TY - INPR A1 - Blanchard, Gilles A1 - Mücke, Nicole T1 - Optimal rates for regularization of statistical inverse learning problems N2 - We consider a statistical inverse learning problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points X_i, superposed with an additional noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependence of the constant factor in the variance of the noise and the radius of the source condition set. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 5 KW - statistical inverse problem KW - minimax rate KW - kernel method Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-89782 SN - 2193-6943 VL - 5 IS - 5 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Brauer, Uwe A1 - Karp, Lavi T1 - Local existence of classical solutions for the Einstein-Euler system using weighted Sobolev spaces of fractional order N2 - We prove the existence of a class of local in time solutions, including static solutions, of the Einstein-Euler system. This result is the relativistic generalisation of a similar result for the Euler-Poisson system obtained by Gamblin [8]. As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article. T3 - Preprint - (2006) 17 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30175 ER - TY - INPR A1 - Brauer, Uwe A1 - Karp, Lavi T1 - Well-posedness of Einstein-Euler systems in asymptotically flat spacetimes N2 - We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein{Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for hyperbolic and elliptic equations in these spaces. The well posedness is obtained in these spaces. T3 - Preprint - (2008) 07 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30347 ER - TY - INPR A1 - Buchholz, Thilo A1 - Schulze, Bert-Wolfgang T1 - Volterra operators and parabolicity : anisotropic pseudo-differential operators N2 - Parabolic equations on manifolds with singularities require a new calculus of anisotropic pseudo-differential operators with operator-valued symbols. The paper develops this theory along the lines of sn abstract wedge calculus with strongly continuous groups of isomorphisms on the involved Banach spaces. The corresponding pseodo-diferential operators are continuous in anisotropic wedge Sobolev spaces, and they form an alegbra. There is then introduced the concept of anisotropic parameter-dependent ellipticity, based on an order reduction variant of the pseudo-differential calculus. The theory is appled to a class of parabolic differential operators, and it is proved the invertibility in Sobolev spaces with exponential weights at infinity in time direction. T3 - Preprint - (1998) 11 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25231 ER - TY - INPR A1 - Bär, Christian T1 - Renormalized integrals and a path integral formula for the heat kernel on a manifold N2 - We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)21 KW - Renormalized integral KW - path integral KW - Feynman-Kac formula KW - generalized Laplace operator KW - Riemannian manifold Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60052 ER - TY - INPR A1 - Bär, Christian T1 - Some properties of solutions to weakly hypoelliptic equations N2 - A linear differential operator L is called weakly hypoelliptic if any local solution u of Lu = 0 is smooth. We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which covers all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p solution must vanish. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)22 KW - Hypoelliptic operators KW - hypoelliptic estimate KW - Montel theorem KW - Vitali theorem KW - Liouville theorem Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60064 ER - TY - INPR A1 - Bär, Christian A1 - Ballmann, Werner T1 - Boundary value problems for elliptic differential operators of first order N2 - We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)18 KW - Elliptic operators KW - elliptic boundary conditions KW - completeness KW - coercivity KW - boundary regularity Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60023 ER - TY - INPR A1 - Bär, Christian A1 - Ginoux, Nicolas T1 - Classical and quantum fields on Lorentzian manifolds N2 - We construct bosonic and fermionic locally covariant quantum fields theories on curved backgrounds for large classes of fields. We investigate the quantum field and n-point functions induced by suitable states. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)15 KW - Wave operator KW - Dirac-type operator KW - globally hyperbolic spacetime KW - Green's operator KW - CCR-algebra Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-59973 ER - TY - INPR A1 - Bär, Christian A1 - Pfäffle, Frank T1 - Wiener measures on Riemannian manifolds and the Feynman-Kac formula N2 - This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)17 KW - Wiener measure KW - conditional Wiener measure KW - Brownian motion KW - Brownian bridge KW - Riemannian manifold Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-59998 ER - TY - INPR A1 - Böckmann, Christine A1 - Biele, Jens A1 - Neuber, Roland A1 - Niebsch, Jenny T1 - Retrieval of multimodal aerosol size distribution by inversion of multiwavelength data N2 - The ill-posed problem of aerosol size distribution determination from a small number of backscatter and extinction measurements was solved successfully with a mollifier method which is advantageous since the ill-posed part is performed on exactly given quantities, the points r where n(r) is evaluated may be freely selected. A new twodimensional model for the troposphere is proposed. T3 - NLD Preprints - 38 KW - Multiwavelength LIDAR KW - aerosol size distribution KW - ill-posed problem KW - inversion KW - mollifier method KW - coated and absorbing aerosols Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-14360 ER - TY - INPR A1 - Böckmann, Christine A1 - Niebsch, Jenny T1 - Examination of the nonlinear LIDAR-operator : the influence of inhomogeneous absorbing spheres on the operator N2 - The determination of the atmospheric aerosol size distribution is an inverse illposed problem. The shape and the material composition of the air-carried particles are two substantial model parameters. Present evaluation algorithms only used an approximation with spherical homogeneous particles. In this paper we propose a new numerically efficient recursive algorithm for inhomogeneous multilayered coated and absorbing particles. Numerical results of real existing particles show that the influence of the two parameters on the model is very important and therefore cannot be ignored. T3 - NLD Preprints - 47 KW - multiwavelength lidar KW - aerosol size distribution KW - inverse ill-posed problem KW - multilayered coated and absorbing aerosol KW - new recursive algorithm Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-14725 ER - TY - INPR A1 - Böckmann, Christine A1 - Sarközi, Janos T1 - The ill-posed inversion of multiwavelength lidar data by a hybrid method of variable projection N2 - The ill-posed problem of aerosol distribution determination from a small number of backscatter and extinction lidar measurements was solved successfully via a hybrid method by a variable dimension of projection with B-Splines. Numerical simulation results with noisy data at different measurement situations show that it is possible to derive a reconstruction of the aerosol distribution only with 4 measurements. T3 - NLD Preprints - 53 KW - Ill-posed problem KW - inversion KW - variable projection method KW - multiwavelength Lidar KW - aerosol distribution Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-14847 ER - TY - INPR A1 - Calin, Ovidium A1 - Der-Chen, Chang T1 - The geometry on a step 3 Grushin model N2 - In this article we study the geometry associated with the sub-elliptic operator ½ (X²1 +X²2), where X1 = ∂x and X2 = x²/2 ∂y are vector fields on R². We show that any point can be connected with the origin by at least one geodesic and we provide an approximate formula for the number of the geodesics between the origin and the points situated outside of the y-axis. We show there are in¯nitely many geodesics between the origin and the points on the y-axis. T3 - Preprint - (2004) 08 KW - Grushin operator KW - subRiemannian geometry KW - geodesics KW - Hamilton-Jacobi theory KW - elliptic functions KW - Euler's theta functions Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26724 ER - TY - INPR A1 - Calvo, D. A1 - Schulze, Bert-Wolfgang T1 - Operators on corner manifolds with exit to infinity N2 - We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ ψ(A), σ ^(A), σ ^(A)), where σ ψ is the interior symbol and σ ^(A)(y, η), (y, η) 2 T*Y \ 0, the (operator-valued) edge symbol of ‘first generation’, cf. [15]. The novelty here is the edge symbol σ^ of ‘second generation’, parametrised by (z, Ϛ) 2 T*Z \ 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone. T3 - Preprint - (2005) 01 KW - Operators on manifolds with edge and conical exit to infinity KW - Sobolev spaces with double weights on singular cones Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29753 ER - TY - INPR A1 - Calvo, D. A1 - Schulze, Bert-Wolfgang T1 - Edge symbolic structures of second generation N2 - Operators on a manifold with (geometric) singularities are degenerate in a natural way. They have a principal symbolic structure with contributions from the different strata of the configuration. We study the calculus of such operators on the level of edge symbols of second generation, based on specific quantizations of the corner-degenerate interior symbols, and show that this structure is preserved under compositions. T3 - Preprint - (2005) 18 KW - Operators on manifolds with second order singularities KW - edge quantizations KW - continuity in Sobolev spaces with double weights Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29940 ER - TY - INPR A1 - Camalès, Renaud T1 - A note on the ramified Cauchy problem N2 - In this paper, the ramified Cauchy problem in C² for operator with multiple characteristics of constant multiplicity and second member ramified around some analytic set is studied. T3 - Preprint - (2003) 21 KW - Ramified Cauchy problem KW - analytic continuation Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26646 ER - TY - INPR A1 - Cattiaux, Patrick A1 - Dai Pra, Paolo A1 - Poelly, Sylvie T1 - A constructive approach to a class of ergodic HJB equations with nonsmooth cost N2 - We consider a class of ergodic Hamilton-Jacobi-Bellman (HJB) equations, related to large time asymptotics of non-smooth multiplicative functional of difusion processes. Under suitable ergodicity assumptions on the underlying difusion, we show existence of these asymptotics, and that they solve the related HJB equation in the viscosity sense. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2007, 02 Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49430 ER - TY - INPR A1 - Cattiaux, Patrick A1 - Fradon, Myriam A1 - Kulik, Alexei Michajlovič A1 - Roelly, Sylvie T1 - Long time behavior of stochastic hard ball systems N2 - We study the long time behavior of a system of two or three Brownian hard balls living in the Euclidean space of dimension at least two, submitted to a mutual attraction and to elastic collisions. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)15 KW - Stochastic differential equations KW - hard core interaction KW - reversible measure KW - normal reflection KW - local time Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-68388 ER - TY - INPR A1 - Champagnat, Nicolas A1 - Roelly, Sylvie T1 - Limit theorems for conditioned multitype Dawson-Watanabe processes N2 - A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every nite time interval, its distribution law is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. The explicit form of the Laplace functional of the conditioned process is used to obtain several results on the long time behaviour of the mass of the conditioned and unconditioned processes. The general case is considered first, where the mutation matrix which modelizes the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are also analysed. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2007, 01 Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49426 ER - TY - INPR A1 - Chen, Hua A1 - Hidetoshi, Tahara T1 - On the holomorphic solution of non-linear totally characteristic equations N2 - The paper deals with a non-linear singular partial differential equation: (E) t∂/∂t = F(t, x, u, ∂u/∂x) in the holomorphic category. When (E) is of Fuchsian type, the existence of the unique holomorphic solution was established by Gérard-Tahara [2]. In this paper, under the assumption that (E) is of totally characteristic type, the authors give a sufficient condition for (E) to have a unique holomorphic solution. The result is extended to higher order case. T3 - Preprint - (1998) 20 KW - Nonlinear KW - singular partial differential equation KW - holomorphic solution Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25333 ER - TY - INPR A1 - Chen, Hua A1 - Li, Jun-Feng A1 - Liu, Wei-An T1 - Behavior of the solution to a chemotaxis model with reproduction term N2 - Contents: 1 Introduction 2 Global existence and blow-up or quenching of the solution 3 Detailed asymptotical behavior of the solution T3 - Preprint - (2008) 02 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30304 ER - TY - INPR A1 - Chen, Hua A1 - Li, Ke T1 - The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations N2 - Let X = (X1,.....,Xm) be an infinitely degenerate system of vector fields, we study the existence and regularity of multiple solutions of Dirichelt problem for a class of semi-linear infinitely degenerate elliptic operators associated with the sum of square operator Δx = ∑m(j=1) Xj* Xj. T3 - Preprint - (2007) 03 KW - degenerate elliptic equations KW - Logarithmic Sobolev inequality Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30247 ER - TY - INPR A1 - Chen, Hua A1 - Li, Wei-Xi A1 - Xu, Chao-Jiang T1 - Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations N2 - This paper studies the Gevrey regularity of weak solutions of a class of linear and semilinear Fokker-Planck equations. T3 - Preprint - (2007) 07 Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30283 ER - TY - INPR A1 - Chen, Hua A1 - Lua, Zhuangehu T1 - On the holomorphic solution of non-linear totally characteristic equations with several space variables N2 - In this paper we study a class of non-linear singular partial differential equation in complex domain Csub(t) x C n sub(x). Under certain assumptions, we prove the existence and uniqueness of holomorphic solution near origin of Csub(t) x C n sub(x). T3 - Preprint - (1998) 06 KW - Non-linear KW - singular partial differential equation KW - holomorphic solution Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25189 ER - TY - INPR A1 - Chen, Hua A1 - Wu, Shaohua T1 - On existence of solutions for some hyperbolic-parabolic type chemotaxis systems N2 - In this paper, we discuss the local and global existence of week solutions for some hyperbolic-parabolic systems modelling chemotaxis. T3 - Preprint - (2006) 04 KW - Hyperbolic-parabolic system KW - KS model KW - Chemotaxis Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30082 ER - TY - INPR A1 - Chen, Hua A1 - Yu, Chun T1 - Asymptotic behaviour of the trace for Schrödinger operator on fractal drums T3 - Preprint - (2001) 32 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26157 ER - TY - INPR A1 - Conforti, Giovanni T1 - Reciprocal classes of continuous time Markov Chains N2 - In this work we study reciprocal classes of Markov walks on graphs. Given a continuous time reference Markov chain on a graph, its reciprocal class is the set of all probability measures which can be represented as a mixture of the bridges of the reference walks. We characterize reciprocal classes with two different approaches. With the first approach we found it as the set of solutions to duality formulae on path space, where the differential operators have the interpretation of the addition of infinitesimal random loops to the paths of the canonical process. With the second approach we look at short time asymptotics of bridges. Both approaches allow an explicit computation of reciprocal characteristics, which are divided into two families, the loop characteristics and the arc characteristics. They are those specific functionals of the generator of the reference chain which determine its reciprocal class. We look at the specific examples such as Cayley graphs, the hypercube and planar graphs. Finally we establish the first concentration of measure results for the bridges of a continuous time Markov chain based on the reciprocal characteristics. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 8 KW - random walks on graphs KW - bridges of random walks KW - reciprocal characteristics KW - Schrödinger problem KW - integration by parts on path space Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-78234 SN - 2193-6943 VL - 4 IS - 8 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Conforti, Giovanni A1 - Dai Pra, Paolo A1 - Roelly, Sylvie T1 - Reciprocal class of jump processes N2 - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set A in R^d. We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of A plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 6 KW - reciprocal processes KW - stochastic bridges KW - jump processes KW - compound Poisson processes Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70776 SN - 2193-6943 VL - 3 IS - 6 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Conforti, Giovanni A1 - Léonard, Christian A1 - Murr, Rüdiger A1 - Roelly, Sylvie T1 - Bridges of Markov counting processes : reciprocal classes and duality formulas N2 - Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 9 KW - counting process KW - bridge KW - reciprocal class KW - duality formula Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71855 SN - 2193-6943 VL - 3 IS - 9 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Conforti, Giovanni A1 - Roelly, Sylvie T1 - Reciprocal class of random walks on an Abelian group N2 - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 1 KW - reciprocal class KW - stochastic bridge KW - random walk on Abelian group Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72604 SN - 2193-6943 VL - 4 IS - 1 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Coriasco, Sandro A1 - Panarese, Paolo T1 - Fourier integral operators defined by classical symbols with exit behaviour N2 - We continue the investigation of the calculus of Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour (SG symbols). Here we analyse what happens when one restricts the choice of amplitude and phase functions to the subclass of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a "classical" counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1); for such operators we refine the known results about the analogous problem for general SG hyperbolic operators. The material contained here will be used in a forthcoming paper to obtain a Weyl formula for a class of operators defined on manifolds with cylindrical ends, improving the results obtained in [9]. T3 - Preprint - (2000) 24 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25896 ER - TY - INPR A1 - Coriasco, Sandro A1 - Schrohe, Elmar A1 - Seiler, Jörg T1 - Bounded imaginary powers of differential operators on manifolds with conical singularities N2 - We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B,1 < p < ∞. Under suitable ellipticity assumptions we can define a family of complex powers A up(z), z ∈ C. We also obtain sufficient information on the resolvent of A to show the boundedness of the pure imaginary powers. Examples concern unique solvability and maximal regularity of the solution of the Cauchy problem u' - Δu = f, u(0) = 0, for the Laplacian on conical manifolds. T3 - Preprint - (2001) 12 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25962 ER - TY - INPR A1 - Coriasco, Sandro A1 - Schulze, Bert-Wolfgang T1 - Edge problems on configurations with model cones of different dimensions N2 - Elliptic equations on configurations W = W1 ∪ ... ∪ Wn with edge Y and components Wj of different dimension can be treated in the frame of pseudo-differential analysis on manifolds with geometric singularities, here, edges. Starting from edge-degenerate operators on Wj, j = 1, ..., N, we construct an algebra with extra "transmission" conditions on Y that satisfy an analogue of the Shapiro-Lopatinskij condition. Ellipticity refers to a two-component symbolic hierarchy with an interior and an edge part; the latter one is operator-valued, operating on the union of different dimensional model cones. We construct parametrices within our calculus, where exchange of information between the various components is encoded in Green and Mellin operators that are smoothing on W\Y. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics. T3 - Preprint - (2002) 26 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26438 ER - TY - INPR A1 - Davis, Simon T1 - On the absence of large-order divergences in superstring theory N2 - The genus-dependence of multi-loop superstring ams is estimated at large orders in perturbation theory using the super-Schottky group parameterization of supermoduli space. Restriction of the integration region to a subset of supermoduli space and a single fundamental domain of the super-modular group suggests an exponential dependence on the genus. Upper bounds for these estimates are obtained for arbitrary N-point superstring scattering amplitudes and are shown to be consistent with exact results obtained for special type II string amplitudes for orbifold or Calabi-Yau compactifications. The genus-dependence is then obtained by considering the effect of the remaining contribution to the superstring amplitudes after the coefficients of the formally divergent parts of the integrals vanish as a result of a sum over spin structures. The introduction of supersymmetry therefore leads to the elimination of large-order divergences in string pertubation theory, a result which is based only on the supersymmetric generalization of the polyakov measure and not the gauge group of the string model. T3 - Preprint - (2002) 28 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26452 ER - TY - INPR A1 - Davis, Simon T1 - Connections and generalized gauge transformations N2 - The derivation of the standard model from a higher-dimensional action suggests a further study of the fibre bundle formulation of gauge theories to determine the variations in the choice of structure group that are allowed in this geometrical setting. The action of transformations on the projection of fibres to their submanifolds are characteristic of theories with fewer gauge vector bosons, and specific examples are given, which may have phenomenological relevance. The spinor space for the three generations of fermions in the standard model is described algebraically. T3 - Preprint - (2002) 29 KW - bundles KW - connections KW - fibre coordinates KW - parallelizable spheres KW - division algebras KW - force unification Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26468 ER - TY - INPR A1 - Davis, Simon T1 - On the existence of a non-zero lower bound for the number of Goldbach partitions of an even integer N2 - The Goldbach partitions of an even number greater than 2, given by the sums of two prime addends, form the non-empty set for all integers 2n with 2 ≤ n ≤ 2 × 1014. It will be shown how to determine by the method of induction the existence of a non-zero lower bound for the number of Goldbach partitions of all even integers greater than or equal to 4. The proof depends on contour arguments for complex functions in the unit disk. T3 - Preprint - (2002) 30 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26474 ER - TY - INPR A1 - Davis, Simon T1 - The quantum cosmological wavefunction at very early times for a quadratic gravity theory N2 - The quantum cosmological wavefunction for a quadratic gravity theory derived from the heterotic string effective action is obtained near the inflationary epoch and during the initial Planck era. Neglecting derivatives with respect to the scalar field, the wavefunction would satisfy a third-order differential equation near the inflationary epoch which has a solution that is singular in the scale factor limit a(t) → 0. When scalar field derivatives are included, a sixth-order differential equation is obtained for the wavefunction and the solution by Mellin transform is regular in the a → 0 limit. It follows that inclusion of the scalar field in the quadratic gravity action is necessary for consistency of the quantum cosmology of the theory at very early times. T3 - Preprint - (2003) 04 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26520 ER - TY - INPR A1 - De Donno, G. A1 - Schulze, Bert-Wolfgang T1 - Meromorphic symbolic structures for boundary value problems on manifolds with edges N2 - We investigate the ideal of Green and Mellin operators with asymtotics for a manifold with edge-corner singularities and boundary which belongs to the structure of parametrices of elliptic boundary value problems on a configuration with corners whose base manifolds have edges. T3 - Preprint - (2003) 10 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26570 ER - TY - INPR A1 - De-Xing, Kong A1 - Hui, Yao T1 - Global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws II N2 - In this paper, by a new constructive method, the authors reprove the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate fields. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C¹ functions. In particular, the authors give the optimal control time of the system. Finally, a new application is also given. T3 - Preprint - (2003) 08 KW - Quasilinear hyperbolic system KW - conservation laws KW - global exact boundary controllability KW - Cauchy problem KW - Goursat problem Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26565 ER - TY - INPR A1 - Debussche, Arnaud A1 - Hoegele, Michael A1 - Imkeller, Peter T1 - The dynamics of nonlinear reaction-diffusion equations with small levy noise preface T2 - Lecture notes in mathematics : a collection of informal reports and seminars T2 - Lecture Notes in Mathematics Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 SN - 0075-8434 VL - 2085 SP - V EP - + PB - Springer CY - Berlin ER - TY - INPR A1 - Debussche, Arnaud A1 - Högele, Michael A1 - Imkeller, Peter T1 - The dynamics of nonlinear reaction-diffusion equations with small levy noise T2 - Lecture notes in mathematics : a collection of informal reports and seminars T2 - Lecture Notes in Mathematics N2 - Our primary interest in this book lies in the study of dynamical properties of reaction-diffusion equations perturbed by Lévy noise of intensity ? in the small noise limit ??0 . Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 U6 - https://doi.org/10.1007/978-3-319-00828-8_1 SN - 0075-8434 VL - 2085 SP - 1 EP - 10 PB - Springer CY - Berlin ER - TY - INPR A1 - Denk, Robert A1 - Krainer, Thomas T1 - R-Boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators N2 - It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class (HT ) and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with R-bounded symbols, yielding by an iteration argument the R-boundedness of λ(A − λ)−1 in R(λ)≥ τ for some τ ∈ IR. To this end, elements of a symbolic and operator calculus of pseudodifferential operators with R-bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on Rd with operator valued coefficients. T3 - Preprint - (2006) 14 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30147 ER - TY - INPR A1 - Dereudre, David A1 - Mazzonetto, Sara A1 - Roelly, Sylvie T1 - Exact simulation of Brownian diffusions with drift admitting jumps N2 - Using an algorithm based on a retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical difficulty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 7 KW - exact simulation method KW - skew Brownian motion KW - skew diffusion KW - Brownian motion with discontinuous drift Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-91049 SN - 2193-6943 VL - 5 IS - 7 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Dereudre, David A1 - Mazzonetto, Sara A1 - Roelly, Sylvie T1 - An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers N2 - In this paper we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover we propose a rejection method to simulate this density in an exact way. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 9 KW - skew Brownian motion KW - semipermeable barriers KW - distorted Brownian motion KW - local time KW - rejection sampling KW - exact simulation Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-80613 SN - 2193-6943 VL - 4 IS - 9 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Dereudre, David A1 - Roelly, Sylvie T1 - Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions N2 - We study the (strong-)Gibbsian character on RZd of the law at time t of an infinitedimensional gradient Brownian diffusion , when the initial distribution is Gibbsian. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 06 KW - infinite-dimensional Brownian diffusion KW - Gibbs measure KW - cluster expansion Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51535 ER - TY - INPR A1 - Dereudre, David A1 - Roelly, Sylvie T1 - Path-dependent infinite-dimensional SDE with non-regular drift : an existence result N2 - We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)11 KW - Infinite-dimensional SDE KW - non-Markov drift KW - non-regular drift KW - variational principle KW - specific entropy Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72084 SN - 2193-6943 VL - 3 IS - 11 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Dicken, Volker T1 - Simultaneous activity and attenuation reconstruction in emission tomography N2 - In single photon emission computed tomography (SPECT) one is interested in reconstructing the activity distribution f of some radiopharmaceutical. The data gathered suffer from attenuation due to the tissue density µ. Each imaged slice incorporates noisy sample values of the nonlinear attenuated Radon transform (formular at this place in the original abstract) Traditional theory for SPECT reconstruction treats µ as a known parameter. In practical applications, however, µ is not known, but either crudely estimated, determined in costly additional measurements or plainly neglected. We demonstrate that an approximation of both f and µ from SPECT data alone is feasible, leading to quantitatively more accurate SPECT images. The result is based on nonlinear Tikhonov regularization techniques for parameter estimation problems in differential equations combined with Gauss-Newton-CG minimization. T3 - NLD Preprints - 50 KW - SPECT KW - tomogrphy KW - attenuated Radon transform KW - nonlinear invers problem KW - Tikhonov regularization KW - nonlinear optimization Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-14747 ER - TY - INPR A1 - Dines, Nicoleta A1 - Harutjunjan, Gohar A1 - Schulze, Bert-Wolfgang T1 - The Zaremba problem in edge Sobolev spaces N2 - Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge Sobolev spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. Our methods from the calculus of boundary value problems on a manifold with edges will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator. T3 - Preprint - (2003) 13 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26615 ER - TY - INPR A1 - Dines, Nicoleta A1 - Liu, X. A1 - Schulze, Bert-Wolfgang T1 - Edge quantisation of elliptic operators N2 - The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy σ = (σψ, σ∧), where the second component takes value in operators on the infinite model cone of the local wedges. In general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the elliptcity of the principal edge symbol σ∧ which includes the (in general not explicitly known) number of additional conditions on the edge of trace and potential type. We focus here on these queations and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems, and we establish relations of elliptic operators for different weights, via the spectral flow of the underlying conormal symbols. T3 - Preprint - (2004) 24 KW - Boundary value problems KW - edge singularities KW - ellipticity KW - spectral flow Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26838 ER - TY - INPR A1 - Dines, Nicoleta A1 - Schulze, Bert-Wolfgang T1 - Mellin-edge representations of elliptic operators N2 - 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. T3 - Preprint - (2003) 18 KW - Pseudo-differential operators KW - edge algebra KW - ellipticity with interface conditions Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26627 ER - TY - INPR A1 - Duduchava, Roland T1 - The Green formula and layer potentials N2 - The Green formula is proved for boundary value problems (BVPs), when "basic" operator is arbitrary partial differential operator with variable matrix coefficients and "boundary" operators are quasi-normal with vector-coeficients. If the system possesses the fundamental solution, representation formula for a solution is derived and boundedness properties of participating layer potentials from function spaces on the boundary (Besov, Zygmund spaces) into appropriate weighted function spaces on the inner and the outer domains are established. Some related problems are discussed in conclusion: traces of functions from weighted spaces, traces of potential-type functions, Plemelji formulae,Calderón projections, restricted smoothness of the underlying surface and coefficients. The results have essential applications in investigations of BVPs by the potential method, in apriori estimates and in asymptotics of solutions. T3 - Preprint - (1999) 26 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25601 ER - TY - INPR A1 - Dyachenko, Evgueniya A1 - Tarkhanov, Nikolai Nikolaevich T1 - Degeneration of boundary layer at singular points N2 - We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)23 KW - Heat equation KW - Dirichlet problem KW - characteristic points KW - boundary layer Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60135 ER - TY - INPR A1 - Dyachenko, Evgueniya A1 - Tarkhanov, Nikolai Nikolaevich T1 - Singular perturbations of elliptic operators N2 - We develop a new approach to the analysis of pseudodifferential operators with small parameter 'epsilon' in (0,1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on 'epsilon'. Instead we consider the cylinder [0,1] x X over X and study pseudodifferential operators on the cylinder which act, by the very nature, on functions depending on 'epsilon' as well. The action in 'epsilon' reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of 'epsilon'. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 1 KW - singular perturbation KW - pseudodifferential operator KW - ellipticity with parameter KW - regularization KW - asymptotics Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69502 SN - 2193-6943 VL - 3 IS - 1 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Egorov, Jurij V. A1 - Kondratiev, V. A. A1 - Schulze, Bert-Wolfgang T1 - On the completeness of root functions of elliptic boundary problems in a domain with conical points on the boundary N2 - Contents: 1 Introduction 2 Definitions 3 Rays of minimal growth 4 Proof of Theorem 2. 5 The growth of the resolvent 6 Proof of Theorem 3. 7 The completeness of root functions 8 Some generalizations T3 - Preprint - (2004) 17 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26773 ER - TY - INPR A1 - Egorov, Yu. A1 - Kondratiev, V. A1 - Schulze, Bert-Wolfgang T1 - On completeness of eigenfunctions of an elliptic operator on a manifold with conical points N2 - Contents: 1 Introduction 2 Definitions 3 Rays of minimal growth 4 Completeness of root functions T3 - Preprint - (2001) 04 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25937 ER - TY - INPR A1 - Elin, Mark A1 - Shoikhet, David A1 - Tarkhanov, Nikolai Nikolaevich T1 - Analytic semigroups of holomorphic mappings and composition operators N2 - In this paper we study the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators. We also provide a brief review around this topic. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 6 KW - nonlinear semigroup KW - composition operator Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-77914 SN - 2193-6943 VL - 4 IS - 6 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Fang, Daoyuan A1 - Xu, Jiang T1 - Asymptotic behavior of solutions to multidimensional nonisentropic hydrodynamic model for semiconductors N2 - In this paper, a global existence result of smooth solutions to the multidimen- sional nonisentropic hydrodynamic model for semiconductors is proved, under the assumption that the initial data is a perturbation of the stationary solutions for the thermal equilibrium state. The resulting evolutionary solutions converge to the stationary solutions in time asymptotically exponentially fast. T3 - Preprint - (2005) 02 KW - Multidimensional nonisentropic hydrodynamic model KW - semiconductors KW - asymptotic behavior KW - global solutions Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29767 ER - TY - INPR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Radó Theorem for the Porous Medium Equation T2 - Preprints des Instituts für Mathematik der Universität Potsdam N2 - We prove that each locally Lipschitz continuous function satisfying the porous medium equation away from the set of its zeroes is actually a weak solution of this equation in the whole domain. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 6 (2017) 1 KW - quasilinear equation KW - removable set KW - porous medium equation Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-102735 VL - 6 IS - 1 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - An index formula for Toeplitz operators N2 - We prove a Fedosov index formula for the index of Toeplitz operators connected with the Hardy space of solutions to an elliptic system of first order partial differential equations in a bounded domain of Euclidean space with infinitely differentiable boundary. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)12 KW - Toeplitz operators KW - Fredholm property KW - index Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72499 SN - 2193-6943 VL - 3 IS - 12 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Class of Toeplitz Operators in Several Variables N2 - We introduce the concept of Toeplitz operator associated with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We characterise those Toeplitz operators which are Fredholm, thus initiating the index theory. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)17 KW - Cauchy data spaces KW - Laplace-Beltrami operator KW - Toeplitz operators KW - Fredholm property Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-68932 SN - 2193-6943 ER - TY - INPR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - Boundary value problems for elliptic complexes N2 - The aim of this paper is to bring together two areas which are of great importance for the study of overdetermined boundary value problems. The first area is homological algebra which is the main tool in constructing the formal theory of overdetermined problems. And the second area is the global calculus of pseudodifferential operators which allows one to develop explicit analysis. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 3 KW - elliptic complexes KW - Fredholm property KW - index Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-86705 SN - 2193-6943 VL - 5 IS - 3 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Fedosov, B. T1 - On a spectral theorem for deformation quantization N2 - We give a construction of an eigenstate for a non-critical level of the Hamiltonian function, and investigate the contribution of Morse critical points to the spectral decomposition. We compare the rigorous result with the series obtained by a perturbation theory. As an example the relation to the spectral asymptotics is discussed. T3 - Preprint - (2006) 16 KW - star-product KW - WKB method KW - spectral theorem Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30161 ER - TY - INPR A1 - Fedosov, Boris T1 - Pseudo-differential operators and deformation quantization N2 - Using the Riemannian connection on a compact manifold X, we show that the algebra of classical pseudo-differential operators on X generates a canonical deformation quantization on the cotangent manifold T*X. The corresponding Abelian connection is calculated explicitly in terms of the of the exponential mapping. We prove also that the index theorem for elliptic operators may be obtained as a consequence of the index theorem for deformation quantization. T3 - Preprint - (1999) 32 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25651 ER - TY - INPR A1 - Fedosov, Boris T1 - Non-Abelian reduction in deformation quantization N2 - We consider a G-invariant star-product algebra A on a symplectic manifold (M,ω) obtained by a canonical construction of deformation quantization. Under assumptions of the classical Marsden-Weinstein theorem we define a reduction of the algebra A with respect to the G-action. The reduced algebra turns out to be isomorphic to a canonical star-product algebra on the reduced phase space B. In other words, we show that the reduction commutes with the canonical G-invariant deformation quantization. A similar statement in the framework of geometric quantization is known as the Guillemin-Sternberg conjecture (by now completely proved). T3 - Preprint - (1997) 26 KW - deformation quantization KW - Hamiltonian group action KW - moment map KW - classical and quantum reduction Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25101 ER - TY - INPR A1 - Fedosov, Boris T1 - Moduli spaces and deformation quantization in infinite dimensions N2 - We construct a deformation quantization on an infinite-dimensional symplectic space of regular connections on an SU(2)-bundle over a Riemannian surface of genus g ≥ 2. The construction is based on the normal form thoerem representing the space of connections as a fibration over a finite-dimensional moduli space of flat connections whose fibre is a cotangent bundle of the infinite-dimensional gauge group. We study the reduction with respect to the gauge groupe both for classical and quantum cases and show that our quantization commutes with reduction. T3 - Preprint - (1998) 27 KW - moduli space of flat connections KW - gauge group KW - star-product KW - Weyl algebras bundle KW - symplectic reduction Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25396 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A general index formula on tropic manifolds with conical points N2 - We solve the index problem for general elliptic pseudodifferential operators on toric manifolds with conical points. T3 - Preprint - (1999) 15 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25501 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A remark on the index of symmetric operators N2 - We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol. T3 - Preprint - (1998) 04 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25169 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The index of elliptic operators on manifolds with conical points N2 - For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero. T3 - Preprint - (1997) 24 KW - manifolds with singularities KW - pseudodifferential operators KW - elliptic operators KW - index Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25096 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The index of higher order operators on singular surfaces N2 - The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol. T3 - Preprint - (1998) 03 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant KW - monodromy matrix Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25127 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - On the index formula for singular surfaces N2 - In the preceding paper we proved an explicit index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points. Apart from the Atiyah-Singer integral, it contains two additional terms, one of the two being the 'eta' invariant defined by the conormal symbol. In this paper we clarify the meaning of the additional terms for differential operators. T3 - Preprint - (1997) 31 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant KW - monodromy matrix Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25116 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - On index theorem for symplectic orbifolds N2 - We give an explicit construction of the trace on the algebra of quantum observables on a symplectic orbifold and propose an index formula. T3 - Preprint - (2003) 07 KW - star product KW - symmetry group KW - G-trace KW - G-index Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26550 ER - TY - INPR A1 - Fedosov, Boris A1 - Tarkhanov, Nikolai Nikolaevich T1 - Deformation quantisation and boundary value problems N2 - We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 5 KW - symplectic manifold KW - star product KW - trace KW - index Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-77150 SN - 2193-6943 VL - 4 IS - 5 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Flad, Heinz-Jürgen A1 - Schneider, Reinhold A1 - Schulze, Bert-Wolfgang T1 - Asymptotic regularity of solutions of Hartree-Fock equations with coulomb potential N2 - We study the asymptotic regularity of solutions of Hartree-Fock equations for Coulomb systems. In order to deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo-differential operators on conical manifolds. First, the non-self-consistent-field case is considered which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator which guarantee ellipticity for the local part of the Fock operator on the open stretched cone R+ × S². This proves existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed-point approach based on Cances and Le Bris analysis of the level-shifting algorithm, we show via another bootstrap argument, that the corresponding self-consistent-field solutions of the Hartree-Fock equation have the same type of asymptotic regularity. T3 - Preprint - (2007) 05 Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30268 ER - TY - INPR A1 - Flandoli, Franco A1 - Högele, Michael T1 - A solution selection problem with small stable perturbations N2 - The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is a general alpha-stable process. It is proved that extremal solutions are selected and the probability of selection is computed. Detailed analysis of the characteristic function of an exit time form on the half-line is performed, with a suitable decomposition in small and large jumps adapted to the singular drift. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 8 KW - stochastic differential equations KW - singular drifts KW - zero-noise limit KW - Peano phenomena KW - non-uniqueness Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71205 SN - 2193-6943 VL - 3 IS - 8 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Fradon, Myriam A1 - Roelly, Sylvie T1 - Infinite system of Brownian balls with interaction : the non-reversible case N2 - We consider an infinite system of hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite- dimensional Stochastic Differential Equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also show that Gibbs measures are reversible measures. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2005, 01 KW - Stochastic Differential Equation KW - local time KW - hard core potential KW - Gibbs measure KW - reversible measure Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51546 ER - TY - INPR A1 - Fradon, Myriam A1 - Roelly, Sylvie T1 - Infinite system of Brownian Balls: Equilibrium measures are canonical Gibbs N2 - We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional Stochastic Differential Equation with a local time term. We prove that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2005, 02 KW - Stochastic Differential Equation KW - hard core potential KW - Canonical Gibbs measure KW - detailed balance equation KW - reversible measure Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51594 ER - TY - INPR A1 - Fradon, Myriam A1 - Roelly, Sylvie T1 - Infinitely many Brownian globules with Brownian radii N2 - We consider an infinite system of non overlaping globules undergoing Brownian motions in R3. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinitedimensional Stochastic Differential Equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2009, 07 Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49552 ER - TY - INPR A1 - Fradon, Myriam A1 - Roelly, Sylvie T1 - Brownian Hard Balls submitted to an infinite rangeinteraction with slow decay N2 - We consider an infinite system of hard balls in Rd undergoing Brownian motions and submitted to a pair potential with infinite range and quasi polynomial decay. It is modelized by an infinite-dimensional Stochastic Differential Equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with deterministic initial condition. We also show that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2006, 01 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49379 ER - TY - INPR A1 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana T1 - Transportation distances and noise sensitivity of multiplicative Lévy SDE with applications N2 - This article assesses the distance between the laws of stochastic differential equations with multiplicative Lévy noise on path space in terms of their characteristics. The notion of transportation distance on the set of Lévy kernels introduced by Kosenkova and Kulik yields a natural and statistically tractable upper bound on the noise sensitivity. This extends recent results for the additive case in terms of coupling distances to the multiplicative case. The strength of this notion is shown in a statistical implementation for simulations and the example of a benchmark time series in paleoclimate. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 2 KW - stochastic differential equations KW - multiplicative Lévy noise KW - Lévy type processes KW - heavy-tailed distributions KW - model selection KW - Wasserstein distance KW - time series Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-86693 SN - 2193-6943 VL - 5 IS - 2 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana A1 - Kulik, Alexei Michajlovič T1 - Coupling distances between Lévy measures and applications to noise sensitivity of SDE N2 - We introduce the notion of coupling distances on the space of Lévy measures in order to quantify rates of convergence towards a limiting Lévy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Lévy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two Lévy diffusions in terms of the couping distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)16 KW - Lévy diffusion approximation KW - coupling methods KW - Skorokhod' s invariance principle KW - statistical model selection Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-68886 ER - TY - INPR A1 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana A1 - Kulik, Alexei Michajlovič T1 - On the calibration of Lévy driven time series with coupling distances : an application in paleoclimate N2 - This article aims at the statistical assessment of time series with large fluctuations in short time, which are assumed to stem from a continuous process perturbed by a Lévy process exhibiting a heavy tail behavior. We propose an easily implementable procedure to estimate efficiently the statistical difference between the noisy behavior of the data and a given reference jump measure in terms of so-called coupling distances. After a short introduction to Lévy processes and coupling distances we recall basic statistical approximation results and derive rates of convergence. In the sequel the procedure is elaborated in detail in an abstract setting and eventually applied in a case study to simulated and paleoclimate data. It indicates the dominant presence of a non-stable heavy-tailed jump Lévy component for some tail index greater than 2. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 2 KW - time series with heavy tails KW - index of stability KW - goodness-of-fit KW - empirical Wasserstein distance Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69781 SN - 2193-6943 VL - 3 IS - 2 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Galstian, Anahit T1 - Lp - Lq decay estimates for the equation with exponentially growing coefficient N2 - Contents: 1 Introduction 1 Representation formulas 2 Consideration in the pseudodifferential zone: e up(t) |ξ| ≤ 1 3 Consideration in he hyperbolic zone: e up(t) |ξ| ≥ 1 T3 - Preprint - (2001) 24 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26079 ER - TY - INPR A1 - Galstian, Anahit A1 - Yagdjian, Karen T1 - Exponential function of pseudo-differential operators N2 - The paper is devoted to the construction of the exponential function of a matrix pseudo-differential operator which do not satisfy any of the known theorems (see, Sec.8 Ch.VIII and Sec.2 Ch.XI of [17]). The applications to the construction of the fundamental solution for the Cauchy problem for the hyperbolic operators with the characteristics of variable multiplicity are given, too. T3 - Preprint - (1997) 13 KW - pseudodifferential operators KW - exponential function KW - Gevrey classes KW - hyperbolic operators KW - multiple characteristics Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-24982 ER - TY - INPR A1 - Gauthier, Paul M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - A covering property of the Riemann zeta-function N2 - For each compact subset K of the complex plane C which does not surround zero, the Riemann surface Sζ of the Riemann zeta function restricted to the critical half-strip 0 < Rs < 1/2 contains infinitely many schlicht copies of K lying ‘over’ K. If Sζ also contains at least one such copy, for some K which surrounds zero, then the Riemann hypothesis fails. T3 - Preprint - (2004) 03 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26683 ER - TY - INPR A1 - Gibali, Aviv A1 - Shoikhet, David A1 - Tarkhanov, Nikolai Nikolaevich T1 - On the convergence of continuous Newton method N2 - In this paper we study the convergence of continuous Newton method for solving nonlinear equations with holomorphic mappings in complex Banach spaces. Our contribution is based on a recent progress in the geometric theory of spirallike functions. We prove convergence theorems and illustrate them by numerical simulations. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015)10 KW - Newton method KW - spirallike function Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-81537 SN - 2193-6943 VL - 4 IS - 10 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Gil, Juan B. A1 - Krainer, Thomas A1 - Mendoza, Gerardo A. T1 - Geometry and spectra of closed extensions of elliptic cone operators N2 - We study the geometry of the set of closed extensions of index 0 of an elliptic cone operator and its model operator in connection with the spectra of the extensions, and give a necessary and sufficient condition for the existence of rays of minimal growth for such operators. T3 - Preprint - (2004) 21 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26815 ER - TY - INPR A1 - Gil, Juan B. A1 - Krainer, Thomas A1 - Mendoza, Gerardo A. T1 - Resolvents of elliptic cone operators N2 - We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary. Special attention is devoted to the clarification of the analytic structure of the resolvent. T3 - Preprint - (2004) 22 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26820 ER - TY - INPR A1 - Gil, Juan B. A1 - Krainer, Thomas A1 - Mendoza, Gerardo A. T1 - On rays of minimal growth for elliptic cone operators N2 - We present an overview of some of our recent results on the existence of rays of minimal growth for elliptic cone operators and two new results concerning the necessity of certain conditions for the existence of such rays. T3 - Preprint - (2006) 02 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30064 ER - TY - INPR A1 - Gilkey, Peter T1 - The heat content asymptotics for variable geometries N2 - We study the heat content asymptotics on a compact manifold with boundary defened by a time dependent family of operators of Laplace type. T3 - Preprint - (1998) 26 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25381 ER - TY - INPR A1 - Glebov, S. G. A1 - Kiselev, O. M. T1 - The forced KdV equation and passage through resonance N2 - We construct a special asymptotic solution for the forced KdV equation. In the frame of the shallow water model this kind of the external driving force is related to the atmospheric disturbance. The perturbation slowly passes through a resonance and it leads to the solution exchange. The detailed asymptotic description of the process is presented. T3 - Preprint - (2005) 21 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29975 ER - TY - INPR A1 - Gosson, Maurice A. de T1 - Extended Weyl calculus and application to the phase-space Schrödinger equation N2 - We show that the Schr¨odinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extendedWeyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians. T3 - Preprint - (2005) 11 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29879 ER - TY - INPR A1 - Gosson, Maurice A. de T1 - Symplectic geometry, Wigner-Weyl-Moyal calculus, and quantum mechanics in phase space N2 - Contents: Part I: Symplectic Geometry Chapter 1: Symplectic Spaces and Lagrangian Planes Chapter 2: The Symplectic Group Chapter 3: Multi-Oriented Symplectic Geometry Chapter 4: Intersection Indices in Lag(n) and Sp(n) Part II: Heisenberg Group, Weyl Calculus, and Metaplectic Representation Chapter 5: Lagrangian Manifolds and Quantization Chapter 6: Heisenberg Group and Weyl Operators Chapter 7: The Metaplectic Group Part III: Quantum Mechanics in Phase Space Chapter 8: The Uncertainty Principle Chapter 9: The Density Operator Chapter 10: A Phase Space Weyl Calculus T3 - Preprint - (2006) 06 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30212 ER - TY - INPR A1 - Gosson, Maurice A. de T1 - On the Weyl representation of metaplectic operators N2 - We study the Weyl representation of metaplectic operators associated to a symplectic matrix having no non-trivial fixed point, and justify a formula suggested in earlier work of Mehlig and Wilkinson. We give precise calculations of the associated Maslov-type indices; these indices intervene in a crucial way in Gutzwiller’s formula of semiclassical mechanics, and are simply related to an index defined by Conley and Zehnder. T3 - Preprint - (2005) 10 KW - Weyl symbol KW - metaplectic operators KW - Maslov and Conley–Zehnder index KW - Gutzwiller formula Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29865 ER - TY - INPR A1 - Grudsky, Serguey A1 - Tarkhanov, Nikolai Nikolaevich T1 - Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary N2 - We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)10 KW - singular integral equations KW - nonsmooth curves KW - boundary value problems Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57745 ER - TY - INPR A1 - Harutjunjan, G. A1 - Schulze, Bert-Wolfgang T1 - Conormal symbols of mixed elliptic problems with singular interfaces N2 - Mixed elliptic problems are characterised by conditions that have a discontinuity on an interface of the boundary of codimension 1. The case of a smooth interface is treated in [3]; the investigation there refers to additional interface conditions and parametrices in standard Sobolev spaces. The present paper studies a necessary structure for the case of interfaces with conical singularities, namely, corner conormal symbols of such operators. These may be interpreted as families of mixed elliptic problems on a manifold with smooth interface. We mainly focus on second order operators and additional interface conditions that are holomorphic in an extra parameter. In particular, for the case of the Zaremba problem we explicitly obtain the number of potential conditions in this context. The inverses of conormal symbols are meromorphic families of pseudo-differential mixed problems referring to a smooth interface. Pointwise they can be computed along the lines [3]. T3 - Preprint - (2005) 12 KW - Corner boundary value problems KW - mixed elliptic problems KW - interfaces with conical singularities KW - Zaremba problem Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29885 ER - TY - INPR A1 - Harutjunjan, G. A1 - Schulze, Bert-Wolfgang T1 - Reduction of orders in boundary value problems without the transmission property N2 - Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We consider smooth symbols and ellipticity without additional boundary conditions which is the relevant case on a manifold with boundary. Starting from a class of symbols that has been investigated before for integer orders in boundary value problems with the transmission property we study operators of arbitrary real orders that play a similar role for operators without the transmission property. Moreover, we show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We finally investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. T3 - Preprint - (2002) 03 KW - Boundary value problems KW - elliptic operators KW - order reduction KW - Volterra symbols Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26220 ER - TY - INPR A1 - Harutjunjan, Gohar A1 - Schulze, Bert-Wolfgang T1 - The Zaremba problem with singular interfaces as a corner boundary value problem N2 - We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y , i.e., Au = f in int X, T±u = g± on int Y±, where Y is subdivided into subsets Y± with an interface Z and boundary conditions T± on Y± that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z ⊂ Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T− Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in [3]. With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z \ {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions. T3 - Preprint - (2004) 26 KW - Zaremba problem KW - corner Sobolev spaces with double weights KW - pseudodifferential boundary value problems Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26855 ER - TY - INPR A1 - Harutjunjan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Asymptotics and relative index on a cylinder with conical cross section N2 - We study pseudodifferential operators on a cylinder IR x B with cross section B that conical singularities. Configurations of that kind are the local model of cornere singularities with base spaces B. Operators A in our calculus are assumed to have symbols α which are meromorphic in the complex covariable with values in the space of all cone operators on B. In case α is dependent of the axial variable t ∈ IR, we show an explicit formula for solutions of the homogeneous equation. Each non-bjectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula. T3 - Preprint - (2002) 27 KW - Meromorphic operator functions KW - relative index formulas KW - parameter-dependent cone operators Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26446 ER - TY - INPR A1 - Harutjunjan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Boundary problems with meromorphic symbols in cylindrical domains N2 - We show relative index formulas for boundary value problems in cylindrical domains and Sobolev spaces with different weigths at ±∞. The amplitude functions are meromorphic in the axial covariable and take values in the space of boundary value problems on the cross section of the cylinder. T3 - Preprint - (2004) 12 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26735 ER - TY - INPR A1 - Harutyunyan, Anahit V. T1 - Toeplitz operators and division theorems in anisotropic spaces of holomorphic functions in the polydisc N2 - This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These operators are bounded in these (Lipshitz and Djrbashian) spaces. As an application, we show a theorem about the division by good-inner functions in the mentioned classes is proved. T3 - Preprint - (2001) 28 KW - Toeplitz operators KW - anisotropic spaces KW - polydisc KW - good-inner function Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26110 ER -