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21	Polynomiale Kollokations-Quadraturverfahren für singuläre Integralgleichungen mit festen Singularitäten Kaiser, Robert 13 October 2017 (has links) Viele Probleme der Riss- und Bruchmechanik sowie der mathematischen Physik lassen sich auf Lösungen von singulären Integralgleichungen über einem Intervall zurückführen. Diese Gleichungen setzen sich im Wesentlichen aus dem Cauchy'schen singulären Integraloperator und zusätzlichen Integraloperatoren mit festen Singularitäten in den jeweiligen Kernen zusammen. Zur numerischen Lösung solcher Gleichungen werden polynomiale Kollokations-Quadraturverfahren betrachet. Als Ansatzfunktionen und Kollokationspunkte werden dabei gewichtete Polynome und Tschebyscheff-Knoten gewählt. Die Gewichte sind so gewählt, dass diese das asymptotische Verhalten der Lösung in den Randpunkten widerspiegeln. Mit Hilfe von C*-Algebra Techniken, werden in dieser Arbeit notwendige und hinreichende Bedingungen für die Stabilität der Kollokations-Quadraturverfahren angegeben. Die theoretischen Resultate werden dabei durch numerische Berechnungen anhand des Problems der angerissenen Halbebene und des angerissenen Loches überprüft. info:eu-repo/classification/ddc/510 ddc:510
22	Koliha–Drazin invertibles form a regularity Smit, Joukje Anneke 10 1900 (has links) The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics) Banach algebra Radical Spectrum Resolvent Quasinilpotent Nilpotent Spectral idempotent Isolated spectral point Accumulation point Regularity Koliha-Drazin invertible Quasipolar KD-spectrum D-spectrum Laurent expansion Poles of the resolvent 511.322 Axiomatic set theory Banach algebras Spectrum analysis Spectral sequences (Mathematics)
23	Weighted inequalities and properties of operators and embeddings on function spaces / Weighted inequalities and properties of operators and embeddings on function spaces Slavíková, Lenka January 2016 (has links) The present thesis is devoted to the study of various properties of Banach func- tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higher-order embeddings of Sobolev-type spaces built upon rearrangement-invariant Banach function spaces. In particular, we show that optimal higher-order Sobolev embeddings follow from isoperimetric inequal- ities. In the second paper we focus on the question when the above-mentioned Sobolev-type space is a Banach algebra with respect to a pointwise multiplica- tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue differentiation theorem in the context of rearrangement- invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the...
24	Koliha–Drazin invertibles form a regularity Smit, Joukje Anneke 10 1900 (has links) The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics) Banach algebra Radical Spectrum Resolvent Quasinilpotent Nilpotent Spectral idempotent Isolated spectral point Accumulation point Regularity Koliha-Drazin invertible Quasipolar KD-spectrum D-spectrum Laurent expansion Poles of the resolvent 511.322 Axiomatic set theory Banach algebras Spectrum analysis Spectral sequences (Mathematics)
25	Transmission problems for Dirac's and Maxwell's equations with Lipschitz interfaces Axelsson, Andreas, kax74@yahoo.se January 2002 (has links) The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwells equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwells equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator. Dirac operator Maxwell's equations transmission problem Hodge decomposition Cauchy integral double layer potential Lipschitz surface singular integral Carleson measure boundary integral method oblique boundary value problem Fredholm theory exterior algebra Clifford analysis Rellich inequality Banach algebra projection operator Toeplitz operator Calderón projection

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