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c_w4zfivnmdfme | A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes. Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic fu... | Formal algebraic geometry |
c_vxamasfdaxpk | In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on co... | Grothendieck-Riemann-Roch theorem |
c_r4e9qbuc5c23 | The Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves. The theorem has been very influential, not least for the developme... | Grothendieck-Riemann-Roch theorem |
c_2jd644zw0psj | Conversely, complex analytic analogues of the Grothendieck–Riemann–Roch theorem can be proved using the index theorem for families. Alexander Grothendieck gave a first proof in a 1957 manuscript, later published. Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck's proof in 1958. Later, Grothendieck... | Grothendieck-Riemann-Roch theorem |
c_441lvmgjg0ek | In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieti... | Fiber product of schemes |
c_i4ati46rj1it | In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f: X → Y {\displaystyle f:X\to Y} , the inverse image functor is a functor from the category of sheaves on Y to the category of sh... | Inverse image functor |
c_p40qsxechht0 | In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod. One can define a ho... | Dimension axiom |
c_nrrq7iiafikw | In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space X × Y {\displaystyle X\times Y} and those of the spaces X {\displaystyle X} and Y {\displaystyle Y} . The theorem first appeared in a 1953 paper... | Eilenberg–Zilber theorem |
c_s9sld69noxot | In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characte... | Euler class |
c_sh8y9woyxosk | In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ri... | Cup product |
c_yerr3lxywm2n | In mathematics, specifically in axiomatic set theory, a Hartogs number is an ordinal number associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardin... | Hartogs number |
c_u7lcfk2x04x1 | In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where f ′ {\displaystyle f'} is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely f ′ , {\displaysty... | Logarithmic differential |
c_bdt7hys00ieb | In mathematics, specifically in category theory and algebraic topology, the Baez–Dolan stabilization hypothesis, proposed in (Baez & Dolan 1995), states that suspension of a weak n-category has no more essential effect after n + 2 times. Precisely, it states that the suspension functor n C a t k → n C a t k + 1 {\displ... | Stabilization hypothesis |
c_8fkshumigh68 | In mathematics, specifically in category theory, Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in 1970 in the general context of enriched functor categories. Day convolution acts as a tensor product for a monoidal cat... | Day convolution |
c_y5nrhou77th9 | In mathematics, specifically in category theory, F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor F, the signature... | F-algebra |
c_bjeu27heyfru | In mathematics, specifically in category theory, a functor F: C → D {\displaystyle F:C\to D} is essentially surjective (or dense) if each object d {\displaystyle d} of D {\displaystyle D} is isomorphic to an object of the form F c {\displaystyle Fc} for some object c {\displaystyle c} of C {\displaystyle C} . Any funct... | Essentially surjective functor |
c_gwgp976io2m2 | In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C... | Preabelian category |
c_1t3vthk1ym6h | In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of m... | Preadditive categories |
c_5biahkcg290g | In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel. Recall that an idempotent morphism p {\displaystyle p} is an endomorphism of an object with the property that p ∘ p = p {\displaystyle p\circ p=p} . Elementary con... | Pseudo-abelian category |
c_0grztnfcs23l | In mathematics, specifically in category theory, a quasi-abelian category is a pre-abelian category in which the pushout of a kernel along arbitrary morphisms is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel. | Quasi-abelian category |
c_mtoobjug93qi | In mathematics, specifically in category theory, a semi-abelian category is a pre-abelian category in which the induced morphism f ¯: coim f → im f {\displaystyle {\overline {f}}:\operatorname {coim} f\rightarrow \operatorname {im} f} is a bimorphism, i.e., a monomorphism and an epimorphism, for every morphism f {\... | Semi-abelian category |
c_fbt1ek4s1zmt | In mathematics, specifically in category theory, an F {\displaystyle F} -coalgebra is a structure defined according to a functor F {\displaystyle F} , with specific properties as defined below. For both algebras and coalgebras, a functor is a convenient and general way of organizing a signature. This has applications i... | F-coalgebra |
c_x7wvnfyl5lus | In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. | Additive category |
c_0owywjz6v4wc | In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined product... | Exponential object |
c_la5pgsz2y4ep | In mathematics, specifically in category theory, an extranatural transformation is a generalization of the notion of natural transformation. | Extranatural transformation |
c_5tx8c48ika7e | In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. | Canonical bifunctor |
c_73tnhqdms5cb | In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms. The initial obj... | Category of small categories |
c_82lhiet6mtnj | The terminal object is the terminal category or trivial category 1 with a single object and morphism.The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”. But it is possible to form ... | Category of small categories |
c_u86aw5baqgvf | In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an element from every non-empty set. | Axiom of global choice |
c_lcbxzmqaf072 | In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting poin... | Normal polytope |
c_4ghcfps6cyu2 | In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial P is given by an ex... | Elementary symmetric polynomial |
c_uo6vrgeq1pml | In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rati... | Power sum symmetric polynomial |
c_sx1ckr9s2u95 | In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk. | Fatou's theorem |
c_i7p3s3leudft | In mathematics, specifically in complex geometry, the Kähler quotient of a Kähler manifold X {\displaystyle X} by a Lie group G {\displaystyle G} acting on X {\displaystyle X} by preserving the Kähler structure and with moment map μ: X → g ∗ {\displaystyle \mu :X\to {\mathfrak {g}}^{*}} (with respect to the Kähler form... | Kähler quotient |
c_yocsilo9rmjb | In mathematics, specifically in computational geometry, geometric nonrobustness is a problem wherein branching decisions in computational geometry algorithms are based on approximate numerical computations, leading to various forms of unreliability including ill-formed output and software failure through crashing or in... | Robust geometric computation |
c_0g656l2cjac9 | For instance, two-dimensional convex hulls can be computed using predicates that test the sign of quadratic polynomials, and therefore may require twice as many bits of precision within these calculations as the input numbers. When integer arithmetic cannot be used (for instance, when the result of a calculation is an ... | Robust geometric computation |
c_ygpnr3s2mgif | In mathematics, specifically in control theory, subspace identification (SID) aims at identifying linear time invariant (LTI) state space models from input-output data. SID does not require that the user parametrizes the system matrices before solving a parametric optimization problem and, as a consequence, SID methods... | Subspace identification method |
c_hpcsidqzmpw8 | In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. | Equilibrium points |
c_vb3d8jw2os7d | In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form g = φ ( d x 1 2 + ⋯ + d x n 2 ) , {\displaystyle g=\va... | Isothermal coordinates |
c_2ud2i7sthij0 | Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally ... | Isothermal coordinates |
c_chc1fbtjv365 | In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse t... | Morse–Bott function |
c_3mgacgse7gba | Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics (critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem... | Morse–Bott function |
c_gj8gdrlujch3 | In mathematics, specifically in differential topology, a Kervaire manifold K 4 n + 2 {\displaystyle K^{4n+2}} is a piecewise-linear manifold of dimension 4 n + 2 {\displaystyle 4n+2} constructed by Michel Kervaire (1960) by plumbing together the tangent bundles of two ( 2 n + 1 ) {\displaystyle (2n+1)} -spheres, and th... | Kervaire manifold |
c_pw1scx6t8fgl | In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. The method is also occasionally known as the "cross your heart" method because lines resemb... | Rule of proportion |
c_jq9mac3jxvuj | In mathematics, specifically in functional analysis and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessarily and sufficient condition for a Hausdorff pre-Hilbert space to be a Hilbert space in terms of the canonical isometry of a pre-Hilbert space into its anti-dual. | Fundamental theorem of Hilbert spaces |
c_9rc8d7h96vzy | In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C {\displaystyle \mathbb {C} } ) to ... | Vector-valued Hahn–Banach theorems |
c_kbmo6w12eo28 | In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X {\displaystyle X} that has a partial order ≤ {\displaystyle \,\leq \,} making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid ... | Topological vector lattice |
c_p1bl5k6gzhe6 | In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) X that has a partial order ≤ making it into an ordered vector space whose positive cone C := { x ∈ X: x ≥ 0 } {\displaystyle C:=\left\{x\in X:x\geq 0... | Ordered topological vector space |
c_12q3gftkpuqy | In mathematics, specifically in functional analysis, a Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are inner (that is of the form a ↦ a . x − x . a {\displaystyle a\mapsto a.x-x.a} for some x {\displaystyle x} in the dual module). An equivalent characterization is that ... | Amenable Banach algebra |
c_s5pdtk7wm1yh | In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: A i... | C* algebra |
c_lqr43zqfxpab | This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras, which culminated in a series of papers on rings of operators. These papers cons... | C* algebra |
c_3sx192h8tvdr | Around 1943, the work of Israel Gelfand and Mark Naimark yielded an abstract characterisation of C*-algebras making no reference to operators on a Hilbert space. C*-algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quant... | C* algebra |
c_d1dkd0ut0th8 | In mathematics, specifically in functional analysis, a family G {\displaystyle {\mathcal {G}}} of subsets a topological vector space (TVS) X {\displaystyle X} is said to be saturated if G {\displaystyle {\mathcal {G}}} contains a non-empty subset of X {\displaystyle X} and if for every G ∈ G , {\displaystyle G\in {\mat... | Saturated family |
c_qcgulcwbo4x9 | In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions ƒ: D → C {\displaystyle \mathbb {C} } ,(where D is the open unit disk in the complex plane C {\displaystyle \mathbb {C} } ) that extend to a continuous function on the c... | Disk algebra |
c_63ygh8ke0lxq | By construction the disc algebra is a closed subalgebra of the Hardy space H∞. In contrast to the stronger requirement that a continuous extension to the circle exists, it is a lemma of Fatou that a general element of H∞ can be radially extended to the circle almost everywhere. == References == | Disk algebra |
c_4wfttsn81rq4 | In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961). Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery... | Algebraic surgery theory |
c_m1h3sbjebjxr | This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood manifold M and perform surgery on it to produce a manifold M′ having some desired property, in such a way that the effects on the homology, homotopy groups, or other invariants of... | Algebraic surgery theory |
c_9rhcoaqem67z | In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a mathematical tree called a hydra where, usually, the goal is to cut off the hydra's "heads" while the hydra simultaneously expands itself. Hydra games can be used to generate large num... | Hydra game |
c_c7wony9t8d0j | In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in which the common order is p are a particular kind of p-group. A group for which ... | Elementary abelian |
c_uoio98beyqg8 | Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript notation means the n-fold direct product of groups.In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p. (Note that in the finite case the direct product and direc... | Elementary abelian |
c_3hzbgv9b1hkm | In mathematics, specifically in group theory, residue-class-wise affine groups are certain permutation groups acting on Z {\displaystyle \mathbb {Z} } (the integers), whose elements are bijective residue-class-wise affine mappings. A mapping f: Z → Z {\displaystyle f:\mathbb {Z} \rightarrow \mathbb {Z} } is called resi... | Residue-class-wise affine group |
c_j0xr2qoooko0 | Many of them act multiply transitively on Z {\displaystyle \mathbb {Z} } or on subsets thereof. A particularly basic type of residue-class-wise affine permutations are the class transpositions: given disjoint residue classes r 1 ( m 1 ) {\displaystyle r_{1}(m_{1})} and r 2 ( m 2 ) {\displaystyle r_{2}(m_{2})} , the cor... | Residue-class-wise affine group |
c_561owpepg71w | In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th roots. The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian grou... | Quasicyclic group |
c_ssxqo5qjmam7 | In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. an... | Semi-direct product |
c_zi6s0g1wly5o | In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In t... | Direct product (group theory) |
c_4olnrm0vki5d | In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but th... | Groups of Lie type |
c_qd35zv3cmgfq | In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer. | Commensurability (group theory) |
c_kedv8jk061rk | In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homo... | Generalized homology theory |
c_an3zojvzcq1a | From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout... | Generalized homology theory |
c_vto1zfs1gmr2 | In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism". The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g... | Coherence theorem |
c_o8uj9epv12q8 | In mathematics, specifically in homotopy theory in the context of a model category M, a fibrant object A of M is an object that has a fibration to the terminal object of the category. | Fibrant object |
c_e6k2ovxall26 | In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifol... | Classifying space |
c_r7bssp6jtrq5 | This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamental group of X is isomorphic to G and ... | Classifying space |
c_1h6qpgwlp65n | In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a complete quadrilateral is ... | Complete quadrilateral |
c_i6909xxd1ptq | In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite was originally proved by Helmut Hasse, and the general result was prove... | Hasse–Arf theorem |
c_7rprr53quggj | In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. | Borel probability measure |
c_j5dxnzk9juyg | In mathematics, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ. | Trivial measure |
c_qeb4sah8uyy6 | In mathematics, specifically in number theory, Newman's conjecture is a conjecture about the behavior of the partition function modulo any integer. Specifically, it states that for any integers m and r such that 0 ≤ r ≤ m − 1 {\displaystyle 0\leq r\leq m-1} , the value of the partition function p ( n ) {\displaystyle p... | Newman's conjecture |
c_jo0puxnl8s78 | In mathematics, specifically in number theory, a Cunningham number is a certain kind of integer named after English mathematician A. J. C. Cunningham. | Cunningham number |
c_gtmud33u5uvn | In mathematics, specifically in number theory, a binomial number is an integer which can be obtained by evaluating a homogeneous polynomial containing two terms. It is a generalization of a Cunningham number. | Binomial number |
c_fr0faijttdbg | In mathematics, specifically in number theory, the extremal orders of an arithmetic function are best possible bounds of the given arithmetic function. Specifically, if f(n) is an arithmetic function and m(n) is a non-decreasing function that is ultimately positive and lim inf n → ∞ f ( n ) m ( n ) = 1 {\displaystyle \... | Extremal orders of an arithmetic function |
c_5ltk62avi070 | In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-... | Baum–Connes conjecture |
c_g1h0as8fso4k | For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for discrete torsion-free groups, and the injectivity is closely related to the Novikov conjecture. The conjecture is also closely related to index theory, as the assembly map μ {\displaystyle \mu } is a sort of index, and it plays a major rol... | Baum–Connes conjecture |
c_0hzncaaxu7fe | In mathematics, specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle A^{*}} on that space according to the rule ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ , {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}... | Hermitian conjugate |
c_tps7yrlk77zw | The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H {\displaystyle H} . The definition has been further extended to include unbounded densely defined operators whose domain is topologically dense in - but not necessarily equal to - H . {\displaystyle H.} | Hermitian conjugate |
c_uurijdfcqb08 | In mathematics, specifically in order theory and functional analysis, a Fréchet lattice is a topological vector lattice that is also a Fréchet space. Fréchet lattices are important in the theory of topological vector lattices. | Fréchet lattice |
c_25jnplaj5kr4 | In mathematics, specifically in order theory and functional analysis, a band in a vector lattice X {\displaystyle X} is a subspace M {\displaystyle M} of X {\displaystyle X} that is solid and such that for all S ⊆ M {\displaystyle S\subseteq M} such that x = sup S {\displaystyle x=\sup S} exists in X , {\displaystyle X... | Band (order theory) |
c_ocq5um4hlb3s | In mathematics, specifically in order theory and functional analysis, a filter F {\displaystyle {\mathcal {F}}} in an order complete vector lattice X {\displaystyle X} is order convergent if it contains an order bounded subset (that is, is contained in an interval of the form := { x ∈ X: a ≤ x and x ≤ b } {\displaysty... | Order convergence |
c_d0c7jfyzufr5 | In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices. | Locally convex vector lattice |
c_shquqfdzm132 | In mathematics, specifically in order theory and functional analysis, a normed lattice is a topological vector lattice that is also a normed space whose unit ball is a solid set. Normed lattices are important in the theory of topological vector lattices. They are closely related to Banach vector lattices, which are nor... | Normed lattice |
c_67qijzakpn1y | In mathematics, specifically in order theory and functional analysis, a sequence of positive elements ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} in a preordered vector space X {\displaystyle X} (that is, x i ≥ 0 {\displaystyle x_{i}\geq 0} for all i {\displaystyle i} ) is called order summable i... | Order summable |
c_v4xzr0w6zb3y | In mathematics, specifically in order theory and functional analysis, a subset S {\displaystyle S} of a vector lattice is said to be solid and is called an ideal if for all s ∈ S {\displaystyle s\in S} and x ∈ X , {\displaystyle x\in X,} if | x | ≤ | s | {\displaystyle |x|\leq |s|} then x ∈ S . {\displaystyle x\in S.} ... | Solid set |
c_ypibu5zrpwlh | If S ⊆ X {\displaystyle S\subseteq X} then the ideal generated by S {\displaystyle S} is the smallest ideal in X {\displaystyle X} containing S . {\displaystyle S.} An ideal generated by a singleton set is called a principal ideal in X . {\displaystyle X.} | Solid set |
c_20abilhi4u9p | In mathematics, specifically in order theory and functional analysis, an abstract L-space, an AL-space, or an abstract Lebesgue space is a Banach lattice ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} whose norm is additive on the positive cone of X.In probability theory, it means the standard probability space. | Abstract L-space |
c_y3gkf3jru98i | In mathematics, specifically in order theory and functional analysis, an abstract m-space or an AM-space is a Banach lattice ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot \|)} whose norm satisfies ‖ sup { x , y } ‖ = sup { ‖ x ‖ , ‖ y ‖ } {\displaystyle \left\|\sup\{x,y\}\right\|=\sup \left\{\|x\|,\|y\|\right\}} for all x an... | Abstract m-space |
c_1p7hbrtfdi1t | In mathematics, specifically in order theory and functional analysis, an element x {\displaystyle x} of a vector lattice X {\displaystyle X} is called a weak order unit in X {\displaystyle X} if x ≥ 0 {\displaystyle x\geq 0} and also for all y ∈ X , {\displaystyle y\in X,} inf { x , | y | } = 0 implies y = 0. {\display... | Weak order unit |
c_ry2byfrt211q | In mathematics, specifically in order theory and functional analysis, an element x {\displaystyle x} of an ordered topological vector space X {\displaystyle X} is called a quasi-interior point of the positive cone C {\displaystyle C} of X {\displaystyle X} if x ≥ 0 {\displaystyle x\geq 0} and if the order interval := ... | Quasi-interior point |
c_fsrvm76p6c13 | In mathematics, specifically in order theory and functional analysis, if C {\displaystyle C} is a cone at 0 in a vector space X {\displaystyle X} such that 0 ∈ C , {\displaystyle 0\in C,} then a subset S ⊆ X {\displaystyle S\subseteq X} is said to be C {\displaystyle C} -saturated if S = C , {\displaystyle S=_{C},} wh... | Cone-saturated |
c_69r2jtmv465l | {\displaystyle S.} If F {\displaystyle {\mathcal {F}}} is a collection of subsets of X {\displaystyle X} then C := { C: F ∈ F } . {\displaystyle \left_{C}:=\left\{_{C}:F\in {\mathcal {F}}\right\}.} | Cone-saturated |
c_hi7ryt90axwt | If T {\displaystyle {\mathcal {T}}} is a collection of subsets of X {\displaystyle X} and if F {\displaystyle {\mathcal {F}}} is a subset of T {\displaystyle {\mathcal {T}}} then F {\displaystyle {\mathcal {F}}} is a fundamental subfamily of T {\displaystyle {\mathcal {T}}} if every T ∈ T {\displaystyle T\in {\mathcal ... | Cone-saturated |
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