TY - INPR A1 - Paneah, Boris A1 - Schulze, Bert-Wolfgang T1 - On the existence of smooth solutions of the dirichlet problem for hyperbolic : differential equations T3 - Preprint - (1998) 05 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25179 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 - THES A1 - Mazzonetto, Sara T1 - On the exact simulation of (skew) Brownian diffusions with discontinuous drift T1 - Über die exakte Simulation (skew) Brownsche Diffusionen mit unstetiger Drift T1 - Simulation exacte de diffusions browniennes (biaisées) avec dérive discontinue N2 - This thesis is focused on the study and the exact simulation of two classes of real-valued Brownian diffusions: multi-skew Brownian motions with constant drift and Brownian diffusions whose drift admits a finite number of jumps. The skew Brownian motion was introduced in the sixties by Itô and McKean, who constructed it from the reflected Brownian motion, flipping its excursions from the origin with a given probability. Such a process behaves as the original one except at the point 0, which plays the role of a semipermeable barrier. More generally, a skew diffusion with several semipermeable barriers, called multi-skew diffusion, is a diffusion everywhere except when it reaches one of the barriers, where it is partially reflected with a probability depending on that particular barrier. Clearly, a multi-skew diffusion can be characterized either as solution of a stochastic differential equation involving weighted local times (these terms providing the semi-permeability) or by its infinitesimal generator as Markov process. In this thesis we first obtain a contour integral representation for the transition semigroup of the multiskew Brownian motion with constant drift, based on a fine analysis of its complex properties. Thanks to this representation we write explicitly the transition densities of the two-skew Brownian motion with constant drift as an infinite series involving, in particular, Gaussian functions and their tails. Then we propose a new useful application of a generalization of the known rejection sampling method. Recall that this basic algorithm allows to sample from a density as soon as one finds an - easy to sample - instrumental density verifying that the ratio between the goal and the instrumental densities is a bounded function. The generalized rejection sampling method allows to sample exactly from densities for which indeed only an approximation is known. The originality of the algorithm lies in the fact that one finally samples directly from the law without any approximation, except the machine's. As an application, we sample from the transition density of the two-skew Brownian motion with or without constant drift. The instrumental density is the transition density of the Brownian motion with constant drift, and we provide an useful uniform bound for the ratio of the densities. We also present numerical simulations to study the efficiency of the algorithm. The second aim of this thesis is to develop an exact simulation algorithm for a Brownian diffusion whose drift admits several jumps. In the literature, so far only the case of a continuous drift (resp. of a drift with one finite jump) was treated. The theoretical method we give allows to deal with any finite number of discontinuities. Then we focus on the case of two jumps, using the transition densities of the two-skew Brownian motion obtained before. Various examples are presented and the efficiency of our approach is discussed. N2 - In dieser Dissertation wird die exakte Simulation zweier Klassen reeller Brownscher Diffusionen untersucht: die multi-skew Brownsche Bewegung mit konstanter Drift sowie die Brownsche Diffusionen mit einer Drift mit endlich vielen Sprüngen. Die skew Brownsche Bewegung wurde in den sechzigern Jahren von Itô and McKean als eine Brownsche Bewegung eingeführt, für die die Richtung ihrer Exkursionen am Ursprung zufällig mit einer gegebenen Wahrscheinlichkeit ausgewürfelt wird. Solche asymmetrischen Prozesse verhalten sich im Wesentlichen wie der Originalprozess außer bei 0, das sich wie eine semipermeable Barriere verhält. Allgemeiner sind skew Diffusionsprozesse mit mehreren semipermeablen Barrieren, auch multi-skew Diffusionen genannt, Diffusionsprozesse mit Ausnahme an den Barrieren, wo sie jeweils teilweise reflektiert wird. Natürlich ist eine multi-skew Diffusion durch eine stochastische Differentialgleichung mit Lokalzeiten (diese bewirken die Semipermeabilität) oder durch ihren infinitesimalen Generator als Markov Prozess charakterisiert. In dieser Arbeit leiten wir zunächst eine Konturintegraldarstellung der Übergangshalbgruppe der multi-skew Brownschen Bewegung mit konstanter Drift durch eine feine Analyse ihrer komplexen Eigenschaften her. Dank dieser Darstellung wird eine explizite Darstellung der Übergangswahrscheinlichkeiten der zweifach-skew Brownschen Bewegung mit konstanter Drift als eine unendliche Reihe Gaußscher Dichten erhalten. Anschlieẞend wird eine nützliche Verallgemeinerung der bekannten Verwerfungsmethode vorgestellt. Dieses grundlegende Verfahren ermöglicht Realisierungen von Zufallsvariablen, sobald man eine leicht zu simulierende Zufallsvariable derart findet, dass der Quotient der Dichten beider Zufallsvariablen beschränkt ist. Die verallgmeinerte Verwerfungsmethode erlaubt eine exakte Simulation für Dichten, die nur approximiert werden können. Die Originalität unseres Verfahrens liegt nun darin, dass wir, abgesehen von der rechnerbedingten Approximation, exakt von der Verteilung ohne Approximation simulieren. In einer Anwendung simulieren wir die zweifach-skew Brownsche Bewegung mit oder ohne konstanter Drift. Die Ausgangsdichte ist dabei die der Brownschen Bewegung mit konstanter Drift, und wir geben gleichmäẞige Schranken des Quotienten der Dichten an. Dazu werden numerische Simulationen gezeigt, um die Leistungsfähigkeit des Verfahrens zu demonstrieren. Das zweite Ziel dieser Arbeit ist die Entwicklung eines exakten Simulationsverfahrens für Brownsche Diffusionen, deren Drift mehrere Sprünge hat. In der Literatur wurden bisher nur Diffusionen mit stetiger Drift bzw. mit einer Drift mit höchstens einem Sprung behandelt. Unser Verfahren erlaubt den Umgang mit jeder endlichen Anzahl von Sprüngen. Insbesondere wird der Fall zweier Sprünge behandelt, da unser Simulationsverfahren mit den bereits erhaltenen Übergangswahrscheinlichkeiten der zweifach-skew Brownschen Bewegung verwandt ist. An mehreren Beispielen demonstrieren wir die Effizienz unseres Ansatzes. KW - exact simulation KW - exakte Simulation KW - skew diffusions KW - Skew Diffusionen KW - local time KW - discontinuous drift KW - diskontinuierliche Drift Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-102399 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 - Nehring, Benjamin A1 - Poghosyan, Suren A1 - Zessin, Hans T1 - On the construction of point processes in statistical mechanics N2 - By means of the cluster expansion method we show that a recent result of Poghosyan and Ueltschi (2009) combined with a result of Nehring (2012) yields a construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose gas of Brownian loops and to the dissolution in R^d of Ginibre's Fermi-Dirac gas of such loops. The latter will be identified as a Gibbs perturbation of the ideal Fermi gas. On generalizing these considerations we will obtain the existence of a large class of Gibbs perturbations of the so-called KMM-processes as they were introduced by Nehring (2012). Moreover, it is shown that certain "limiting Gibbs processes" are Gibbs in the sense of Dobrushin, Lanford and Ruelle if the underlying potential is positive. And finally, Gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 5 KW - Levy measure KW - cluster expansion KW - Gibbs perturbation KW - DLR equation Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64080 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 - 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 - Krainer, Thomas T1 - On the calculus of pseudodifferential operators with an anisotropic analytic parameter N2 - We introduce the Volterra calculus of pseudodifferential operators with an anisotropic analytic parameter based on "twisted" operator-valued Volterra symbols. We establish the properties of the symbolic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side. In particular, we investigate the kernel cut-off operator via direct oscillatory integral techniques purely on symbolic level. We discuss the notion of parabolic for the calculus of Volterra operators, and construct Volterra parametrices for parabolic operators within the calculus. T3 - Preprint - (2002) 01 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26200 ER - TY - INPR A1 - Myslivets, Simona T1 - On the boundary behaviour of the logarithmic residue integral N2 - A formula of multidimensional logarithmic residue is proved for holomorphic maps with zeroes on the boundary of a bounded domain in Cn. T3 - Preprint - (2000) 07 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25733 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 -