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harish-chandra's c-function | In mathematics, Harish-Chandra's c-function is a function related to the intertwining operator between two principal series representations, that appears in the Plancherel measure for semisimple Lie groups. Harish-Chandra (1958a, 1958b) introduced a special case of it defined in terms of the asymptotic behavior of a zo... | wikipedia |
hooley's delta function | In mathematics, Hooley's delta function ( Δ ( n ) {\displaystyle \Delta (n)} ), also called Erdős--Hooley delta-function, defines the maximum number of divisors of n {\displaystyle n} in {\displaystyle } for all u {\displaystyle u} , where e {\displaystyle e} is the Euler's number. The first few terms of this sequence... | wikipedia |
euclidean hurwitz algebra | In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into ... | wikipedia |
euclidean hurwitz algebra | The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwit... | wikipedia |
christiaan huygens | In mathematics, Huygens mastered the methods of ancient Greek geometry, particularly the work of Archimedes, and was an adept user of the analytic geometry and infinitesimal techniques of Descartes and Fermat. His mathematical style can be best described as geometrical infinitesimal analysis of curves and of motion. Dr... | wikipedia |
christiaan huygens | Huygens brought this type of geometrical analysis to a close, as more mathematicians turned away from classical geometry to the calculus for handling infinitesimals, limit processes, and motion.Huygens was moreover able to fully employ mathematics to answer questions of physics. Often this entailed introducing a simple... | wikipedia |
christiaan huygens | Huygens favoured axiomatic presentations of his results, which require rigorous methods of geometric demonstration: although he allowed levels of uncertainty in the selection of primary axioms and hypotheses, the proofs of theorems derived from these could never be in doubt. Huygens's style of publication exerted an in... | wikipedia |
christiaan huygens | In demanding such mathematical tractability and precision, Huygens set an example for eighteenth-century scientists such as Johann Bernoulli, Jean le Rond d'Alembert, and Charles-Augustin de Coulomb.Although never intended for publication, Huygens made use of algebraic expressions to represent physical entities in a ha... | wikipedia |
serre's conjecture ii (algebra) | In mathematics, Jean-Pierre Serre conjectured the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group. Namely, he conjectured that if G is such a group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H1(F, G) is zero. A convers... | wikipedia |
serre's conjecture ii (algebra) | (Note that such fields do indeed have cohomological dimension at most 2.) The conjecture also holds when F is finitely generated over the complex numbers and has transcendence degree at most 2.The conjecture is also known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Susl... | wikipedia |
jordan operator algebra | In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algeb... | wikipedia |
jordan operator algebra | Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are cal... | wikipedia |
jordan operator algebra | Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and... | wikipedia |
algebraic de rham cohomology | In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calcul... | wikipedia |
lady windermere's fan (mathematics) | In mathematics, Lady Windermere's Fan is a telescopic identity employed to relate global and local error of a numerical algorithm. The name is derived from Oscar Wilde's 1892 play Lady Windermere's Fan, A Play About a Good Woman. | wikipedia |
landau's function | In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn. Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation of n elements can be... | wikipedia |
landau's function | An element of order 6 in the group S5 can be written in cycle notation as (1 2) (3 4 5). Note that the same argument applies to the number 6, that is, g(6) = 6. There are arbitrarily long sequences of consecutive numbers n, n + 1, …, n + m on which the function g is constant.The integer sequence g(0) = 1, g(1) = 1, g(2... | wikipedia |
landau's function | Equivalently (using little-o notation), g ( n ) = e ( 1 + o ( 1 ) ) n ln n {\displaystyle g(n)=e^{(1+o(1)){\sqrt {n\ln n}}}} . The statement that ln g ( n ) < L i − 1 ( n ) {\displaystyle \ln g(n)<{\sqrt {\mathrm {Li} ^{-1}(n)}}} for all sufficiently large n, where Li−1 denotes the inverse of the logarithmic integr... | wikipedia |
lie algebra cohomology | In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by Claude Chevalley and S... | wikipedia |
liouville's theorem (differential algebra) | In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. These are called n... | wikipedia |
liouville's theorem (differential algebra) | Other examples include the functions sin ( x ) x {\displaystyle {\frac {\sin(x)}{x}}} and x x . {\displaystyle x^{x}.} Liouville's theorem states that elementary antiderivatives, if they exist, are in the same differential field as the function, plus possibly a finite number of applications of the logarithm function. | wikipedia |
mathieu differential equation | In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation d 2 y d x 2 + ( a − 2 q cos ( 2 x ) ) y = 0 , {\displaystyle {\frac {d^{2}y}{dx^{2}}}+(a-2q\cos(2x))y=0,} where a, q are real-valued parameters. Since we may add π/2 to x to change the sign ... | wikipedia |
matsumoto zeta function | In mathematics, Matsumoto zeta functions are a type of zeta function introduced by Kohji Matsumoto in 1990. They are functions of the form ϕ ( s ) = ∏ p 1 A p ( p − s ) {\displaystyle \phi (s)=\prod _{p}{\frac {1}{A_{p}(p^{-s})}}} where p is a prime and Ap is a polynomial. | wikipedia |
minkowski's question-mark function | In mathematics, Minkowski's question-mark function, denoted ? (x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the bin... | wikipedia |
miser algorithm | In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand ... | wikipedia |
sweedler's hopf algebra | In mathematics, Moss E. Sweedler (1969, p. 89–90) introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra H4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative. | wikipedia |
neville's schema | In mathematics, Neville's algorithm is an algorithm used for polynomial interpolation that was derived by the mathematician Eric Harold Neville in 1934. Given n + 1 points, there is a unique polynomial of degree ≤ n which goes through the given points. Neville's algorithm evaluates this polynomial. Neville's algorithm ... | wikipedia |
owen's t function | In mathematics, Owen's T function T(h, a), named after statistician Donald Bruce Owen, is defined by T ( h , a ) = 1 2 π ∫ 0 a e − 1 2 h 2 ( 1 + x 2 ) 1 + x 2 d x ( − ∞ < h , a < + ∞ ) . {\displaystyle T(h,a)={\frac {1}{2\pi }}\int _{0}^{a}{\frac {e^{-{\frac {1}{2}}h^{2}(1+x^{2})}}{1+x^{2}}}dx\quad \left(-\infty | wikipedia |
painleve equations | In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions. They were discovered by Émile Picard (... | wikipedia |
joseph petzval | In mathematics, Petzval stressed practical applicability. He said, "Mankind does not exist for science's sake, but science should be used to improve the conditions of mankind." He worked on applications of the Laplace transformation. His work was very thorough, but not completely satisfying, since he could not use an e... | wikipedia |
joseph petzval | Petzval wrote a paper in two volumes as well as a long work on this subject. A controversy with the student Simon Spritzer, who accused Petzval of plagiarism of Pierre-Simon Laplace, led the Spritzer-influenced mathematicians George Boole and Jules Henri Poincaré to later name the transformation after Laplace. Petzval ... | wikipedia |
pfaffian function | In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally introduced by Askold Khovanskii in the 1970s, but are named after German mathematician Johann Pfaff. | wikipedia |
philo of byzantium | In mathematics, Philo tackled the problem of doubling the cube. The doubling of the cube was necessitated by the following problem: given a catapult, construct a second catapult that is capable of firing a projectile twice as heavy as the projectile of the first catapult. His solution was to find the point of intersect... | wikipedia |
pontryagin dual | In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive grou... | wikipedia |
pontryagin dual | The subject is named after Lev Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the groups being second-countable and either compact or discrete. This was improved to cover the genera... | wikipedia |
probabilistic number theory | In mathematics, Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions about the integers and integer-valued functions. One basic idea underlying it is that different prime numbers are, in some serious sense, like independent random variables. This however is n... | wikipedia |
rathjen's psi function | In mathematics, Rathjen's ψ {\displaystyle \psi } psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals M {\displaystyle M} to generate large countable ordinals. A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below M {\displaystyle... | wikipedia |
absolute differential calculus | In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus),... | wikipedia |
absolute differential calculus | Tensors and tensor fields can be expressed in terms of their components, and operations on tensors and tensor fields can be expressed in terms of operations on their components. The description of tensor fields and operations on them in terms of their components is the focus of the Ricci calculus. This notation allows ... | wikipedia |
absolute differential calculus | While much of the notation may be applied with any tensors, operations relating to a differential structure are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays. | wikipedia |
absolute differential calculus | A tensor may be expressed as a linear sum of the tensor product of vector and covector basis elements. The resulting tensor components are labelled by indices of the basis. Each index has one possible value per dimension of the underlying vector space. | wikipedia |
absolute differential calculus | The number of indices equals the degree (or order) of the tensor. For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generall... | wikipedia |
s2s (mathematics) | In mathematics, S2S is the monadic second order theory with two successors. It is one of the most expressive natural decidable theories known, with many decidable theories interpretable in S2S. Its decidability was proved by Rabin in 1969. | wikipedia |
bochner's theorem (riemannian geometry) | In mathematics, Salomon Bochner proved in 1946 that any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero. Consequently the isometry group of the manifold must be finite. | wikipedia |
schubert's enumerative calculus | In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in ... | wikipedia |
schubert's enumerative calculus | Even more generally, "Schubert calculus" is often understood to encompass the study of analogous questions in generalized cohomology theories. The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective ... | wikipedia |
schubert's enumerative calculus | The intersection theory of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points, which are potentially concrete answers to enumer... | wikipedia |
schur algebra | In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of those two groups. Their use was promoted by the influential monograph of J. A. G... | wikipedia |
skew schur function | In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible represe... | wikipedia |
schwartz function | In mathematics, Schwartz space S {\displaystyle {\mathcal {S}}} is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for e... | wikipedia |
spence's function | In mathematics, Spence's function, or dilogarithm, denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: Li 2 ( z ) = − ∫ 0 z ln ( 1 − u ) u d u , z ∈ C {\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-... | wikipedia |
spence's function | Alternatively, the dilogarithm function is sometimes defined as ∫ 1 v ln t 1 − t d t = Li 2 ( 1 − v ) . {\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v).} In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal simplex. | wikipedia |
spence's function | Specifically, a simplex whose vertices have cross ratio z has hyperbolic volume D ( z ) = Im Li 2 ( z ) + arg ( 1 − z ) log | z | . {\displaystyle D(z)=\operatorname {Im} \operatorname {Li} _{2}(z)+\arg(1-z)\log |z|.} The function D(z) is sometimes called the Bloch-Wigner function. | wikipedia |
spence's function | Lobachevsky's function and Clausen's function are closely related functions. William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century. He was at school with John Galt, who later wrote a biographical essay on Spence. | wikipedia |
representation theorem for boolean algebras | In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone.... | wikipedia |
fundamental theorem of topos theory | In mathematics, The fundamental theorem of topos theory states that the slice E / X {\displaystyle \mathbf {E} /X} of a topos E {\displaystyle \mathbf {E} } over any one of its objects X {\displaystyle X} is itself a topos. Moreover, if there is a morphism f: A → B {\displaystyle f:A\rightarrow B} in E {\displaystyle \... | wikipedia |
geometrisation conjecture | In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface ca... | wikipedia |
geometrisation conjecture | The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since... | wikipedia |
geometrisation conjecture | Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at the arxiv.org preprint server. Perelman's papers were studied by several independent groups that produced books and online manuscripts filling in the complete details of his arguments. ... | wikipedia |
thābit ibn qurra | In mathematics, Thābit derived an equation for determining amicable numbers. His proof of this rule is presented in the Treatise on the Derivation of the Amicable Numbers in an Easy Way. This was done while writing on the theory of numbers, extending their use to describe the ratios between geometrical quantities, a st... | wikipedia |
thābit ibn qurra | He provided a strengthened extension of Pythagoras' proof which included the knowledge of Euclid's fifth postulate. This postulate states that the intersection between two straight line segments combine to create two interior angles which are less than 180 degrees. The method of reduction and composition used by Thābit... | wikipedia |
thābit ibn qurra | Thābit believed that geometry was tied with the equality and differences of magnitudes of lines and angles, as well as that ideas of motion (and ideas taken from physics more widely) should be integrated in geometry.The continued work done on geometric relations and the resulting exponential series allowed Thābit to ca... | wikipedia |
thābit ibn qurra | His work with conic sections and the calculation of a paraboloid shape (cupola) show his proficiency as an Archimedean geometer. This is further embossed by Thābit's use of the Archimedean property in order to produce a rudimentary approximation of the volume of a paraboloid. The use of uneven sections, while relativel... | wikipedia |
tonelli's theorem (functional analysis) | In mathematics, Tonelli's theorem in functional analysis is a fundamental result on the weak lower semicontinuity of nonlinear functionals on Lp spaces. As such, it has major implications for functional analysis and the calculus of variations. Roughly, it shows that weak lower semicontinuity for integral functionals is... | wikipedia |
ulugh beg | In mathematics, Ulugh Beg wrote accurate trigonometric tables of sine and tangent values correct to at least eight decimal places. | wikipedia |
vinberg's algorithm | In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. Conway (1983) used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech... | wikipedia |
volterra's function | In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties: V is differentiable everywhere The derivative V ′ is bounded everywhere The derivative is not Riemann-integrable. | wikipedia |
weber's theorem (algebraic curves) | In mathematics, Weber's theorem, named after Heinrich Martin Weber, is a result on algebraic curves. It states the following. Consider two non-singular curves C and C′ having the same genus g > 1. If there is a rational correspondence φ between C and C′, then φ is a birational transformation. | wikipedia |
weingarten function | In mathematics, Weingarten functions are rational functions indexed by partitions of integers that can be used to calculate integrals of products of matrix coefficients over classical groups. They were first studied by Weingarten (1978) who found their asymptotic behavior, and named by Collins (2003), who evaluated the... | wikipedia |
quantization condition | In mathematics, a (classical) gauge field is defined as a connection over a principal G-bundle over spacetime. G is the gauge group, and it acts on each fiber of the bundle separately. A connection on a G-bundle tells you how to glue fibers together at nearby points of M. It starts with a continuous symmetry group G th... | wikipedia |
quantization condition | In physics, the connection is the fundamental physical object. One of the fundamental observations in the theory of characteristic classes in algebraic topology is that many homotopical structures of nontrivial principal bundles may be expressed as an integral of some polynomial over any connection over it. Note that a... | wikipedia |
quantization condition | If spacetime is ℝ4 the space of all possible connections of the G-bundle is connected. But consider what happens when we remove a timelike worldline from spacetime. The resulting spacetime is homotopically equivalent to the topological sphere S2. | wikipedia |
quantization condition | A principal G-bundle over S2 is defined by covering S2 by two charts, each homeomorphic to the open 2-ball such that their intersection is homeomorphic to the strip S1×I. 2-balls are homotopically trivial and the strip is homotopically equivalent to the circle S1. So a topological classification of the possible connect... | wikipedia |
quantization condition | The transition function maps the strip to G, and the different ways of mapping a strip into G are given by the first homotopy group of G. So in the G-bundle formulation, a gauge theory admits Dirac monopoles provided G is not simply connected, whenever there are paths that go around the group that cannot be deformed to... | wikipedia |
quantization condition | The total magnetic flux is none other than the first Chern number of the principal bundle, and depends only upon the choice of the principal bundle, and not the specific connection over it. In other words, it is a topological invariant. This argument for monopoles is a restatement of the lasso argument for a pure U(1) ... | wikipedia |
quantization condition | It generalizes to d + 1 dimensions with d ≥ 2 in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension d − 3. Another way is to examine the type of topological singularity at a point with the homotopy group πd−2(G). | wikipedia |
leibniz algebra | In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear product satisfying the Leibniz identity , c ] = ] + , b ] . {\displaystyle ,c]=]+,b].\,} In other words, right multiplic... | wikipedia |
leibniz algebra | Conversely any Lie algebra is obviously a Leibniz algebra. In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. The investigation of which theorems and properties of Lie algebras are still valid for Leibniz algebras is a recurrent theme in the literature. | wikipedia |
leibniz algebra | For instance, it has been shown that Engel's theorem still holds for Leibniz algebras and that a weaker version of Levi-Malcev theorem also holds.The tensor module, T(V) , of any vector space V can be turned into a Loday algebra such that = a 1 ⊗ ⋯ a n ⊗ x for a 1 , … , a n , x ∈ V . {\displaystyle =a_{1}\otimes \cdot... | wikipedia |
leibniz algebra | They attracted interest after Jean-Louis Loday noticed that the classical Chevalley–Eilenberg boundary map in the exterior module of a Lie algebra can be lifted to the tensor module which yields a new chain complex. In fact this complex is well-defined for any Leibniz algebra. The homology HL(L) of this chain complex i... | wikipedia |
leibniz algebra | If L is the Lie algebra of (infinite) matrices over an associative R-algebra A then Leibniz homology of L is the tensor algebra over the Hochschild homology of A. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. It has defining identity: ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) + a ∘ ( c ∘ b ) . {\displaystyle (... | wikipedia |
*-algebra | In mathematics, a *-ring is a ring with a map *: A → A that is an antiautomorphism and an involution. More precisely, * is required to satisfy the following properties: (x + y)* = x* + y* (x y)* = y* x* 1* = 1 (x*)* = xfor all x, y in A. This is also called an involutive ring, involutory ring, and ring with involution.... | wikipedia |
*-algebra | One can define a sesquilinear form over any *-ring. Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: x ∈ I ⇒ x* ∈ I and so on. *-rings are unrelated to star semirings in the theory of computation. | wikipedia |
banach bundle (non-commutative geometry) | 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. | wikipedia |
barnes zeta function | 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. | wikipedia |
batalin–vilkovisky formalism | 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 |\De... | wikipedia |
beurling zeta function | 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 tha... | wikipedia |
finitary boolean function | 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 func... | wikipedia |
finitary boolean function | 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 {\displays... | wikipedia |
finitary boolean function | . . , x k {\displaystyle x_{1},...,x_{k}} , and two propositional formulas are logically equivalent if and only if they express the same Boolean function. | wikipedia |
wightman functional | 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 funct... | wikipedia |
borel algebra | 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 ... | wikipedia |
borel algebra | 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. | wikipedia |
borel algebra | 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... | wikipedia |
bose–mesner algebra | 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... | wikipedia |
brauer algebra | 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. | wikipedia |
buekenhout geometry | 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). | wikipedia |
cantor algebra | 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... | wikipedia |
cantor algebra | 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 ... | wikipedia |
cartan algebra | 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 {\displaysty... | wikipedia |
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