Mathematics. Dictionary
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In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy ‖ x y ‖ ≤ ‖ x ‖ ‖ y ‖ for all x , y ∈ A . {\displaystyle \|x\,y\|\ \leq \|x\|\,\|y\|\quad {\text{ for all }}x,y\in A.} This ensures that the multiplication operation is continuous with respect to the metric topology.…
The lowest side of a triangle or other polygon, or the lowest face of a cone, pyramid or other polyhedron laid flat.
In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ∂ w ∂ z ¯ = μ ∂ w ∂ z . {\displaystyle {\frac {\partial w}{\partial {\bar {z}}}}=\mu {\frac {\partial w}{\partial z}}.} for w {\displaystyle w} a complex distribution of the complex variable z {\displaystyle z} in some open set U {\displaystyle U} , with derivatives that are locally L2, and where μ {\displaystyle \mu } is a given complex func…
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.
In mathematics, Bernstein's theorem may refer to: Bernstein's theorem about the Sato–Bernstein polynomial Bernstein's problem about minimal surfaces Bernstein's theorem on monotone functions Bernstein's theorem (approximation theory) Bernstein's theorem (polynomials) Bernstein's lethargy theorem Bernstein–von Mises theorem Cantor–Bernstein theorem in set theory and order theory Cantor–Bernstein–Schroeder theorem in set theory Bernstein–Kushnirenk…
In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x {\displaystyle x} in a Hilbert space with respect to an orthonormal sequence. The inequality is named for F. W. Bessel, who derived a special case of it in 1828. Conceptually, the inequality is a generalization of the Pythagorean theorem to infinite-dimensional spaces. It states that the "energy" of a vector x {\displaystyle x} , given by ‖ x ‖ 2 {\displaystyle \|x\|^{2}} , is greater than or equal to the sum of the energies of its projections onto a set of perpendicular basis directions.…
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden "qualitative" or topological change in its behavior.…
A function of two arguments from the same vector space which maps onto a field of scalars, which acts like a linear form with respect to either one of its arguments when the other one is held constant.
Billion is a word (in several languages) for a large number with two distinct definitions: 1,000,000,000, i.e. one thousand million, or 109 (ten to the ninth power), as defined on the short scale (also known as a milliard on the long scale). This is now the most common sense of the word in all varieties of English; it has long been established in American English and has since become common in Britain and other English-speaking countries as well. 1,000,000,000,000, i.e.…
a coefficient of any of the terms in the expansion of the binomial (x+y)n, defined by {n \choose k} = \frac{n!}{k!(n-k)!}, read as "n choose k"
The discrete probability distribution of the number of successes in a sequence of n independent trials, each of which yields success with probability p.
A root-finding method that applies to any continuous functions for which one knows two values with opposite signs, and consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root.
A line or curve that bisects or divides a line segment, angle, or other figure into two equal parts.
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as blocks, chosen such that number of occurrences of each element satisfies certain conditions making the collection of blocks exhibit symmetry (balance). Block designs have applications in many areas, including experimental design, finite geometry, physical chemistry, software testing, cryptography, and algebraic geometry.…
In geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should imply a stronger, topological notion (namely, homeomorphism).
A rectangular object in any number of dimensions.
BPP may refer to:
In mathematics, the braid group on n strands (denoted B n {\displaystyle B_{n}} ), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem…
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.
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often itself called "Brownian motion", even in mathematical sources. This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume.…
In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) G = B W B {\displaystyle G=BWB} of certain algebraic groups G = B W B {\displaystyle G=BWB} into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this.…
In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of isotropic reductive linear algebraic groups over arbitrary fields. The more specialized theory of Bruhat–Tits bu…
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory, and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants that differ in the additional conditions imposed on the orders of the group elements (see bounded and restricted below).…
Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, or the orbit-counting theorem, is a result in group theory that is often useful in taking account of symmetry when counting mathematical objects. It was discovered by Augustin Louis Cauchy and Ferdinand Georg Frobenius, and became well known after William Burnside quoted it. The result enumerates orbits of a symmetry group acting on some objects: that…
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