analysis, a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert...

multiplication operator and as a direct integral, since L 2 ( σ ( A ) , μ ) {\displaystyle L^{2}(\sigma (A),\mu )} is just a direct integral in which each Hilbert...

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process...

Unlike the multiplication-operator version of the spectral theorem, the direct-integral version is unique in the sense that the measure equivalence class of...

methods include: direct integration of a complex-valued function along a curve in the complex plane application of the Cauchy integral formula application...

scalar operators. A finite von Neumann algebra is one which is the direct integral of finite factors (meaning the von Neumann algebra has a faithful normal...

by spectral multiplicity theory and requires the use of direct integrals. Using direct integral theory, it can be shown that the abelian von Neumann algebras...

we provide a general example of projection-valued measure based on direct integrals. Suppose (X, M, μ) is a measure space and let {Hx}x ∈ X be a μ-measurable...

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that...

In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied...

calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of...

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function...

mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear...

antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function f is a differentiable function F whose...

of G is of type I, then L2(G) as a unitary representation of G is a direct integral of irreducible representations. It is parametrized therefore by the...

the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form...

rays, cannot penetrate Earth's atmosphere, and direct observations must be made from space. INTEGRAL is an observatory, scientists can propose for observing...

infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes...

calculus), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially...

improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context...

The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces...

In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration...

vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary...

derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general...

An integral, or intrinsic, membrane protein (IMP) is a type of membrane protein that is permanently attached to the biological membrane. All transmembrane...

proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949...

several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of...

Stratonovich integral can readily be expressed in terms of the Itô integral, and vice versa. The main benefit of the Stratonovich integral is that it obeys...

In mathematics, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} is the area of the region in the xy-plane bounded by the...

calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the...