TY - THES A1 - Wallenta, Daniel T1 - Sequences of compact curvature T1 - Sequenzen mit kompakter Krümmung N2 - By perturbing the differential of a (cochain-)complex by "small" operators, one obtains what is referred to as quasicomplexes, i.e. a sequence whose curvature is not equal to zero in general. In this situation the cohomology is no longer defined. Note that it depends on the structure of the underlying spaces whether or not an operator is "small." This leads to a magical mix of perturbation and regularisation theory. In the general setting of Hilbert spaces compact operators are "small." In order to develop this theory, many elements of diverse mathematical disciplines, such as functional analysis, differential geometry, partial differential equation, homological algebra and topology have to be combined. All essential basics are summarised in the first chapter of this thesis. This contains classical elements of index theory, such as Fredholm operators, elliptic pseudodifferential operators and characteristic classes. Moreover we study the de Rham complex and introduce Sobolev spaces of arbitrary order as well as the concept of operator ideals. In the second chapter, the abstract theory of (Fredholm) quasicomplexes of Hilbert spaces will be developed. From the very beginning we will consider quasicomplexes with curvature in an ideal class. We introduce the Euler characteristic, the cone of a quasiendomorphism and the Lefschetz number. In particular, we generalise Euler's identity, which will allow us to develop the Lefschetz theory on nonseparable Hilbert spaces. Finally, in the third chapter the abstract theory will be applied to elliptic quasicomplexes with pseudodifferential operators of arbitrary order. We will show that the Atiyah-Singer index formula holds true for those objects and, as an example, we will compute the Euler characteristic of the connection quasicomplex. In addition to this we introduce geometric quasiendomorphisms and prove a generalisation of the Lefschetz fixed point theorem of Atiyah and Bott. N2 - Die Theorie der Sequenzen mit kompakter Krümmung, sogenannter Quasikomplexe, ist eine Verallgemeinerung der Theorie der Fredholm Komplexe. Um ein Verständnis für (Quasi-)Komplexe zu gewinnen, müssen Inhalte aus verschiedenen Teilgebieten der Mathematik kombiniert werden. Alle hierfür wesentlichen Grundlagen sind im ersten Kapitel dieser Dissertation zusammengefasst. Dies betrifft unter anderem gewisse Elemente der Funktionalanalysis und der Differentialgeometrie, sowie die Theorie der klassischen Pseudodifferentialoperatoren. Im zweiten Kapitel wird anschließend die abstrakte Theorie der Quasikomplexe und zugehöriger Quasimorphismen im Kontext der Funktionalanalysis entwickelt. Dabei werden verschiedene Typen von Quasikomplexen und Quasimorphismen klassifiziert, deren Eigenschaften analysiert und Beispiele betrachtet. Ein zentraler Punkt hierbei ist die Lösung des Problems, für welche dieser Objekte sich eine besondere charakteristische Zahl, die sogenannte Lefschetz-Zahl, definieren lässt. Die dargestellten Resultate zeigen, dass die in dieser Arbeit gegebene Definition eine natürliche Erweiterung der klassischen Lefschetz-Zahl darstellt. Abschließend wird die entwickelte Theorie im dritten Kapitel auf elliptische Quasikomplexe von Pseudodifferentialoperatoren angewendet. Dabei werden insbesondere Verallgemeinerungen der berühmten Atiyah-Singer-Index-Formel und des Lefschetz-Fixpunkt-Theorems von Atiyah and Bott bewiesen. KW - Index Theorie KW - Fredholm Komplexe KW - Elliptische Komplexe KW - Index theory KW - Elliptic complexes KW - Fredholm complexes Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-87489 ER - TY - THES A1 - Di Gesù, Giacomo T1 - Semiclassical spectral analysis of discrete Witten Laplacians T1 - Semiklassische Spektraltheorie von diskreten Witten-Laplace-Operatoren N2 - A discrete analogue of the Witten Laplacian on the n-dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low-lying spectrum. Our proof, inspired by work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of metastable Markov processes on the lattice. N2 - In dieser Arbeit wird auf dem n-dimensionalen Gitter der ganzen Zahlen ein Analogon des Witten-Laplace-Operatoren eingeführt. Nach geeigneter Skalierung des Gitters und des Operatoren analysieren wir den Tunneleffekt zwischen verschiedenen Potentialtöpfen und erhalten vollständige Aymptotiken für das tiefliegende Spektrum. Der Beweis (nach Methoden, die von B. Helffer, M. Klein und F. Nier im Falle des kontinuierlichen Witten-Laplace-Operatoren entwickelt wurden) basiert auf der Konstruktion eines diskreten Witten-Komplexes und der Analyse des zugehörigen Witten-Laplace-Operatoren auf 1-Formen. Das Resultat kann im Kontext von metastabilen Markov Prozessen auf dem Gitter reformuliert werden und ermöglicht scharfe Aussagen über metastabile Austrittszeiten. KW - Semiklassische Spektralasymptotik KW - Metastabilität KW - diskreter Witten-Laplace-Operator KW - Eyring-Kramers Formel KW - Tunneleffekt KW - semiclassical spectral asymptotics KW - metastability KW - low-lying eignvalues KW - discrete Witten complex KW - rescaled lattice Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-65286 ER - TY - THES A1 - Hohberger, Horst T1 - Semiclassical asymptotics for the scattering amplitude in the presence of focal points at infinity T1 - Semiklassische Asymptotik der Streuamplitude bei unendlich fernen Fokalpunkten N2 - We consider scattering in $\R^n$, $n\ge 2$, described by the Schr\"odinger operator $P(h)=-h^2\Delta+V$, where $V$ is a short-range potential. With the aid of Maslov theory, we give a geometrical formula for the semiclassical asymptotics as $h\to 0$ of the scattering amplitude $f(\omega_-,\omega_+;\lambda,h)$ $\omega_+\neq\omega_-$) which remains valid in the presence of focal points at infinity (caustics). Crucial for this analysis are precise estimates on the asymptotics of the classical phase trajectories and the relationship between caustics in euclidean phase space and caustics at infinity. N2 - Wir betrachten Streuung in $\R^n$, $n\ge 2$, beschrieben durch den Schr\"odinger operator $P(h)=-h^2\Delta+V$, wo $V$ ein kurzreichweitiges Potential ist. Mit Hilfe von Maslov Theorie erhalten wir eine geometrische Formel fuer die semiklassische Asymptotik ($h\to 0$) der Streuamplitude $f(\omega_-,\omega_+;\lambda,h)$ ($\omega_+\neq\omega_-$) welche auch bei Vorhandensein von Fokalpunkten bei Unendlich (Kaustiken) gueltig bleibt. KW - Mathematik KW - Physik KW - Streutheorie KW - Streuamplitude KW - Semiklassik KW - mathematics KW - physics KW - scattering theory KW - semiclassics KW - scattering amplitude Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-11574 ER - TY - JOUR A1 - Sukiasyan, Hayk A1 - Melkonyan, Tatev T1 - Semi-recursive algorithm of piecewise linear approximation of two-dimensional function by the method of worst segment dividing JF - Lectures in pure and applied mathematics KW - random point processes KW - statistical mechanics KW - stochastic analysis Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471982 SN - 978-3-86956-485-2 SN - 2199-4951 SN - 2199-496X IS - 6 SP - 35 EP - 44 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Kiselev, Oleg M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Scattering of autoresonance trajectories upon a separatrix N2 - We study asymptotic properties of solutions to the primary resonance equation with large amplitude on a long time interval. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 2 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56880 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Root functions of elliptic boundary problems in domains with conic points of the boundary N2 - We prove the completeness of the system of eigen and associated functions (i.e., root functions) of an elliptic boundary value problem in a domain whose boundary is a smooth surface away from a finite number of points, each of them possesses a neighbourhood where the boundary is a conical surface. T3 - Preprint - (2005) 07 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29812 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 - Klein, Markus A1 - Zitt, Pierre-André T1 - Resonances for a diffusion with small noise N2 - We study resonances for the generator of a diffusion with small noise in R(d) : L = -∈∆ + ∇F * ∇, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F. We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the "quick growth" case, and whose imaginary parts are small. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2008, 02 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49448 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 - Krainer, Thomas T1 - Resolvents of elliptic boundary problems on conic manifolds N2 - We prove the existence of sectors of minimal growth for realizations of boundary value problems on conic manifolds under natural ellipticity conditions. Special attention is devoted to the clarification of the analytic structure of the resolvent. T3 - Preprint - (2005) 03 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29773 ER -