@article{BeckusEliaz2021, author = {Beckus, Siegfried and Eliaz, Latif}, title = {Eigenfunctions growth of R-limits on graphs}, series = {Journal of spectral theory / European Mathematical Society}, volume = {11}, journal = {Journal of spectral theory / European Mathematical Society}, number = {4}, publisher = {EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut f{\"u}r Mathematik, Technische Universit{\"a}t}, address = {Berlin}, issn = {1664-039X}, doi = {10.4171/JST/389}, pages = {1895 -- 1933}, year = {2021}, abstract = {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.}, language = {en} } @article{BeckusPinchover2020, author = {Beckus, Siegfried and Pinchover, Yehuda}, title = {Shnol-type theorem for the Agmon ground state}, series = {Journal of spectral theory}, volume = {10}, journal = {Journal of spectral theory}, number = {2}, publisher = {EMS Publishing House}, address = {Z{\"u}rich}, issn = {1664-039X}, doi = {10.4171/JST/296}, pages = {355 -- 377}, year = {2020}, abstract = {LetH be a Schrodinger operator defined on a noncompact Riemannianmanifold Omega, and let W is an element of L-infinity (Omega; R). Suppose that the operator H + W is critical in Omega, and let phi be the corresponding Agmon ground state. We prove that if u is a generalized eigenfunction ofH satisfying vertical bar u vertical bar <= C-phi in Omega for some constant C > 0, then the corresponding eigenvalue is in the spectrum of H. The conclusion also holds true if for some K is an element of Omega the operator H admits a positive solution in (Omega) over bar = Omega \ K, and vertical bar u vertical bar <= C psi in (Omega) over bar for some constant C > 0, where psi is a positive solution of minimal growth in a neighborhood of infinity in Omega. Under natural assumptions, this result holds also in the context of infinite graphs, and Dirichlet forms.}, language = {en} } @phdthesis{Berner2016, author = {Berner, Nadine}, title = {Deciphering multiple changes in complex climate time series using Bayesian inference}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-100065}, school = {Universit{\"a}t Potsdam}, pages = {xvi, 135}, year = {2016}, abstract = {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.}, language = {en} } @article{BaerMazzeo2021, author = {B{\"a}r, Christian and Mazzeo, Rafe}, title = {Manifolds with many Rarita-Schwinger fields}, series = {Communications in mathematical physics}, volume = {384}, journal = {Communications in mathematical physics}, number = {1}, publisher = {Springer}, address = {Berlin}, issn = {0010-3616}, doi = {10.1007/s00220-021-04030-0}, pages = {533 -- 548}, year = {2021}, abstract = {The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kahler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.}, language = {en} } @unpublished{BoeckmannBieleNeuberetal.1997, author = {B{\"o}ckmann, Christine and Biele, Jens and Neuber, Roland and Niebsch, Jenny}, title = {Retrieval of multimodal aerosol size distribution by inversion of multiwavelength data}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14360}, year = {1997}, abstract = {The ill-posed problem of aerosol size distribution determination from a small number of backscatter and extinction measurements was solved successfully with a mollifier method which is advantageous since the ill-posed part is performed on exactly given quantities, the points r where n(r) is evaluated may be freely selected. A new twodimensional model for the troposphere is proposed.}, language = {en} } @unpublished{BoeckmannNiebsch1998, author = {B{\"o}ckmann, Christine and Niebsch, Jenny}, title = {Examination of the nonlinear LIDAR-operator : the influence of inhomogeneous absorbing spheres on the operator}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14725}, year = {1998}, abstract = {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.}, language = {en} } @article{Clavier2021, author = {Clavier, Pierre J.}, title = {Borel-{\´E}calle resummation of a two-point function}, series = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincar{\´e} and by the Swiss Physical Society}, volume = {22}, journal = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincar{\´e} and by the Swiss Physical Society}, number = {6}, publisher = {Springer}, address = {Cham}, issn = {1424-0637}, doi = {10.1007/s00023-021-01057-w}, pages = {2103 -- 2136}, year = {2021}, abstract = {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.}, language = {en} } @article{DereudreHoudebert2018, author = {Dereudre, David and Houdebert, Pierre}, title = {Sharp phase transition for the continuum Widom-Rowlinson model}, series = {Annales de l'Institut Henri Poincar{\´e}. B, Probability and statistics}, volume = {57}, journal = {Annales de l'Institut Henri Poincar{\´e}. B, Probability and statistics}, number = {1}, publisher = {Association des Publications de l'Institut Henri Poincar{\´e}}, address = {Bethesda, Md.}, issn = {0246-0203}, doi = {10.1214/20-AIHP1082}, pages = {387 -- 407}, year = {2018}, abstract = {The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in R-d with the formal Hamiltonian defined as the volume of Ux epsilon omega B1(x), where. is a locally finite configuration of points and B-1(x) denotes the unit closed ball centred at x. The model is also tuned by two other parameters: the activity z > 0 related to the intensity of the process and the inverse temperature beta >= 0 related to the strength of the interaction. In the present paper we investigate the phase transition of the model in the point of view of percolation theory and the liquid-gas transition. First, considering the graph connecting points with distance smaller than 2r > 0, we show that for any beta >= 0, there exists 0 <(similar to a)(zc) (beta, r) < +infinity such that an exponential decay of connectivity at distance n occurs in the subcritical phase (i.e. z <(similar to a)(zc) (beta, r)) and a linear lower bound of the connection at infinity holds in the supercritical case (i.e. z >(similar to a)(zc) (beta, r)). These results are in the spirit of recent works using the theory of randomised tree algorithms (Probab. Theory Related Fields 173 (2019) 479-490, Ann. of Math. 189 (2019) 75-99, Duminil-Copin, Raoufi and Tassion (2018)). Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states depending on the parameters z, beta. Old results (Phys. Rev. Lett. 27 (1971) 1040-1041, J. Chem. Phys. 52 (1970) 1670-1684) claim that a non-uniqueness regime occurs for z = beta large enough and it is conjectured that the uniqueness should hold outside such an half line ( z = beta >= beta(c) > 0). We solve partially this conjecture in any dimension by showing that for beta large enough the non-uniqueness holds if and only if z = beta. We show also that this critical value z = beta corresponds to the percolation threshold (similar to a)(zc) (beta, r) = beta for beta large enough, providing a straight connection between these two notions of phase transition.}, language = {en} } @unpublished{Dicken1998, author = {Dicken, Volker}, title = {Simultaneous activity and attenuation reconstruction in emission tomography}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14747}, year = {1998}, abstract = {In single photon emission computed tomography (SPECT) one is interested in reconstructing the activity distribution f of some radiopharmaceutical. The data gathered suffer from attenuation due to the tissue density µ. Each imaged slice incorporates noisy sample values of the nonlinear attenuated Radon transform (formular at this place in the original abstract) Traditional theory for SPECT reconstruction treats µ as a known parameter. In practical applications, however, µ is not known, but either crudely estimated, determined in costly additional measurements or plainly neglected. We demonstrate that an approximation of both f and µ from SPECT data alone is feasible, leading to quantitatively more accurate SPECT images. The result is based on nonlinear Tikhonov regularization techniques for parameter estimation problems in differential equations combined with Gauss-Newton-CG minimization.}, language = {en} } @article{GottwaldReich2021, author = {Gottwald, Georg A. and Reich, Sebastian}, title = {Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {31}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {10}, publisher = {AIP}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/5.0066080}, pages = {8}, year = {2021}, abstract = {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.}, language = {en} }