Composition operators on spaces of real analytic functions

Domański, Paweł; Langenbruch, Michael

doi:10.1002/mana.200310053

Cited by 18 publications

(8 citation statements)

References 48 publications

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“…Here ρ 0 is a constant density 1 , v is a velocity, m is a 1-D strain, n is the direction of the plane wave propagation, I is a 3 × 3 identity matrix and B is a 3 × 3 symmetric matrix, which we restrict to consist of zero and first order terms in strains Λ and Ψ, respectively. Alternatively, in components,…”

Section: Plane Wave Equationsmentioning

confidence: 99%

“…In the examples below we consider a cubic crystal of class m3m in which the strain energy W is characterized by three second order and six third order elastic constants (see [1,2,3,4]) c 12 , c 44 , c 111 , c 112 , c 144 , c 123 , c 166 , c 456 ).…”

Section: Examplesmentioning

confidence: 99%

“…Here we present the case where the propagation direction of the plane wave is a two-fold symmetry axis n = 1 √ 2 [1 1 0] which is not an acoustic axis. The speeds of shear waves are (see [1,2,3,4])…”

Section: Examplementioning

confidence: 99%

“…Let us consider the system of equations describing plane waves propagating in n direction in an arbitrary nonlinear anisotropic hyperelastic material in which both geometrical and physical nonlinearities are taken into account. This system of equations can be written as follows [1,2,3,4]:…”

Section: Plane Wave Equationsmentioning

confidence: 99%

“…It is known that transverse elastic waves cannot interact with each other on the quadratically nonlinear level in an isotropic material [5]. On the other hand it is also known that in anisotropic materials this is not the case [1]. There are certain directions in crystals for which quadratically nonlinear interactions do occur between (quasi-) shear elastic waves.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Domanski¹,

Norris²

2008

AIP Conference Proceedings

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Transverse elastic waves behave differently in nonlinear isotropic and anisotropic media. While in the former the quadratically nonlinear coupling in the evolution equations for wave amplitudes is not possible, such a coupling may occur for certain directions in anisotropic materials. We identify the expression responsible for the coupling and we derive coupled canonical evolution equations for transverse wave amplitudes in the case of the two-fold and three-fold symmetry acoustic axes. We illustrate our considerations by examples for a cubic crystal.

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Section: Plane Wave Equationsmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Plane Wave Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Domanski¹,

Norris²

2008

AIP Conference Proceedings

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Continuity of Some Standard Geometric Operations

Lewis

2023

Lecture Notes in Mathematics

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A non-trivial Fr�chet quotient of the space of real analytic functions

Domański

Vogt

2003

Archiv der Mathematik

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We prove that the space of real analytic functions A( ) on an arbitrary open set R d has a Fréchet infinite dimensional quotient space with a continuous norm.In the paper [10] (comp.[11]) we proved that every Fréchet quotient of A ( ) has the very restrictive property ( ). This was a crucial point in the proof that A ( ) has no basis. It was quite annoying (see the remark after Theorem 5.2 in [10]) that up to now the only infinite dimensional Fréchet quotients known to us were isomorphic to the space C N of all sequences. That might have suggested that our analysis of Fréchet quotients of A ( ) contained in [10] was far from being optimal. Here we show that there are plenty of pairwise non-isomorphic infinite dimensional Fréchet quotients of A ( ), even Köthe sequence spaces with continuous norms are among them. Theorem 1. There is a Köthe sequence Fréchet space λ 1 (B) with a continuous norm such that for every open set R d the space λ 1 (B) is a quotient space of A ( ). Let us note that the paper [7] contains the full description of LB and Fréchet subspaces of A ( ) as isomorphic respectively to subspaces of H (D d ) and H (D d ), if is connected. In general, Fréchet subspaces are isomorphic to subspaces of H (D d ) p where p is the cardinality of the set of connected components of . It follows easily from [7] that LB quotients (or LB complemented subspaces) of A ( ) are exactly quotients (or complemented subspaces) of H (D d ). By [10] we know that Fréchet complemented subspaces of A ( ) are finite dimensional. The problem of a description of Fréchet quotients of A ( ) remains still open.The present paper is a step toward the structural theory of the space A ( ). We believe that it is worthwhile to develop such a theory because an analogous theory for Fréchet spaces has proved to be very useful in the study of classical operators on them and in solving problems Mathematics Subject Classification (2000): Primary 46E10; Secondary 46A04, 46A45, 46A63.

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Composition operators on spaces of real analytic functions

Cited by 18 publications

References 48 publications

Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Continuity of Some Standard Geometric Operations

A non-trivial Fr�chet quotient of the space of real analytic functions

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