TY - JOUR A1 - Malass, Ihsane A1 - Tarkhanov, Nikolaj Nikolaevič T1 - A perturbation of the de Rham complex T1 - Возмущение комплекса де Рама JF - Journal of Siberian Federal University : Mathematics & Physics JF - Žurnal Sibirskogo Federalʹnogo Universiteta : Matematika i fizika N2 - We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings. N2 - Рассмотрим возмущение комплекса де Рама на компактном многообразии с краем. Это возмущение выходит за рамки комплексов, и поэтому когомологии к нему не относятся. С другой стороны, его кривизна "мала", поэтому существует естественный способ ввести характеристику Эйлера и разработать теорию Лефшеца для возмущения. Данная работа предназначена для попытки разработать теорию когомологий для произвольных последовательностей линейных отображений. KW - de Rham complex KW - cohomology KW - Hodge theory KW - Neumann problem KW - комплекс де Рама KW - когомологии KW - теория Ходжа KW - проблема Неймана Y1 - 2020 U6 - https://doi.org/10.17516/1997-1397-2020-13-5-519-532 SN - 1997-1397 SN - 2313-6022 VL - 13 IS - 5 SP - 519 EP - 532 PB - Siberian Federal University CY - Krasnojarsk ER - TY - JOUR A1 - Hanisch, Florian A1 - Ludewig, Matthias T1 - A rigorous construction of the supersymmetric path integral associated to a compact spin manifold JF - Communications in mathematical physics N2 - We give a rigorous construction of the path integral in N = 1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Guneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah-Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaume, Atiyah, Bismut and Witten. Y1 - 2022 U6 - https://doi.org/10.1007/s00220-022-04336-7 SN - 0010-3616 SN - 1432-0916 VL - 391 IS - 3 SP - 1209 EP - 1239 PB - Springer CY - Berlin ; Heidelberg ER - TY - GEN A1 - Wiljes, Jana de A1 - Tong, Xin T. T1 - Analysis of a localised nonlinear ensemble Kalman Bucy filter with complete and accurate observations T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1221 KW - data assimilation KW - stability and accuracy KW - dimension independent bound KW - localisation KW - high dimensional KW - filter KW - nonlinear Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-540417 SN - 1866-8372 VL - 33 IS - 9 SP - 4752 EP - 4782 PB - IOP Publ. CY - Bristol ER - TY - JOUR A1 - Wiljes, Jana de A1 - Tong, Xin T. T1 - Analysis of a localised nonlinear ensemble Kalman Bucy filter with complete and accurate observations JF - Nonlinearity N2 - Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests. KW - data assimilation KW - stability and accuracy KW - dimension independent bound KW - localisation KW - high dimensional KW - filter KW - nonlinear Y1 - 2020 U6 - https://doi.org/10.1088/1361-6544/ab8d14 SN - 0951-7715 SN - 1361-6544 VL - 33 IS - 9 SP - 4752 EP - 4782 PB - IOP Publ. CY - Bristol ER - TY - JOUR A1 - Clavier, Pierre J. T1 - Borel-Écalle resummation of a two-point function JF - Annales Henri Poincaré : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincaré and by the Swiss Physical Society N2 - We provide an overview of the tools and techniques of resurgence theory used in the Borel-ecalle resummation method, which we then apply to the massless Wess-Zumino model. Starting from already known results on the anomalous dimension of the Wess-Zumino model, we solve its renormalisation group equation for the two-point function in a space of formal series. We show that this solution is 1-Gevrey and that its Borel transform is resurgent. The Schwinger-Dyson equation of the model is then used to prove an asymptotic exponential bound for the Borel transformed two-point function on a star-shaped domain of a suitable ramified complex plane. This proves that the two-point function of the Wess-Zumino model is Borel-ecalle summable. Y1 - 2021 U6 - https://doi.org/10.1007/s00023-021-01057-w SN - 1424-0637 SN - 1424-0661 VL - 22 IS - 6 SP - 2103 EP - 2136 PB - Springer CY - Cham ER - TY - JOUR A1 - Gottwald, Georg A. A1 - Reich, Sebastian T1 - Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We present a supervised learning method to learn the propagator map of a dynamical system from partial and noisy observations. In our computationally cheap and easy-to-implement framework, a neural network consisting of random feature maps is trained sequentially by incoming observations within a data assimilation procedure. By employing Takens's embedding theorem, the network is trained on delay coordinates. We show that the combination of random feature maps and data assimilation, called RAFDA, outperforms standard random feature maps for which the dynamics is learned using batch data. Y1 - 2021 U6 - https://doi.org/10.1063/5.0066080 SN - 1054-1500 SN - 1089-7682 VL - 31 IS - 10 PB - AIP CY - Melville 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 - JOUR A1 - Beckus, Siegfried A1 - Eliaz, Latif T1 - Eigenfunctions growth of R-limits on graphs JF - Journal of spectral theory / European Mathematical Society N2 - A characterization of the essential spectrum of Schrodinger operators on infinite graphs is derived involving the concept of R-limits. This concept, which was introduced previously for operators on N and Z(d) as "right-limits," captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in sigma(ss)(H) corresponds to a bounded generalized eigenfunction of a corresponding R-limit of H. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds. KW - Essential spectrum KW - Schrodinger operators KW - graphs KW - right limits KW - generalized eigenfunctions Y1 - 2021 U6 - https://doi.org/10.4171/JST/389 SN - 1664-039X SN - 1664-0403 VL - 11 IS - 4 SP - 1895 EP - 1933 PB - EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut für Mathematik, Technische Universität CY - Berlin ER - TY - INPR A1 - Böckmann, Christine A1 - Niebsch, Jenny T1 - Examination of the nonlinear LIDAR-operator : the influence of inhomogeneous absorbing spheres on the operator N2 - The determination of the atmospheric aerosol size distribution is an inverse illposed problem. The shape and the material composition of the air-carried particles are two substantial model parameters. Present evaluation algorithms only used an approximation with spherical homogeneous particles. In this paper we propose a new numerically efficient recursive algorithm for inhomogeneous multilayered coated and absorbing particles. Numerical results of real existing particles show that the influence of the two parameters on the model is very important and therefore cannot be ignored. T3 - NLD Preprints - 47 KW - multiwavelength lidar KW - aerosol size distribution KW - inverse ill-posed problem KW - multilayered coated and absorbing aerosol KW - new recursive algorithm Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-14725 ER - TY - JOUR A1 - Lewandowski, Max T1 - Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes JF - Journal of mathematical physics N2 - According to Radzikowski’s celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF−G˜F+G˜A−G˜R) in the sense of Duistermaat and Hörmander [Acta Math. 128, 183–269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by Bär, Ginoux, and Pfäffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Zürich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF−G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied. Y1 - 2022 U6 - https://doi.org/10.1063/5.0055753 SN - 0022-2488 SN - 1089-7658 VL - 63 IS - 1 PB - American Institute of Physics CY - Melville ER -