TY - INPR A1 - Kytmanov, Aleksandr A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Zeta-function of a nonlinear system N2 - Given a system of entire functions in Cn with at most countable set of common zeros, we introduce the concept of zeta-function associated with the system. Under reasonable assumptions on the system, the zeta-function is well defined for all s ∈ Zn with sufficiently large components. Using residue theory we get an integral representation for the zeta-function which allows us to construct an analytic extension of the zeta-function to an infinite cone in Cn. T3 - Preprint - (2004) 19 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26795 ER - TY - JOUR A1 - Figari, Rodolfo A1 - Teta, Alessandro T1 - Zero-range hamiltonians for three quantum particles 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-opus4-472189 SN - 978-3-86956-485-2 SN - 2199-4951 SN - 2199-496X IS - 6 SP - 175 EP - 184 PB - Universitätsverlag Potsdam CY - Potsdam 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 - JOUR A1 - Lykov, Alexander A1 - Malyshev, Vadim T1 - When bounded chaos becomes unbounded 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-opus4-472060 SN - 978-3-86956-485-2 SN - 2199-4951 SN - 2199-496X IS - 6 SP - 97 EP - 106 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Maniccia, L. A1 - Mughetti, M. T1 - Weyl calculus for a class of subelliptic operators N2 - Weyl-Hörmander calculus is used to get a parametrix in OPS¹-m sub(½, ½)(Ω)for a class of subelliptic pseudodifferential operators in OPS up(m)sub(1, 0)(Ω) with real non-negative principal symbol. T3 - Preprint - (2001) 19 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26038 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 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - Weak boundary values of solutions of Lagrangian problems N2 - We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 2 KW - nonlinear equations KW - Lagrangian system KW - weak boundary values KW - quasilinear Fredholm operator KW - mapping degree Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72617 SN - 2193-6943 VL - 4 IS - 2 PB - Universitätsverlag Potsdam CY - Potsdam 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 - Liu, Weian A1 - Yang, Yin A1 - Lu, Gang T1 - Viscosity solutions of fully nonlinear parabolic systems N2 - In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3]. T3 - Preprint - (2002) 02 KW - Viscosity solutions KW - systems of partial differential equations KW - fully non-linear degenerate parabolic equations KW - Perron's method KW - coupled solution Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26215 ER - TY - JOUR A1 - Jansen, Sabine A1 - Kuna, Tobias A1 - Tsagkarogiannis, Dimitrios T1 - Virial inversion for inhomogeneous systems 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-opus4-472111 SN - 978-3-86956-485-2 SN - 2199-4951 SN - 2199-496X IS - 6 SP - 135 EP - 144 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - THES A1 - Hanisch, Florian T1 - Variational problems on supermanifolds T1 - Variationsprobleme auf Supermannigfaltigkeiten N2 - In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anticommutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components. We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields. N2 - In dieser Dissertation wird die Formulierung von Variationsproblemen auf Supermannigfaltigkeiten diskutiert. Supermannigfaltigkeiten enthalten sowohl bosonische als auch fermionische Freiheitsgrade. Fermionische Felder nehmen Werte im ungeraden Teil einer Grassmannalgebra an, sie antikommutieren deshalb untereinander. Eine systematische Behandlung dieser Grassmann-Parameter erfordert jedoch die Beschreibung von Räumen durch Funktoren, z.B. von der Kategorie der Grassmannalgebren in diejenige der Mengen (der topologischen Räume, Mannigfaltigkeiten, ...). Nach einer Einführung in das allgemeine Konzept dieses Zugangs verwenden wir es um eine Beschreibung der resultierenden Supermannigfaltigkeit der Felder bzw. Abbildungen anzugeben. Wir zeigen, dass jede Abbildung eindeutig durch eine Familie von Differentialoperatoren geeigneter Ordnung charakterisiert wird. Darüber hinaus beweisen wir, dass jede solche Abbildung eineindeutig durch ihre Komponentenfelder, d.h. durch die Koeffizienten einer Taylorentwickelung bzgl. von ungeraden Koordinaten bestimmt ist. Im Allgemeinen sind Komponentenfelder nur lokal definiert. Wir stellen einen Weg vor, der diese Einschränkung umgeht: Durch das Vergrößern der betreffenden Supermannigfaltigkeit ist es immer möglich, mit globalen Koordinaten zu arbeiten. Schließlich wenden wir diesen Formalismus an, um Variationsprobleme zu untersuchen, genauer betrachten wir eine super-Version der Geodäte und eine Verallgemeinerung von harmonischen Abbildungen auf Supermannigfaltigkeiten. Bewegungsgleichungen werden von Energiefunktionalen abgeleitet und wir zeigen, wie sie sich in Komponenten zerlegen lassen. Schließlich kann in Spezialfällen die Existenz von kritischen Punkten gezeigt werden, indem das Problem auf Gleichungen der gewöhnlichen geometrischen Analysis reduziert wird. Es kann dann gezeigt werden, dass die Lösungen dieser Gleichungen sich zu kritischen Punkten im betreffenden Funktor-Raum der Felder zusammensetzt. KW - Supergeometrie KW - Variationsrechnung KW - Differentialoperatoren KW - Funktorgeometrie KW - supergeometry KW - variational calculus KW - differential operators KW - functor geometry Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-59757 ER - TY - INPR A1 - Alsaedy, Ammar T1 - Variational primitive of a differential form N2 - In this paper we specify the Dirichlet to Neumann operator related to the Cauchy problem for the gradient operator with data on a part of the boundary. To this end, we consider a nonlinear relaxation of this problem which is a mixed boundary problem of Zaremba type for the p-Laplace equation. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 4 KW - Dirichlet-to-Neumann operator KW - Cauchy problem KW - p-Laplace operator KW - calculus of variations Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-89223 SN - 2193-6943 VL - 5 IS - 4 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - THES A1 - Vu, Dinh Phuong T1 - Using video study to investigate eighth-grade mathematics classrooms in Vietnam T1 - Die Nutzung von Videostudien zur Untersuchung des Mathematikunterrichts in der 8. Klasse in Vietnam N2 - The International Project for the Evaluation of Educational Achievement (IEA) was formed in the 1950s (Postlethwaite, 1967). Since that time, the IEA has conducted many studies in the area of mathematics, such as the First International Mathematics Study (FIMS) in 1964, the Second International Mathematics Study (SIMS) in 1980-1982, and a series of studies beginning with the Third International Mathematics and Science Study (TIMSS) which has been conducted every 4 years since 1995. According to Stigler et al. (1999), in the FIMS and the SIMS, U.S. students achieved low scores in comparison with students in other countries (p. 1). The TIMSS 1995 “Videotape Classroom Study” was therefore a complement to the earlier studies conducted to learn “more about the instructional and cultural processes that are associated with achievement” (Stigler et al., 1999, p. 1). The TIMSS Videotape Classroom Study is known today as the TIMSS Video Study. From the findings of the TIMSS 1995 Video Study, Stigler and Hiebert (1999) likened teaching to “mountain ranges poking above the surface of the water,” whereby they implied that we might see the mountaintops, but we do not see the hidden parts underneath these mountain ranges (pp. 73-78). By watching the videotaped lessons from Germany, Japan, and the United States again and again, they discovered that “the systems of teaching within each country look similar from lesson to lesson. At least, there are certain recurring features [or patterns] that typify many of the lessons within a country and distinguish the lessons among countries” (pp. 77-78). They also discovered that “teaching is a cultural activity,” so the systems of teaching “must be understood in relation to the cultural beliefs and assumptions that surround them” (pp. 85, 88). From this viewpoint, one of the purposes of this dissertation was to study some cultural aspects of mathematics teaching and relate the results to mathematics teaching and learning in Vietnam. Another research purpose was to carry out a video study in Vietnam to find out the characteristics of Vietnamese mathematics teaching and compare these characteristics with those of other countries. In particular, this dissertation carried out the following research tasks: - Studying the characteristics of teaching and learning in different cultures and relating the results to mathematics teaching and learning in Vietnam - Introducing the TIMSS, the TIMSS Video Study and the advantages of using video study in investigating mathematics teaching and learning - Carrying out the video study in Vietnam to identify the image, scripts and patterns, and the lesson signature of eighth-grade mathematics teaching in Vietnam - Comparing some aspects of mathematics teaching in Vietnam and other countries and identifying the similarities and differences across countries - Studying the demands and challenges of innovating mathematics teaching methods in Vietnam – lessons from the video studies Hopefully, this dissertation will be a useful reference material for pre-service teachers at education universities to understand the nature of teaching and develop their teaching career. N2 - Das International Project for the Evaluation of Educational Achievement (IEA) wurde in den 1950er Jahren gegründet. Seitdem führte das IEA viele Studien in Bereich mathematischer Bildung durch, insbesondere die First International Mathematics Study (FIMS) im Jahre 1964, die Second International Mathematics Study (SIMS) in den Jahren 1980–1982 und eine Reihe von Studien, die mit der Third International Mathematics and Science Study (TIMSS) begann und seit 1995 alle vier Jahre durchgeführt wird. Nach Stigler et al. (1999) erreichten US-amerikanische Studenten bei FIMS und SIMS niedrigere Ergebnisse als Schüler anderer Länder (S. 1). Daher wurde TIMSS 1995 erweitert um eine ‘Videotape Classroom Study’ mit dem Ziel, „mehr über die unterrichtlichen und kulturellen Prozesse, die mit Leistung zusammenhängen“, zu erfahren (S. 1; Übersetzung vom engl. Original). Von den Ergebnissen der TIMMS 1995 Video Study ausgehend verglichen Stigler und Hiebert (1999) Unterricht mit „Gebirgszügen, die die Wasseroberfläche durchstoßen“, womit sie ausdrücken sollten, was die Bergspitzen sichtbar, große Teile des Gebirges aber unter dem Wasser verborgen sind (S. 73–78; Übersetzung vom engl. Original). Durch die wiederholte Analyse videographierter Unterrichtsstunden aus Deutschland, Japan und den USA entdeckten sie, dass „die Arten des Unterrichts innerhalb jedes Landes von Stunde zu Stunde ähnlich sind. Zumindest gibt es bestimmte wiederkehrende Aspekte [oder Skripte], welche für viele Stunden eines Landes typisch sind und die Stunden gegenüber anderen Ländern abgrenzen“ (S. 77f.). Sie entdeckten außerdem, dass Unterricht eine „kulturelle Aktivität“ ist, Unterrichtsarten also „verstanden werden müssen in Relation zu den kulturellen Überzeugungen und Annahmen, die sie umgeben“ (S. 85, 88). Hierauf aufbauend war es ein Ziel der Dissertation, kulturelle Aspekte des Mathematikunterricht zu untersuchen und die Ergebnisse mit Mathematikunterricht in Vietnam zu vergleichen. Ein weiteres Ziel war die Erhebung der Charakteristika vietnamesischen Mathematikunterricht durch eine Videostudie in Vietnam und der anschließende Vergleich dieser Charakteristika mit denen anderer Länder. Im Einzelnen befasste sich diese Dissertation mit den folgenden Forschungszielen: - Untersuchung der Charakteristika von Lehren und Lernen in unterschiedlichen Kulturen und vorläufiger Vergleich der Resultate mit dem Lehren und Lernen von Mathematik in Vietnam - Einführung der TIMSS und der TIMSS Video Study und der methodologischen Vorteile von Videostudien für die Untersuchung von Mathematikunterricht in Vietnam - Durchführung der Videostudie in Vietnam, um Unterrichtsskripte des Mathematikunterrichts in 8. Klassen in Vietnam zu identifizieren - Vergleich ausgewählter Aspekte des Mathematikunterrichts in Vietnam mit denen anderer Länder auf der Grundlage der Videostudie in Vietnam und Diskussion von Ähnlichkeiten und Unterschieden zwischen Ländern - Untersuchung der Herausforderungen für eine Innovation der Unterrichtsmethoden im Mathematikunterricht Vietnams Diese Dissertation entstand in der Hoffnung, dass sie eine nützliche Referenz für Lehramtsstudenten zum Verständnis der Natur des Unterrichts und zur Entwicklung der eigenen Lehrerpersönlichkeit darstellen möge. KW - Videostudie KW - Mathematikunterricht KW - Unterrichtsmethode KW - TIMSS KW - Kulturelle Aktivität KW - video study KW - mathematics education KW - teaching methods KW - TIMSS KW - Vietnam Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72464 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Unitary solutions of partial differential equations N2 - We give an explicit construction of a fundamental solution for an arbitrary non-degenerate partial differential equation with smooth coefficients. T3 - Preprint - (2005) 09 KW - fundamental solution KW - geometric optics approximation Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29852 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Qin, Yuming T1 - Uniform compact attractors for a nonlinear non-autonomous equation of viscoelasticity N2 - In this paper we establish the regularity, exponential stability of global (weak) solutions and existence of uniform compact attractors of semiprocesses, which are generated by the global solutions, of a two-parameter family of operators for the nonlinear 1-d non-autonomous viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions. T3 - Preprint - (2005) 13 KW - exponential stability KW - semiprocess KW - absorbing set KW - C0−semigroup KW - uniform compact attractor Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29892 ER - TY - INPR A1 - Klein, Markus A1 - Rosenberger, Elke T1 - Tunneling for a class of difference operators N2 - We analyze a general class of difference operators containing a multi-well potential and a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we treat the eigenvalue problem as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix similar to the analysis for the Schrödinger operator, and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 5 KW - semi-classical difference operator KW - tunneling KW - interaction matrix Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56989 ER - TY - JOUR A1 - Zagrebnov, Valentin T1 - Trotter product formula on Hilbert and Banach spaces for operator-norm convergence 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-opus4-471971 SN - 978-3-86956-485-2 SN - 2199-4951 SN - 2199-496X SP - 23 EP - 34 PB - Universitätsverlag Potsdam CY - Potsdam 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 - Schulze, Bert-Wolfgang T1 - Toeplitz operators, and ellipticity of boundary value problems with global projection conditions N2 - Ellipticity of (pseudo-) differential operators A on a compact manifold X with boundary (or with edges) Y is connected with boundary (or edge) conditions of trace and potential type, formulated in terms of global projections on Y together with an additional symbolic structure. This gives rise to operator block matrices A with A in the upper left corner. We study an algebra of such operators, where ellipticity is equivalent to the Fredhom property in suitable scales of spaces: Sobolev spaces on X plus closed subspaces of Sobolev spaces on Y which are the range of corresponding pseudo-differential projections. Moreover, we express parametrices of elliptic elements within our algebra and discuss spectral boundary value problems for differential operators. T3 - Preprint - (2003) 03 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26510 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 - TY - INPR A1 - Louis, Pierre-Yves A1 - Rouquier, Jean-Baptiste T1 - Time-to-Coalescence for interacting particle systems : parallel versus sequential updating N2 - Studying the influence of the updating scheme for MCMC algorithm on spatially extended models is a well known problem. For discrete-time interacting particle systems we study through simulations the effectiveness of a synchronous updating scheme versus the usual sequential one. We compare the speed of convergence of the associated Markov chains from the point of view of the time-to-coalescence arising in the coupling-from-the-past algorithm. Unlike the intuition, the synchronous updating scheme is not always the best one. The distribution of the time-to-coalescence for these spatially extended models is studied too. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2009, 03 Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49454 ER - TY - THES A1 - Angwenyi, David T1 - Time-continuous state and parameter estimation with application to hyperbolic SPDEs T1 - Zeitkontinuierliche Zustands- und Parameterschätzung bei Anwendung auf hyperbolische SPDEs N2 - Data assimilation has been an active area of research in recent years, owing to its wide utility. At the core of data assimilation are filtering, prediction, and smoothing procedures. Filtering entails incorporation of measurements' information into the model to gain more insight into a given state governed by a noisy state space model. Most natural laws are governed by time-continuous nonlinear models. For the most part, the knowledge available about a model is incomplete; and hence uncertainties are approximated by means of probabilities. Time-continuous filtering, therefore, holds promise for wider usefulness, for it offers a means of combining noisy measurements with imperfect model to provide more insight on a given state. The solution to time-continuous nonlinear Gaussian filtering problem is provided for by the Kushner-Stratonovich equation. Unfortunately, the Kushner-Stratonovich equation lacks a closed-form solution. Moreover, the numerical approximations based on Taylor expansion above third order are fraught with computational complications. For this reason, numerical methods based on Monte Carlo methods have been resorted to. Chief among these methods are sequential Monte-Carlo methods (or particle filters), for they allow for online assimilation of data. Particle filters are not without challenges: they suffer from particle degeneracy, sample impoverishment, and computational costs arising from resampling. The goal of this thesis is to:— i) Review the derivation of Kushner-Stratonovich equation from first principles and its extant numerical approximation methods, ii) Study the feedback particle filters as a way of avoiding resampling in particle filters, iii) Study joint state and parameter estimation in time-continuous settings, iv) Apply the notions studied to linear hyperbolic stochastic differential equations. The interconnection between Itô integrals and stochastic partial differential equations and those of Stratonovich is introduced in anticipation of feedback particle filters. With these ideas and motivated by the variants of ensemble Kalman-Bucy filters founded on the structure of the innovation process, a feedback particle filter with randomly perturbed innovation is proposed. Moreover, feedback particle filters based on coupling of prediction and analysis measures are proposed. They register a better performance than the bootstrap particle filter at lower ensemble sizes. We study joint state and parameter estimation, both by means of extended state spaces and by use of dual filters. Feedback particle filters seem to perform well in both cases. Finally, we apply joint state and parameter estimation in the advection and wave equation, whose velocity is spatially varying. Two methods are employed: Metropolis Hastings with filter likelihood and a dual filter comprising of Kalman-Bucy filter and ensemble Kalman-Bucy filter. The former performs better than the latter. N2 - Die Datenassimilation war in den letzten Jahren aufgrund ihres breiten Nutzens ein aktives Forschungsgebiet. Im Zentrum der Datenassimilation stehen Filter-, Vorhersage- und Glättungsverfahren. Die Filterung beinhaltet die Einbeziehung von Messinformationen in das Modell, um einen besseren Einblick in einen gegebenen Zustand zu erhalten, der durch ein verrauschtes Zustandsraummodell gesteuert wird. Die meisten Naturgesetze werden von zeitkontinuierlichen nichtlinearen Modellen bestimmt. Das verfügbare Wissen über ein Modell ist größtenteils unvollständig; und daher werden Unsicherheiten mittels Wahrscheinlichkeiten angenähert. Die zeitkontinuierliche Filterung verspricht daher eine größere Nützlichkeit, denn sie bietet die Möglichkeit, verrauschte Messungen mit einem unvollkommenen Modell zu kombinieren, um mehr Einblick in einen bestimmten Zustand zu erhalten. Das Problem der zeitkontinuierlichen nichtlinearen Gaußschen Filterung wird durch die Kushner-Stratonovich-Gleichung gelöst. Leider fehlt der Kushner-Stratonovich-Gleichung eine geschlossene Lösung. Darüber hinaus sind die numerischen Näherungen, die auf der Taylor-Erweiterung über der dritten Ordnung basieren, mit rechnerischen Komplikationen behaftet. Aus diesem Grund wurde auf numerische Methoden zurückgegriffen, die auf Monte-Carlo-Methoden basieren. Die wichtigsten dieser Methoden sind sequentielle Monte-Carlo-Methoden (oder Partikelfilter), da sie die Online-Assimilation von Daten ermöglichen. Partikelfilter sind nicht unproblematisch: Sie leiden unter Partikelentartung, Probenverarmung und Rechenkosten, die sich aus der Neuabtastung ergeben. Das Ziel dieser Arbeit ist es, i) die Ableitung der Kushner-Stratonovich-Gleichung aus den ersten Prinzipien und ihre vorhandenen numerischen Approximationsmethoden zu überprüfen, ii) die Rückkopplungs-Partikelfilter zu untersuchen, um eine Neuabtastung in Partikelfiltern zu vermeiden, iii) Studieren Sie die Zustands- und Parameterschätzung in zeitkontinuierlichen Einstellungen, iv) Wenden Sie die untersuchten Begriffe auf lineare hyperbolische stochastische Differentialgleichungen an. Die Verbindung zwischen Itô Integralen und stochastischen partiellen Differentialgleichungen und denen von Stratonovich wird in Erwartung von Rückkopplungs-Partikelfiltern eingeführt. Mit diesen Ideen und motiviert durch die Varianten von Kalman-Bucy-Filtern, die auf der Struktur des Innovationsprozesses gegründet, wird ein Feedback-Partikelfilter mit zufällig gestörter Innovation vorgeschlagen. Darüber hinaus werden Rückkopplungspartikelfilter basierend auf der Kopplung von Vorhersage- und Analysemaßnahmen vorgeschlagen. Diese Feedback-Partikelfiltern haben eine bessere Leistung als der Bootstrap-Partikelfilter bei niedrigeren Ensemble-Größen. Wir untersuchen gemeinsame Zustands- und Parameterschätzungen, sowohl durch erweiterte Zustandsräume als auch durch Verwendung von Doppelfiltern. Rückkopplungs-Partikelfilter scheinen in beiden Fällen gut zu funktionieren. Schließlich wenden wir eine gemeinsame Zustands- und Parameterschätzung in der Advektions-und Wellengleichung an, deren Geschwindigkeit räumlich variiert. Es werden zwei Verfahren verwendet: Metropolis-Hastings mit Filterwahrscheinlichkeit und ein Doppelfilter bestehend aus Kalman-Bucy-Filter und Ensemble-Kalman-Bucy-Filter. Ersteres schneidet besser ab als letzteres. KW - state estimation KW - filtering KW - parameter estimation KW - Zustandsschätzung KW - Filterung KW - Parameter Schätzung Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436542 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 - 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 - Schulze, Bert-Wolfgang T1 - The structure of operators on manifolds with polyhedral singularities N2 - We discuss intuitive ideas and historical background of concepts in the analysis on configurations with singularities, here in connection with our iterative approach for higher singularities. T3 - Preprint - (2006) 05 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30099 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The Riemann-Roch theorem for manifolds with conical singularities N2 - The classical Riemann-Roch theorem is extended to solutions of elliptic equations on manifolds with conical points. T3 - Preprint - (1997) 18 KW - manifolds with singularities KW - elliptic operators KW - divisors Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25051 ER - TY - INPR A1 - Schrohe, Elmar A1 - Seiler, Jörg T1 - The resolvent of closed extensions of cone differential operators N2 - We study an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted Lp-space. Under suitable conditions we show that the resolvent (λ - A )-¹ exists in a sector of the complex plane and decays like 1/|λ| as |λ| -> ∞. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A. As an application we treat the Laplace-Beltrami operator for a metric with striaght conical degeneracy and establish maximal regularity for the Cauchy problem u - Δu = f, u(0) = 0. T3 - Preprint - (2002) 19 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26378 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 - Martin, C.-I. A1 - Schulze, Bert-Wolfgang T1 - The quantisation of edge symbols N2 - We investigate operators on manifolds with edges from the point of view of the symbolic calculus induced by the singularities. We discuss new aspects of the quantisation of edge-degenerate symbols which lead to continuous operators in weighted edge spaces. T3 - Preprint - (2005) 19 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29959 ER - TY - JOUR A1 - Jursenas, Rytis T1 - The peak model for finite rank supersingular perturbations 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-opus4-472090 IS - 6 SP - 117 EP - 126 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Mera, Azal A1 - Tarkhanov, Nikolai Nikolaevich T1 - The Neumann problem after Spencer N2 - When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 6 KW - elliptic complex KW - manifold with boundary KW - Hodge theory KW - Neumann problem Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-90631 SN - 2193-6943 VL - 5 IS - 6 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - THES A1 - Mera, Azal Jaafar Musa T1 - The Navier-Stokes equations for elliptic quasicomplexes T1 - Die Navier–Stokes–Gleichungen für elliptische Quasikomplexe N2 - The classical Navier-Stokes equations of hydrodynamics are usually written in terms of vector analysis. More promising is the formulation of these equations in the language of differential forms of degree one. In this way the study of Navier-Stokes equations includes the analysis of the de Rham complex. In particular, the Hodge theory for the de Rham complex enables one to eliminate the pressure from the equations. The Navier-Stokes equations constitute a parabolic system with a nonlinear term which makes sense only for one-forms. A simpler model of dynamics of incompressible viscous fluid is given by Burgers' equation. This work is aimed at the study of invariant structure of the Navier-Stokes equations which is closely related to the algebraic structure of the de Rham complex at step 1. To this end we introduce Navier-Stokes equations related to any elliptic quasicomplex of first order differential operators. These equations are quite similar to the classical Navier-Stokes equations including generalised velocity and pressure vectors. Elimination of the pressure from the generalised Navier-Stokes equations gives a good motivation for the study of the Neumann problem after Spencer for elliptic quasicomplexes. Such a study is also included in the work.We start this work by discussion of Lamé equations within the context of elliptic quasicomplexes on compact manifolds with boundary. The non-stationary Lamé equations form a hyperbolic system. However, the study of the first mixed problem for them gives a good experience to attack the linearised Navier-Stokes equations. On this base we describe a class of non-linear perturbations of the Navier-Stokes equations, for which the solvability results still hold. N2 - Die klassischen Navier–Stokes–Differentialgleichungen der Hydrodynamik werden in der Regel im Rahmen der Vektoranalysis formuliert. Mehr versprechend ist die Formulierung dieser Gleichungen in Termen von Differentialformen vom Grad 1. Auf diesem Weg beinhaltet die Untersuchung der Navier–Stokes–Gleichungen die Analyse des de Rhamschen Komplexes. Insbesondere ermöglicht die Hodge–Theorie für den de Rham–Komplex den Druck aus den Gleichungen zu eliminieren. Die Navier–Stokes–Gleichungen bilden ein parabolisches System mit einem nichtlinearen Term, welcher Sinn nur für die Pfaffschen Formen (d.h Formen vom Grad 1) hat. Ein einfacheres Modell für Dynamik der inkompressiblen viskosen Flüssigkeit wird von der Burgers–Gleichungen gegeben. Diese Arbeit richtet sich an das Studium der invarianten Struktur der Navier–Stokes–Gleichungen, die eng mit der algebraischen Struktur des de Rham–Komplexes im schritt 1 zusammen steht. Zu diesem Zweck stellen wir vor die Navier–Stokes–Gleichungen im Zusammenhang mit jedem elliptischen Quasikomplex von Differentialoperatoren der ersten Ordnung. So ähneln die Gleichungen den klassischen Navier–Stokes–Gleichungen, einschließlich allgemeiner Geschwindigkeit– und Druckvektoren. Elimination des Drucks aus den verallgemeinerten Navier–Stokes–Gleichungen gibt eine gute Motivation für die Untersuchung des Neumann–Problems nach Spencer für elliptische Quasikomplexe. Eine solche Untersuchung ist auch in der Arbeit mit der Erörterung der Lamé-Gleichungen im Kontext der elliptischen Quasikomplexe auf kompakten Mannigfaltigkeiten mit Rand. Die nichtstationären Lamé-Gleichungen bilden ein hyperbolisches System. Allerdings gibt die Studie des ersten gemischten Problems für sie eine gute Erfahrung, um die linearisierten Navier–Stokes–Gleichungen anzugreifen. Auf dieser Basis beschreiben wir eine Klasse von nichtlinearen Störungen der Navier–Stokes–Gleichungen, für welche die Lösungsresultate noch gelten. KW - Navier-Stokes-Gleichungen KW - elliptische Quasi-Komplexe KW - Navier-Stoks equations KW - elliptic quasicomplexes Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-398495 ER - TY - INPR A1 - Tepoyan, Liparit T1 - The mixed problem for a degenerate operator equation N2 - We consider a mixed problem for a degenerate differentialoperator equation of higher order. We establish some embedding theorems in weighted Sobolev spaces and show existence and uniqueness of the generalized solution of this problem. We also give a description of the spectrum for the corresponding operator. T3 - Preprint - (2008) 06 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30334 ER - TY - THES A1 - Lopez Valencia, Diego Andres T1 - The Milnor-Moore and Poincaré-Birkhoff-Witt theorems in the locality set up and the polar structure of Shintani zeta functions T1 - Die Milnor-Moore und Poincaré-Birkhoff-Witt Theoreme in der Lokalität und die polare Struktur der Shintani-Zeta-Abbildungen N2 - This thesis bridges two areas of mathematics, algebra on the one hand with the Milnor-Moore theorem (also called Cartier-Quillen-Milnor-Moore theorem) as well as the Poincaré-Birkhoff-Witt theorem, and analysis on the other hand with Shintani zeta functions which generalise multiple zeta functions. The first part is devoted to an algebraic formulation of the locality principle in physics and generalisations of classification theorems such as Milnor-Moore and Poincaré-Birkhoff-Witt theorems to the locality framework. The locality principle roughly says that events that take place far apart in spacetime do not infuence each other. The algebraic formulation of this principle discussed here is useful when analysing singularities which arise from events located far apart in space, in order to renormalise them while keeping a memory of the fact that they do not influence each other. We start by endowing a vector space with a symmetric relation, named the locality relation, which keeps track of elements that are "locally independent". The pair of a vector space together with such relation is called a pre-locality vector space. This concept is extended to tensor products allowing only tensors made of locally independent elements. We extend this concept to the locality tensor algebra, and locality symmetric algebra of a pre-locality vector space and prove the universal properties of each of such structures. We also introduce the pre-locality Lie algebras, together with their associated locality universal enveloping algebras and prove their universal property. We later upgrade all such structures and results from the pre-locality to the locality context, requiring the locality relation to be compatible with the linear structure of the vector space. This allows us to define locality coalgebras, locality bialgebras, and locality Hopf algebras. Finally, all the previous results are used to prove the locality version of the Milnor-Moore and the Poincaré-Birkhoff-Witt theorems. It is worth noticing that the proofs presented, not only generalise the results in the usual (non-locality) setup, but also often use less tools than their counterparts in their non-locality counterparts. The second part is devoted to study the polar structure of the Shintani zeta functions. Such functions, which generalise the Riemman zeta function, multiple zeta functions, Mordell-Tornheim zeta functions, among others, are parametrised by matrices with real non-negative arguments. It is known that Shintani zeta functions extend to meromorphic functions with poles on afine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coeficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound. N2 - Diese Arbeit schlägt eine Brücke zwischen zwei Bereichen der Mathematik, einerseits der Algebra mit dem Milnor-Moore-Theorem (auch Cartier-Quillen-Milnor-Moore-Theorem genannt) sowie dem Poincaré-Birkhoff-Witt-Theorem und andererseits der Analysis mit den Shintani-Zetafunktionen, die eine Verallgemeinerung der Mehrfach-Zetafunktionen darstellen. Der erste Teil ist einer algebraischen Formulierung des Lokalitätsprinzips in der Physik und Verallgemeinerungen von Klassifikationstheoremen wie dem Milnor-Moore- und dem Poincaré-Birkhoff-Witt-Theorem auf den Lokalitätsrahmen gewidmet. Das Lokalitätsprinzip besagt grob, dass Ereignisse, die in der Raumzeit weit voneinander entfernt stattfinden, sich nicht gegenseitig beeinflussen. Die hier erörterte algebraische Formulierung dieses Prinzips ist nützlich bei der Analyse von Singularitäten, die aus weit voneinander entfernten Ereignissen im Raum entstehen, um sie zu renormalisieren und dabei die Tatsache im Gedächtnis zu behalten, dass sie sich nicht gegenseitig beeinflussen. Wir beginnen damit, dass wir einen Vektorraum mit einer symmetrischen Relation, der so genannten Lokalitätsrelation, ausstatten, die die "lokal unabhängigen" Elemente festhält. Das Paar aus einem Vektorraum und einer solchen Relation wird als Vorlokalitäts-Vektorraum bezeichnet. Dieses Konzept wird auf Tensorprodukte erweitert, die nur Tensoren aus lokal unabhängigen Elementen zulassen. Wir erweitern dieses Konzept auf die Lokalitäts-Tensor-Algebra und die symmetrische Lokalitäts-Algebra eines Vorlokalitäts-Vektorraums und beweisen die universellen Eigenschaften jeder dieser Strukturen. Wir führen auch die Vorlokalitäts-Lie-Algebren zusammen mit den zugehörigen universellen Hüllalgebren der Lokalität ein und beweisen ihre universelle Eigenschaft. Später übertragen wir alle diese Strukturen und Ergebnisse aus dem Kontext der Vorlokalität in den Kontext der Lokalität, wobei die Lokalitätsbeziehung mit der linearen Struktur des Vektorraums kompatibel sein muss. Auf diese Weise können wir Lokalitäts-Kohlengebren, Lokalitäts-Bialgebren und Lokalitäts-Hopf-Algebren definieren. Schließlich werden alle vorherigen Ergebnisse verwendet, um die Lokalitätsversionen des Milnor-Moore- und des Poincaré-Birkhoff-Witt-Theorems zu beweisen. Es ist erwähnenswert, dass die vorgestellten Beweise nicht nur die Ergebnisse im üblichen (Nichtlokalitäts-) Aufbau verallgemeinern, sondern auch oft weniger Hilfsmittel verwenden als ihre Gegenstücke in ihren Nichtlokalitäts-Gegenstücken. Der zweite Teil ist der Untersuchung der polaren Struktur der Shintani-Zeta-Funktionen gewidmet. Diese Funktionen, die u.a. die Riemman-Zetafunktion, die multiplen Zetafunktionen und die Mordell-Tornheim-Zetafunktionen verallgemeinern, werden durch Matrizen mit reellen, nicht-negativen Argumenten parametrisiert. Es ist bekannt, dass Shintani-Zetafunktionen sich zu meromorphen Funktionen mit Polen auf affinen Hyperebenen erweitern. Wir verfeinern dieses Ergebnis, indem wir zeigen, dass die Pole auf Hyperebenen liegen, die parallel zu den Facetten bestimmter konvexer Polyeder verlaufen, die mit der Definitionsmatrix für die Shintani-Zeta-Funktion assoziiert sind. Letztere sind explizit die Newton-Polytope der Polynome, die durch die Spalten der zugrunde liegenden Matrix induziert werden. Wir beweisen dann, dass die Koeffizienten der Gleichung, die die Hyperebenen in der kanonischen Basis beschreibt, entweder Null oder Eins sind, ähnlich wie die Pole, die bei der Renormierung generischer Feynman-Amplituden entstehen. Zu diesem Zweck führen wir einen Algorithmus ein, um die Gewichte über einen Graphen so zu verteilen, dass das Gewicht an jedem Knoten eine gegebene untere Schranke erfüllt. KW - locality principle KW - multizeta functions KW - meromorphic continuation KW - Milnor Moore theorem KW - Poincaré Birkhoff Witt theorem KW - Newton polytopes KW - Satz von Milnor Moore KW - Newton Polytope KW - Satz von Poincaré Birkhoff Witt KW - Lokalitätsprinzip KW - meromorphe Fortsetzung KW - Multizeta-Abbildungen Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-594213 ER - TY - INPR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - The method of Fischer-Riesz equations for elliptic boundary value problems N2 - We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)24 KW - Boundary value problems for first order systems KW - Green formula KW - Fischer-Riesz equations KW - regularisation Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-61792 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich A1 - Wallenta, Daniel T1 - The Lefschetz number of sequences of trace class curvature N2 - For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 3 KW - Perturbed complexes KW - curvature KW - Lefschetz number Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56969 ER - TY - INPR A1 - Schulze, Bert-Wolfgang T1 - The iterative structure of corner operators N2 - We give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference “Elliptic and Hyperbolic Equations on Singular Spaces”, October 27 - 31, 2008, at the MSRI, University of Berkeley. T3 - Preprint - (2008) 08 KW - Categories of stratified spaces KW - ellipticity of corners operators KW - principal symbolic hierarchies KW - boundary value problems Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30353 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Nazaikinskii, Vladimir A1 - Sternin, Boris T1 - The index of quantized contact transformations on manifolds with conical singularities N2 - The quantization of contact transformations of the cosphere bundle over a manifold with conical singularities is described. The index of Fredholm operators given by this quantization is calculated. The answer is given in terms of the Epstein-Melrose contact degree and the conormal symbol of the corresponding operator. T3 - Preprint - (1998) 16 KW - manifolds with conical singularities KW - contact transformations KW - quantization KW - ellipticity KW - Fredholm operators KW - regularizers KW - index formulas Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25276 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 - 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 - 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 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Savin, Anton T1 - The homotopy classification and the index of boundary value problems for general elliptic operators N2 - We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition. T3 - Preprint - (1999) 20 KW - elliptic boundary value problems KW - Atiyah-Bott condition KW - index theory KW - K-theory KW - homotopy classification Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25568 ER - TY - INPR A1 - Pfäffle, Frank A1 - Stephan, Christoph A. T1 - The Holst action by the spectral action principle N2 - We investigate the Holst action for closed Riemannian 4-manifolds with orthogonal connections. For connections whose torsion has zero Cartan type component we show that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on left-handed spinor fields. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)19 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60032 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 - 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 - 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 - 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 - Makhmudov, Olimdjan A1 - Tarkhanov, Nikolai Nikolaevich T1 - The first mixed problem for the nonstationary Lamé system N2 - We find an adequate interpretation of the Lamé operator within the framework of elliptic complexes and study the first mixed problem for the nonstationary Lamé system. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)10 KW - Lamé system KW - evolution equation KW - first boundary value problem Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71923 SN - 2193-6943 VL - 3 IS - 10 PB - Universitätsverlag Potsdam CY - Potsdam 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 - THES A1 - Branding, Volker T1 - The evolution equations for Dirac-harmonic Maps T1 - Die Evolutionsgleichungen für Dirac-harmonische Abbildungen N2 - This thesis investigates the gradient flow of Dirac-harmonic maps. Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points of this energy functional couple the equation for harmonic maps with spinor fields. At present, many analytical properties of Dirac-harmonic maps are known, but a general existence result is still missing. In this thesis the existence question is studied using the evolution equations for a regularized version of Dirac-harmonic maps. Since the energy functional for Dirac-harmonic maps is unbounded from below the method of the gradient flow cannot be applied directly. Thus, we first of all consider a regularization prescription for Dirac-harmonic maps and then study the gradient flow. Chapter 1 gives some background material on harmonic maps/harmonic spinors and summarizes the current known results about Dirac-harmonic maps. Chapter 2 introduces the notion of Dirac-harmonic maps in detail and presents a regularization prescription for Dirac-harmonic maps. In Chapter 3 the evolution equations for regularized Dirac-harmonic maps are introduced. In addition, the evolution of certain energies is discussed. Moreover, the existence of a short-time solution to the evolution equations is established. Chapter 4 analyzes the evolution equations in the case that the domain manifold is a closed curve. Here, the existence of a smooth long-time solution is proven. Moreover, for the regularization being large enough, it is shown that the evolution equations converge to a regularized Dirac-harmonic map. Finally, it is discussed in which sense the regularization can be removed. In Chapter 5 the evolution equations are studied when the domain manifold is a closed Riemmannian spin surface. For the regularization being large enough, the existence of a global weak solution, which is smooth away from finitely many singularities is proven. It is shown that the evolution equations converge weakly to a regularized Dirac-harmonic map. In addition, it is discussed if the regularization can be removed in this case. N2 - Die vorliegende Dissertation untersucht den Gradientenfluss von Dirac-harmonischen Abbildungen. Dirac-harmonische Abbildungen sind kritische Punkte eines Energiefunktionals, welches aus supersymmetrischen Feldtheorien motiviert ist. Die kritischen Punkte dieses Energiefunktionals koppeln die Gleichung für harmonische Abbildungen mit Spinorfeldern. Viele analytische Eigenschaften von Dirac-harmonischen Abbildungen sind bereits bekannt, ein allgemeines Existenzresultat wurde aber noch nicht erzielt. Diese Dissertation untersucht das Existenzproblem, indem der Gradientenfluss von einer regularisierten Version Dirac-harmonischer Abbildungen untersucht wird. Die Methode des Gradientenflusses kann nicht direkt angewendet werden, da das Energiefunktional für Dirac-harmonische Abbildungen nach unten unbeschränkt ist. Daher wird zunächst eine Regularisierungsvorschrift für Dirac-harmonische Abbildungen eingeführt und dann der Gradientenfluss betrachtet. Kapitel 1 stellt für die Arbeit wichtige Resultate über harmonische Abbildungen/harmonische Spinoren zusammen. Außerdem werden die zur Zeit bekannten Resultate über Dirac-harmonische Abbildungen zusammengefasst. In Kapitel 2 werden Dirac-harmonische Abbildungen im Detail eingeführt, außerdem wird eine Regularisierungsvorschrift präsentiert. Kapitel 3 führt die Evolutionsgleichungen für regularisierte Dirac-harmonische Abbildungen ein. Zusätzlich wird die Evolution von verschiedenen Energien diskutiert. Schließlich wird die Existenz einer Kurzzeitlösung bewiesen. In Kapitel 4 werden die Evolutionsgleichungen für den Fall analysiert, dass die Ursprungsmannigfaltigkeit eine geschlossene Kurve ist. Die Existenz einer Langzeitlösung der Evolutionsgleichungen wird bewiesen. Es wird außerdem gezeigt, dass die Evolutionsgleichungen konvergieren, falls die Regularisierung groß genug gewählt wurde. Schließlich wird diskutiert, ob die Regularisierung wieder entfernt werden kann. Kapitel 5 schlussendlich untersucht die Evolutionsgleichungen für den Fall, dass die Ursprungsmannigfaltigkeit eine geschlossene Riemannsche Spin Fläche ist. Es wird die Existenz einer global schwachen Lösung bewiesen, welche bis auf endlich viele Singularitäten glatt ist. Die Lösung konvergiert im schwachen Sinne gegen eine regularisierte Dirac-harmonische Abbildung. Auch hier wird schließlich untersucht, ob die Regularisierung wieder entfernt werden kann. KW - Dirac-harmonische Abbildungen KW - Gradientenfluss KW - Wärmefluss KW - Spin Geometrie KW - nichtlineare partielle Differentialgleichung KW - Dirac-harmonic maps KW - Gradient flow KW - Heat Flow KW - Spin Geometry KW - nonlinear partial differential equations Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64204 ER -