TY - INPR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Hilbert boundary value problem for generalised Cauchy-Riemann equations N2 - We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed problems, and construct an explicit formula for approximate solutions. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 1 KW - Dirac operator KW - Clifford algebra KW - Riemann-Hilbert problem KW - Fredholm operator Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-86109 SN - 2193-6943 VL - 5 IS - 1 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - Normally solvable nonlinear boundary value problems N2 - We study a boundary value problem for an overdetermined elliptic system of nonlinear first order differential equations with linear boundary operators. Such a problem is solvable for a small set of data, and so we pass to its variational formulation which consists in minimising the discrepancy. The Euler-Lagrange equations for the variational problem are far-reaching analogues of the classical Laplace equation. Within the framework of Euler-Lagrange equations we specify an operator on the boundary whose zero set consists precisely of those boundary data for which the initial problem is solvable. The construction of such operator has much in common with that of the familiar Dirichlet to Neumann operator. In the case of linear problems we establish complete results. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)11 KW - Nonlinear Laplace operator KW - boundary value problem KW - Dirichlet to Neumann operator Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-65077 SN - 2193-6943 ER - TY - INPR A1 - Anders, Martin T1 - Martingale, Amarts und das starke Gesetz der Grossen Zahlen N2 - Aus dem Inhalt: Einleitung Kapitel 1. Starke Gesetze der Grossen Zahlen 1. SGGZ unter Wachstumsbedingungen an die p-ten Momente 2. SGGZ für identisch verteilte Zufallsvariablen 3. SGGZ für Prozesse mit *-mixing-Eigenschaft Kapitel 2. Einführung zu diskreten (Sub-,Super-)Martingalen 1. Vorhersagbarkeit 2. gestoppte (Sub-,Super-)Martingale 3. Upcrossings 4. Konvergenzsätze 5. Doob-Zerlegung 6. Eine äquivalente Definition eines (Sub-)Martingals Kapitel 3. Martingale und gleichgradige Integrierbarkeit 1. Gleichmäßige(-f¨ormige,-gradige) Integrierbarkeit 2. gleichgradig integrierbare Martingale Kapitel 4. Martingale und das SGGZ Kapitel 5.”reversed“ (Sub-,Super-)Martingale 1. Konvergenzsätze Kapitel 6. (Sub-,Super-)Martingale mit gerichteter Indexmenge 1. Äquivalente Formulierung eines (Sub-)Martingals 2. Konvergenzsätze Kapitel 7. Quasimartingale,Amarts und Semiamarts 1. Konvergenzsätze 2. Riesz-Zerlegung 3. Doob-Zerlegung Kapitel 8. Amarts und das SGGZ Kapitel 9.”reversed“ Amarts und Semiamarts 1. Konvergenzsätze 2.”Aufwärts“- gegen ”Abwärts“-Adaptiertheit 3. Riesz-Zerlegung 4. Stabilitätsanalyse Kapitel 10. Amarts mit gerichteter Indexmenge 1. Konvergenzsätze 2. Riesz-Zerlegung Anhang A. zur Existenz einer Folge unabhängiger Zufallsvariablen B. Konvergenz T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2009, 01 Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49494 ER - TY - INPR A1 - Antoniouk, Alexandra Viktorivna A1 - Kiselev, Oleg A1 - Stepanenko, Vitaly A1 - Tarkhanov, Nikolai Nikolaevich T1 - Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point N2 - The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)25 KW - Heat equation KW - the first boundary value problem KW - characteristic boundary point KW - cusp Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-61987 ER - TY - INPR A1 - Bagderina, Yulia Yu. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Differential invariants of a class of Lagrangian systems with two degrees of freedom N2 - We consider systems of Euler-Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lie's infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 2 KW - equivalence KW - invariant KW - Euler-Lagrange equations Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63129 ER - TY - INPR A1 - Bagdonavičius, Vilijandas B. A1 - Levuliene, Ruta A1 - Nikulin, Mikhail S. A1 - Zdorova-Cheminade, Olga T1 - Tests for homogeneity of survival distributions against non-location alternatives and analysis of the gastric cancer data N2 - The two and k-sample tests of equality of the survival distributions against the alternatives including cross-effects of survival functions, proportional and monotone hazard ratios, are given for the right censored data. The asymptotic power against approaching alternatives is investigated. The tests are applied to the well known chemio and radio therapy data of the Gastrointestinal Tumor Study Group. The P-values for both proposed tests are much smaller then in the case of other known tests. Differently from the test of Stablein and Koutrouvelis the new tests can be applied not only for singly but also to randomly censored data. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 03 KW - Censoring KW - Cross-effects KW - Kolmogorov-Smirnov type tests KW - Logrank test KW - Non-proportional hazards KW - Proportional hazards KW - Two-sample tests Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51527 ER - TY - INPR A1 - Berman, Gennady A1 - Tarkhanov, Nikolai Nikolaevich T1 - Quantum dynamics in the Fermi-Pasta-Ulam problem N2 - We study the dynamics of four wave interactions in a nonlinear quantum chain of oscillators under the "narrow packet" approximation. We determine the set of times for which the evolution of decay processes is essentially specified by quantum effects. Moreover, we highlight the quantum increment of instability. T3 - Preprint - (2004) 05 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26695 ER - TY - INPR A1 - Blanchard, Gilles A1 - Delattre, Sylvain A1 - Roquain, Étienne T1 - Testing over a continuum of null hypotheses N2 - We introduce a theoretical framework for performing statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses. This extends the standard statistical setting for multiple hypotheses testing, which is restricted to a finite set. This work is motivated by numerous modern applications where the observed signal is modeled by a stochastic process over a continuum. As a measure of type I error, we extend the concept of false discovery rate (FDR) to this setting. The FDR is defined as the average ratio of the measure of two random sets, so that its study presents some challenge and is of some intrinsic mathematical interest. Our main result shows how to use the p-value process to control the FDR at a nominal level, either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting, the latter one leading to a less conservative procedure. The interest of this approach is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables. Conceptually, an interesting feature of the setting advocated here is that it focuses directly on the intrinsic hypothesis space associated with a testing model on a random process, without referring to an arbitrary discretization. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 1 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56877 ER - TY - INPR A1 - Blanchard, Gilles A1 - Krämer, Nicole T1 - Convergence rates of kernel conjugate gradient for random design regression N2 - We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 8 KW - nonparametric regression KW - reproducing kernel Hilbert space KW - conjugate gradient KW - partial least squares KW - minimax convergence rates Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-94195 SN - 2193-6943 VL - 5 IS - 8 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Blanchard, Gilles A1 - Mathé, Peter T1 - Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration N2 - The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which takes into account both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration this modification is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 7 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57117 ER - 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 -