In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm. The norm is required to satisfy
In mathematics, when X is a finite set of at least two elements, the permutations of X fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity of a permutation of X can be defined as the parity of the number of inversions for σ, i.e., of pairs of elements x, y of X such that and .
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.
In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. It is essentially a chi distribution with two degrees of freedom.
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.
In mathematics, the braid group on n strands, also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids, and whose group operation is composition of braids. Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids ; in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation ; and in monodromy invariants of algebraic geometry.
In mathematics, a filtration is an indexed set of subobjects of a given algebraic structure , with the index running over some index set that is a totally ordered set, subject to the condition that
The spectrum of a linear operator that operates on a Banach space consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts:
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different categories of spectra, but they all determine the same homotopy category, known as the stable homotopy category.
In measure theory, Carathéodory's extension theorem states that any pre-measure defined on a given ring R of subsets of a given set Ω can be extended to a measure on the σ-algebra generated by R, and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure.
The possibility that there might be more than one dimension of time has occasionally been discussed in physics and philosophy.
In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of
In convex geometry, Gordan's lemma states that the semigroup of integral points in the dual cone of a rational convex polyhedral cone is finitely generated. In algebraic geometry, the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety; thus, the lemma says an affine toric variety is indeed an algebraic variety. The lemma is named after the German mathematician Paul Gordan (1837–1912).
Difference algebra is a branch of mathematics concerned with the study of difference equations from the algebraic point of view. Difference algebra is analogous to differential algebra but concerned with difference equations rather than differential equations. As an independent subject it was initiated by Joseph Ritt and his student Richard Cohn.
In mathematics, in the subfield of ring theory, a ring R is a polynomial identity ring if there is, for some N > 0, an element P other than 0 of the free algebra, Z⟨X1, X2, ..., XN⟩, over the ring of integers in N variables X1, X2, ..., XN such that for all N-tuples r1, r2, ..., rN taken from R it happens that
Nonuniform sampling is a branch of sampling theory involving results related to the Nyquist–Shannon sampling theorem. Nonuniform sampling is based on Lagrange interpolation and the relationship between itself and the (uniform) sampling theorem. Nonuniform sampling is a generalisation of the Whittaker–Shannon–Kotelnikov (WSK) sampling theorem.
