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1	Symbolic calculus for boundary value problems on manifolds with edges Kapanadze, David, Schulze, Bert-Wolfgang January 2001 (has links) Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour. Mathematics
2	Mellin-edge representations of elliptic operators Dines, Nicoleta, Schulze, Bert-Wolfgang January 2003 (has links) We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator-valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator As in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of As as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices Ps of As, interpreted as Mellin-edge representations of P. Pseudo-differential operators edge algebra ellipticity with interface conditions Mathematics
3	Modular curvature for toric noncommutative manifolds Liu, Yang January 2015 (has links) No description available. Mathematics
4	Affine Processes and Pseudo-Differential Operators with Unbounded Coefficients Schwarzenberger, Michael 04 October 2016 (has links) (PDF) The concept of pseudo-differential operators allows one to study stochastic processes through their symbol. This approach has generated many new insights in recent years. However, most results are based on the assumption of bounded coefficients. In this thesis, we study Levy-type processes with unbounded coefficients and, especially, affine processes. In particular, we establish a connection between pseudo-differential operators and affine processes which are well-known from mathematical finance. Affine processes are an interesting example in this field since they have linearly growing and hence unbounded coefficients. New techniques and tools are developed to handle the affine case and then expanded to general Levy-type processes. In this way, the convergence of a simulation scheme based on a Markov chain approximation, results on path properties, and necessary conditions for the symmetry of operators were proven. Affiner Prozess Pseudodifferentialoperator Symbol Simulation Affine process pseudo-differential operator symbol simulation ddc:510 rvk:SK 820
5	Caracterização de espaços de potência fracionária por meio de operadores pseudodiferenciais / Characterization of fractional power spaces by pseudo-differential operators Macedo, Bruno Vicente Marchi de 22 March 2016 (has links) Neste trabalho mostramos uma caracterização para os espaços de potência fracionária associados ao operador 1 - Δp, em que Δp representa o fecho do operador laplaciano em Lp(Rn), usando o fato de que o mesmo pode ser visto como um operador pseudodiferencial com símbolo a(ξ) = 1+4π2\|ξ\|2. No processo para obter essa caracterização representamos de maneira concreta a solução abstrata u : [0;+ ∞) → Lp(Rn), obtida através da teoria de operadores setoriais e semigrupos analíticos, da equação u - Δpu = 0 em (0;+∞) com condição inicial u(0) = f ∈ Lp(Rn). / In this work we show a characterization for the fractional power spaces associated with the operator 1 - Δp, where Δp, represents the closure of the Laplacian operator in Lp(Rn), using the fact that the operator may be seen as a pseudo-differential operator with symbol a(ξ) = 1+4π2\|ξ\|2. In the process for this characterization we represent of concrete way the abstract solution u : [0;+∞) Lp(Rn), obtained through the theory of sector operators and analytic semigroups, of the equation u - Δpu = 0 in (0;+∞) with initial condition u(0) = f ∈ Lp(Rn). Espaços de potência fracionária Fractional power spaces Operador pseudodiferencial Pseudo-differential operator
6	Continuity and compositions of operators with kernels in ultra-test function and ultra-distribution spaces Chen, Yuanyuan January 2016 (has links) In this thesis we consider continuity and positivity properties of pseudo-differential operators in Gelfand-Shilov and Pilipović spaces, and their distribution spaces. We also investigate composition property of pseudo-differential operators with symbols in quasi-Banach modulation spaces. We prove that positive elements with respect to the twisted convolutions, possesing Gevrey regularity of certain order at origin, belong to the Gelfand-Shilov space of the same order. We apply this result to positive semi-definite pseudo-differential operators, as well as show that the strongest Gevrey irregularity of kernels to positive semi-definite operators appear at the diagonals. We also prove that any linear operator with kernel in a Pilipović or Gelfand-Shilov space can be factorized by two operators in the same class. We give links on numerical approximations for such compositions and apply these composition rules to deduce estimates of singular values and establish Schatten-von Neumann properties for such operators. Furthermore, we derive sufficient and necessary conditions for continuity of the Weyl product with symbols in quasi-Banach modulation spaces. Composition modulation spaces positivity pseudo-differential operators Schatten-von Neumann operators twisted convolutions ultra-distributions
7	Pseudo-differential crack theory Kapanadze, David, Schulze, Bert-Wolfgang January 2000 (has links) Crack problems are regarded as elements in a pseudo-differential algbra, where the two sdes int S± of the crack S are treated as interior boundaries and the boundary Y of the crack as an edge singularity. We employ the pseudo-differential calculus of boundary value problems with the transmission property near int S± and the edge pseudo-differential calculus (in a variant with Douglis-Nirenberg orders) to construct parametrices od elliptic crack problems (with extra trace and potential conditions along Y) and to characterise asymptotics of solutions near Y (expressed in the framework of continuous asymptotics). Our operator algebra with boundary and edge symbols contains new weight and order conventions that are necessary also for the more general calculus on manifolds with boundary and edges. Crack theory conormal asymptotics Mathematics
8	Boundary-contact problems for domains with edge singularities Kapanadze, David, Schulze, B.-Wolfgang January 2005 (has links) We study boundary-contact problems for elliptic equations (and systems) with interfaces that have edge singularities. Such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics. Ellipticity is formulated in terms of a principal symbolic hierarchy, containing interior, transmission, and edge symbols. We construct parametrices, show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary-contact problems for the Laplacian with jumping coefficients. Boundary-contact problems pseudo-differential operators edge spaces asymptotics of solutions Mathematics
9	Caracterização de espaços de potência fracionária por meio de operadores pseudodiferenciais / Characterization of fractional power spaces by pseudo-differential operators Bruno Vicente Marchi de Macedo 22 March 2016 (has links) Neste trabalho mostramos uma caracterização para os espaços de potência fracionária associados ao operador 1 - Δp, em que Δp representa o fecho do operador laplaciano em Lp(Rn), usando o fato de que o mesmo pode ser visto como um operador pseudodiferencial com símbolo a(ξ) = 1+4π2\|ξ\|2. No processo para obter essa caracterização representamos de maneira concreta a solução abstrata u : [0;+ ∞) → Lp(Rn), obtida através da teoria de operadores setoriais e semigrupos analíticos, da equação u - Δpu = 0 em (0;+∞) com condição inicial u(0) = f ∈ Lp(Rn). / In this work we show a characterization for the fractional power spaces associated with the operator 1 - Δp, where Δp, represents the closure of the Laplacian operator in Lp(Rn), using the fact that the operator may be seen as a pseudo-differential operator with symbol a(ξ) = 1+4π2\|ξ\|2. In the process for this characterization we represent of concrete way the abstract solution u : [0;+∞) Lp(Rn), obtained through the theory of sector operators and analytic semigroups, of the equation u - Δpu = 0 in (0;+∞) with initial condition u(0) = f ∈ Lp(Rn). Espaços de potência fracionária Operador pseudodiferencial Fractional power spaces Pseudo-differential operator
10	The Symbol of a Markov Semimartingale Schnurr, Alexander 10 June 2009 (has links) (PDF) We prove that every (nice) Feller process is an It^o process in the sense of Cinlar, Jacod, Protter and Sharpe (1980). Next we generalize the notion of the symbol and define it for this larger class of processes. As examples the solutions of stochastic differential equations are considered. The symbol is then used to derive a quick approach to the semimartingale characteristics as well as the generator of the process under consideration. Finally we give some examples of how our methods work for processes used in mathematical finance. / Wir haben gezeigt, dass jeder (nette) Feller Prozess ein It^o Prozess im Sinne von Cinlar, Jacod, Protter und Sharpe (1980) ist. Es stellt sich heraus, dass man den Begriff des Symbols, der für Feller Prozesse bekannt ist, auf diese größere Klasse verallgemeinern kann. Dieses Symbol haben wir für die Lösungen verschiedener stochastischer Differentialgleichungen berechnet. Außerdem haben wir gezeigt, dass das Symbol einen schnellen Zugang zur Berechnung der Semimartingal-Charakteristiken und des Erzeugers eines It^o Prozesses liefert. Zuletzt wurden die Ergebnisse auf Prozesse angewendet, die in der Finanzmathematik gebräuchlich sind. - (Die Dissertation ist veröffentlicht im Shaker Verlag GmbH, Postfach 101818, 52018 Aachen, Deutschland, http://www.shaker.de, ISBN: 978-3-8322-8244-8) Markov process semimartingale Itô process Feller semigroup generator symbol pseudo differential operator stochastic differential equation COGARCH process Markov Prozess Semimartingal Itô Prozess Feller Halbgruppe Erzeuger Symbol Pseudo-Differential-Operator stochastische Differentialgleichung COGARCH Prozess ddc:510 rvk:SK 820

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