Home  About us  Contact
 

Extension Theorems (extension + theorem)

Distribution by Scientific Domains
Mathematics and Statistics88%

Selected Abstracts

Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2002
N. S. Bakhvalov
Abstract We prove extension theorems in the norms described by Stokes and Lamé operators for the three-dimensional case with periodic boundary conditions. For the Lamé equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem. Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and fictitious domain methods, and in analysis of finite element methods. We consider an application of established extension theorems to an efficient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coefficients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coefficient problem. Such preconditioner allows the use of well-known fast algorithms for preconditioning. Under some natural assumptions on smoothness and topological properties of subdomains with small coefficients, we prove convergence of the simplest Richardson method uniform in the jump of coefficients. For the Lamé equations, the convergence is also uniform in the incompressible limit. Our preliminary numerical results for two-dimensional diffusion problems show fast convergence uniform in the jump and in the mesh size parameter. Copyright © 2002 John Wiley & Sons, Ltd. [source]

On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circle

MATHEMATISCHE NACHRICHTEN, Issue 1-2 2007
Lev Aizenberg
Abstract Let D be a unit disk andM be an open arc of the unit circle whose Lebesgue measure satisfies 0 < l (M) < 2,. Our first result characterizes the restriction of the holomorphic functions f , ,(D), which are in the Hardy class ,1 near the arcM and are denoted by ,, ,1M(,,), to the open arcM. This result is a direct consequence of the complete description of the space of holomorphic functions in the unit disk which are represented by the Carleman formulas on the open arc M. As an application of the above characterization, we present an extension theorem for a function f , L1(M) from any symmetric sub-arc L , M of the unit circle, such that , M, to a function f , ,, ,1L(,,). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Opérateurs d'extension linéaires explicites dans des intersections de classes ultradifférentiables

MATHEMATISCHE NACHRICHTEN, Issue 12 2006
Pascal Beaugendre
Abstract B. S. Mityagin a montré que les polynômes de Tchebyshev forment une base de Schauder de l'espace des fonctions de classe C, sur l'intervalle [,1,1]. Il en déduit un opérateur linéaire continu d'extension explicite. Ces résultats ont été étendus, par A. Goncharov, à des compacts ne satisfaisant pas la propriété de Markov. A contrario, M. Tidten a donné des exemples de compacts pour lesquels il n'y a pas d'opérateur linéaire continu d'extension. Dans cet article, on généralise ces travaux à des classes de fonctions ultradifférentiables construites sur le modèle de l'intersection des classes de Gevrey non quasi-analytiques. On obtient notamment un théorème d'extension linéaire dans des classes de Beurling assez grandes. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) B. S. Mityagin proved that the Chebyshev polynomials form a Schauder basis of the space of C, functions on the interval [,1,1]. Whereof he deduced an explicit continuous linear extension operator. These results were extended, by A. Goncharov, to compact sets without Markov's property. On the reverse, M. Tidten gave examples of compact sets for which there is no continuous linear extension operator. In this paper, we generalize these works to the intersections of ultradifferentiable classes of functions built on the model of the non quasianalytic intersection of Gevrey classes. We get, among other things, a Whitney linear extension theorem for ultradifferentiable jets of Beurling type. [source]

Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2002
N. S. Bakhvalov
Abstract We prove extension theorems in the norms described by Stokes and Lamé operators for the three-dimensional case with periodic boundary conditions. For the Lamé equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem. Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and fictitious domain methods, and in analysis of finite element methods. We consider an application of established extension theorems to an efficient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coefficients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coefficient problem. Such preconditioner allows the use of well-known fast algorithms for preconditioning. Under some natural assumptions on smoothness and topological properties of subdomains with small coefficients, we prove convergence of the simplest Richardson method uniform in the jump of coefficients. For the Lamé equations, the convergence is also uniform in the incompressible limit. Our preliminary numerical results for two-dimensional diffusion problems show fast convergence uniform in the jump and in the mesh size parameter. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Biregular extendability via isotonic Clifford analysis

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2010
Ricardo Abreu Blaya
Abstract We use the so-called isotonic functions to obtain extension theorems in the framework of biregular functions of Clifford analysis. In this context we also prove the Plemelj,Sokhotski formulae for the Bochner,Martinelli integral and an expression for the square of its singular version. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Two constructive embedding-extension theorems with applications to continuity principles and to Banach-Mazur computability

MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4-5 2004
Andrej Bauer
Abstract We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to , extends to a sequentially continuous function from X to ,. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between "continuity principles" asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle "all functions from X to , are continuous", when X is an inhabited CSM without isolated points, and when X is an inhabited locally non-compact CSM. One situation in which the latter case applies is in models based on "domain realizability", in which the failure of the continuity principle for any inhabited locally non-compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1]. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space X, there exists a Banach-Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2002
N. S. Bakhvalov
Abstract We prove extension theorems in the norms described by Stokes and Lamé operators for the three-dimensional case with periodic boundary conditions. For the Lamé equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem. Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and fictitious domain methods, and in analysis of finite element methods. We consider an application of established extension theorems to an efficient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coefficients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coefficient problem. Such preconditioner allows the use of well-known fast algorithms for preconditioning. Under some natural assumptions on smoothness and topological properties of subdomains with small coefficients, we prove convergence of the simplest Richardson method uniform in the jump of coefficients. For the Lamé equations, the convergence is also uniform in the incompressible limit. Our preliminary numerical results for two-dimensional diffusion problems show fast convergence uniform in the jump and in the mesh size parameter. Copyright © 2002 John Wiley & Sons, Ltd. [source]