TY - JOUR A1 - Mera, Azal A1 - Shlapunov, Alexander A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Navier-Stokes Equations for Elliptic Complexes JF - Journal of Siberian Federal University. Mathematics & Physics N2 - We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations. KW - Navier-Stokes equations KW - classical solution Y1 - 2019 U6 - https://doi.org/10.17516/1997-1397-2019-12-1-3-27 SN - 1997-1397 SN - 2313-6022 VL - 12 IS - 1 SP - 3 EP - 27 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - JOUR A1 - Nüsken, Nikolas A1 - Pavhotis, Grigorios A. T1 - Constructing Sampling Schemes via Coupling BT - Markov Semigroups and Optimal Transport JF - SIAM ASA journal on uncertainty quantification / Society for Industrial and Applied Mathematics ; American Statistical Association N2 - In this paper we develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincare inequality. KW - sampling KW - optimal transport KW - particle methods KW - Markov semigroups KW - MCMC Y1 - 2019 U6 - https://doi.org/10.1137/18M119896X SN - 2166-2525 VL - 7 IS - 1 SP - 324 EP - 382 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Katz-Samuels, Julian A1 - Blanchard, Gilles A1 - Scott, Clayton T1 - Decontamination of Mutual Contamination Models JF - Journal of machine learning research N2 - Many machine learning problems can be characterized by mutual contamination models. In these problems, one observes several random samples from different convex combinations of a set of unknown base distributions and the goal is to infer these base distributions. This paper considers the general setting where the base distributions are defined on arbitrary probability spaces. We examine three popular machine learning problems that arise in this general setting: multiclass classification with label noise, demixing of mixed membership models, and classification with partial labels. In each case, we give sufficient conditions for identifiability and present algorithms for the infinite and finite sample settings, with associated performance guarantees. KW - multiclass classification with label noise KW - classification with partial labels KW - mixed membership models KW - topic modeling KW - mutual contamination models Y1 - 2019 UR - http://arxiv.org/abs/1710.01167 SN - 1532-4435 VL - 20 PB - Microtome Publishing CY - Cambridge, Mass. ER - TY - JOUR A1 - Malass, Ihsane A1 - Tarkhanov, Nikolai Nikolaevich T1 - The de Rham Cohomology through Hilbert Space Methods JF - Journal of Siberian Federal University. Mathematics & physics N2 - We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer. KW - De Rham complex KW - cohomology KW - Hodge theory KW - Neumann problem Y1 - 2019 U6 - https://doi.org/10.17516/1997-1397-2019-12-4-455-465 SN - 1997-1397 SN - 2313-6022 VL - 12 IS - 4 SP - 455 EP - 465 PB - Sibirskij Federalʹnyj Universitet CY - Krasnoyarsk ER - TY - JOUR A1 - Dombrowsky, Charlotte A1 - Und, Myriam Fradon A1 - Roelly, Sylvie T1 - Packungen aus Kreisscheiben BT - Wie eine wahrscheinlichkeitstheoretische Sichtweise eine geometrische Analyse vervollständigen kann JF - Elemente der Mathematik N2 - Der englische Seefahrer Sir Walter Raleigh fragte sich einst, wie er in seinem Schiffsladeraum moeglichst viele Kanonenkugeln stapeln koennte. Johannes Kepler entwickelte daraufhin 1611 eine Vermutung ueber die optimale Anordnung der Kugeln. Diese Vermutung sollte sich als eine der haertesten mathematischen Nuesse der Geschichte erweisen. Selbst in der Ebene sind dichteste Packungen kongruenter Kreise eine Herausforderung. 1892 und 1910 veroeffentlichte Axel Thue (kritisierte) Beweise, dass die hexagonale Kreispackung optimal sei. Erst 1940 lieferte Laszlo Fejes Toth schliesslich einen wasserdichten Beweis fuer diese Tatsache. Eine Variante des Problems verlangt, Packungen mit endlich vielen kongruenten Kugeln zu finden, die eine gewisse quadratische Energie minimieren: Diese spannende geometrische Aufgabe wurde 1967 von Toth gestellt. Sie ist auch heute noch nicht vollstaendig gelaest. In diesem Beitrag schlagen die Autorinnen eine originelle wahrscheinlichkeitstheoretische Methode vor, um in der Ebene Näherungen der Lösung zu konstruieren. Y1 - 2019 U6 - https://doi.org/10.4171/EM/381 SN - 0013-6018 SN - 1420-8962 VL - 74 IS - 2 SP - 45 EP - 62 PB - EMS Publ. CY - Zürich ER - TY - JOUR A1 - Klein, Markus A1 - Rosenberger, Elke T1 - Tunneling for a class of difference operators BT - Complete asymptotics JF - Annales Henri Poincaré : a journal of theoretical and mathematical physics N2 - We analyze a general class of difference operators Hε=Tε+Vε on ℓ2((εZ)d), where Vε is a multi-well potential and ε is a small parameter. We derive full asymptotic expansions of the prefactor of the exponentially small eigenvalue splitting due to interactions between two “wells” (minima) of the potential energy, i.e., for the discrete tunneling effect. We treat both the case where there is a single minimal geodesic (with respect to the natural Finsler metric induced by the leading symbol h0(x,ξ) of Hε) connecting the two minima and the case where the minimal geodesics form an ℓ+1 dimensional manifold, ℓ≥1. These results on the tunneling problem are as sharp as the classical results for the Schrödinger operator in Helffer and Sjöstrand (Commun PDE 9:337–408, 1984). Technically, our approach is pseudo-differential and we adapt techniques from Helffer and Sjöstrand [Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique), Mémoires de la S.M.F., 2 series, tome 34, pp 1–113, 1988)] and Helffer and Parisse (Ann Inst Henri Poincaré 60(2):147–187, 1994) to our discrete setting. Y1 - 2018 U6 - https://doi.org/10.1007/s00023-018-0732-0 SN - 1424-0637 SN - 1424-0661 VL - 19 IS - 11 SP - 3511 EP - 3559 PB - Springer International Publishing CY - Cham ER - TY - JOUR A1 - Paycha, Sylvie T1 - When the market wins over research and higher education JF - Sustainable Futures for Higher Education : the Making of Knowledge Makers N2 - In this chapter, an overview of systematic eradication of basic science foci in European universities in the last two decades is given. This happens under the slogan of optimisation of the university education to the needs and demands of the society. It is pointed out that reliance on “market demands” brings with it long-term deficiencies in the maintenance of basic and advanced knowledge construction in societies necessary for long-term future technological advances. University policies that claim improvement of higher education towards more immediate efficiency may end up with the opposite effect of affecting its quality and long term expected positive impact on society. Y1 - 2018 SN - 978-3-319-96035-7 SN - 978-3-319-96034-0 U6 - https://doi.org/10.1007/978-3-319-96035-7_2 SN - 2364-6799 VL - 7 SP - 23 EP - 28 PB - Springer CY - Cham ER - TY - JOUR A1 - Bachoc, Francois A1 - Blanchard, Gilles A1 - Neuvial, Pierre T1 - On the post selection inference constant under restricted isometry properties JF - Electronic journal of statistics N2 - Uniformly valid confidence intervals post model selection in regression can be constructed based on Post-Selection Inference (PoSI) constants. PoSI constants are minimal for orthogonal design matrices, and can be upper bounded in function of the sparsity of the set of models under consideration, for generic design matrices. In order to improve on these generic sparse upper bounds, we consider design matrices satisfying a Restricted Isometry Property (RIP) condition. We provide a new upper bound on the PoSI constant in this setting. This upper bound is an explicit function of the RIP constant of the design matrix, thereby giving an interpolation between the orthogonal setting and the generic sparse setting. We show that this upper bound is asymptotically optimal in many settings by constructing a matching lower bound. KW - Inference post model-selection KW - confidence intervals KW - PoSI constants KW - linear regression KW - high-dimensional inference KW - sparsity KW - restricted isometry property Y1 - 2018 U6 - https://doi.org/10.1214/18-EJS1490 SN - 1935-7524 VL - 12 IS - 2 SP - 3736 EP - 3757 PB - Institute of Mathematical Statistics CY - Cleveland ER - TY - JOUR A1 - Conforti, Giovanni A1 - Pra, Paolo Dai A1 - Roelly, Sylvie T1 - Reciprocal Class of Jump Processes JF - Journal of theoretical probability N2 - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set . We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state. KW - Reciprocal processes KW - Stochastic bridges KW - Jump processes KW - Compound Poisson processes Y1 - 2015 U6 - https://doi.org/10.1007/s10959-015-0655-3 SN - 0894-9840 SN - 1572-9230 VL - 30 SP - 551 EP - 580 PB - Springer CY - New York ER - TY - JOUR A1 - Nüsken, Nikolas A1 - Reich, Sebastian A1 - Rozdeba, Paul J. T1 - State and parameter estimation from observed signal increments JF - Entropy : an international and interdisciplinary journal of entropy and information studies N2 - The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems. KW - parameter estimation KW - continuous-time data assimilation KW - ensemble Kalman filter KW - correlated noise KW - multi-scale diffusion processes Y1 - 2019 U6 - https://doi.org/10.3390/e21050505 SN - 1099-4300 VL - 21 IS - 5 PB - MDPI CY - Basel ER - TY - JOUR A1 - Mera, Azal Jaafar Musa A1 - Tarchanov, Nikolaj Nikolaevič T1 - The Neumann Problem after Spencer JF - Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University : Matematika i fizika = Mathematics & physics 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. KW - elliptic complexes KW - manifolds with boundary KW - Hodge theory KW - Neumann problem Y1 - 2017 U6 - https://doi.org/10.17516/1997-1397-2017-10-4-474-493 SN - 1997-1397 SN - 2313-6022 VL - 10 SP - 474 EP - 493 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - JOUR A1 - Rœlly, Sylvie A1 - Zass, Alexander T1 - Marked Gibbs point processes with unbounded interaction BT - An existence result JF - Journal of statistical physics N2 - We construct marked Gibbs point processes in R-d under quite general assumptions. Firstly, we allow for interaction functionals that may be unbounded and whose range is not assumed to be uniformly bounded. Indeed, our typical interaction admits an a.s. finite but random range. Secondly, the random marks-attached to the locations in R-d-belong to a general normed space G. They are not bounded, but their law should admit a super-exponential moment. The approach used here relies on the so-called entropy method and large-deviation tools in order to prove tightness of a family of finite-volume Gibbs point processes. An application to infinite-dimensional interacting diffusions is also presented. KW - Marked Gibbs process KW - Infinite-dimensional interacting diffusion KW - Specific entropy KW - DLR equation Y1 - 2020 U6 - https://doi.org/10.1007/s10955-020-02559-3 SN - 0022-4715 SN - 1572-9613 VL - 179 IS - 4 SP - 972 EP - 996 PB - Springer CY - New York ER - TY - JOUR A1 - Sanchez, Sabrina A1 - Wicht, Johannes A1 - Bärenzung, Julien T1 - Predictions of the geomagnetic secular variation based on the ensemble sequential assimilation of geomagnetic field models by dynamo simulations JF - Earth, planets and space N2 - The IGRF offers an important incentive for testing algorithms predicting the Earth's magnetic field changes, known as secular variation (SV), in a 5-year range. Here, we present a SV candidate model for the 13th IGRF that stems from a sequential ensemble data assimilation approach (EnKF). The ensemble consists of a number of parallel-running 3D-dynamo simulations. The assimilated data are geomagnetic field snapshots covering the years 1840 to 2000 from the COV-OBS.x1 model and for 2001 to 2020 from the Kalmag model. A spectral covariance localization method, considering the couplings between spherical harmonics of the same equatorial symmetry and same azimuthal wave number, allows decreasing the ensemble size to about a 100 while maintaining the stability of the assimilation. The quality of 5-year predictions is tested for the past two decades. These tests show that the assimilation scheme is able to reconstruct the overall SV evolution. They also suggest that a better 5-year forecast is obtained keeping the SV constant compared to the dynamically evolving SV. However, the quality of the dynamical forecast steadily improves over the full assimilation window (180 years). We therefore propose the instantaneous SV estimate for 2020 from our assimilation as a candidate model for the IGRF-13. The ensemble approach provides uncertainty estimates, which closely match the residual differences with respect to the IGRF-13. Longer term predictions for the evolution of the main magnetic field features over a 50-year range are also presented. We observe the further decrease of the axial dipole at a mean rate of 8 nT/year as well as a deepening and broadening of the South Atlantic Anomaly. The magnetic dip poles are seen to approach an eccentric dipole configuration. KW - Earth's magnetic field KW - Geomagnetic secular variation KW - Dynamo KW - simulations KW - Data assimilation Y1 - 2020 U6 - https://doi.org/10.1186/s40623-020-01279-y SN - 1880-5981 VL - 72 IS - 1 PB - Springer CY - New York ER - TY - JOUR A1 - Garbuno-Inigo, Alfredo A1 - Nüsken, Nikolas A1 - Reich, Sebastian T1 - Affine invariant interacting Langevin dynamics for Bayesian inference JF - SIAM journal on applied dynamical systems N2 - We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem. KW - Langevin dynamics KW - interacting particle systems KW - Bayesian inference KW - gradient flow KW - multiplicative noise KW - affine invariance KW - gradient-free Y1 - 2020 U6 - https://doi.org/10.1137/19M1304891 SN - 1536-0040 VL - 19 IS - 3 SP - 1633 EP - 1658 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Froyland, Gary A1 - Koltai, Peter A1 - Stahn, Martin T1 - Computation and optimal perturbation of finite-time coherent sets for aperiodic flows without trajectory integration JF - SIAM journal on applied dynamical systems N2 - Understanding the macroscopic behavior of dynamical systems is an important tool to unravel transport mechanisms in complex flows. A decomposition of the state space into coherent sets is a popular way to reveal this essential macroscopic evolution. To compute coherent sets from an aperiodic time-dependent dynamical system we consider the relevant transfer operators and their infinitesimal generators on an augmented space-time manifold. This space-time generator approach avoids trajectory integration and creates a convenient linearization of the aperiodic evolution. This linearization can be further exploited to create a simple and effective spectral optimization methodology for diminishing or enhancing coherence. We obtain explicit solutions for these optimization problems using Lagrange multipliers and illustrate this technique by increasing and decreasing mixing of spatial regions through small velocity field perturbations. KW - coherent set KW - mixing KW - transfer operator KW - infinitesimal generator KW - unsteady flow KW - mixing optimization Y1 - 2020 U6 - https://doi.org/10.1137/19M1261791 SN - 1536-0040 VL - 19 IS - 3 SP - 1659 EP - 1700 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Vasiliev, Sergey B. A1 - Tarchanov, Nikolaj Nikolaevič T1 - Construction of series of perfect lattices by layer superposition JF - Journal of Siberian Federal University : Mathematics & physics JF - Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University : Serija Matematika i fizika = Mathematics & physics N2 - We construct a new series of perfect lattices in n dimensions by the layer superposition method of Delaunay-Barnes. KW - lattice packing and covering KW - polyhedra and polytopes KW - regular figures KW - division of spaces Y1 - 2017 U6 - https://doi.org/10.17516/1997-1397-2017-10-3-353-361 SN - 1997-1397 SN - 2313-6022 VL - 10 IS - 3 SP - 353 EP - 361 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - GEN A1 - Bandara, Menaka Lashitha A1 - Rosén, Andreas T1 - Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions T2 - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe N2 - On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator in depends Riesz continuously on perturbations of local boundary conditions The Lipschitz bound for the map depends on Lipschitz smoothness and ellipticity of and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 758 KW - boundary value problems KW - Dirac operator KW - functional calculus KW - real-variable harmonic analysis KW - Riesz continuity KW - spectral flow Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-434078 SN - 1866-8372 IS - 758 SP - 1253 EP - 1284 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 - GEN A1 - Molahajloo, Shahla A1 - Pirhayati, Mohammad T1 - Traces of pseudo-differential operators on compact and Hausdorff groups T2 - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe N2 - We give a characterization of and a trace formula for trace class pseudo-differential operators on compact Hausdorff groups. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 893 KW - pseudo-differential operators KW - compact groups Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436371 SN - 1866-8372 IS - 893 SP - 361 EP - 369 ER - TY - JOUR A1 - Dereudre, David A1 - Mazzonetto, Sara A1 - Roelly, Sylvie T1 - Exact simulation of Brownian diffusions with drift admitting jumps JF - SIAM journal on scientific computing N2 - In this paper, using an algorithm based on the retrospective rejection sampling scheme introduced in [A. Beskos, O. Papaspiliopoulos, and G. O. Roberts,Methodol. Comput. Appl. Probab., 10 (2008), pp. 85-104] and [P. Etore and M. Martinez, ESAIM Probab.Stat., 18 (2014), pp. 686-702], we propose an exact simulation of a Brownian di ff usion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical di ffi culty due to the presence of t w o jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift. KW - exact simulation methods KW - skew Brownian motion KW - skew diffusions KW - Brownian motion with discontinuous drift Y1 - 2017 U6 - https://doi.org/10.1137/16M107699X SN - 1064-8275 SN - 1095-7197 VL - 39 IS - 3 SP - A711 EP - A740 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - GEN A1 - Devchand, Chandrashekar A1 - Nuyts, Jean A1 - Weingart, Gregor T1 - Matryoshka of special democratic forms T2 - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe N2 - Special p-forms are forms which have components fµ1…µp equal to +1, -1 or 0 in some orthonormal basis. A p-form ϕ ∈ � pRd is called democratic if the set of nonzero components {ϕμ1...μp} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1, . . . , d}. Knowledge of these symmetry groups allows us to define mappings of special democratic p-forms in d dimensions to special democratic P-forms in D dimensions for successively higher P = p and D = d. In particular, we display a remarkable nested structure of special forms including a U(3)-invariant 2-form in six dimensions, a G2-invariant 3-form in seven dimensions, a Spin(7)-invariant 4-form in eight dimensions and a special democratic 6-form O in ten dimensions. The latter has the remarkable property that its contraction with one of five distinct bivectors, yields, in the orthogonal eight dimensions, the Spin(7)-invariant 4-form. We discuss various properties of this ten dimensional form. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 841 KW - commutator subgroup KW - transitive action KW - cycle decomposition KW - democratic form KW - special holonomy Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-429002 SN - 1866-8372 IS - 841 SP - 545 EP - 562 ER - TY - JOUR A1 - Rastogi, Abhishake T1 - Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems JF - Communications on Pure and Applied Analysis N2 - In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces. KW - Statistical inverse problem KW - Tikhonov regularization KW - Hilbert Scales KW - reproducing kernel Hilbert space KW - minimax convergence rates Y1 - 2020 U6 - https://doi.org/10.3934/cpaa.2020183 SN - 1534-0392 SN - 1553-5258 VL - 19 IS - 8 SP - 4111 EP - 4126 PB - American Institute of Mathematical Sciences CY - Springfield ER - TY - JOUR A1 - Chauhan, Ankit A1 - Friedrich, Tobias A1 - Rothenberger, Ralf T1 - Greed is good for deterministic scale-free networks JF - Algorithmica : an international journal in computer science N2 - Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306-1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical NP-hard optimization problems on PLB networks. It is known that on general graphs with maximum degree Delta, a greedy algorithm, which chooses nodes in the order of their degree, only achieves a Omega (ln Delta)-approximation forMinimum Vertex Cover and Minimum Dominating Set, and a Omega(Delta)-approximation forMaximum Independent Set. We prove that the PLB-U property with beta>2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are APX-hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless P = NP. Furthermore, we prove that all three problems are in MAX SNP if the PLB-U property holds. KW - random graphs KW - deterministic properties KW - power-law KW - approximation KW - APX-hardness Y1 - 2020 U6 - https://doi.org/10.1007/s00453-020-00729-z SN - 0178-4617 SN - 1432-0541 VL - 82 IS - 11 SP - 3338 EP - 3389 PB - Springer CY - New York ER - TY - JOUR A1 - Klein, Markus A1 - Rosenberger, Elke T1 - The tunneling effect for Schrödinger operators on a vector bundle JF - Analysis and mathematical physics N2 - In the semiclassical limit (h) over bar -> 0, we analyze a class of self-adjoint Schrodinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V center dot id(E) acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m(1),... m(r) is an element of M, called potential wells. Using quasimodes of WKB-type near m(j) for eigenfunctions associated with the low lying eigenvalues of H-(h) over bar, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension l + 1. This dimension l determines the polynomial prefactor for exponentially small eigenvalue splitting. KW - Laplace-type operator KW - Vector bundle KW - WKB-expansion KW - Quasimodes KW - Tunneling KW - Spectral gap KW - Complete asymptotics Y1 - 2021 U6 - https://doi.org/10.1007/s13324-021-00485-5 SN - 1664-2368 SN - 1664-235X VL - 11 IS - 2 PB - Springer International Publishing AG CY - Cham (ZG) ER - TY - JOUR A1 - Schlosser, Rainer T1 - Heuristic mean-variance optimization in Markov decision processes using state-dependent risk aversion JF - IMA journal of management mathematics / Institute of Mathematics and Its Applications N2 - In dynamic decision problems, it is challenging to find the right balance between maximizing expected rewards and minimizing risks. In this paper, we consider NP-hard mean-variance (MV) optimization problems in Markov decision processes with a finite time horizon. We present a heuristic approach to solve MV problems, which is based on state-dependent risk aversion and efficient dynamic programming techniques. Our approach can also be applied to mean-semivariance (MSV) problems, which particularly focus on the downside risk. We demonstrate the applicability and the effectiveness of our heuristic for dynamic pricing applications. Using reproducible examples, we show that our approach outperforms existing state-of-the-art benchmark models for MV and MSV problems while also providing competitive runtimes. Further, compared to models based on constant risk levels, we find that state-dependent risk aversion allows to more effectively intervene in case sales processes deviate from their planned paths. Our concepts are domain independent, easy to implement and of low computational complexity. KW - risk aversion KW - mean-variance optimization KW - Markov decision process; KW - dynamic programming KW - dynamic pricing KW - heuristics Y1 - 2021 U6 - https://doi.org/10.1093/imaman/dpab009 SN - 1471-678X SN - 1471-6798 VL - 33 IS - 2 SP - 181 EP - 199 PB - Oxford Univ. Press CY - Oxford ER - TY - JOUR A1 - Gräter, Joachim T1 - Free division rings of fractions of crossed products of groups with Conradian left-orders JF - Forum mathematicum N2 - Let D be a division ring of fractions of a crossed product F[G, eta, alpha], where F is a skew field and G is a group with Conradian left-order <=. For D we introduce the notion of freeness with respect to <= and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space F((G)) of all formal power series in G over F with respect to <=. From this we obtain that all division rings of fractions of F[G, eta, alpha] which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, F[G, eta, alpha] possesses a division ring of fraction which is free in this sense if and only if the rational closure of F[G, eta, alpha] in the endomorphism ring of the corresponding right F-vector space F((G)) is a skew field. KW - crossed product KW - group ring KW - ordered group KW - Conradian left-order KW - locally indicable group KW - division ring of fractions KW - Hughes-free KW - formal KW - power series Y1 - 2020 U6 - https://doi.org/10.1515/forum-2019-0264 SN - 0933-7741 SN - 1435-5337 VL - 32 IS - 3 SP - 739 EP - 772 PB - De Gruyter CY - Berlin ER - TY - THES A1 - Busaman, Saofee T1 - Hyperequational theory for partial algebras T1 - Hyperequationale Theorie für partielle Algebren N2 - Our work goes in two directions. At first we want to transfer definitions, concepts and results of the theory of hyperidentities and solid varieties from the total to the partial case. (1) We prove that the operators chi^A_RNF and chi^E_RNF are only monotone and additive and we show that the sets of all fixed points of these operators are characterized only by three instead of four equivalent conditions for the case of closure operators. (2) We prove that V is n − SF-solid iff clone^SF V is free with respect to itself, freely generated by the independent set {[fi(x_1, . . . , x_n)]Id^SF_n V | i \in I}. (3) We prove that if V is n-fluid and ~V |P(V ) =~V −iso |P(V ) then V is kunsolid for k >= n (where P(V ) is the set of all V -proper hypersubstitutions of type \tau ). (4) We prove that a strong M-hyperquasi-equational theory is characterized by four equivalent conditions. The second direction of our work is to follow ideas which are typical for the partial case. (1) We characterize all minimal partial clones which are strongly solidifyable. (2)We define the operator Chi^A_Ph where Ph is a monoid of regular partial hypersubstitutions.Using this concept, we define the concept of a Phyp_R(\tau )-solid strong regular variety of partial algebras and we prove that a PHyp_R(\tau )-solid strong regular variety satisfies four equivalent conditions. KW - partial algebras KW - hyperequational theory Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-12048 ER - TY - JOUR A1 - Schirmer, Philipp A1 - Papenbrock, Thorsten A1 - Koumarelas, Ioannis A1 - Naumann, Felix T1 - Efficient discovery of matching dependencies JF - ACM transactions on database systems : TODS N2 - Matching dependencies (MDs) are data profiling results that are often used for data integration, data cleaning, and entity matching. They are a generalization of functional dependencies (FDs) matching similar rather than same elements. As their discovery is very difficult, existing profiling algorithms find either only small subsets of all MDs or their scope is limited to only small datasets. We focus on the efficient discovery of all interesting MDs in real-world datasets. For this purpose, we propose HyMD, a novel MD discovery algorithm that finds all minimal, non-trivial MDs within given similarity boundaries. The algorithm extracts the exact similarity thresholds for the individual MDs from the data instead of using predefined similarity thresholds. For this reason, it is the first approach to solve the MD discovery problem in an exact and truly complete way. If needed, the algorithm can, however, enforce certain properties on the reported MDs, such as disjointness and minimum support, to focus the discovery on such results that are actually required by downstream use cases. HyMD is technically a hybrid approach that combines the two most popular dependency discovery strategies in related work: lattice traversal and inference from record pairs. Despite the additional effort of finding exact similarity thresholds for all MD candidates, the algorithm is still able to efficiently process large datasets, e.g., datasets larger than 3 GB. KW - matching dependencies KW - functional dependencies KW - dependency discovery KW - data profiling KW - data matching KW - entity resolution KW - similarity measures Y1 - 2020 U6 - https://doi.org/10.1145/3392778 SN - 0362-5915 SN - 1557-4644 VL - 45 IS - 3 PB - Association for Computing Machinery CY - New York ER - TY - JOUR A1 - Maoutsa, Dimitra A1 - Reich, Sebastian A1 - Opper, Manfred T1 - Interacting particle solutions of Fokker–Planck equations through gradient–log–density estimation JF - Entropy N2 - Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control. KW - stochastic systems KW - Fokker-Planck equation KW - interacting particles KW - multiplicative noise KW - gradient flow KW - stochastic differential equations Y1 - 2020 U6 - https://doi.org/10.3390/e22080802 SN - 1099-4300 VL - 22 IS - 8 PB - MDPI CY - Basel ER - TY - JOUR A1 - Bandara, Lashi A1 - Bryan, Paul T1 - Heat kernels and regularity for rough metrics on smooth manifolds JF - Mathematische Nachrichten N2 - We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Holder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces. KW - heat kernel KW - parabolic Harnack estimate KW - rough metrics Y1 - 2020 U6 - https://doi.org/10.1002/mana.201800459 SN - 0025-584X SN - 1522-2616 VL - 293 IS - 12 SP - 2255 EP - 2270 PB - Wiley-VCH CY - Weinheim ER - TY - JOUR A1 - Azzali, Sara A1 - Paycha, Sylvie T1 - Spectral zeta-invariants lifted to coverings JF - Transactions of the American Mathematical Society N2 - The canonical trace and the Wodzicki residue on classical pseudo-differential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral zeta-invariants using lifted defect formulae which express discrepancies of zeta-regularised traces in terms of Wodzicki residues. We derive Atiyah's L-2-index theorem as an instance of the Z(2)-graded generalisation of the canonical lift of spectral zeta-invariants and we show that certain lifted spectral zeta-invariants for geometric operators are integrals of Pontryagin and Chern forms. Y1 - 2020 U6 - https://doi.org/10.1090/tran/8067 SN - 0002-9947 SN - 1088-6850 VL - 373 IS - 9 SP - 6185 EP - 6226 PB - American Mathematical Society CY - Providence, RI ER - TY - JOUR A1 - Bilò, Davide A1 - Lenzner, Pascal T1 - On the tree conjecture for the network creation game JF - Theory of computing systems N2 - Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al. (2003) is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all alpha and that for alpha >= n all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edge-price alpha and employ it to improve on the best known bound for the latter conjecture. In particular we show that for alpha > 4n - 13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees. KW - network creation games KW - price of anarchy KW - tree conjecture KW - algorithmic KW - game theory Y1 - 2019 U6 - https://doi.org/10.1007/s00224-019-09945-9 SN - 1432-4350 SN - 1433-0490 VL - 64 IS - 3 SP - 422 EP - 443 PB - Springer CY - New York ER - TY - JOUR A1 - Blanchard, Gilles A1 - Mücke, Nicole T1 - Kernel regression, minimax rates and effective dimensionality BT - beyond the regular case JF - Analysis and applications N2 - We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales. KW - Kernel regression KW - minimax optimality KW - eigenvalue decay Y1 - 2020 U6 - https://doi.org/10.1142/S0219530519500258 SN - 0219-5305 SN - 1793-6861 VL - 18 IS - 4 SP - 683 EP - 696 PB - World Scientific CY - New Jersey ER - TY - JOUR A1 - Perera, Upeksha A1 - Böckmann, Christine T1 - Solutions of Sturm-Liouville problems JF - Mathematics N2 - This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm-Liouville problems. Next, a concrete implementation to the inverse Sturm-Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm-Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm-Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions. KW - Sturm-Liouville problems of higher order KW - singular Sturm-Liouville KW - problems KW - inverse Sturm-Liouville problems Y1 - 2020 U6 - https://doi.org/10.3390/math8112074 SN - 2227-7390 VL - 8 IS - 11 PB - MDPI CY - Basel ER - TY - JOUR A1 - Hyde, Stephen T. A1 - Evans, Myfanwy E. T1 - Symmetric tangled Platonic polyhedra JF - Proceedings of the National Academy of Sciences of the United States of America N2 - Conventional embeddings of the edge-graphs of Platonic polyhedra, {f,z}, where f,z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S-2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S-1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway's two-dimensional (2D) orbifold notation (equivalent to Schonflies symbols I-h, O-h, and T-d). Tangled Platonic {f,z} polyhedra-which cannot lie on the sphere without edge-crossings-are constructed as windings of helices with three, five, seven,... strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the "theta(z)" polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo. KW - regular polyhedra KW - compound polyhedra KW - helicates KW - metal-organic KW - frameworks KW - clathrin Y1 - 2022 U6 - https://doi.org/10.1073/pnas.2110345118 SN - 0027-8424 SN - 1091-6490 VL - 119 IS - 1 PB - National Acad. of Sciences CY - Washington ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Eta invariant and parity conditions N2 - We give a formula for the η-invariant of odd order operators on even-dimensional manifolds, and for even order operators on odd-dimensional manifolds. Geometric second order operators are found with nontrivial η-invariants. This solves a problem posed by P. Gilkey. T3 - Preprint - (2000) 21 KW - eta invariant KW - parity conditions KW - K-theory KW - linking coefficients KW - Dirac operators KW - spectral flow KW - elliptic operators Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25869 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Sternin, Boris T1 - On surgery in elliptic theory N2 - We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities. T3 - Preprint - (2000) 22 KW - elliptic operators KW - index theory KW - surgery KW - relative index KW - manifold with singularities Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25873 ER - TY - INPR A1 - Coriasco, Sandro A1 - Panarese, Paolo T1 - Fourier integral operators defined by classical symbols with exit behaviour N2 - We continue the investigation of the calculus of Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour (SG symbols). Here we analyse what happens when one restricts the choice of amplitude and phase functions to the subclass of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a "classical" counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1); for such operators we refine the known results about the analogous problem for general SG hyperbolic operators. The material contained here will be used in a forthcoming paper to obtain a Weyl formula for a class of operators defined on manifolds with cylindrical ends, improving the results obtained in [9]. T3 - Preprint - (2000) 24 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25896 ER - TY - INPR A1 - Kytmanov, Aleksandr A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Removable singularities of CR functions on singular boundaries N2 - The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature o singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied. T3 - Preprint - (2000) 18 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25836 ER - TY - INPR A1 - Tepoyan, Liparit T1 - Degenerated operator equations of higher order N2 - Content: 1 Introduction 2 The one-dimensional case 2.1 The space Wm sub (α) 2.2 Self-adjoint Equation 2.3 Non-selfadjoint Equation 3 Operator Equation T3 - Preprint - (2000) 23 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25888 ER - TY - INPR A1 - Krainer, Thomas T1 - The calculus of Volterra Mellin pseudodifferential operators with operator-valued symbols N2 - We introduce the calculus of Mellin pseudodifferential operators parameters based on "twisted" operator-valued Volterra symbols as well aas the abstract Mellin calclus with holomorphic symbols. We establish the properties of the symblic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side, e. g., for the Leibniz-product, kernel cut-off, and Mellin quantization. Moreover, we introduce the notion of parabolicity for the calculi of Volterra Mellin operators, and construct Volterra parametrices for parabolic operators within the calculi. T3 - Preprint - (2001) 35 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26185 ER - TY - INPR A1 - Xiaochun, Liu A1 - Witt, Ingo T1 - Pseudodifferential calculi on the half-line respecting prescribed asymptotic types N2 - Contents: 1. Introduction 2. Preliminaries 3. Basic Elements of the Calculus 4. Further Elements of the Calculus T3 - Preprint - (2002) 06 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26255 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Anisotropic edge problems N2 - We investigate elliptic pseudodifferential operators which degenerate in an anisotropic way on a submanifold of arbitrary codimension. To find Fredholm problems for such operators we adjoint to them boundary and coboundary conditions on the submanifold.The algebra obtained this way is a far reaching generalisation of Boutet de Monvel's algebra of boundary value problems with transmission property. We construct left and right regularisers and prove theorems on hypoellipticity and local solvability. T3 - Preprint - (2002) 09 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26280 ER - TY - INPR A1 - Witt, Ingo T1 - A calculus for a class of finitely degenerate pseudodifferential operators N2 - For a class of degenerate pseudodifferential operators, local parametrices are constructed. This is done in the framework of a pseudodifferential calculus upon adding conditions of trace and potential type, respectively, along the boundary on which the operators degenerate. T3 - Preprint - (2002) 05 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26246 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 - INPR A1 - Schrohe, Elmar A1 - Seiler, Jörg T1 - Ellipticity and invertibility in the cone algebra on Lp-Sobolev spaces N2 - Given a manifold B with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale of Lp-Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces, it turns out to be independent of the choice of p. We then show that the cone algebra is closed under inversion: whenever an operator is invertible between the associated Sobolev spaces, its inverse belongs to the calculus. We use these results to analyze the behaviour of these operators on Lp(B). T3 - Preprint - (1999) 28 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25621 ER - TY - INPR A1 - Fedosov, Boris T1 - Pseudo-differential operators and deformation quantization N2 - Using the Riemannian connection on a compact manifold X, we show that the algebra of classical pseudo-differential operators on X generates a canonical deformation quantization on the cotangent manifold T*X. The corresponding Abelian connection is calculated explicitly in terms of the of the exponential mapping. We prove also that the index theorem for elliptic operators may be obtained as a consequence of the index theorem for deformation quantization. T3 - Preprint - (1999) 32 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25651 ER - TY - INPR A1 - Sadykov, Timour T1 - Hypergeometric systems of differential equations and amoebas of rational functions N2 - We study the approach to the theory of hypergeometric functions in several variables via a generalization of the Horn system of differential equations. A formula for the dimension of its solution space is given. Using this formula we construct an explicit basis in the space of holomorphic solutions to the generalized Horn system under some assumptions on its parameters. These results are applied to the problem of describing the complement of the amoeba of a rational function, which was posed in [12]. T3 - Preprint - (1999) 33 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25665 ER - TY - INPR A1 - Shlapunov, Alexander T1 - On Iterations of double layer potentials N2 - We prove the existence of Hp(D)-limit of iterations of double layer potentials constructed with the use of Hodge parametrix on a smooth compact manifold X, D being an open connected subset of X. This limit gives us an orthogonal projection from Sobolev space Hp(D) to a closed subspace of Hp(D)-solutions of an elliptic operator P of order p ≥ 1. Using this result we obtain formulae for Sobolev solutions to the equation Pu = f in D whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of double layer potentials. Similar regularization is constructed also for a P-Neumann problem in D. T3 - Preprint - (2000) 02 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25687 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 - Coriasco, Sandro A1 - Schrohe, Elmar A1 - Seiler, Jörg T1 - Bounded imaginary powers of differential operators on manifolds with conical singularities N2 - We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B,1 < p < ∞. Under suitable ellipticity assumptions we can define a family of complex powers A up(z), z ∈ C. We also obtain sufficient information on the resolvent of A to show the boundedness of the pure imaginary powers. Examples concern unique solvability and maximal regularity of the solution of the Cauchy problem u' - Δu = f, u(0) = 0, for the Laplacian on conical manifolds. T3 - Preprint - (2001) 12 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25962 ER - TY - INPR A1 - Xiaochun, Liu A1 - Witt, Ingo T1 - Asymptotic expansions for bounded solutions to semilinear Fuchsian equations N2 - It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions. T3 - Preprint - (2001) 01 KW - Calculus of conormal symbols KW - conormal asymptotic expansions KW - discrete saymptotic types KW - weighted Sobolev spaces with discrete saymptotics Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25912 ER - TY - INPR A1 - Ma, Li A1 - Xu, Xingwang T1 - Positive solutions of a logistic equation on unbounded intervals N2 - In this paper, we study the existence of positive solutions of a one-parameter family of logistic equations on R+ or on R. These equations are stationary versions of the Fisher equations and the KPP equations. We also study the blow up region of a sequence of the solutions when the parameter approachs a critical value and the nonexistence of positive solutions beyond the critical value. We use the direct method and the sub and super solution method. T3 - Preprint - (2001) 17 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26015 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Harmonic integrals on domains with edges N2 - We study the Neumann problem for the de Rham complex in a bounded domain of Rn with singularities on the boundary. The singularities may be general enough, varying from Lipschitz domains to domains with cuspidal edges on the boundary. Following Lopatinskii we reduce the Neumann problem to a singular integral equation of the boundary. The Fredholm solvability of this equation is then equivalent to the Fredholm property of the Neumann problem in suitable function spaces. The boundary integral equation is explicitly written and may be treated in diverse methods. This way we obtain, in particular, asymptotic expansions of harmonic forms near singularities of the boundary. T3 - Preprint - (2004) 20 KW - domains with singularities KW - de Rham complex KW - Neumann problem KW - Hodge theory Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26800 ER - TY - INPR A1 - Krainer, Thomas T1 - On the inverse of parabolic boundary value problems for large times N2 - We construct algebras of Volterra pseudodifferential operators that contain, in particular, the inverses of the most natural classical systems of parabolic boundary value problems of general form. Parabolicity is determined by the invertibility of the principal symbols, and as a result is equivalent to the invertibility of the operators within the calculus. Existence, uniqueness, regularity, and asymptotics of solutions as t → ∞ are consquences of the mapping properties of the operators in exponentially weighted Sobolev spaces and subspaces with asymptotics. An important aspect of this work is that the microlocal and global kernel structure of the inverse operator (solution operator) of a parabolic boundary value problem for large times is clarified. Moreover, our approach naturally yields qualitative pertubation results for the solvability theory of parabolic boundary value problems. To achieve these results, we assign t = ∞ the meaning of a conical point and treat the operators as totally characteristic pseudodifferential boundary value problems. T3 - Preprint - (2002) 12 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26310 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Sternin, Boris T1 - Relative elliptic theory N2 - This paper is a survey of relative elliptic theory (i.e. elliptic theory in the category of smooth embeddings), closely related to the Sobolev problem, first studied by Sternin in the 1960s. We consider both analytic aspects to the theory (the structure of the algebra of morphismus, ellipticity, Fredholm property) and topological aspects (index formulas and Riemann-Roch theorems). We also study the algebra of Green operators arising as a subalgebra of the algebra of morphisms. T3 - Preprint - (2002) 23 KW - Sobolev problem KW - elliptic morphism KW - (co)boundary operator KW - Green operator KW - index KW - Riemann-Roch theorem Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26400 ER - TY - INPR A1 - Davis, Simon T1 - On the absence of large-order divergences in superstring theory N2 - The genus-dependence of multi-loop superstring ams is estimated at large orders in perturbation theory using the super-Schottky group parameterization of supermoduli space. Restriction of the integration region to a subset of supermoduli space and a single fundamental domain of the super-modular group suggests an exponential dependence on the genus. Upper bounds for these estimates are obtained for arbitrary N-point superstring scattering amplitudes and are shown to be consistent with exact results obtained for special type II string amplitudes for orbifold or Calabi-Yau compactifications. The genus-dependence is then obtained by considering the effect of the remaining contribution to the superstring amplitudes after the coefficients of the formally divergent parts of the integrals vanish as a result of a sum over spin structures. The introduction of supersymmetry therefore leads to the elimination of large-order divergences in string pertubation theory, a result which is based only on the supersymmetric generalization of the polyakov measure and not the gauge group of the string model. T3 - Preprint - (2002) 28 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26452 ER - TY - INPR A1 - Junker, Wolfgang A1 - Schrohe, Elmar T1 - Adiabatic vacuum states on general spacetime manifolds : definition, construction, and physical properties N2 - Adiabatic vacuum states are a well-known class of physical states for linear quantum fields n Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetme manifolds (factoriality, quasiequivalence, local definteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface. T3 - Preprint - (2001) 27 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26100 ER - TY - INPR A1 - Yagdjian, Karen T1 - Geometric optics for the nonlinear hyperbolic systems of Kirchhoff-type N2 - Contents: 1 Introduction 2 Main result 3 Construction of the asymptotic solutions 3.1 Derivation of the equations for the profiles 3.2 Exsistence of the principal profile 3.3 Determination of Usub(2) and the remaining profiles 4 Stability of the samll global solutions. Justification of One Phase Nonlinear Geometric Optics for the Kirchhoff-type equations 4.1 Stability of the global solutions to the Kirchhoff-type symmetric hyperbolic systems 4.2 The nonlinear system of ordinary differential equations with the parameter 4.3 Some energies estimates 4.4 The dependence of the solution W(t, ξ) on the function s(t) 4.5 The oscillatory integrals of the bilinear forms of the solutions 4.6 Estimates for the basic bilinear form Γsub(s)(t) 4.7 Contraction mapping 4.8 Stability of the global solution 4.9 Justification of One Phase Nonlinear Geometric Optics for the Kirchhoff-type equations T3 - Preprint - (2001) 22 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26059 ER - TY - INPR A1 - Yihong, Du A1 - Li, Ma T1 - Some remarks related to De Giorgi's conjecture N2 - For several classes of functions including the special case f(u) = u − u³, we obtain boundedness and symmetry results for solutions of the problem −Δu = f(u) defined on R up(n). Our results complement a number of recent results related to a conjecture of De Giorgi. T3 - Preprint - (2001) 18 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26027 ER - TY - INPR A1 - Galstian, Anahit T1 - Lp - Lq decay estimates for the equation with exponentially growing coefficient N2 - Contents: 1 Introduction 1 Representation formulas 2 Consideration in the pseudodifferential zone: e up(t) |ξ| ≤ 1 3 Consideration in he hyperbolic zone: e up(t) |ξ| ≥ 1 T3 - Preprint - (2001) 24 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26079 ER - TY - INPR A1 - 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 - 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 - Nazaikinskii, Vladimir A1 - Sternin, Boris T1 - Some problems of control of semiclassical states for the Schrödinger equation N2 - Contents: Introduction Controlled Quantum Systems The Asymptotic Controllability Problem The Stabilization Problem Unitarily Nonlinear Equations The Quantum Problem The Stabilization Problem for the Schrödinger Equation with a Unitarily Non-linear Control T3 - Preprint - (2001) 30 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26130 ER - TY - INPR A1 - Witt, Ingo T1 - Asymptotic algebras N2 - The concept of asymptotic type that primarily appears in singular and asymptotic analysis is developed. Especially, asymptotic algebras are introduced. T3 - Preprint - (2001) 23 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26069 ER - TY - INPR A1 - Rozenblum, G. T1 - On some analytical index formulas related to operator-valued symbols N2 - For several classes of pseudodifferential operators with operator-valued symbol analytic index formulas are found. The common feature is that uasual index formulas are not valid for these operators. Applications are given to pseudodifferential operators on singular manifolds. T3 - Preprint - (2000) 16 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25811 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 - 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 - Gauthier, Paul M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - A covering property of the Riemann zeta-function N2 - For each compact subset K of the complex plane C which does not surround zero, the Riemann surface Sζ of the Riemann zeta function restricted to the critical half-strip 0 < Rs < 1/2 contains infinitely many schlicht copies of K lying ‘over’ K. If Sζ also contains at least one such copy, for some K which surrounds zero, then the Riemann hypothesis fails. T3 - Preprint - (2004) 03 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26683 ER - TY - INPR A1 - Kytmanov, Alexander A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Holomorphic Lefschetz formula for manifolds with boundary N2 - The classical Lefschetz fixed point formula expresses the number of fixed points of a continuous map f : M -> M in terms of the transformation induced by f on the cohomology of M. In 1966 Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they presented a holomorphic Lefschtz formula for compact complex manifolds without boundary, a result, in the framework of algebraic geometry due to Eichler (1957) for holomorphic curves. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, hence the Atiyah-Bott theory is no longer applicable. To get rid of the difficulties related to the boundary behaviour of the Dolbeault cohomology, Donelli and Fefferman (1986) derived a fixed point formula for the Bergman metric. The purpose of this paper is to present a holomorphic Lefschtz formula on a compact complex manifold with boundary T3 - Preprint - (2002) 17 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26354 ER - TY - INPR A1 - De-Xing, Kong A1 - Hui, Yao T1 - Global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws II N2 - In this paper, by a new constructive method, the authors reprove the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate fields. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C¹ functions. In particular, the authors give the optimal control time of the system. Finally, a new application is also given. T3 - Preprint - (2003) 08 KW - Quasilinear hyperbolic system KW - conservation laws KW - global exact boundary controllability KW - Cauchy problem KW - Goursat problem Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26565 ER - TY - INPR A1 - Huicheng, Yin A1 - Witt, Ingo T1 - Global singularity structure of weak solutions to 3-D semilinear dispersive wave equations with discontinuous initial data T3 - Preprint - (2002) 21 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26395 ER - TY - INPR A1 - Camalès, Renaud T1 - A note on the ramified Cauchy problem N2 - In this paper, the ramified Cauchy problem in C² for operator with multiple characteristics of constant multiplicity and second member ramified around some analytic set is studied. T3 - Preprint - (2003) 21 KW - Ramified Cauchy problem KW - analytic continuation Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26646 ER - TY - RPRT A1 - Sultanow, Eldar A1 - Volkov, Denis A1 - Cox, Sean T1 - Introducing a Finite State Machine for processing Collatz Sequences N2 - The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine’s word transformation is equivalent to the standard Collatz number transformation and subsequently discuss the possibilities for use of this approach at solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation. KW - Collatz Conjecture KW - State Machine KW - Graph KW - Double Colored Edges Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-399223 ET - 1st version ER - TY - INPR A1 - Sultanov, Oskar A1 - Kalyakin, Leonid A1 - Tarkhanov, Nikolai Nikolaevich T1 - Elliptic perturbations of dynamical systems with a proper node N2 - The paper is devoted to asymptotic analysis of the Dirichlet problem for a second order partial differential equation containing a small parameter multiplying the highest order derivatives. It corresponds to a small perturbation of a dynamical system having a stationary solution in the domain. We focus on the case where the trajectories of the system go into the domain and the stationary solution is a proper node. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 4 KW - dynamical system KW - singular perturbation KW - asymptotic methods Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70460 SN - 2193-6943 VL - 3 IS - 4 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - THES A1 - Etzold, Heiko T1 - Neue Zugänge zum Winkelbegriff T1 - New Ways to the Angle Concept BT - Fachdidaktische Entwicklungsforschung zur Ausbildung des Winkelfeldbegriffs bei Schülerinnen und Schülern der vierten Klassenstufe N2 - Die Vielfältigkeit des Winkelbegriffs ist gleichermaßen spannend wie herausfordernd in Hinblick auf seine Zugänge im Mathematikunterricht der Schule. Ausgehend von verschiedenen Vorstellungen zum Winkelbegriff wird in dieser Arbeit ein Lehrgang zur Vermittlung des Winkelbegriffs entwickelt und letztlich in konkrete Umsetzungen für den Schulunterricht überführt. Dabei erfolgt zunächst eine stoffdidaktische Auseinandersetzung mit dem Winkelbegriff, die von einer informationstheoretischen Winkeldefinition begleitet wird. In dieser wird eine Definition für den Winkelbegriff unter der Fragestellung entwickelt, welche Informationen man über einen Winkel benötigt, um ihn beschreiben zu können. So können die in der fachdidaktischen Literatur auftretenden Winkelvorstellungen aus fachmathematischer Perspektive erneut abgeleitet und validiert werden. Parallel dazu wird ein Verfahren beschrieben, wie Winkel – auch unter dynamischen Aspekten – informationstechnisch verarbeitet werden können, so dass Schlussfolgerungen aus der informationstheoretischen Winkeldefinition beispielsweise in dynamischen Geometriesystemen zur Verfügung stehen. Unter dem Gesichtspunkt, wie eine Abstraktion des Winkelbegriffs im Mathematikunterricht vonstatten gehen kann, werden die Grundvorstellungsidee sowie die Lehrstrategie des Aufsteigens vom Abstrakten zum Konkreten miteinander in Beziehung gesetzt. Aus der Verknüpfung der beiden Theorien wird ein grundsätzlicher Weg abgeleitet, wie im Rahmen der Lehrstrategie eine Ausgangsabstraktion zu einzelnen Winkelaspekten aufgebaut werden kann, was die Generierung von Grundvorstellungen zu den Bestandteilen des jeweiligen Winkelaspekts und zum Operieren mit diesen Begriffsbestandteilen ermöglichen soll. Hierfür wird die Lehrstrategie angepasst, um insbesondere den Übergang von Winkelsituationen zu Winkelkontexten zu realisieren. Explizit für den Aspekt des Winkelfeldes werden, anhand der Untersuchung der Sichtfelder von Tieren, Lernhandlungen und Forderungen an ein Lernmodell beschrieben, die Schülerinnen und Schüler bei der Begriffsaneignung unterstützen. Die Tätigkeitstheorie, der die genannte Lehrstrategie zuzuordnen ist, zieht sich als roter Faden durch die weitere Arbeit, wenn nun theoriebasiert Designprinzipien generiert werden, die in die Entwicklung einer interaktiven Lernumgebung münden. Hierzu wird u. a. das Modell der Artifact-Centric Activity Theory genutzt, das das Beziehungsgefüge aus Schülerinnen und Schülern, dem mathematischen Gegenstand und einer zu entwickelnden App als vermittelndes Medium beschreibt, wobei der Einsatz der App im Unterrichtskontext sowie deren regelgeleitete Entwicklung Bestandteil des Modells sind. Gemäß dem Ansatz der Fachdidaktischen Entwicklungsforschung wird die Lernumgebung anschließend in mehreren Zyklen erprobt, evaluiert und überarbeitet. Dabei wird ein qualitatives Setting angewandt, das sich der Semiotischen Vermittlung bedient und untersucht, inwiefern sich die Qualität der von den Schülerinnen und Schülern gezeigten Lernhandlungen durch die Designprinzipien und deren Umsetzung erklären lässt. Am Ende der Arbeit stehen eine finale Version der Designprinzipien und eine sich daraus ergebende Lernumgebung zur Einführung des Winkelfeldbegriffs in der vierten Klassenstufe. N2 - The diversity of the concept »angle« can be both exciting and challenging when looking at how to access it in mathematics education in schools. In this thesis, based on different ideas of the angle concept, a training course for conveying the concept will be developed and translated into concrete implementations for school teaching. First, there will be a didactical subject matter discussion of the angle concept, which will be accompanied by an angle definition from information theory. Through the didactical subject matter discussion, a definition for the angle concept will be developed which is guided by the question of what kind of information about an angle is needed in order to describe it. This way, the diverse ideas of the angle concept discussed in mathematics didactics literature can be once again derived and validated from a mathematical point of view. In parallel, a method will be described of how an angle - even one with dynamic aspects - can be handled in terms of information technology, so that conclusions can be drawn from a definition from information theory for dynamic geometry environments for instance. Considering how abstraction of the angle concept can take place in mathematics education, the Idea of Grundvorstellungen will then be connected to the structural principle of the Ascent From the Abstract to the Concrete. Based on the connection of these two theories, a training course will be developed that aims to construct an initial abstract of certain aspects of the angle concept which, in turn, aims at enabling the generating of Grundvorstellungen towards components of the angle concept and at operating with it. For this, the structural principle will be adapted – specifically to realize the transition from angle situations to angle contexts. For one aspect, the angular field, there will be a description of learning actions and demands on a learning model that supports students’ concept acquisition. The angular field, in this step, will be represented by vision fields of animals. Activity theory, on which the structural principle is based, depicts the recurring theme throughout this thesis when generating design principles that lead towards the development of an interactive learning environment. For this, the Artifact-Centric Activity Theory model will be used in order to describe connections between students, the mathematical topic and the to-be-created app. The use of the app in classroom situations, as well as its rule-governed development, are components of the model. Following a Design-Based Research approach, this learning environment will then go through several cycles of test, evaluation and revision. For this purpose, a qualitative setting will be applied using Semiotic Mediation. It will be used to investigate how far design principles, as well as their implementation, impacts on the quality of student’s learning actions. As an outcome of this thesis, a final version of the design principles and an ensuing learning environment that introduces the concept of »angular field« in grade four teaching will be created. KW - Winkel KW - Tätigkeitstheorie KW - Digitale Werkzeuge KW - Digital Tools KW - Activity Theory KW - Angle Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-504187 ER - TY - THES A1 - Zass, Alexander T1 - A multifaceted study of marked Gibbs point processes T1 - Facetten von markierten Gibbsschen Punktprozessen N2 - This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation. We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight. We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution. Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation. Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry. N2 - Diese Arbeit konzentriert sich auf die Untersuchung von markierten Gibbs-Punkt-Prozessen und stellt insbesondere einige Ergebnisse zu deren Existenz und Eindeutigkeit vor. Dabei werden Ideen und Techniken aus verschiedenen Bereichen der statistischen Mechanik verwendet: die Entropie-Methode aus der Theorie der großen Abweichungen, die Cluster-Expansion und die Kirkwood-Salsburg-Gleichungen, das Dobrushin-Kontraktionsprinzip und die Disagreement-Perkolation. Wir präsentieren zunächst ein Existenzergebnis für unendlich-volumige markierte Gibbs-Punkt-Prozesse. Genauer gesagt verwenden wir die sogenannte Entropie-Methode (und Werkzeuge der großen Abweichung), um markierte Gibbs-Punkt-Prozesse in R^d unter möglichst allgemeinen Annahmen zu konstruieren. Insbesondere gehören die zufälligen Markierungen zu einem allgemeinen normierten Raum und sind nicht beschränkt. Außerdem lassen wir Interaktionsfunktionale zu, die unbeschränkt sein können und deren Reichweite endlich, aber zufällig ist. Die Entropie-Methode beruht darauf, zu zeigen, dass eine Familie von endlich-volumigen Gibbs-Punkt-Prozessen zu sequentiell kompakten Entropie-Niveau-Mengen gehört, und daher dicht ist. Wir stellen dann unendlich-dimensionale Langevin-Diffusionen vor, die wir über eine Gibbssche Beschreibung in Wechselwirkung setzen. In dieser Umgebung sind wir in der Lage, das vorangehend vorgestellte allgemeine Ergebnis anzupassen, um die Existenz des zugehörigen unendlich-dimensionalen Maßes zu zeigen. Wir untersuchen auch seine Korrelationsfunktionen über Cluster-Expansions Techniken und erhalten die Eindeutigkeit des Gibbs-Prozesses für alle inversen Temperaturen β und Aktivitäten z unterhalb einer bestimmten Schwelle. Diese Methode beruht darauf, zunächst zu zeigen, dass die Korrelationsfunktionen des Prozesses eine so genannte Ruelle-Schranke erfüllen, um diese dann zur Lösung eines Fixpunktproblems in einem geeigneten Banach-Raum zu verwenden. Der Eindeutigkeitsbereich, den wir erhalten, wird dann aus den Modellparametern z und β definiert, für die ein solches Problem genau eine Lösung hat. Schließlich untersuchen wir die Frage nach der Eindeutigkeit von unendlich-volumigen Gibbs-Punkt-Prozessen auf R^d im unmarkierten Fall weiter. Im Zusammenhang mit repulsiven Wechselwirkungen basierend auf einer Hartkernkomponente stellen wir einen neuen Ansatz zur Eindeutigkeit vor, indem wir das diskrete Dobrushin-Kriterium im kontinuierlichen Rahmen anwenden. Wir legen zunächst einen Diskretisierungsparameter a>0 fest und untersuchen dann das Verhalten des Bereichs der Eindeutigkeit, wenn a gegen 0 geht. Mit dieser Technik sind wir in der Lage, explizite Schwellenwerte für die Parameter z und β zu erhalten, die wir dann mit bestehenden Ergebnissen aus den verschiedenen Methoden der Cluster-Expansion und der Disagreement-Perkolation vergleichen. In dieser Arbeit illustrieren wir unsere theoretischen Ergebnisse mit verschiedenen Beispielen sowohl aus der klassischen statistischen Mechanik als auch aus der stochastischen Geometrie. KW - marked Gibbs point processes KW - Langevin diffusions KW - Dobrushin criterion KW - Entropy method KW - Cluster expansion KW - Kirkwood--Salsburg equations KW - DLR equations KW - Markierte Gibbs-Punkt-Prozesse KW - Entropiemethode KW - Cluster-Expansion KW - DLR-Gleichungen KW - Dobrushin-Kriterium KW - Kirkwood-Salsburg-Gleichungen KW - Langevin-Diffusions Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-512775 ER - TY - THES A1 - Berner, Nadine T1 - Deciphering multiple changes in complex climate time series using Bayesian inference T1 - Bayes'sche Inferenz als diagnostischer Ansatz zur Untersuchung multipler Übergänge in komplexen Klimazeitreihen N2 - Change points in time series are perceived as heterogeneities in the statistical or dynamical characteristics of the observations. Unraveling such transitions yields essential information for the understanding of the observed system’s intrinsic evolution and potential external influences. A precise detection of multiple changes is therefore of great importance for various research disciplines, such as environmental sciences, bioinformatics and economics. The primary purpose of the detection approach introduced in this thesis is the investigation of transitions underlying direct or indirect climate observations. In order to develop a diagnostic approach capable to capture such a variety of natural processes, the generic statistical features in terms of central tendency and dispersion are employed in the light of Bayesian inversion. In contrast to established Bayesian approaches to multiple changes, the generic approach proposed in this thesis is not formulated in the framework of specialized partition models of high dimensionality requiring prior specification, but as a robust kernel-based approach of low dimensionality employing least informative prior distributions. First of all, a local Bayesian inversion approach is developed to robustly infer on the location and the generic patterns of a single transition. The analysis of synthetic time series comprising changes of different observational evidence, data loss and outliers validates the performance, consistency and sensitivity of the inference algorithm. To systematically investigate time series for multiple changes, the Bayesian inversion is extended to a kernel-based inference approach. By introducing basic kernel measures, the weighted kernel inference results are composed into a proxy probability to a posterior distribution of multiple transitions. The detection approach is applied to environmental time series from the Nile river in Aswan and the weather station Tuscaloosa, Alabama comprising documented changes. The method’s performance confirms the approach as a powerful diagnostic tool to decipher multiple changes underlying direct climate observations. Finally, the kernel-based Bayesian inference approach is used to investigate a set of complex terrigenous dust records interpreted as climate indicators of the African region of the Plio-Pleistocene period. A detailed inference unravels multiple transitions underlying the indirect climate observations, that are interpreted as conjoint changes. The identified conjoint changes coincide with established global climate events. In particular, the two-step transition associated to the establishment of the modern Walker-Circulation contributes to the current discussion about the influence of paleoclimate changes on the environmental conditions in tropical and subtropical Africa at around two million years ago. N2 - Im Allgemeinen stellen punktuelle Veränderungen in Zeitreihen (change points) eine Heterogenität in den statistischen oder dynamischen Charakteristika der Observablen dar. Das Auffinden und die Beschreibung solcher Übergänge bietet grundlegende Informationen über das beobachtete System hinsichtlich seiner intrinsischen Entwicklung sowie potentieller externer Einflüsse. Eine präzise Detektion von Veränderungen ist daher für die verschiedensten Forschungsgebiete, wie den Umweltwissenschaften, der Bioinformatik und den Wirtschaftswissenschaften von großem Interesse. Die primäre Zielsetzung der in der vorliegenden Doktorarbeit vorgestellten Detektionsmethode ist die Untersuchung von direkten als auch indirekten Klimaobservablen auf Veränderungen. Um die damit verbundene Vielzahl an möglichen natürlichen Prozessen zu beschreiben, werden im Rahmen einer Bayes’schen Inversion die generischen statistischen Merkmale Zentraltendenz und Dispersion verwendet. Im Gegensatz zu etablierten Bayes’schen Methoden zur Analyse von multiplen Übergängen, die im Rahmen von Partitionsmodellen hoher Dimensionalität formuliert sind und die Spezifikation von Priorverteilungen erfordern, wird in dieser Doktorarbeit ein generischer, Kernel-basierter Ansatz niedriger Dimensionalität mit minimal informativen Priorverteilungen vorgestellt. Zunächst wird ein lokaler Bayes’scher Inversionsansatz entwickelt, der robuste Rückschlüsse auf die Position und die generischen Charakteristika einer einzelnen Veränderung erlaubt. Durch die Analyse von synthetischen Zeitreihen die dem Einfluss von Veränderungen unterschiedlicher Signifikanz, Datenverlust und Ausreißern unterliegen wird die Leistungsfähigkeit, Konsistenz und Sensitivität der Inversionmethode begründet. Um Zeitreihen auch auf multiple Veränderungen systematisch untersuchen zu können, wird die Methode der Bayes’schen Inversion zu einem Kernel-basierten Ansatz erweitert. Durch die Einführung grundlegender Kernel-Maße können die Kernel-Resultate zu einer gewichteten Wahrscheinlichkeit kombiniert werden die als Proxy einer Posterior-Verteilung multipler Veränderungen dient. Der Detektionsalgorithmus wird auf reale Umweltmessreihen vom Nil-Fluss in Aswan und von der Wetterstation Tuscaloosa, Alabama, angewendet, die jeweils dokumentierte Veränderungen enthalten. Das Ergebnis dieser Analyse bestätigt den entwickelten Ansatz als eine leistungsstarke diagnostische Methode zur Detektion multipler Übergänge in Zeitreihen. Abschließend wird der generische Kernel-basierte Bayes’sche Ansatz verwendet, um eine Reihe von komplexen terrigenen Staubdaten zu untersuchen, die als Klimaindikatoren der afrikanischen Region des Plio-Pleistozän interpretiert werden. Eine detaillierte Untersuchung deutet auf multiple Veränderungen in den indirekten Klimaobservablen hin, von denen einige als gemeinsame Übergänge interpretiert werden. Diese gemeinsam auftretenden Ereignisse stimmen mit etablierten globalen Klimaereignissen überein. Insbesondere der gefundene Zwei-Stufen-Übergang, der mit der Ausbildung der modernen Walker-Zirkulation assoziiert wird, liefert einen wichtigen Beitrag zur aktuellen Diskussion über den Einfluss von paläoklimatischen Veränderungen auf die Umweltbedingungen im tropischen und subtropischen Afrika vor circa zwei Millionen Jahren. KW - kernel-based Bayesian inference KW - multi-change point detection KW - direct and indirect climate observations KW - Plio-Pleistocene KW - (sub-) tropical Africa KW - terrigenous dust KW - kernel-basierte Bayes'sche Inferenz KW - Detektion multipler Übergänge KW - direkte und indirekte Klimaobservablen KW - Plio-Pleistozän KW - (sub-) tropisches Afrika KW - terrigener Staub Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-100065 ER - TY - THES A1 - Hecher, Markus T1 - Advanced tools and methods for treewidth-based problem solving N2 - In the last decades, there was a notable progress in solving the well-known Boolean satisfiability (Sat) problem, which can be witnessed by powerful Sat solvers. One of the reasons why these solvers are so fast are structural properties of instances that are utilized by the solver’s interna. This thesis deals with the well-studied structural property treewidth, which measures the closeness of an instance to being a tree. In fact, there are many problems parameterized by treewidth that are solvable in polynomial time in the instance size when parameterized by treewidth. In this work, we study advanced treewidth-based methods and tools for problems in knowledge representation and reasoning (KR). Thereby, we provide means to establish precise runtime results (upper bounds) for canonical problems relevant to KR. Then, we present a new type of problem reduction, which we call decomposition-guided (DG) that allows us to precisely monitor the treewidth when reducing from one problem to another problem. This new reduction type will be the basis for a long-open lower bound result for quantified Boolean formulas and allows us to design a new methodology for establishing runtime lower bounds for problems parameterized by treewidth. Finally, despite these lower bounds, we provide an efficient implementation of algorithms that adhere to treewidth. Our approach finds suitable abstractions of instances, which are subsequently refined in a recursive fashion, and it uses Sat solvers for solving subproblems. It turns out that our resulting solver is quite competitive for two canonical counting problems related to Sat. N2 - In den letzten Jahrzehnten konnte ein beachtlicher Fortschritt im Bereich der Aussagenlogik verzeichnet werden. Dieser äußerte sich dadurch, dass für das wichtigste Problem in diesem Bereich, genannt „Sat“, welches sich mit der Fragestellung befasst, ob eine gegebene aussagenlogische Formel erfüllbar ist oder nicht, überwältigend schnelle Computerprogramme („Solver“) entwickelt werden konnten. Interessanterweise liefern diese Solver eine beeindruckende Leistung, weil sie oft selbst Probleminstanzen mit mehreren Millionen von Variablen spielend leicht lösen können. Auf der anderen Seite jedoch glaubt man in der Wissenschaft weitgehend an die Exponentialzeithypothese (ETH), welche besagt, dass man im schlimmsten Fall für das Lösen einer Instanz in diesem Bereich exponentielle Laufzeit in der Anzahl der Variablen benötigt. Dieser vermeintliche Widerspruch ist noch immer nicht vollständig geklärt, denn wahrscheinlich gibt es viele ineinandergreifende Gründe für die Schnelligkeit aktueller Sat Solver. Einer dieser Gründe befasst sich weitgehend mit strukturellen Eigenschaften von Probleminstanzen, die wohl indirekt und intern von diesen Solvern ausgenützt werden. Diese Dissertation beschäftigt sich mit solchen strukturellen Eigenschaften, nämlich mit der sogenannten Baumweite. Die Baumweite ist sehr gut erforscht und versucht zu messen, wie groß der Abstand von Probleminstanzen zu Bäumen ist (Baumnähe). Allerdings ist dieser Parameter sehr generisch und bei Weitem nicht auf Problemstellungen der Aussagenlogik beschränkt. Tatsächlich gibt es viele weitere Probleme, die parametrisiert mit Baumweite in polynomieller Zeit gelöst werden können. Interessanterweise gibt es auch viele Probleme in der Wissensrepräsentation (KR), von denen man davon ausgeht, dass sie härter sind als das Problem Sat, die bei beschränkter Baumweite in polynomieller Zeit gelöst werden können. Ein prominentes Beispiel solcher Probleme ist das Problem QSat, welches sich für die Gültigkeit einer gegebenen quantifizierten, aussagenlogischen Formel (QBF), das sind aussagenlogische Formeln, wo gewisse Variablen existenziell bzw. universell quantifiziert werden können, befasst. Bemerkenswerterweise wird allerdings auch im Zusammenhang mit Baumweite, ähnlich zu Methoden der klassischen Komplexitätstheorie, die tatsächliche Komplexität (Härte) solcher Problemen quantifiziert, wo man die exakte Laufzeitabhängigkeit beim Problemlösen in der Baumweite (Stufe der Exponentialität) beschreibt. Diese Arbeit befasst sich mit fortgeschrittenen, Baumweite-basierenden Methoden und Werkzeugen für Probleme der Wissensrepräsentation und künstlichen Intelligenz (AI). Dabei präsentieren wir Methoden, um präzise Laufzeitresultate (obere Schranken) für prominente Fragmente der Antwortmengenprogrammierung (ASP), welche ein kanonisches Paradigma zum Lösen von Problemen der Wissensrepräsentation darstellt, zu erhalten. Unsere Resultate basieren auf dem Konzept der dynamischen Programmierung, die angeleitet durch eine sogenannte Baumzerlegung und ähnlich dem Prinzip „Teile-und-herrsche“ funktioniert. Solch eine Baumzerlegung ist eine konkrete, strukturelle Zerlegung einer Probleminstanz, die sich stark an der Baumweite orientiert. Des Weiteren präsentieren wir einen neuen Typ von Problemreduktion, den wir als „decomposition-guided (DG)“, also „zerlegungsangeleitet“, bezeichnen. Dieser Reduktionstyp erlaubt es, Baumweiteerhöhungen und -verringerungen während einer Problemreduktion von einem bestimmten Problem zu einem anderen Problem präzise zu untersuchen und zu kontrollieren. Zusätzlich ist dieser neue Reduktionstyp die Basis, um ein lange offen gebliebenes Resultat betreffend quantifizierter, aussagenlogischer Formeln zu zeigen. Tatsächlich sind wir damit in der Lage, präzise untere Schranken, unter der Annahme der Exponentialzeithypothese, für das Problem QSat bei beschränkter Baumweite zu zeigen. Genauer gesagt können wir mit diesem Konzept der DG Reduktionen zeigen, dass das Problem QSat, beschränkt auf Quantifizierungsrang ` und parametrisiert mit Baumweite k, im Allgemeinen nicht besser als in einer Laufzeit, die `-fach exponentiell in der Baumweite und polynomiell in der Instanzgröße ist1, lösen. Dieses Resultat hebt auf nicht-inkrementelle Weise ein bekanntes Ergebnis für Quantifizierungsrang 2 auf beliebige Quantifizierungsränge, allerdings impliziert es auch sehr viele weitere Konsequenzen. Das Resultat über die untere Schranke des Problems QSat erlaubt es, eine neue Methodologie zum Zeigen unterer Schranken einer Vielzahl von Problemen der Wissensrepräsentation und künstlichen Intelligenz, zu etablieren. In weiterer Konsequenz können wir damit auch zeigen, dass die oberen Schranken sowie die DG Reduktionen dieser Arbeit unter der Hypothese ETH „eng“ sind, d.h., sie können wahrscheinlich nicht mehr signifikant verbessert werden. Die Ergebnisse betreffend der unteren Schranken für QSat und die dazugehörige Methodologie konstituieren in gewisser Weise eine Hierarchie von über Baumweite parametrisierte Laufzeitklassen. Diese Laufzeitklassen können verwendet werden, um die Härte von Problemen für das Ausnützen von Baumweite zu quantifizieren und diese entsprechend ihrer Laufzeitabhängigkeit bezüglich Baumweite zu kategorisieren. Schlussendlich und trotz der genannten Resultate betreffend unterer Schranken sind wir im Stande, eine effiziente Implementierung von Algorithmen basierend auf dynamischer Programmierung, die entlang einer Baumzerlegung angeleitet wird, zur Verfügung zu stellen. Dabei funktioniert unser Ansatz dahingehend, indem er probiert, passende Abstraktionen von Instanzen zu finden, die dann im Endeffekt sukzessive und auf rekursive Art und Weise verfeinert und verbessert werden. Inspiriert durch die enorme Effizienz und Effektivität der Sat Solver, ist unsere Implementierung ein hybrider Ansatz, weil sie den starken Gebrauch von Sat Solvern zum Lösen diverser Subprobleme, die während der dynamischen Programmierung auftreten, pflegt. Dabei stellt sich heraus, dass der resultierende Solver unserer Implementierung im Bezug auf Effizienz beim Lösen von zwei kanonischen, Sat-verwandten Zählproblemen mit bestehenden Solvern locker mithalten kann. Tatsächlich sind wir im Stande, Instanzen, wo die oberen Schranken von Baumweite 260 übersteigen, zu lösen. Diese überraschende Beobachtung zeigt daher, dass Baumweite ein wichtiger Parameter sein könnte, der wohl in modernen Designs von Solvern berücksichtigt werden sollte. KW - Treewidth KW - Dynamic Programming KW - Knowledge Representation and Reasoning KW - Artificial Intelligence KW - Computational Complexity KW - Parameterized Complexity KW - Answer Set Programming KW - Exponential Time Hypothesis KW - Lower Bounds KW - Algorithms KW - Algorithmen KW - Antwortmengenprogrammierung KW - Künstliche Intelligenz KW - Komplexitätstheorie KW - Dynamische Programmierung KW - Exponentialzeit Hypothese KW - Wissensrepräsentation und Schlussfolgerung KW - Untere Schranken KW - Parametrisierte Komplexität KW - Baumweite Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-512519 ER - TY - GEN A1 - Goychuk, Igor A1 - Kharchenko, Vasyl O. T1 - Rocking subdiffusive ratchets BT - origin, optimization and efficiency T2 - Mathematical Modelling of Natural Phenomena N2 - We study origin, parameter optimization, and thermodynamic efficiency of isothermal rocking ratchets based on fractional subdiffusion within a generalized non-Markovian Langevin equation approach. A corresponding multi-dimensional Markovian embedding dynamics is realized using a set of auxiliary Brownian particles elastically coupled to the central Brownian particle (see video on the journal web site). We show that anomalous subdiffusive transport emerges due to an interplay of nonlinear response and viscoelastic effects for fractional Brownian motion in periodic potentials with broken space-inversion symmetry and driven by a time-periodic field. The anomalous transport becomes optimal for a subthreshold driving when the driving period matches a characteristic time scale of interwell transitions. It can also be optimized by varying temperature, amplitude of periodic potential and driving strength. The useful work done against a load shows a parabolic dependence on the load strength. It grows sublinearly with time and the corresponding thermodynamic efficiency decays algebraically in time because the energy supplied by the driving field scales with time linearly. However, it compares well with the efficiency of normal diffusion rocking ratchets on an appreciably long time scale. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 622 KW - anomalous Brownian motion KW - generalized Langevin equation KW - memory effects KW - viscoelasticity KW - ratchet transport KW - stochastic Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-416138 SN - 1866-8372 IS - 622 ER - TY - THES A1 - Oancea, Marius-Adrian T1 - Spin Hall effects in general relativity T1 - Spin Hall Effekte in der Allgemeinen Relativitätstheorie N2 - The propagation of test fields, such as electromagnetic, Dirac or linearized gravity, on a fixed spacetime manifold is often studied by using the geometrical optics approximation. In the limit of infinitely high frequencies, the geometrical optics approximation provides a conceptual transition between the test field and an effective point-particle description. The corresponding point-particles, or wave rays, coincide with the geodesics of the underlying spacetime. For most astrophysical applications of interest, such as the observation of celestial bodies, gravitational lensing, or the observation of cosmic rays, the geometrical optics approximation and the effective point-particle description represent a satisfactory theoretical model. However, the geometrical optics approximation gradually breaks down as test fields of finite frequency are considered. In this thesis, we consider the propagation of test fields on spacetime, beyond the leading-order geometrical optics approximation. By performing a covariant Wentzel-Kramers-Brillouin analysis for test fields, we show how higher-order corrections to the geometrical optics approximation can be considered. The higher-order corrections are related to the dynamics of the spin internal degree of freedom of the considered test field. We obtain an effective point-particle description, which contains spin-dependent corrections to the geodesic motion obtained using geometrical optics. This represents a covariant generalization of the well-known spin Hall effect, usually encountered in condensed matter physics and in optics. Our analysis is applied to electromagnetic and massive Dirac test fields, but it can easily be extended to other fields, such as linearized gravity. In the electromagnetic case, we present several examples where the gravitational spin Hall effect of light plays an important role. These include the propagation of polarized light rays on black hole spacetimes and cosmological spacetimes, as well as polarization-dependent effects on the shape of black hole shadows. Furthermore, we show that our effective point-particle equations for polarized light rays reproduce well-known results, such as the spin Hall effect of light in an inhomogeneous medium, and the relativistic Hall effect of polarized electromagnetic wave packets encountered in Minkowski spacetime. N2 - Unser grundlegendes Verständnis des Universums basiert auf Einsteins allgemeiner Relativitätstheorie, die eine Beschreibung in Form einer vierdimensional gekrümmten Raumzeit liefert, in der die Anziehungskraft der Gravitation in der Krümmung der Raumzeit kodiert ist. Die überwiegende Mehrheit der experimentellen Tests, die Einsteins allgemeine Relativitätstheorie bestätigt haben, basiert auf der Beobachtung elektromagnetischer Strahlung, die von entfernten astrophysikalischen Quellen wie Sternen oder Galaxien stammt. Daher ist ein tiefgreifendes Verständnis der Dynamik der sich in der Raumzeit ausbreitenden elektromagnetischen Strahlung von entscheidender Bedeutung. Elektromagnetische Phänomene werden durch Maxwell-Gleichungen beschrieben. Die Ausbreitung elektromagnetischer Strahlung in der Raumzeit ist jedoch sehr komplexe, und es ist im Allgemeinen nützlich, Näherungen zu betrachten, welche eine vereinfachte Beschreibung liefern. Auf diese Weise können die Haupteigenschaften des Systems in einem reduzierten Gleichungssystem codiert und die Gültigkeit der Näherung quantitativ kontrolliert werden. Beispielsweise kann die Ausbreitung elektromagnetischer Strahlung in der Raumzeit durch Anwendung der geometrischen Optik auf die Maxwell-Gleichungen beschrieben werden. Diese liefert ein Modell für die Ausbreitung elektromagnetischer Strahlung in Form von Lichtstrahlen, die sich auf dem kürzesten Weg zwischen zwei Punkten ausbreiten. Im Kontext von Einsteins allgemeiner Relativitätstheorie entsprechen dise Lichtstrahlen den Nullgeodäten der zugrunde liegenden gekrümmten Raumzeit. Für die meisten astrophysikalischen Anwendungen von Interesse, wie die Beobachtung von Himmelskörpern oder Gravitationslinsen, stellen die Näherungen der geometrischen Optik und damit die Beschreibung der Ausbreitung elektromagnetischer Strahlung durch Lichtstrahlen ein zufriedenstellendes theoretisches Modell dar. In dieser Arbeit untersuchen wir mögliche Korrekturen der Ausbreitung elektromagnetischer Strahlung in der Raumzeit, welche durch die Näherung der geometrischen Optik nicht erfasst werden. Solche Korrekturen sind aus der Optik bekannt, wo beobachtet wurde, dass die Ausbreitung von Lichtstrahlen in bestimmten Materialien durch die Polarisation des Lichts beeinflusst werden kann. Diese Korrekturen sind als Spin-Hall-Effekt von Licht bekannt. In dieser Arbeit wird gezeigt, dass ein ähnlicher Effekt für elektromagnetische Strahlung auftreten kann, welche sich in gekrümmter Raumzeit in der Nähe massiver astrophysikalischer Objekte wie Schwarzer Löcher oder Sterne ausbreitet. Darüber hinaus präsentieren wir, basierend auf der Dirac-Gleichung, eine ähnliche Analyse für die Bewegung von Elektronen in gekrümmten Raumzeiten. KW - spin Hall effect KW - gravitation KW - black hole KW - Schwarzes Loch KW - Gravitation KW - Spin Hall effekte Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-502293 ER - TY - INPR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Duality by reproducing kernels N2 - Let A be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write Ssub(A)(D) for the space of solutions to thesystem Au = 0 in a domain D ⊂ X. Using reproducing kernels related to various Hilbert structures on subspaces of Ssub(A)(D) we show explicit identifications of the dual spaces. To prove the "regularity" of reproducing kernels up to the boundary of D we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the ∂-Neumann problem. The duality itself takes place only for those domains D which possess certain convexity properties with respect to A. T3 - Preprint - (2001) 26 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26095 ER - TY - GEN A1 - Reich, Sebastian T1 - On a geometrical interpretation of differential-algebraic equations N2 - The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE. and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lowsr-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 157 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-46683 ER - TY - GEN A1 - Champagnat, Nicolas A1 - Roelly, Sylvie T1 - Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions N2 - A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process - the conditioned multitype Feller branching diffusion - are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too . T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 065 KW - multitype measure-valued branching processes KW - conditioned KW - critical and subcritical Dawson-Watanabe process KW - conditioned Feller diffusion Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-18610 ER - TY - INPR A1 - Conforti, Giovanni A1 - Léonard, Christian A1 - Murr, Rüdiger A1 - Roelly, Sylvie T1 - Bridges of Markov counting processes : reciprocal classes and duality formulas N2 - Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 9 KW - counting process KW - bridge KW - reciprocal class KW - duality formula Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71855 SN - 2193-6943 VL - 3 IS - 9 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - GEN A1 - Reich, Sebastian T1 - Momentum conserving symplectic integrators N2 - In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n). T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 044 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-16824 ER - TY - INPR A1 - Fradon, Myriam A1 - Roelly, Sylvie T1 - Infinite system of Brownian balls with interaction : the non-reversible case N2 - We consider an infinite system of hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite- dimensional Stochastic Differential Equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also show that Gibbs measures are reversible measures. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2005, 01 KW - Stochastic Differential Equation KW - local time KW - hard core potential KW - Gibbs measure KW - reversible measure Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51546 ER - TY - GEN A1 - Imkeller, Peter A1 - Roelly, Sylvie T1 - Die Wiederentdeckung eines Mathematikers: Wolfgang Döblin N2 - "Considerons une particule mobile se mouvant aleatoirement sur la droite (ou sur un segment de droite). Supposons qu'il existe une probabilite F(x,y;s,t) bien definie pour que la particule se trouvant a l'instant s dans la position x se trouve a l'instant t (> s) a gauche de y, probabilite independante du mouvement anterieur de la particule...." Mit diesen Worten beginnt eines der berühmtesten mathematischen Manuskripte des letzten Jahrhunderts. Es stammt vom Soldaten Wolfgang Döblin, Sohn des deutschen Schriftstellers Alfred Döblin, und trägt den Titel "Sur l'equation de Kolmogoroff". Seine Veröffentlichung verbindet sich mit einer unglaublichen Geschichte. Wolfgang Döblin, stationiert mit seiner Einheit in den Ardennen im Winter 1939/1940, arbeitete an diesem Manuskript. Er entschloss sich, es als versiegeltes Manuskript an die Academie des Sciences in Paris zu schicken. Aber er kehrte nie aus diesem Krieg zurück. Sein Manuskript blieb 60 Jahre unter Verschluss im Archiv, und wurde erst im Jahre 2000 geöffnet. Wie weit Döblin damit seiner Zeit voraus war, wurde erkannt, nachdem es von Bernard Bru und Marc Yor ausgewertet worden war. Im ersten Satz umschreibt W. Döblin gleichzeitig das Programm des Manuskripts: "Wir betrachten ein bewegliches Teilchen, das sich zufällig auf der Geraden (oder einem Teil davon) bewegt." Er widmet sich damit der Aufgabe, die Fundamente eines Gebiets zu legen, das wir heute als stochastische Analysis bezeichnen. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 035 KW - Kolmogorov-Gleichung KW - Stochastische Analysis KW - Döblin KW - Wolfgang KW - Doblin KW - Vincent KW - Doeblin KW - Wolfgang Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-16397 ER - TY - INPR A1 - Siegert, Sabine T1 - Das Sankt-Petersburg-Paradoxon N2 - Aus dem Inhalt: 1 Einleitung 2 Historische Lösungsansätze 3 Martingal-Ansatz 4 Markovketten-Ansatz 5 Asymptotische Interpretationen 6 Bezug zur Praxis 7 Résumé Anhang Literaturverzeichnis T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2010, 05 Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49595 ER - TY - INPR A1 - Pra, Paolo Dai A1 - Louis, Pierre-Yves A1 - Minelli, Ida G. T1 - Complete monotone coupling for Markov processes N2 - We formalize and analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuoustime but not in discrete-time. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2008, 01 KW - Markov processes KW - coupling KW - partial ordering KW - monotonicity conditions KW - monotone random KW - dynamical system representation Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-18286 ER - TY - INPR A1 - Paneah, Boris T1 - Dynamic methods in the general theory of cauchy type functional equations N2 - Contents: 1 Introduction. Denfitions and Discussions 2 Solvability of the Cauchy Type Functional Equations 2.1 The Case of a P-configuration 2.2 The Case of a Z-configuration 2.3 Multiplicative Cauchy type functional equations 3 Problems in Analysis Reducing to Cauchy Type Functional Equations 3.1 Some problems in Integral Geometry and Cauchy Functional Equations 3.2 First Boundary Problem for Hyperbolic Differential Equations and Cauchy Type Functional Equations 4 Functional Equations Determining Polynomials T3 - Preprint - (2002) 10 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26295 ER - TY - INPR A1 - Karp, Lavi T1 - On the well-posedness of the vacuum Einstein's equations N2 - The Cauchy problem of the vacuum Einstein's equations aims to find a semimetric g(αβ) of a spacetime with vanishing Ricci curvature Rα,β and prescribed initial data. Under the harmonic gauge condition, the equations Rα,β = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein's equations are a proper Riemannian metric h(αβ) and a second fundamental form K(αβ). A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h(αβ),K(αβ)) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of our work is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces. T3 - Preprint - (2009) 06 Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-36593 ER - TY - INPR A1 - Klein, Markus A1 - Léonard, Christian A1 - Rosenberger, Elke T1 - Agmon-type estimates for a class of jump processes N2 - In the limit we analyze the generators of families of reversible jump processes in the n-dimensional space associated with a class of symmetric non-local Dirichlet forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of certain eikonal equation. Fine results are sensitive to the rate functions being twice differentiable or just Lipschitz. Our estimates are similar to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 6 KW - finsler distance KW - decay of eigenfunctions KW - jump process KW - Dirichlet form Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56995 ER - TY - INPR A1 - Kunze, Simone T1 - Das Sammelbilderproblem N2 - Aus dem Inhalt: 1 Einleitung 2 Entwicklung der Lösungsansätze 3 Martingalansatz 4 Markov-Ketten Ansatz 5 Einbettung in Poisson Prozesse 6 Kombinatorische Ansätze 7 Zusammenfassung und Ausblick Literaturverzeichnis T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2010, 12 Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51646 ER - TY - THES A1 - Hain, Tobias Martin T1 - Structure formation and identification in geometrically driven soft matter systems T1 - Strukturbildung und Identifikation in geometrisch getriebenen weiche Materie-Systemen N2 - Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces. Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system's constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness. In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system "forced" to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate. We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase. The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy. N2 - Die Unterteilung eines Raums durch Grenzflächen führt zu Raumaufteilungen, die für die Selbstorganisation weicher Materie relevant sind. Bekannte Beispiele sind zelluläre Medien, wie z.B. Seifenschaum, der aus Luftblasen besteht, getrennt durch Wände aus Wasser und Seife, und komplexere Partitionen, wie sie z.B. durch bikontinuierliche Minimalflächen erzeugt werden. In dieser Arbeit werden mit Hilfe von Computersimulationen Systeme weicher Materie in Bezug auf den Zusammenhang zwischen dem im System vorherrschenden, physikalischen Kräften und der Struktur der resultierenden Grenzflächen oder Partitionen untersucht. Der Schwerpunkt liegt hierbei auf zwei Systemen, eine Copolymerschmelze und das sogenannte Quantizer Problem, bei denen der treibende Faktor der Strukturbildung, nämlich die Minimierung der freien Energie, aus einem Zusammenspiel der Minimierung der Oberfläche der Grenzflächen und der gleichzeitigen Minimierung der Elastizitätsenergie besteht. Unter diesen Gegebenheiten bevorzugen solche Systeme Zellen gleichmäßiger Größe. Im ersten Teil der Arbeit befassen wir uns mit der Bildung von scharfen Grenzflächen in Systemen von Copolymeren. Wir analysieren die zylindrische Phase eines Copolymersystems, das gezwungen wird, sich auf einer kugelförmigen Oberfläche zu organisieren. Die Topologie dieser Oberfläche erlaubt es der optimalen Konfiguration, dem Sechseckgitter, nicht, sich zu bilden. Für dreiarmige Copolymere wird die entstehende Struktur durch Lösungen des sogenannten Thomson Problems beschrieben. Letzteres sucht nach der Konfigurationen von abstoßenden Ladungen auf einer Kugeloberfläche mit minimaler Energie. Auf einem Substrat haben wir eine Kombination aus drei ineinandergreifende Lösungen des Thomson Problems gefunden, wobei der Typ der Lösungen statistisch von dem Radius des Substrates abhängt. Anschließend untersuchen wir die Bildung von amorphen und kristallinen Strukturen im Quantizersystem, einem teilchenbasierenden Modell, dessen Energiefunktional keine Oberflächenspannung enthält und möglichst kugelförmige Zellen gleicher Größe begünstigt. Wird das System quasistatisch im thermodynamischen Gleichgewicht abgekühlt, kristallisiert das Quantizersystem in den geordneten BCC Grundzustand. Wird das System allerdings zu schnell abgekühlt, sodass es sich nicht mehr im thermodynamischen Gleichgewicht befindet, bildet sich eine amorphe, annähernd hyperuniforme Struktur aus. Wir konnten zeigen, dass diese Struktur bemerkenswert unabhängig von den Ausganszuständen, sowie der Art der Energieminimierung zu sein scheint. Im Ausblick erweitern wir das Quantizersystem, indem wir Oberflächenspannung einführen. Unsere Ergebnisse deutet darauf hin, dass dieses so erweiterte Modell Mizellenphasen in Polymersystem modellieren kann. Wir beobachten einen Phasenübergang von einer ungeordneten, flüssigen Phase hin zu einer festen Frank-Caspar-Phase. Der zweite Teil der Arbeit behandelt bikontinuierliche Grenzflächen, die den Raum in zwei netzwerkartige Domänen aufteilen. Wir führen eine Open-Source Software ein, das die Identifizierung von Strukturen anhand derer Mikroskopaufnahmen ermöglicht. Hierzu erweitern und verbessern wir eine Methode, die durch den Abgleich experimentell zugänglicher Projektionen in Mikroskopaufnahmen mit berechneten Projektionen potenzieller Strukturen basiert. Dieses Verfahren wurde erstmal von Deng und Mieczkowski (1998) eingeführt. Die simulierten Strukturen basieren auf einer Nodalflächenmodellierung von dreifach-periodischen Flächen konstanter mittlerer Krümmung. Wir führen eine Fallstudie an Zellmembranen von Etioplasten, den Vorläufern von Chloroplasten, durch. Wir konnten die Struktur dieser Etioplasten als die Diamond-Struktur identifizieren. Als Ausblick automatisieren wir den Identifizierungsproyess mit Hilfe von Deep-Learning-Methoden. Erste Ergebnisse zeigen, dass mit diesem Ansatz die Identifizierung von Strukturen mit ausgezeichneter Genauigkeit gelingt. KW - soft matter KW - geometry KW - self-assembly KW - structure formation KW - quantizer KW - polymer KW - Geometrie KW - Polymere KW - Quantizer KW - Selbstassemblierung KW - weiche Materie KW - Strukturbildung Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-558808 N1 - Parts of this publication are reproduced with permission from the Royal Society of Chemistry and AIP Publishing. ER - TY - INPR A1 - Champagnat, Nicolas A1 - Roelly, Sylvie T1 - Limit theorems for conditioned multitype Dawson-Watanabe processes N2 - A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every nite time interval, its distribution law is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. The explicit form of the Laplace functional of the conditioned process is used to obtain several results on the long time behaviour of the mass of the conditioned and unconditioned processes. The general case is considered first, where the mutation matrix which modelizes the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are also analysed. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2007, 01 Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49426 ER - TY - INPR A1 - Fradon, Myriam A1 - Roelly, Sylvie T1 - Infinite system of Brownian Balls: Equilibrium measures are canonical Gibbs N2 - We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional Stochastic Differential Equation with a local time term. We prove that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2005, 02 KW - Stochastic Differential Equation KW - hard core potential KW - Canonical Gibbs measure KW - detailed balance equation KW - reversible measure Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51594 ER - TY - INPR A1 - Cattiaux, Patrick A1 - Dai Pra, Paolo A1 - Poelly, Sylvie T1 - A constructive approach to a class of ergodic HJB equations with nonsmooth cost N2 - We consider a class of ergodic Hamilton-Jacobi-Bellman (HJB) equations, related to large time asymptotics of non-smooth multiplicative functional of difusion processes. Under suitable ergodicity assumptions on the underlying difusion, we show existence of these asymptotics, and that they solve the related HJB equation in the viscosity sense. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2007, 02 Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49430 ER - TY - INPR A1 - Läuter, Henning A1 - Liero, Hannelore T1 - Nonparametric estimation and testing in survival models N2 - The aim of this paper is to demonstrate that nonparametric smoothing methods for estimating functions can be an useful tool in the analysis of life time data. After stating some basic notations we will present a data example. Applying standard parametric methods to these data we will see that this approach fails - basic features of the underlying functions are not reflected by their estimates. Our proposal is to use nonparametric estimation methods. These methods are explained in section 2. Nonparametric approaches are better in the sense that they are more flexible, and misspecifications of the model are avoided. But, parametric models have the advantage that the parameters can be interpreted. So, finally, we will formulate a test procedure to check whether a parametric or a nonparametric model is appropriate. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 05 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51586 ER - TY - INPR A1 - Liero, Hannelore A1 - Liero, Matthias T1 - Testing the acceleration function in life time models N2 - The accelerated life time model is considered. First, test procedures for testing the parameter of a parametric acceleration function is investigated; this is done under the assumption of parametric and nonparametric baseline distribution. Further, based on nonparametric estimators for regression functions tests are proposed for checking whether a parametric acceleration function is appropriate to model the influence of the covariates. Resampling procedures are discussed for the realization of these methods. Simulations complete the considerations. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2006, 03 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49405 ER -