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In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an n-sphere.
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https://en.wikipedia.org/wiki/3-sphere
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In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step group in these two cases is slightly different.
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https://en.wikipedia.org/wiki/3-step_group
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In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are important in physics because in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.
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https://en.wikipedia.org/wiki/4-manifold
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In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected compact 5-manifolds were first classified by Stephen Smale and then in full generality by Dennis Barden, while another proof was later given by Aleksey V. Zhubr. This turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Thurston geometrisation conjecture, and the 4-dimensional case was solved by Michael Freedman (1982) in the topological case, but is a very hard unsolved problem in the smooth case.
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https://en.wikipedia.org/wiki/5-manifold
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In dimension 5, the smooth classification of simply connected manifolds is governed by classical algebraic topology. Namely, two simply connected, smooth 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving the linking form and the second Stiefel–Whitney class. Moreover, any such isomorphism in second homology is induced by some diffeomorphism. It is undecidable if a given 5-manifold is homeomorphic to S 5 {\displaystyle S^{5}} , the 5-sphere.
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https://en.wikipedia.org/wiki/5-manifold
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In mathematics, a Baer group is a group in which every cyclic subgroup is subnormal. Every Baer group is locally nilpotent.Baer groups are named after Reinhold Baer. == References ==
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https://en.wikipedia.org/wiki/Baer_group
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In mathematics, a Bailey pair is a pair of sequences satisfying certain relations, and a Bailey chain is a sequence of Bailey pairs. Bailey pairs were introduced by W. N. Bailey (1947, 1948) while studying the second proof Rogers 1917 of the Rogers–Ramanujan identities, and Bailey chains were introduced by Andrews (1984).
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https://en.wikipedia.org/wiki/Bailey_pair
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In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the same as Borel measures that are finite on compact sets. In general Baire sets and Borel sets need not be the same. In spaces with non-Baire Borel sets, Baire measures are used because they connect to the properties of continuous functions more directly.
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https://en.wikipedia.org/wiki/Baire_measure
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In mathematics, a Banach bundle is a fiber bundle over a topological Hausdorff space, such that each fiber has the structure of a Banach space.
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https://en.wikipedia.org/wiki/Banach_bundle_(non-commutative_geometry)
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In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.
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https://en.wikipedia.org/wiki/Banach_bundle
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In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds to infinite dimensions. A further generalisation is to Fréchet manifolds, replacing Banach spaces by Fréchet spaces. On the other hand, a Hilbert manifold is a special case of a Banach manifold in which the manifold is locally modeled on Hilbert spaces.
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https://en.wikipedia.org/wiki/Banach_manifold
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In mathematics, a Barlow surface is one of the complex surfaces introduced by Rebecca Barlow (1984, 1985). They are simply connected surfaces of general type with pg = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points. The Hodge diamond for the Barlow surfaces is:
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https://en.wikipedia.org/wiki/Barlow_surface
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In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by Ernest William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The integral is usually taken along a contour which is a deformation of the imaginary axis passing to the right of all poles of factors of the form Γ(a + s) and to the left of all poles of factors of the form Γ(a − s).
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https://en.wikipedia.org/wiki/Barnes_integral
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In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by E. W. Barnes (1901). It is further generalized by the Shintani zeta function.
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https://en.wikipedia.org/wiki/Barnes_zeta_function
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In mathematics, a Batalin–Vilkovisky algebra is a graded supercommutative algebra (with a unit 1) with a second-order nilpotent operator Δ of degree −1. More precisely, it satisfies the identities | a b | = | a | + | b | {\displaystyle |ab|=|a|+|b|} (The product has degree 0) | Δ ( a ) | = | a | − 1 {\displaystyle |\Delta (a)|=|a|-1} (Δ has degree −1) ( a b ) c = a ( b c ) {\displaystyle (ab)c=a(bc)} (The product is associative) a b = ( − 1 ) | a | | b | b a {\displaystyle ab=(-1)^{|a||b|}ba} (The product is (super-)commutative) Δ 2 = 0 {\displaystyle \Delta ^{2}=0} (Nilpotency (of order 2)) Δ ( a b c ) − Δ ( a b ) c + Δ ( a ) b c − ( − 1 ) | a | a Δ ( b c ) − ( − 1 ) ( | a | + 1 ) | b | b Δ ( a c ) + ( − 1 ) | a | a Δ ( b ) c + ( − 1 ) | a | + | b | a b Δ ( c ) − Δ ( 1 ) a b c = 0 {\displaystyle \Delta (abc)-\Delta (ab)c+\Delta (a)bc-(-1)^{|a|}a\Delta (bc)-(-1)^{(|a|+1)|b|}b\Delta (ac)+(-1)^{|a|}a\Delta (b)c+(-1)^{|a|+|b|}ab\Delta (c)-\Delta (1)abc=0} (The Δ operator is of second order)One often also requires normalization: Δ ( 1 ) = 0 {\displaystyle \Delta (1)=0} (normalization)
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https://en.wikipedia.org/wiki/Batalin–Vilkovisky_formalism
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In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926. Rayleigh's theorem, named after Lord Rayleigh, states that the complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number. Beatty sequences can also be used to generate Sturmian words.
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https://en.wikipedia.org/wiki/Beatty_sequence
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In mathematics, a Beauville surface is one of the surfaces of general type introduced by Arnaud Beauville (1996, exercise X.13 (4)). They are examples of "fake quadrics", with the same Betti numbers as quadric surfaces.
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https://en.wikipedia.org/wiki/Beauville_surface
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In mathematics, a Benz plane is a type of 2-dimensional geometrical structure, named after the German mathematician Walter Benz. The term was applied to a group of objects that arise from a common axiomatization of certain structures and split into three families, which were introduced separately: Möbius planes, Laguerre planes, and Minkowski planes.
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https://en.wikipedia.org/wiki/Benz_plane
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In mathematics, a Berkovich space, introduced by Berkovich (1990), is a version of an analytic space over a non-Archimedean field (e.g. p-adic field), refining Tate's notion of a rigid analytic space.
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https://en.wikipedia.org/wiki/Berkovich_spectrum
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In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process.
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https://en.wikipedia.org/wiki/Bessel_process
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In mathematics, a Beurling zeta function is an analogue of the Riemann zeta function where the ordinary primes are replaced by a set of Beurling generalized primes: any sequence of real numbers greater than 1 that tend to infinity. These were introduced by Beurling (1937). A Beurling generalized integer is a number that can be written as a product of Beurling generalized primes. Beurling generalized the usual prime number theorem to Beurling generalized primes. He showed that if the number N(x) of Beurling generalized integers less than x is of the form N(x) = Ax + O(x log−γx) with γ > 3/2 then the number of Beurling generalized primes less than x is asymptotic to x/log x, just as for ordinary primes, but if γ = 3/2 then this conclusion need not hold.
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https://en.wikipedia.org/wiki/Beurling_zeta_function
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In mathematics, a Bianchi group is a group of the form P S L 2 ( O d ) {\displaystyle PSL_{2}({\mathcal {O}}_{d})} where d is a positive square-free integer. Here, PSL denotes the projective special linear group and O d {\displaystyle {\mathcal {O}}_{d}} is the ring of integers of the imaginary quadratic field Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} . The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of P S L 2 ( C ) {\displaystyle PSL_{2}(\mathbb {C} )} , now termed Kleinian groups. As a subgroup of P S L 2 ( C ) {\displaystyle PSL_{2}(\mathbb {C} )} , a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space H 3 {\displaystyle \mathbb {H} ^{3}} .
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https://en.wikipedia.org/wiki/Bianchi_group
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The quotient space M d = P S L 2 ( O d ) ∖ H 3 {\displaystyle M_{d}=PSL_{2}({\mathcal {O}}_{d})\backslash \mathbb {H} ^{3}} is a non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi orbifold. An exact formula for the volume, in terms of the Dedekind zeta function of the base field Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} , was computed by Humbert as follows. Let D {\displaystyle D} be the discriminant of Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} , and Γ = S L 2 ( O d ) {\displaystyle \Gamma =SL_{2}({\mathcal {O}}_{d})} , the discontinuous action on H {\displaystyle {\mathcal {H}}} , then vol ( Γ ∖ H ) = | D | 3 / 2 4 π 2 ζ Q ( − d ) ( 2 ) .
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https://en.wikipedia.org/wiki/Bianchi_group
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{\displaystyle \operatorname {vol} (\Gamma \backslash \mathbb {H} )={\frac {|D|^{3/2}}{4\pi ^{2}}}\zeta _{\mathbb {Q} ({\sqrt {-d}})}(2)\ .} The set of cusps of M d {\displaystyle M_{d}} is in bijection with the class group of Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} . It is well known that every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.
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https://en.wikipedia.org/wiki/Bianchi_group
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In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}). Alternative names are switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory.A Boolean function takes the form f: { 0 , 1 } k → { 0 , 1 } {\displaystyle f:\{0,1\}^{k}\to \{0,1\}} , where { 0 , 1 } {\displaystyle \{0,1\}} is known as the Boolean domain and k {\displaystyle k} is a non-negative integer called the arity of the function. In the case where k = 0 {\displaystyle k=0} , the function is a constant element of { 0 , 1 } {\displaystyle \{0,1\}} .
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https://en.wikipedia.org/wiki/Finitary_boolean_function
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A Boolean function with multiple outputs, f: { 0 , 1 } k → { 0 , 1 } m {\displaystyle f:\{0,1\}^{k}\to \{0,1\}^{m}} with m > 1 {\displaystyle m>1} is a vectorial or vector-valued Boolean function (an S-box in symmetric cryptography).There are 2 2 k {\displaystyle 2^{2^{k}}} different Boolean functions with k {\displaystyle k} arguments; equal to the number of different truth tables with 2 k {\displaystyle 2^{k}} entries. Every k {\displaystyle k} -ary Boolean function can be expressed as a propositional formula in k {\displaystyle k} variables x 1 , .
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https://en.wikipedia.org/wiki/Finitary_boolean_function
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. . , x k {\displaystyle x_{1},...,x_{k}} , and two propositional formulas are logically equivalent if and only if they express the same Boolean function.
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https://en.wikipedia.org/wiki/Finitary_boolean_function
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In mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the two-element Boolean algebra is used, the Boolean matrix is called a logical matrix. (In some contexts, particularly computer science, the term "Boolean matrix" implies this restriction.) Let U be a non-trivial Boolean algebra (i.e. with at least two elements).
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https://en.wikipedia.org/wiki/Boolean_matrix
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Intersection, union, complementation, and containment of elements is expressed in U. Let V be the collection of n × n matrices that have entries taken from U. Complementation of such a matrix is obtained by complementing each element. The intersection or union of two such matrices is obtained by applying the operation to entries of each pair of elements to obtain the corresponding matrix intersection or union. A matrix is contained in another if each entry of the first is contained in the corresponding entry of the second.
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https://en.wikipedia.org/wiki/Boolean_matrix
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The product of two Boolean matrices is expressed as follows: According to one author, "Matrices over an arbitrary Boolean algebra β satisfy most of the properties over β0 = {0, 1}. The reason is that any Boolean algebra is a sub-Boolean algebra of β 0 S {\displaystyle \beta _{0}^{S}} for some set S, and we have an isomorphism from n × n matrices over β 0 S to β n S . {\displaystyle \beta _{0}^{S}\ {\text{to}}\ \beta _{n}^{S}.} "
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https://en.wikipedia.org/wiki/Boolean_matrix
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In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R, that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨, which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra, George Boole.
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https://en.wikipedia.org/wiki/Boolean_rings
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In mathematics, a Borchers algebra or Borchers–Uhlmann algebra or BU-algebra is the tensor algebra of a vector space, often a space of smooth test functions. They were studied by H. J. Borchers (1962), who showed that the Wightman distributions of a quantum field could be interpreted as a state, called a Wightman functional, on a Borchers algebra. A Borchers algebra with a state can often be used to construct an O*-algebra. The Borchers algebra of a quantum field theory has an ideal called the locality ideal, generated by elements of the form ab−ba for a and b having spacelike-separated support. The Wightman functional of a quantum field theory vanishes on the locality ideal, which is equivalent to the locality axiom for quantum field theory.
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https://en.wikipedia.org/wiki/Wightman_functional
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In mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borel subset of X × X (in the product topology).
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https://en.wikipedia.org/wiki/Borel_equivalence_relation
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In mathematics, a Borel isomorphism is a measurable bijective function between two standard Borel spaces. By Souslin's theorem in standard Borel spaces (which says that a set that is both analytic and coanalytic is necessarily Borel), the inverse of any such measurable bijective function is also measurable. Borel isomorphisms are closed under composition and under taking of inverses. The set of Borel isomorphisms from a space to itself clearly forms a group under composition. Borel isomorphisms on standard Borel spaces are analogous to homeomorphisms on topological spaces: both are bijective and closed under composition, and a homeomorphism and its inverse are both continuous, instead of both being only Borel measurable.
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https://en.wikipedia.org/wiki/Borel_isomorphism
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In mathematics, a Borel measure μ on n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of R n {\displaystyle \mathbb {R} ^{n}} and 0 < λ < 1, one has μ ( λ A + ( 1 − λ ) B ) ≥ μ ( A ) λ μ ( B ) 1 − λ , {\displaystyle \mu (\lambda A+(1-\lambda )B)\geq \mu (A)^{\lambda }\mu (B)^{1-\lambda },} where λ A + (1 − λ) B denotes the Minkowski sum of λ A and (1 − λ) B.
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https://en.wikipedia.org/wiki/Log-concave_measure
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In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).
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https://en.wikipedia.org/wiki/Borel_algebra
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Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure.
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https://en.wikipedia.org/wiki/Borel_algebra
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Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces.
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https://en.wikipedia.org/wiki/Borel_algebra
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In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together with the usual set of rules for combining (forming the products of) those matrices, such that they form an associative algebra, or, more precisely, a unitary commutative algebra. Among these rules are: the result of a product is also within the set of matrices, there is an identity matrix in the set, and taking products is commutative.Bose–Mesner algebras have applications in physics to spin models, and in statistics to the design of experiments. They are named for R. C. Bose and Dale Marsh Mesner.
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https://en.wikipedia.org/wiki/Bose–Mesner_algebra
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In mathematics, a Bost–Connes system is a quantum statistical dynamical system related to an algebraic number field, whose partition function is related to the Dedekind zeta function of the number field. Bost & Connes (1995) introduced Bost–Connes systems by constructing one for the rational numbers. Connes, Marcolli & Ramachandran (2005) extended the construction to imaginary quadratic fields. Such systems have been studied for their connection with Hilbert's Twelfth Problem. In the case of a Bost–Connes system over Q, the absolute Galois group acts on the ground states of the system.
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https://en.wikipedia.org/wiki/Bost–Connes_system
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In mathematics, a Bratteli diagram is a combinatorial structure: a graph composed of vertices labelled by positive integers ("level") and unoriented edges between vertices having levels differing by one. The notion was introduced by Ola Bratteli in 1972 in the theory of operator algebras to describe directed sequences of finite-dimensional algebras: it played an important role in Elliott's classification of AF-algebras and the theory of subfactors. Subsequently Anatoly Vershik associated dynamical systems with infinite paths in such graphs.
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https://en.wikipedia.org/wiki/Bratteli_diagram
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In mathematics, a Bratteli–Veršik diagram is an ordered, essentially simple Bratteli diagram (V, E) with a homeomorphism on the set of all infinite paths called the Veršhik transformation. It is named after Ola Bratteli and Anatoly Vershik.
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https://en.wikipedia.org/wiki/Bratteli–Vershik_diagram
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In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality.
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https://en.wikipedia.org/wiki/Brauer_algebra
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In mathematics, a Brieskorn manifold or Brieskorn–Phạm manifold, introduced by Egbert Brieskorn (1966, 1966b), is the intersection of a small sphere around the origin with the singular, complex hypersurface x 1 k 1 + ⋯ + x n k n = 0 {\displaystyle x_{1}^{k_{1}}+\cdots +x_{n}^{k_{n}}=0} studied by Frédéric Pham (1965). Brieskorn manifolds give examples of exotic spheres.
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https://en.wikipedia.org/wiki/Brieskorn_sphere
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In mathematics, a Brieskorn–Grothendieck resolution is a resolution conjectured by Alexander Grothendieck, that in particular gives a resolution of the universal deformation of a Kleinian singularity. Egbert Brieskorn (1971) announced the construction of this resolution, and Peter Slodowy (1980) published the details of Brieskorn's construction.
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https://en.wikipedia.org/wiki/Brieskorn–Grothendieck_resolution
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In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in Brjuno (1971).
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https://en.wikipedia.org/wiki/Brjuno_number
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In mathematics, a Buekenhout geometry or diagram geometry is a generalization of projective spaces, Tits buildings, and several other geometric structures, introduced by Buekenhout (1979).
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https://en.wikipedia.org/wiki/Buekenhout_geometry
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In mathematics, a Burniat surface is one of the surfaces of general type introduced by Pol Burniat (1966).
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https://en.wikipedia.org/wiki/Burniat_surface
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It is a special case of the Bing–Borsuk conjecture. The Busemann conjecture is known to be true for dimensions 1 to 4. == References ==
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https://en.wikipedia.org/wiki/Busemann_G-space
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In mathematics, a Butler group is a group that is the image of a completely decomposable abelian group of finite rank. They were introduced by M. C. R. Butler (1965).
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https://en.wikipedia.org/wiki/Butler_group
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In mathematics, a Bézout matrix (or Bézoutian or Bezoutiant) is a special square matrix associated with two polynomials, introduced by James Joseph Sylvester (1853) and Arthur Cayley (1857) and named after Étienne Bézout. Bézoutian may also refer to the determinant of this matrix, which is equal to the resultant of the two polynomials. Bézout matrices are sometimes used to test the stability of a given polynomial.
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https://en.wikipedia.org/wiki/Bézout_matrix
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In mathematics, a Böhmer integral is an integral introduced by Böhmer (1939) generalizing the Fresnel integrals. There are two versions, given by C ( x , α ) = ∫ x ∞ t α − 1 cos ( t ) d t {\displaystyle \displaystyle C(x,\alpha )=\int _{x}^{\infty }t^{\alpha -1}\cos(t)\,dt} S ( x , α ) = ∫ x ∞ t α − 1 sin ( t ) d t {\displaystyle \displaystyle S(x,\alpha )=\int _{x}^{\infty }t^{\alpha -1}\sin(t)\,dt} Consequently, Fresnel integrals can be expressed in terms of the Böhmer integrals as S ( y ) = 1 2 − 1 2 π ⋅ S ( 1 2 , y 2 ) {\displaystyle \operatorname {S} (y)={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {S} \left({\frac {1}{2}},y^{2}\right)} C ( y ) = 1 2 − 1 2 π ⋅ C ( 1 2 , y 2 ) {\displaystyle \operatorname {C} (y)={\frac {1}{2}}-{\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {C} \left({\frac {1}{2}},y^{2}\right)} The sine integral and cosine integral can also be expressed in terms of the Böhmer integrals Si ( x ) = π 2 − S ( x , 0 ) {\displaystyle \operatorname {Si} (x)={\frac {\pi }{2}}-\operatorname {S} (x,0)} Ci ( x ) = π 2 − C ( x , 0 ) {\displaystyle \operatorname {Ci} (x)={\frac {\pi }{2}}-\operatorname {C} (x,0)}
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https://en.wikipedia.org/wiki/Böhmer_integral
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In mathematics, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations. Formally, a strongly continuous semigroup is a representation of the semigroup (R+, +) on some Banach space X that is continuous in the strong operator topology. Thus, strictly speaking, a strongly continuous semigroup is not a semigroup, but rather a continuous representation of a very particular semigroup.
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https://en.wikipedia.org/wiki/C0-semigroup
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In mathematics, a CAT ( k ) {\displaystyle \mathbf {\operatorname {\textbf {CAT}} } (k)} space, where k {\displaystyle k} is a real number, is a specific type of metric space. Intuitively, triangles in a CAT ( k ) {\displaystyle \operatorname {CAT} (k)} space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k {\displaystyle k} . In a CAT ( k ) {\displaystyle \operatorname {CAT} (k)} space, the curvature is bounded from above by k {\displaystyle k} .
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https://en.wikipedia.org/wiki/CAT_space
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A notable special case is k = 0 {\displaystyle k=0} ; complete CAT ( 0 ) {\displaystyle \operatorname {CAT} (0)} spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard. Originally, Aleksandrov called these spaces “ R k {\displaystyle {\mathfrak {R}}_{k}} domain”. The terminology CAT ( k ) {\displaystyle \operatorname {CAT} (k)} was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications).
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https://en.wikipedia.org/wiki/CAT_space
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In mathematics, a CAT(k) group is a group that acts discretely, cocompactly and isometrically on a CAT(k) space.
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https://en.wikipedia.org/wiki/CAT(k)_group
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In mathematics, a CH-quasigroup, introduced by Manin (1986, definition 1.3), is a symmetric quasigroup in which any three elements generate an abelian quasigroup. "CH" stands for cubic hypersurface.
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https://en.wikipedia.org/wiki/CH-quasigroup
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In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field. The abbreviation "CM" was introduced by (Shimura & Taniyama 1961).
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https://en.wikipedia.org/wiki/CM-field
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In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. Formally, a CR manifold is a differentiable manifold M together with a preferred complex distribution L, or in other words a complex subbundle of the complexified tangent bundle C T M = T M ⊗ R C {\displaystyle \mathbb {C} TM=TM\otimes _{\mathbb {R} }\mathbb {C} } such that ⊆ L {\displaystyle \subseteq L} (L is formally integrable) L ∩ L ¯ = { 0 } {\displaystyle L\cap {\bar {L}}=\{0\}} .The subbundle L is called a CR structure on the manifold M. The abbreviation CR stands for "Cauchy–Riemann" or "Complex-Real".
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https://en.wikipedia.org/wiki/CR_manifold
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In mathematics, a Caccioppoli set is a set whose boundary is measurable and has (at least locally) a finite measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation.
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https://en.wikipedia.org/wiki/Caccioppoli_set
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In mathematics, a Cannon–Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group on those spaces. The notion originated from a seminal 1980s preprint of James Cannon and William Thurston "Group-invariant Peano curves" (eventually published in 2007) about fibered hyperbolic 3-manifolds.Cannon–Thurston maps provide many natural geometric examples of space-filling curves.
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https://en.wikipedia.org/wiki/Cannon–Thurston_map
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In mathematics, a Cantor algebra, named after Georg Cantor, is one of two closely related Boolean algebras, one countable and one complete. The countable Cantor algebra is the Boolean algebra of all clopen subsets of the Cantor set. This is the free Boolean algebra on a countable number of generators. Up to isomorphism, this is the only nontrivial Boolean algebra that is both countable and atomless.
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https://en.wikipedia.org/wiki/Cantor_algebra
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The complete Cantor algebra is the complete Boolean algebra of Borel subsets of the reals modulo meager sets (Balcar & Jech 2006). It is isomorphic to the completion of the countable Cantor algebra. (The complete Cantor algebra is sometimes called the Cohen algebra, though "Cohen algebra" usually refers to a different type of Boolean algebra.) The complete Cantor algebra was studied by von Neumann in 1935 (later published as (von Neumann 1998)), who showed that it is not isomorphic to the random algebra of Borel subsets modulo measure zero sets.
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https://en.wikipedia.org/wiki/Cantor_algebra
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In mathematics, a Cantor cube is a topological group of the form {0, 1}A for some index set A. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given the discrete topology). If A is a countably infinite set, the corresponding Cantor cube is a Cantor space. Cantor cubes are special among compact groups because every compact group is a continuous image of one, although usually not a homomorphic image.
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https://en.wikipedia.org/wiki/Cantor_cube
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(The literature can be unclear, so for safety, assume all spaces are Hausdorff.) Topologically, any Cantor cube is: homogeneous; compact; zero-dimensional; AE(0), an absolute extensor for compact zero-dimensional spaces. (Every map from a closed subset of such a space into a Cantor cube extends to the whole space.
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https://en.wikipedia.org/wiki/Cantor_cube
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)By a theorem of Schepin, these four properties characterize Cantor cubes; any space satisfying the properties is homeomorphic to a Cantor cube. In fact, every AE(0) space is the continuous image of a Cantor cube, and with some effort one can prove that every compact group is AE(0). It follows that every zero-dimensional compact group is homeomorphic to a Cantor cube, and every compact group is a continuous image of a Cantor cube.
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https://en.wikipedia.org/wiki/Cantor_cube
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In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space.
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https://en.wikipedia.org/wiki/Cantor_space
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In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone. Other uses of Carleman matrices occur in the theory of probability generating functions, and Markov chains.
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https://en.wikipedia.org/wiki/Carleman_matrix
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In mathematics, a Carleson measure is a type of measure on subsets of n-dimensional Euclidean space Rn. Roughly speaking, a Carleson measure on a domain Ω is a measure that does not vanish at the boundary of Ω when compared to the surface measure on the boundary of Ω. Carleson measures have many applications in harmonic analysis and the theory of partial differential equations, for instance in the solution of Dirichlet problems with "rough" boundary. The Carleson condition is closely related to the boundedness of the Poisson operator. Carleson measures are named after the Swedish mathematician Lennart Carleson.
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https://en.wikipedia.org/wiki/Carleson_measure
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In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation; it is named after Thomas Carlyle. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.
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https://en.wikipedia.org/wiki/Carlyle_circle
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In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigenspace is called horizontal. On a Carnot group, any norm on the horizontal subbundle gives rise to a Carnot–Carathéodory metric. Carnot–Carathéodory metrics have metric dilations; they are asymptotic cones (see Ultralimit) of finitely-generated nilpotent groups, and of nilpotent Lie groups, as well as tangent cones of sub-Riemannian manifolds.
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https://en.wikipedia.org/wiki/Carnot_group
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In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle {\mathfrak {g}}} that is self-normalising (if ∈ h {\displaystyle \in {\mathfrak {h}}} for all X ∈ h {\displaystyle X\in {\mathfrak {h}}} , then Y ∈ h {\displaystyle Y\in {\mathfrak {h}}} ). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra g {\displaystyle {\mathfrak {g}}} over a field of characteristic 0 {\displaystyle 0} .
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https://en.wikipedia.org/wiki/Cartan_algebra
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In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., C {\displaystyle \mathbb {C} } ), a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism ad ( x ): g → g {\displaystyle \operatorname {ad} (x):{\mathfrak {g}}\to {\mathfrak {g}}} is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.pg 231In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over an algebraically closed field of characteristic zero, a Cartan subalgebra can also be defined as a maximal toral subalgebra. Kac–Moody algebras and generalized Kac–Moody algebras also have subalgebras that play the same role as the Cartan subalgebras of semisimple Lie algebras (over a field of characteristic zero).
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https://en.wikipedia.org/wiki/Cartan_algebra
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In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. More generally, Casimir elements can be used to refer to any element of the center of the universal enveloping algebra. The algebra of these elements is known to be isomorphic to a polynomial algebra through the Harish-Chandra isomorphism. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931.
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https://en.wikipedia.org/wiki/Casimir_invariant
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In mathematics, a Catalan pseudoprime is an odd composite number n satisfying the congruence ( − 1 ) n − 1 2 ⋅ C n − 1 2 ≡ 2 ( mod n ) , {\displaystyle (-1)^{\frac {n-1}{2}}\cdot C_{\frac {n-1}{2}}\equiv 2{\pmod {n}},} where Cm denotes the m-th Catalan number. The congruence also holds for every odd prime number n that justifies the name pseudoprimes for composite numbers n satisfying it.
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https://en.wikipedia.org/wiki/Catalan_pseudoprime
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In mathematics, a Catalan solid, or Archimedean dual, is a polyhedron that is dual to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan solids are all convex.
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https://en.wikipedia.org/wiki/Catalan_solid
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They are face-transitive but not vertex-transitive. This is because the dual Archimedean solids are vertex-transitive and not face-transitive. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons.
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https://en.wikipedia.org/wiki/Catalan_solid
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However, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Being face-transitive, Catalan solids are isohedra. Additionally, two of the Catalan solids are edge-transitive: the rhombic dodecahedron and the rhombic triacontahedron.
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https://en.wikipedia.org/wiki/Catalan_solid
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These are the duals of the two quasi-regular Archimedean solids. Just as prisms and antiprisms are generally not considered Archimedean solids, bipyramids and trapezohedra are generally not considered Catalan solids, despite being face-transitive. Two of the Catalan solids are chiral: the pentagonal icositetrahedron and the pentagonal hexecontahedron, dual to the chiral snub cube and snub dodecahedron. These each come in two enantiomorphs. Not counting the enantiomorphs, bipyramids, and trapezohedra, there are a total of 13 Catalan solids.
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https://en.wikipedia.org/wiki/Catalan_solid
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In mathematics, a Catanese surface is one of the surfaces of general type introduced by Fabrizio Catanese (1981).
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https://en.wikipedia.org/wiki/Catanese_surface
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In mathematics, a Cauchy (French: ) boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition. It is named after the prolific 19th-century French mathematical analyst Augustin-Louis Cauchy.
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https://en.wikipedia.org/wiki/Cauchy_initial_value_problem
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In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form a i j = 1 x i − y j ; x i − y j ≠ 0 , 1 ≤ i ≤ m , 1 ≤ j ≤ n {\displaystyle a_{ij}={\frac {1}{x_{i}-y_{j}}};\quad x_{i}-y_{j}\neq 0,\quad 1\leq i\leq m,\quad 1\leq j\leq n} where x i {\displaystyle x_{i}} and y j {\displaystyle y_{j}} are elements of a field F {\displaystyle {\mathcal {F}}} , and ( x i ) {\displaystyle (x_{i})} and ( y j ) {\displaystyle (y_{j})} are injective sequences (they contain distinct elements). The Hilbert matrix is a special case of the Cauchy matrix, where x i − y j = i + j − 1. {\displaystyle x_{i}-y_{j}=i+j-1.\;} Every submatrix of a Cauchy matrix is itself a Cauchy matrix.
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https://en.wikipedia.org/wiki/Cauchy_determinant
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In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences.It is not sufficient for each term to become arbitrarily close to the preceding term. For instance, in the sequence of square roots of natural numbers: the consecutive terms become arbitrarily close to each other – their differences tend to zero as the index n grows.
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https://en.wikipedia.org/wiki/Cauchy_sequence
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However, with growing values of n, the terms a n {\displaystyle a_{n}} become arbitrarily large. So, for any index n and distance d, there exists an index m big enough such that a m − a n > d . {\displaystyle a_{m}-a_{n}>d.}
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https://en.wikipedia.org/wiki/Cauchy_sequence
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As a result, no matter how far one goes, the remaining terms of the sequence never get close to each other; hence the sequence is not Cauchy. The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination. Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets.
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https://en.wikipedia.org/wiki/Cauchy_sequence
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In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous functions have the useful property that they can always be (uniquely) extended to the Cauchy completion of their domain.
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https://en.wikipedia.org/wiki/Cauchy-continuous_function
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In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs.
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https://en.wikipedia.org/wiki/Cayley_graph
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In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance" where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and subsequent books and papers. The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics.
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https://en.wikipedia.org/wiki/Cayley_metric
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In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups. The Chevalley basis is the Cartan-Weyl basis, but with a different normalization.
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https://en.wikipedia.org/wiki/Chevalley_basis
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The generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives ± α i {\displaystyle \pm \alpha _{i}} . The Cartan-Weyl basis may be written as = 0 {\displaystyle =0} = α i E α {\displaystyle =\alpha _{i}E_{\alpha }} Defining the dual root or coroot of α {\displaystyle \alpha } as α ∨ = 2 α ( α , α ) {\displaystyle \alpha ^{\vee }={\frac {2\alpha }{(\alpha ,\alpha )}}} One may perform a change of basis to define H α i = ( α i ∨ , H ) {\displaystyle H_{\alpha _{i}}=(\alpha _{i}^{\vee },H)} The Cartan integers are A i j = ( α i , α j ∨ ) {\displaystyle A_{ij}=(\alpha _{i},\alpha _{j}^{\vee })} The resulting relations among the generators are the following: = 0 {\displaystyle =0} = A j i E α j {\displaystyle =A_{ji}E_{\alpha _{j}}} = H α i {\displaystyle =H_{\alpha _{i}}} = ± ( p + 1 ) E β + γ {\displaystyle =\pm (p+1)E_{\beta +\gamma }} where in the last relation p {\displaystyle p} is the greatest positive integer such that γ − p β {\displaystyle \gamma -p\beta } is a root and we consider E β + γ = 0 {\displaystyle E_{\beta +\gamma }=0} if β + γ {\displaystyle \beta +\gamma } is not a root. For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if β ≺ γ {\displaystyle \beta \prec \gamma } then β + α ≺ γ + α {\displaystyle \beta +\alpha \prec \gamma +\alpha } provided that all four are roots.
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https://en.wikipedia.org/wiki/Chevalley_basis
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We then call ( β , γ ) {\displaystyle (\beta ,\gamma )} an extraspecial pair of roots if they are both positive and β {\displaystyle \beta } is minimal among all β 0 {\displaystyle \beta _{0}} that occur in pairs of positive roots ( β 0 , γ 0 ) {\displaystyle (\beta _{0},\gamma _{0})} satisfying β 0 + γ 0 = β + γ {\displaystyle \beta _{0}+\gamma _{0}=\beta +\gamma } . The sign in the last relation can be chosen arbitrarily whenever ( β , γ ) {\displaystyle (\beta ,\gamma )} is an extraspecial pair of roots. This then determines the signs for all remaining pairs of roots.
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https://en.wikipedia.org/wiki/Chevalley_basis
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In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford (1845–1879). The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras.
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https://en.wikipedia.org/wiki/Clifford_multiplication
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In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian manifold M which is called the Clifford bundle of M.
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https://en.wikipedia.org/wiki/Clifford_bundle
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In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra C is a central simple algebra over some field extension L of the field K over which the quadratic form Q defining C is defined. The abstract theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bott and Arnold S. Shapiro. A fundamental result on Clifford modules is that the Morita equivalence class of a Clifford algebra (the equivalence class of the category of Clifford modules over it) depends only on the signature p − q (mod 8). This is an algebraic form of Bott periodicity.
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https://en.wikipedia.org/wiki/Clifford_module
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In mathematics, a Clifford–Klein form is a double coset space Γ\G/H,where G is a reductive Lie group, H a closed subgroup of G, and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous space G/H. A suitable discrete subgroup Γ may or may not exist, for a given G and H. If Γ exists, there is the question of whether Γ\G/H can be taken to be a compact space, called a compact Clifford–Klein form. When H is itself compact, classical results show that a compact Clifford–Klein form exists. Otherwise it may not, and there are a number of negative results.
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https://en.wikipedia.org/wiki/Clifford–Klein_form
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In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways. They are named for Francis Sowerby Macaulay (1916), who proved the unmixedness theorem for polynomial rings, and for Irvin Cohen (1946), who proved the unmixedness theorem for formal power series rings.
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https://en.wikipedia.org/wiki/Cohen–Macaulay_ring
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All Cohen–Macaulay rings have the unmixedness property. For Noetherian local rings, there is the following chain of inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings
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https://en.wikipedia.org/wiki/Cohen–Macaulay_ring
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In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this. Such a multiplication of distributions has long been believed to be impossible because of L. Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one only wants to preserve the product of smooth functions instead such a construction becomes possible, as demonstrated first by Colombeau.
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https://en.wikipedia.org/wiki/Colombeau_algebra
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As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microlocal analysis and general relativity so far. Colombeau algebras are named after French mathematician Jean François Colombeau.
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https://en.wikipedia.org/wiki/Colombeau_algebra
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In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary element method, to mesh problem domains into elements. Coons patches are named after Steven Anson Coons, and date to 1967.
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https://en.wikipedia.org/wiki/Coons_surface
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