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c_0xapjd5iuj72 | In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form d x d t ( t ) ∈ F ( t , x ( t ) ) , {\displaystyle {\frac {dx}{dt}}(t)\in F(t,x(t)),} where F is a multivalued map, i.e. F(t, x) is a set rather than a single point in R d {\displaystyle \mathbb {R}... | Differential inclusion |
c_eywuk10b648g | In differential inclusion, we not only take a set-valued map at the right hand side but also we can take a subset of a Euclidean space R N {\displaystyle \mathbb {R} ^{N}} for some N ∈ N {\displaystyle N\in \mathbb {N} } as following way. Let n ∈ N {\displaystyle n\in \mathbb {N} } and E ⊂ R n × n ∖ { 0 } . {\displayst... | Differential inclusion |
c_kmiet8zt1e83 | In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms. Given a complex manifold M, a differential of the first kind ω is the... | Differential of the first kind |
c_0fjax7weuolh | They include for example the hyperelliptic integrals of type ∫ x k d x Q ( x ) {\displaystyle \int {\frac {x^{k}\,dx}{\sqrt {Q(x)}}}} where Q is a square-free polynomial of any given degree > 4. The allowable power k has to be determined by analysis of the possible pole at the point at infinity on the corresponding hyp... | Differential of the first kind |
c_2igeudbat2tv | In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebr... | Differential (mathematics) |
c_307dkntq3utj | In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of... | Differential Topology |
c_aejcgldxmn73 | The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the (connected) manifolds in each dimension separately: In dimensi... | Differential Topology |
c_yvuwdkp6ch8g | This is the famous classification of closed surfaces. Already in dimension two the classification of non-compact surfaces becomes difficult, due to the existence of exotic spaces such as Jacob's ladder. In dimension 3, William Thurston's geometrization conjecture, proven by Grigori Perelman, gives a partial classificat... | Differential Topology |
c_vcg0z2zk8dcc | Included in this theorem is the Poincaré conjecture, which states that any closed, simply connected three-manifold is homeomorphic (and in fact diffeomorphic) to the 3-sphere.Beginning in dimension 4, the classification becomes much more difficult for two reasons. Firstly, every finitely presented group appears as the ... | Differential Topology |
c_5i8m4oiuxase | By the word problem for groups, which is equivalent to the halting problem, it is impossible to classify such groups, so a full topological classification is impossible. Secondly, beginning in dimension four it is possible to have smooth manifolds that are homeomorphic, but with distinct, non-diffeomorphic smooth struc... | Differential Topology |
c_t8i26d9gg15n | This means that the study of differential topology in dimensions 4 and higher must use tools genuinely outside the realm of the regular continuous topology of topological manifolds. One of the central open problems in differential topology is the four-dimensional smooth Poincaré conjecture, which asks if every smooth 4... | Differential Topology |
c_ibpz4ehrkhlp | This conjecture is true in dimensions 1, 2, and 3, by the above classification results, but is known to be false in dimension 7 due to the Milnor spheres. Important tools in studying the differential topology of smooth manifolds include the construction of smooth topological invariants of such manifolds, such as de Rha... | Differential Topology |
c_fq98nd6li0i4 | Oftentimes more geometric or analytical techniques may be used, by equipping a smooth manifold with a Riemannian metric or by studying a differential equation on it. Care must be taken to ensure that the resulting information is insensitive to this choice of extra structure, and so genuinely reflects only the topologic... | Differential Topology |
c_hw9b1x74qdp1 | For example, the Hodge theorem provides a geometric and analytical interpretation of the de Rham cohomology, and gauge theory was used by Simon Donaldson to prove facts about the intersection form of simply connected 4-manifolds. In some cases techniques from contemporary physics may appear, such as topological quantum... | Differential Topology |
c_aiozjk1ta08e | In mathematics, digital Morse theory is a digital adaptation of continuum Morse theory for scalar volume data. This is not about the Samuel Morse's Morse code of long and short clicks or tones used in manual electric telegraphy. The term was first promulgated by DB Karron based on the work of JL Cox and DB Karron. The ... | Digital Morse theory |
c_bz07fugf6vku | In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most re... | Dimension theory (algebra) |
c_snqk9h2p26vj | In this case, which is the algebraic counterpart of the case of affine algebraic sets, most of the definitions of the dimension are equivalent. For general commutative rings, the lack of geometric interpretation is an obstacle to the development of the theory; in particular, very little is known for non-noetherian ring... | Dimension theory (algebra) |
c_4igo128jdpv1 | In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariant... | Directed algebraic topology |
c_t40wm3fdofxt | For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces. Directed algebraic topology, like algebraic topology, is motivated by the need to describe qualitative properties of complex systems in terms of algebraic properties of stat... | Directed algebraic topology |
c_5gxi7glewd59 | In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of classical discrepancy theory, namely distributing points in some space such that they are evenly distributed wi... | Discrepancy theory |
c_hn1dxu2vyl4t | The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one. Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepa... | Discrepancy theory |
c_e5h1q2dmwxsz | In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gram. They were later found to be applicable to various algebraic properties of spin angular momentum. | Gram polynomial |
c_uqbnw8qrr1gc | In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation.Divided differences is a recursive division process. Given a seq... | Divided differences |
c_d8yynw3nmiv2 | In mathematics, division by infinity is division where the divisor (denominator) is infinity. In ordinary arithmetic, this does not have a well-defined meaning, since infinity is a mathematical concept that does not correspond to a specific number, and moreover, there is no nonzero real number that, when added to itsel... | Division by infinity |
c_a6pedjkfm6rj | For example, on the extended real number line, dividing any real number by infinity yields zero, while in the surreal number system, dividing 1 by the infinite number ω {\displaystyle \omega } yields the infinitesimal number ϵ {\displaystyle \epsilon } . : 12 In floating-point arithmetic, any finite number divided by ±... | Division by infinity |
c_q29ui0b609qb | The challenges of providing a rigorous meaning of "division by infinity" are analogous to those of defining division by zero. Within the domain of mathematical discourse, the contemplation of dividing infinity by itself gives rise to a proposition of interest. Specifically, the assertion that the result of dividing inf... | Division by infinity |
c_64o7p4z5f2lv | A logical journey unveils the underpinnings of this concept and its mathematical validity. Consider a parameter denoted as "y," which, for the sake of analysis, is assigned the value 10. | Division by infinity |
c_uzaxv373yg1g | The crux of the matter rests in the equation ∞ ÷ y = ∞, where the introduction of y introduces an essential condition. To render the equation coherent, y must assume a magnitude that is sufficiently vast to accommodate the division operation involving infinity. This requirement reflects the conceptual intricacies assoc... | Division by infinity |
c_azz2tjg2w9qr | However, the narrative takes a noteworthy turn as we transition to the equation y × ∞ = ∞. This equation signifies a transformation of the division operation into one of multiplication. In essence, this transition underscores a relationship where division of infinity finds equivalence through multiplication with an app... | Division by infinity |
c_tvgm1am1cu4e | Moreover, it's worth mentioning that if we carry forward the same line of thinking, something fascinating emerges. When we take infinity and divide it by a regular number like 10, the result still holds true: it's infinity. This adds another layer of insight to our mathematical journey, underscoring the depth of what w... | Division by infinity |
c_fxjwziwowz3b | In mathematics, division by two or halving has also been called mediation or dimidiation. The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose multiplication algorithm used division by two as one of its fundamental steps. Some mathema... | Division by two |
c_se8ewnkkpwh0 | In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as a 0 {\textstyle {\tfrac {a}{0}}} , where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number that, when multiplied by 0, gives a (ass... | Division by zero |
c_1hht3ljrttws | Since any number multiplied by zero is zero, the expression 0 0 {\displaystyle {\tfrac {0}{0}}} is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to a 0 {\textstyle {\tfrac {a}{0}}}... | Division by zero |
c_05k0pbymzjai | In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone, generalizes the well-known Stone dua... | Duality theory for distributive lattices |
c_8nhqkurpalc4 | The spectral space (X, τ+) is called the prime spectrum of L. The map φ+ is a lattice isomorphism from L onto the lattice of all compact open subsets of (X,τ+). In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice.Similarly, if φ−(a) = {x∈ X: a ∉ x} and τ− denotes the ... | Duality theory for distributive lattices |
c_fjo0o3v6u6sn | The pairwise Stone space (X,τ+,τ−) is called the bitopological dual of L. Each pairwise Stone space is bi-homeomorphic to the bitopological dual of some bounded distributive lattice.Finally, let ≤ be set-theoretic inclusion on the set of prime filters of L and let τ = τ+∨ τ−. Then (X,τ,≤) is a Priestley space. Moreover... | Duality theory for distributive lattices |
c_39fwfv81tt02 | The Priestley space (X,τ,≤) is called the Priestley dual of L. Each Priestley space is isomorphic to the Priestley dual of some bounded distributive lattice.Let Dist denote the category of bounded distributive lattices and bounded lattice homomorphisms. Then the above three representations of bounded distributive latti... | Duality theory for distributive lattices |
c_k93do5wgpwcx | In mathematics, dynamic equation can refer to: difference equation in discrete time differential equation in continuous time time scale calculus in combined discrete and continuous time | Dynamic equation |
c_jw65svulzuz3 | In mathematics, economics and computer science, particularly in the fields of combinatorics, game theory and algorithms, the stable-roommate problem (SRP) is the problem of finding a stable matching for an even-sized set. A matching is a separation of the set into disjoint pairs ("roommates"). The matching is stable if... | Stable roommates problem |
c_m6tsyy82o06s | It is commonly stated as: In a given instance of the stable-roommates problem (SRP), each of 2n participants ranks the others in strict order of preference. A matching is a set of n disjoint pairs of participants. A matching M in an instance of SRP is stable if there are no two participants x and y, each of whom prefer... | Stable roommates problem |
c_nq5oodzu81ds | In mathematics, economics, and computer science, the Gale–Shapley algorithm (also known as the deferred acceptance algorithm or propose-and-reject algorithm) is an algorithm for finding a solution to the stable matching problem, named for David Gale and Lloyd Shapley. It takes polynomial time, and the time is linear in... | Gale–Shapley algorithm |
c_4f44om045kaf | In mathematics, economics, and computer science, the lattice of stable matchings is a distributive lattice whose elements are stable matchings. For a given instance of the stable matching problem, this lattice provides an algebraic description of the family of all solutions to the problem. It was originally described i... | Lattice of stable matchings |
c_hmdr4u7iezjb | The family of all rotations and their partial order can be constructed in polynomial time, leading to polynomial time solutions for other problems on stable matching including the minimum or maximum weight stable matching. The Gale–Shapley algorithm can be used to construct two special lattice elements, its top and bot... | Lattice of stable matchings |
c_97q4cnr05tuo | Every finite distributive lattice can be represented as a lattice of stable matchings. The number of elements in the lattice can vary from an average case of e − 1 n ln n {\displaystyle e^{-1}n\ln n} to a worst-case of exponential. Computing the number of elements is #P-complete. | Lattice of stable matchings |
c_bifu4aqtlub6 | In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem or SMP) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. A matching is a bijection from the elements of one set to the eleme... | Stable marriage problem |
c_5c2k35558c7a | The stable marriage problem has been stated as follows: Given n men and n women, where each person has ranked all members of the opposite sex in order of preference, marry the men and women together such that there are no two people of opposite sex who would both rather have each other than their current partners. When... | Stable marriage problem |
c_55hlplr38h2i | In mathematics, economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions to an instance of the stable matching problem. | Stable matching polytope |
c_qcigl7sb0o8c | In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notions of effective dimension) of which the most common is effective Hausdorff dimension. Dimension, in mathematics, is a par... | Effective dimension |
c_h8slm5t93gy0 | Hausdorff dimension generalizes the well-known integer dimensions assigned to points, lines, planes, etc. by allowing one to distinguish between objects of intermediate size between these integer-dimensional objects. For example, fractal subsets of the plane may have intermediate dimension between 1 and 2, as they are ... | Effective dimension |
c_h4frhjo5147e | In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms. | Elliptic cohomology |
c_gq5snc15pj52 | In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin the same year. The algori... | Elliptic curve primality |
c_ux6ylrlcuxvm | Primality testing is a field that has been around since the time of Fermat, in whose time most algorithms were based on factoring, which become unwieldy with large input; modern algorithms treat the problems of determining whether a number is prime and what its factors are separately. It became of practical importance ... | Elliptic curve primality |
c_p47xm7m3dbcz | In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. They were introduced by Gilles Robert in 1973, and were used by John Coates and Andrew Wiles in their work on the Birch a... | Elliptic unit |
c_a9lobyx19skh | In mathematics, endoscopic groups of reductive algebraic groups were introduced by Robert Langlands (1979, 1983) in his work on the stable trace formula. Roughly speaking, an endoscopic group H of G is a quasi-split group whose L-group is the connected component of the centralizer of a semisimple element of the L-group... | Endoscopic group |
c_pq18wmqvj2hm | In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory. | Clemens conjecture |
c_7bb8tg15i0ao | In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between A and B is written A = B, and pronounced "A equals B". The symbol "="... | Equality (mathematics) |
c_slfirlu9w8q1 | For example: x = y {\displaystyle x=y} means that x and y denote the same object. The identity ( x + 1 ) 2 = x 2 + 2 x + 1 {\displaystyle (x+1)^{2}=x^{2}+2x+1} means that if x is any number, then the two expressions have the same value. This may also be interpreted as saying that the two sides of the equals sign repres... | Equality (mathematics) |
c_kuxeuqbd78bi | { x ∣ P ( x ) } = { x ∣ Q ( x ) } {\displaystyle \{x\mid P(x)\}=\{x\mid Q(x)\}} if and only if P ( x ) ⇔ Q ( x ) . {\displaystyle P(x)\Leftrightarrow Q(x).} | Equality (mathematics) |
c_df404sxrq91u | This assertion, which uses set-builder notation, means that if the elements satisfying the property P ( x ) {\displaystyle P(x)} are the same as the elements satisfying Q ( x ) , {\displaystyle Q(x),} then the two uses of the set-builder notation define the same set. This property is often expressed as "two sets that h... | Equality (mathematics) |
c_l18309jlqeht | In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry), a notion (for example, ellipse or minimal surface) may have more than one definition. These definitions are equivalent in the context of a given mathematical ... | Equivalent definitions of mathematical structures |
c_3rq54rzxiunq | In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition if and only if it satisfies the other definition. In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is m... | Equivalent definitions of mathematical structures |
c_8b8ai0cqnhbn | In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, app... | Equivariant morphism |
c_zpoyydnvvq9z | The value of an equivariant map is often (imprecisely) called an invariant. In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is a central object of study in equi... | Equivariant morphism |
c_flen1sn6pvtd | In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space... | Equivariant cohomology ring |
c_s701i7i2hv0v | If G {\displaystyle G} is the trivial group, this is the ordinary cohomology ring of X {\displaystyle X} , whereas if X {\displaystyle X} is contractible, it reduces to the cohomology ring of the classifying space B G {\displaystyle BG} (that is, the group cohomology of G {\displaystyle G} when G is finite.) If G acts ... | Equivariant cohomology ring |
c_5d0qjrrix1r0 | In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps f: X → Y {\displaystyle f:X\to Y} , and while equivariant topology also considers such maps, there is the additional constraint that each map "resp... | Equivariant algebraic topoloy |
c_x97oaq99dqk7 | In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subg... | Ergodic flow |
c_1v3g19grabii | In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory ... | Unique ergodicity |
c_60xzrmgchwq2 | Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, ... | Unique ergodicity |
c_5in2kigta2og | Ergodic systems capture the common-sense, every-day notions of randomness, such that smoke might come to fill all of a smoke-filled room, or that a block of metal might eventually come to have the same temperature throughout, or that flips of a fair coin may come up heads and tails half the time. A stronger concept tha... | Unique ergodicity |
c_yur73msti48k | In mathematics, error analysis is the study of kind and quantity of error, or uncertainty, that may be present in the solution to a problem. This issue is particularly prominent in applied areas such as numerical analysis and statistics. | Error analysis (mathematics) |
c_se6xzq1z6nxy | In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra. | Categorical Algebra |
c_p2j7jc92p8v9 | In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Many of the problems posed here first appeared in the Loops (Prague) conferenc... | List of problems in loop theory and quasigroup theory |
c_u2sxsbxk77qv | In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass. | Bass conjecture |
c_mav4xxeq3gt8 | In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein and Deligne or BBD decomposition theorem is a set of results concerning the cohomology of algebraic varieties. It was originally conjectured by Gelfand and MacPherson. | Decomposition theorem of Beilinson, Bernstein and Deligne |
c_f33lfgw474tk | In mathematics, especially convex analysis, the recession cone of a set A {\displaystyle A} is a cone containing all vectors such that A {\displaystyle A} recedes in that direction. That is, the set extends outward in all the directions given by the recession cone. | Recession cone |
c_rzxleuvgg8t9 | In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds,... | Cotangent manifold |
c_hn3w1qgmt0ud | In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x {\displaystyle x} in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828.Let H {\displaystyle H} be a Hilbert space, and suppose that e 1 , e 2... | Bessel's inequality |
c_zdaaicamelan | . . {\displaystyle e_{1},e_{2},...} is an orthonormal sequence in H {\displaystyle H} . | Bessel's inequality |
c_fw3qk1daekn6 | In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metr... | Banach ring |
c_puvx6ivotqpz | Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering A e {\displaystyle A_{e}} and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in ... | Banach ring |
c_p6i4xb54og77 | The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements. Banach algebras can also be defined over fields of... | Banach ring |
c_0e7jli6tnqyb | In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra A {\displaystyle A} over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation ( a , b ) ↦ a ∗ b {\displaystyle (a,b)\mapsto a... | Fréchet algebra |
c_ijpy0xkf8s1y | In that case, by rescaling the seminorms, we may also take C n = 1 {\displaystyle C_{n}=1} for each n {\displaystyle n} and the seminorms are said to be submultiplicative: ‖ a b ‖ n ≤ ‖ a ‖ n ‖ b ‖ n {\displaystyle \|ab\|_{n}\leq \|a\|_{n}\|b\|_{n}} for all a , b ∈ A . {\displaystyle a,b\in A.} m {\displaystyle m} -con... | Fréchet algebra |
c_u0bfgpjo17o9 | In mathematics, especially functional analysis, a bornology B {\displaystyle {\mathcal {B}}} on a vector space X {\displaystyle X} over a field K , {\displaystyle \mathbb {K} ,} where K {\displaystyle \mathbb {K} } has a bornology ℬ F {\displaystyle \mathbb {F} } , is called a vector bornology if B {\displaystyle {\mat... | Vector bornology |
c_sdynodif6oip | In mathematics, especially functional analysis, a bornology on a set X is a collection of subsets of X satisfying axioms that generalize the notion of boundedness. One of the key motivations behind bornologies and bornological analysis is the fact that bornological spaces provide a convenient setting for homological al... | Bornology |
c_z5d6f8vrrooq | In mathematics, especially functional analysis, a hypercyclic operator on a Banach space X is a bounded linear operator T: X → X such that there is a vector x ∈ X such that the sequence {Tn x: n = 0, 1, 2, …} is dense in the whole space X. In other words, the smallest closed invariant subset containing x is the whole s... | Hypercyclic operator |
c_h6cv3riiizbm | The hypercyclicity is a special case of broader notions of topological transitivity (see topological mixing), and universality. Universality in general involves a set of mappings from one topological space to another (instead of a sequence of powers of a single operator mapping from X to X), but has a similar meaning t... | Hypercyclic operator |
c_tlaw7bg6hn9n | In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN* = N*N.Normal operators are important because the spectral theorem holds for them. The class of normal operators is well unders... | Normal operator |
c_36rq8aql4klc | In mathematics, especially functional analysis, a quasitrace is a not necessarily additive tracial functional on a C*-algebra. An additive quasitrace is called a trace. It is a major open problem if every quasitrace is a trace. | Quasitrace |
c_eexp4jv1q143 | In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space X {\displaystyle X} is a nested sequence of compact subsets K i {\displaystyle K_{i}} of X {\displaystyle X} (i.e. K 1 ⊆ K 2 ⊆ K 3 ⊆ ⋯ {\displaystyle K_{1}\subseteq K_{2}\subseteq K_{3}\subseteq \cdots } ), su... | Exhaustion by compact sets |
c_zmwba21ewj2o | In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products. It i... | Knit product |
c_dzwxtkc7hwbx | Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904). | Knit product |
c_i08d67rukypb | In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set C G ( S ) {\displaystyle \operatorname {C} _{G}(S)} of elements of G that commute with every element of S, or equivalently, such that conjugation by g {\displaystyle g} leaves each element of S fixed... | Normalizer (group theory) |
c_gd06dmcwmpcx | In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring R is a subring of R. This article also deals with centralizers and normalizers in a Lie algebra. The idealizer in a semigroup or ring is another co... | Normalizer (group theory) |
c_5s9r6d4kocz7 | In mathematics, especially group theory, two elements a {\displaystyle a} and b {\displaystyle b} of a group are conjugate if there is an element g {\displaystyle g} in the group such that b = g a g − 1 . {\displaystyle b=gag^{-1}.} This is an equivalence relation whose equivalence classes are called conjugacy classes. | Class number (group theory) |
c_fsg9lxu76qca | In other words, each conjugacy class is closed under b = g a g − 1 {\displaystyle b=gag^{-1}} for all elements g {\displaystyle g} in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abeli... | Class number (group theory) |
c_6p2for853ku6 | In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also works in the category of groups, for example. The five lemma can be thought ... | Five Lemma |
c_y7anfv8g8ecr | In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h ... | Short five lemma |
c_19ce86vxfhzi | In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded Z {\displaystyle \mathbb {Z} } -module. In detail, this means that Hom ( A , B ) {\display... | Differential graded category |
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