uy3949wysnpa | 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. | Dehn filling |
c_knzjhx32kz1j | 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 ... | Dehn filling |
c_gxmaplqqes0o | 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 e... | Dehn filling |
c_hrw6238tzxt9 | 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 v... | Dehn filling |
c_5upatqf2dn9h | 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. | Dehn filling |
c_k7oo8d7jdnw0 | 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 }} . | Dehn filling |
c_x7huw233ao1m | 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. | Dehn filling |
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