Integral decomposition for the solutions of the generalized Cattaneo equation

AbstractIt is proved that for a second-order, homogeneous, random process on a globally symmetric space a filter, that is a closed linear operator which is invariant under a group of isometries of the space, may be fully described through a response function, that is that it has a direct integral decomposition into components which are scalar multiples of the identity.

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The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-067-4 ◽

1971 ◽

Vol 23 (4) ◽

pp. 598-607 ◽

Cited By ~ 2

Author(s):

Ole A. Nielsen

Keyword(s):

Von Neumann Algebra ◽

Von Neumann Algebras ◽

Direct Integral ◽

Von Neumann ◽

Ratio Set ◽

Almost All ◽

Direct Integral Decomposition ◽

Integral Decomposition ◽

Remarkable Progress

The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.

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The Plancherel formula for the horocycle space and generalizations

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700000069 ◽

1996 ◽

Vol 61 (1) ◽

pp. 42-56 ◽

Cited By ~ 2

Author(s):

Ronald L. Lipsman

Keyword(s):

Harmonic Analysis ◽

Regular Representation ◽

Plancherel Formula ◽

Intertwining Operators ◽

Direct Integral ◽

Regular Representations ◽

Spectral Distributions ◽

Theoretic Technique ◽

Direct Integral Decomposition ◽

Integral Decomposition

AbstractThe Plancherel formula for the horocycle space, and several generalizations, is derived within the framework of quasi-regular representations which have monomial spectrum. The proof uses only machinery from the Penney-Fujiwara distribution-theoretic technique; no special semisimple harmonic analysis is needed. The Plancherel formulas obtained include the spectral distributions and the intertwining operators that effect the direct integral decomposition of the quasi-regular representation.

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Bloch wave homogenisation of quasiperiodic media

European Journal of Applied Mathematics ◽

10.1017/s0956792520000352 ◽

2020 ◽

pp. 1-21

Author(s):

SISTA SIVAJI GANESH ◽

VIVEK TEWARY

Keyword(s):

Bloch Wave ◽

Periodic Operator ◽

Almost Periodic ◽

Periodic Media ◽

Direct Integral ◽

Bloch Waves ◽

Direct Integral Decomposition ◽

Quasiperiodic Media ◽

Integral Decomposition ◽

Periodic Operators

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

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