@book{SavinSchulzeSternin1998, author = {Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris Ju.}, title = {On the invariant index formulas for spectral boundary value problems}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 18b}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {23 S.}, year = {1998}, language = {en} } @book{Schulze1999, author = {Schulze, Bert-Wolfgang}, title = {Operator algebras with symbol hierarchies on manifolds with singularities}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {31 S.}, year = {1999}, language = {en} } @book{BuchholzSchulze1998, author = {Buchholz, Thilo and Schulze, Bert-Wolfgang}, title = {Volterra operators and parabolicity : anisotropic pseudo-differential operators}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 11}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {57 S.}, year = {1998}, language = {en} } @book{SchulzeTarchanov1998, author = {Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {Elliptic complexes of pseudodifferential operators on manifolds with edges}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 14}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {150 S.}, year = {1998}, language = {en} } @article{Schulze1999, author = {Schulze, Bert-Wolfgang}, title = {Iterated asymptotics on corner manifolds}, year = {1999}, language = {en} } @book{NacinovichSchulzeTarchanov1998, author = {Nacinovich, Mauro and Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {Carleman formulas for the Dolbeault cohomology}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 10}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {13 Bl.}, year = {1998}, language = {en} } @article{BurenkovSchulzeTarchanov1998, author = {Burenkov, V. and Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {Extension operators for sobolev spaces commuting with a given transform}, year = {1998}, language = {en} } @article{FedosovSchulzeTarchanov1998, author = {Fedosov, Boris V. and Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {On the index of elliptice operators on a wedge}, year = {1998}, language = {en} } @article{SchulzeTarchanov1998, author = {Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {Green pseudodifferential operators on manifolds with edges}, year = {1998}, language = {en} } @book{FedosovSchulzeTarchanov1998, author = {Fedosov, Boris V. and Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {The index of higher order operators on singular surfaces}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 03}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {27 S.}, year = {1998}, language = {en} } @book{PaneahSchulze1998, author = {Paneah, Boris and Schulze, Bert-Wolfgang}, title = {On the existence of smooth solutions of the Dirichlet problem for hyperbolic differential equations}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 05}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {11 S.}, year = {1998}, language = {en} } @book{FedosovSchulzeTarchanov1998, author = {Fedosov, Boris V. and Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {A remark on the index of symmetric operators}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 04}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {12 S.}, year = {1998}, language = {en} } @book{RabinovichSchulzeTarchanov1998, author = {Rabinovich, Vladimir and Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {Boundary value problems in cuspidal wedges}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 24}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {66 S.}, year = {1998}, language = {en} } @book{Schulze1998, author = {Schulze, Bert-Wolfgang}, title = {Differential equations on singular manifolds : semiclassical theory and operator algebras}, series = {Mathematical topics}, volume = {15}, journal = {Mathematical topics}, publisher = {Wiley-VCH}, address = {Berlin [u.a.]}, pages = {376 S.}, year = {1998}, language = {en} } @book{SchulzeTarchanov1998, author = {Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {Euler solutions of pseudodifferential equations}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 09}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {32 S.}, year = {1998}, language = {en} } @book{Schulze1998, author = {Schulze, Bert-Wolfgang}, title = {Boundary value problems and singular pseudo-differential operators}, publisher = {Wiley-VCH}, address = {Chichester [u.a.]}, pages = {XVI, 637 S.}, year = {1998}, language = {en} } @article{SchulzeSterninSatalov1998, author = {Schulze, Bert-Wolfgang and Sternin, Boris Ju. and Satalov, Viktor E.}, title = {On some global aspects of the theory of partial differential equations on manifolds with singularities}, year = {1998}, language = {en} } @article{SchroheSchulze1998, author = {Schrohe, Elmar and Schulze, Bert-Wolfgang}, title = {Mellin operators in a pseudodifferential calculus for boundary value problems on manifolds with edges}, year = {1998}, language = {en} } @article{NazajkinskijSchulzeSterninetal.1998, author = {Nazajkinskij, Vladimir E. and Schulze, Bert-Wolfgang and Sternin, Boris Ju. and Satalov, Viktor E.}, title = {Spectral boundary value problems and elliptic equations on singular manifolds}, year = {1998}, language = {en} } @article{SchulzeSterninSatalov1998, author = {Schulze, Bert-Wolfgang and Sternin, Boris Ju. and Satalov, Viktor E.}, title = {Differential equations on manifolds with singularities in classes of resurgent functions}, year = {1998}, language = {en} } @book{NazajkinskijSchulzeSternin1998, author = {Nazajkinskij, Vladimir E. and Schulze, Bert-Wolfgang and Sternin, Boris Ju.}, title = {The index of quantized contact transformations on manifolds with conical singularities}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, volume = {1998, 16}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik}, publisher = {Univ.}, address = {Potsdam}, pages = {31 S.}, year = {1998}, language = {en} } @article{ChangKhalilSchulze2021, author = {Chang, Der-Chen and Khalil, Sara and Schulze, Bert-Wolfgang}, title = {Analysis on regular corner spaces}, series = {The journal of geometric analysis}, volume = {31}, journal = {The journal of geometric analysis}, number = {9}, publisher = {Springer}, address = {New York}, issn = {1050-6926}, doi = {10.1007/s12220-021-00614-3}, pages = {9199 -- 9240}, year = {2021}, abstract = {We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel's calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11-51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2 x 2 operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11-51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind.}, language = {en} } @article{RungrottheeraSchulze2014, author = {Rungrottheera, Wannarut and Schulze, Bert-Wolfgang}, title = {Weighted spaces on corner manifolds}, series = {Complex variables and elliptic equations}, volume = {59}, journal = {Complex variables and elliptic equations}, number = {12}, publisher = {Routledge, Taylor \& Francis Group}, address = {Abingdon}, issn = {1747-6933}, doi = {10.1080/17476933.2013.876416}, pages = {1706 -- 1738}, year = {2014}, abstract = {We study spaces on manifolds with double weights and iterated discrete and continuous asymptotics, and their relationship with corner pseudo-differential operators.}, language = {en} } @article{FladHarutyunyanSchneideretal.2011, author = {Flad, Heinz-J{\"u}rgen and Harutyunyan, Gohar and Schneider, Reinhold and Schulze, Bert-Wolfgang}, title = {Explicit Green operators for quantum mechanical Hamiltonians}, series = {Manuscripta mathematica}, volume = {135}, journal = {Manuscripta mathematica}, number = {3-4}, publisher = {Springer}, address = {New York}, issn = {0025-2611}, doi = {10.1007/s00229-011-0429-x}, pages = {497 -- 519}, year = {2011}, abstract = {We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic structure theory in order to establish accurate and efficient models for numerical simulations. Within our approach, coalescence points of particles are treated as embedded geometric singularities in the configuration space of electrons. Based on a general singular pseudo-differential calculus, we provide a recursive scheme for the calculation of the parametrix and corresponding Green operator of a nonrelativistic Hamiltonian. In our singular calculus, the Green operator encodes all the asymptotic information of the eigenfunctions. Explicit calculations and an asymptotic representation for the Green operator of the hydrogen atom and isoelectronic ions are presented.}, language = {en} } @article{MaghsoudiCescaHainzletal.2015, author = {Maghsoudi, Samira and Cesca, Simone and Hainzl, Sebastian and Dahm, Torsten and Z{\"o}ller, Gert and Kaiser, Diethelm}, title = {Maximum Magnitude of Completeness in a Salt Mine}, series = {Bulletin of the Seismological Society of America}, volume = {105}, journal = {Bulletin of the Seismological Society of America}, number = {3}, publisher = {Seismological Society of America}, address = {Albany}, issn = {0037-1106}, doi = {10.1785/0120140039}, pages = {1491 -- 1501}, year = {2015}, abstract = {In this study, we analyze acoustic emission (AE) data recorded at the Morsleben salt mine, Germany, to assess the catalog completeness, which plays an important role in any seismicity analysis. We introduce the new concept of a magnitude completeness interval consisting of a maximum magnitude of completeness (M-c(max)) in addition to the well-known minimum magnitude of completeness. This is required to describe the completeness of the catalog, both for the smallest events (for which the detection performance may be low) and for the largest ones (which may be missed because of sensors saturation). We suggest a method to compute the maximum magnitude of completeness and calculate it for a spatial grid based on (1) the prior estimation of saturation magnitude at each sensor, (2) the correction of the detection probability function at each sensor, including a drop in the detection performance when it saturates, and (3) the combination of detection probabilities of all sensors to obtain the network detection performance. The method is tested using about 130,000 AE events recorded in a period of five weeks, with sources confined within a small depth interval, and an example of the spatial distribution of M-c(max) is derived. The comparison between the spatial distribution of M-c(max) and of the maximum possible magnitude (M-max), which is here derived using a recently introduced Bayesian approach, indicates that M-max exceeds M-c(max) in some parts of the mine. This suggests that some large and important events may be missed in the catalog, which could lead to a bias in the hazard evaluation.}, language = {en} } @phdthesis{Mauerberger2022, author = {Mauerberger, Stefan}, title = {Correlation based Bayesian modeling}, doi = {10.25932/publishup-53782}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-537827}, school = {Universit{\"a}t Potsdam}, pages = {x, 128}, year = {2022}, abstract = {The motivation for this work was the question of reliability and robustness of seismic tomography. The problem is that many earth models exist which can describe the underlying ground motion records equally well. Most algorithms for reconstructing earth models provide a solution, but rarely quantify their variability. If there is no way to verify the imaged structures, an interpretation is hardly reliable. The initial idea was to explore the space of equivalent earth models using Bayesian inference. However, it quickly became apparent that the rigorous quantification of tomographic uncertainties could not be accomplished within the scope of a dissertation. In order to maintain the fundamental concept of statistical inference, less complex problems from the geosciences are treated instead. This dissertation aims to anchor Bayesian inference more deeply in the geosciences and to transfer knowledge from applied mathematics. The underlying idea is to use well-known methods and techniques from statistics to quantify the uncertainties of inverse problems in the geosciences. This work is divided into three parts: Part I introduces the necessary mathematics and should be understood as a kind of toolbox. With a physical application in mind, this section provides a compact summary of all methods and techniques used. The introduction of Bayesian inference makes the beginning. Then, as a special case, the focus is on regression with Gaussian processes under linear transformations. The chapters on the derivation of covariance functions and the approximation of non-linearities are discussed in more detail. Part II presents two proof of concept studies in the field of seismology. The aim is to present the conceptual application of the introduced methods and techniques with moderate complexity. The example about traveltime tomography applies the approximation of non-linear relationships. The derivation of a covariance function using the wave equation is shown in the example of a damped vibrating string. With these two synthetic applications, a consistent concept for the quantification of modeling uncertainties has been developed. Part III presents the reconstruction of the Earth's archeomagnetic field. This application uses the whole toolbox presented in Part I and is correspondingly complex. The modeling of the past 1000 years is based on real data and reliably quantifies the spatial modeling uncertainties. The statistical model presented is widely used and is under active development. The three applications mentioned are intentionally kept flexible to allow transferability to similar problems. The entire work focuses on the non-uniqueness of inverse problems in the geosciences. It is intended to be of relevance to those interested in the concepts of Bayesian inference.}, language = {en} } @misc{BoeckmannRitterCappelletti2018, author = {B{\"o}ckmann, Christine and Ritter, Christoph and Cappelletti, David}, title = {Mathematical tool for a closure study of aerosol microphysical property retrieval using lidar and photometer data}, series = {IGARSS 2018 - 2018 IEEE International Geoscience and Remote Sensing Symposium}, journal = {IGARSS 2018 - 2018 IEEE International Geoscience and Remote Sensing Symposium}, publisher = {IEEE}, address = {New York}, isbn = {978-1-5386-7150-4}, issn = {2153-6996}, doi = {10.1109/IGARSS.2018.8518674}, pages = {5575 -- 5578}, year = {2018}, abstract = {We present a project combining lidar, photometer and particle counter data with a regularization software tool for a closure study of aerosol microphysical property retrieval. In a first step only lidar data are used to retrieve the particle size distribution (PSD). Secondly, photometer data are added, which results in a good consistency of the retrieved PSDs. Finally, those retrieved PSDs may be compared with the measured PSD from a particle counter. The data here were taken in Ny Alesund, Svalbard, as an example.}, language = {en} } @article{DubeBoeckmannRitter2022, author = {Dube, Jonas and B{\"o}ckmann, Christine and Ritter, Christoph}, title = {Lidar-Derived Aerosol Properties from Ny-{\AA}lesund, Svalbard during the MOSAiC Spring 2020}, series = {Remote sensing / Molecular Diversity Preservation International (MDPI)}, volume = {14}, journal = {Remote sensing / Molecular Diversity Preservation International (MDPI)}, number = {11}, publisher = {MDPI}, address = {Basel}, issn = {2072-4292}, doi = {10.3390/rs14112578}, pages = {17}, year = {2022}, abstract = {In this work, we present Raman lidar data (from a Nd:YAG operating at 355 nm, 532 nm and 1064 nm) from the international research village Ny-Alesund for the time period of January to April 2020 during the Arctic haze season of the MOSAiC winter. We present values of the aerosol backscatter, the lidar ratio and the backscatter Angstrom exponent, though the latter depends on wavelength. The aerosol polarization was generally below 2\%, indicating mostly spherical particles. We observed that events with high backscatter and high lidar ratio did not coincide. In fact, the highest lidar ratios (LR > 75 sr at 532 nm) were already found by January and may have been caused by hygroscopic growth, rather than by advection of more continental aerosol. Further, we performed an inversion of the lidar data to retrieve a refractive index and a size distribution of the aerosol. Our results suggest that in the free troposphere (above approximate to 2500 m) the aerosol size distribution is quite constant in time, with dominance of small particles with a modal radius well below 100 nm. On the contrary, below approximate to 2000 m in altitude, we frequently found gradients in aerosol backscatter and even size distribution, sometimes in accordance with gradients of wind speed, humidity or elevated temperature inversions, as if the aerosol was strongly modified by vertical displacement in what we call the "mechanical boundary layer". Finally, we present an indication that additional meteorological soundings during MOSAiC campaign did not necessarily improve the fidelity of air backtrajectories.}, language = {en} } @article{PornsawadSapsakulBoeckmann2019, author = {Pornsawad, Pornsarp and Sapsakul, Nantawan and B{\"o}ckmann, Christine}, title = {A modified asymptotical regularization of nonlinear ill-posed problems}, series = {Mathematics}, volume = {7}, journal = {Mathematics}, edition = {5}, publisher = {MDPI}, address = {Basel, Schweiz}, issn = {2227-7390}, doi = {10.3390/math7050419}, pages = {19}, year = {2019}, abstract = {In this paper, we investigate the continuous version of modified iterative Runge-Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of โˆฅ๐น(๐‘ฅ๐›ฟ(๐‘‡))-๐‘ฆ๐›ฟโˆฅ=๐œ๐›ฟ+ for some ๐›ฟ+>๐›ฟ, and an appropriate source condition. We yield the optimal rate of convergence.}, language = {en} } @unpublished{PornsawadBoeckmann2014, author = {Pornsawad, Pornsarp and B{\"o}ckmann, Christine}, title = {Modified iterative Runge-Kutta-type methods for nonlinear ill-posed problems}, volume = {3}, number = {7}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70834}, pages = {30}, year = {2014}, abstract = {This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under H{\"o}lder-type source-wise condition if the Fr{\ยดe}chet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt and Radau methods.}, language = {en} } @article{PornsawadBoeckmannPanitsupakamon2022, author = {Pornsawad, Pornsarp and B{\"o}ckmann, Christine and Panitsupakamon, Wannapa}, title = {The Levenberg-Marquardt regularization for the backward heat equation with fractional derivative}, series = {Electronic transactions on numerical analysis - ETNA}, volume = {57}, journal = {Electronic transactions on numerical analysis - ETNA}, publisher = {Kent State University}, address = {Kent}, isbn = {978-3-7001-8258-0}, issn = {1068-9613}, doi = {10.1553/etna_vol57s67}, pages = {67 -- 79}, year = {2022}, abstract = {The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg-Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a H{\"o}lder-type source condition. Numerical examples for one and two dimensions are provided.}, language = {en} } @misc{PereraBoeckmann2019, author = {Perera, Upeksha and B{\"o}ckmann, Christine}, title = {Solutions of direct and inverse even-order Sturm-Liouville problems using Magnus expansion}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1336}, issn = {1866-8372}, doi = {10.25932/publishup-47341}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-473414}, pages = {24}, year = {2019}, abstract = {In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm-Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.}, language = {en} } @misc{PornsawadSapsakulBoeckmann2019, author = {Pornsawad, Pornsarp and Sapsakul, Nantawan and B{\"o}ckmann, Christine}, title = {A modified asymptotical regularization of nonlinear ill-posed problems}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1335}, issn = {1866-8372}, doi = {10.25932/publishup-47343}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-473433}, pages = {19}, year = {2019}, abstract = {In this paper, we investigate the continuous version of modified iterative Runge-Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of โˆฅ๐น(๐‘ฅ๐›ฟ(๐‘‡))-๐‘ฆ๐›ฟโˆฅ=๐œ๐›ฟ+ for some ๐›ฟ+>๐›ฟ, and an appropriate source condition. We yield the optimal rate of convergence.}, language = {en} } @article{PornsawadD'AmicoBoeckmannetal.2012, author = {Pornsawad, Pornsarp and D'Amico, Giuseppe and B{\"o}ckmann, Christine and Amodeo, Aldo and Pappalardo, Gelsomina}, title = {Retrieval of aerosol extinction coefficient profiles from Raman lidar data by inversion method}, series = {Applied optics}, volume = {51}, journal = {Applied optics}, number = {12}, publisher = {Optical Society of America}, address = {Washington}, issn = {1559-128X}, doi = {10.1364/AO.51.002035}, pages = {2035 -- 2044}, year = {2012}, abstract = {We regard the problem of differentiation occurring in the retrieval of aerosol extinction coefficient profiles from inelastic Raman lidar signals by searching for a stable solution of the resulting Volterra integral equation. An algorithm based on a projection method and iterative regularization together with the L-curve method has been performed on synthetic and measured lidar signals. A strategy to choose a suitable range for the integration within the framework of the retrieval of optical properties is proposed here for the first time to our knowledge. The Monte Carlo procedure has been adapted to treat the uncertainty in the retrieval of extinction coefficients.}, language = {en} } @article{WormellReich2021, author = {Wormell, Caroline L. and Reich, Sebastian}, title = {Spectral convergence of diffusion maps}, series = {SIAM journal on numerical analysis / Society for Industrial and Applied Mathematics}, volume = {59}, journal = {SIAM journal on numerical analysis / Society for Industrial and Applied Mathematics}, number = {3}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {0036-1429}, doi = {10.1137/20M1344093}, pages = {1687 -- 1734}, year = {2021}, abstract = {Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on the approximation error are, however, generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localized compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace-Beltrami operator's associated PDE and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretization. We also introduce an alternative normalization for diffusion maps based on Sinkhorn weights. This normalization approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalization on flat domains, and we present a highly efficient rigorous algorithm to compute the Sinkhorn weights.}, language = {en} } @article{KellerLiuPeyerimhoff2021, author = {Keller, Matthias and Liu, Shiping and Peyerimhoff, Norbert}, title = {A note on eigenvalue bounds for non-compact manifolds}, series = {Mathematische Nachrichten}, volume = {294}, journal = {Mathematische Nachrichten}, number = {6}, publisher = {Wiley-VCH}, address = {Weinheim}, issn = {0025-584X}, doi = {10.1002/mana.201900209}, pages = {1134 -- 1139}, year = {2021}, abstract = {In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.}, language = {en} } @article{SaynischWagnerBaerenzungHornschildetal.2021, author = {Saynisch-Wagner, Jan and B{\"a}renzung, Julien and Hornschild, Aaron and Irrgang, Christopher and Thomas, Maik}, title = {Tide-induced magnetic signals and their errors derived from CHAMP and Swarm satellite magnetometer observations}, series = {Earth, planets and space : EPS}, volume = {73}, journal = {Earth, planets and space : EPS}, number = {1}, publisher = {Springer}, address = {Heidelberg}, issn = {1880-5981}, doi = {10.1186/s40623-021-01557-3}, pages = {11}, year = {2021}, abstract = {Satellite-measured tidal magnetic signals are of growing importance. These fields are mainly used to infer Earth's mantle conductivity, but also to derive changes in the oceanic heat content. We present a new Kalman filter-based method to derive tidal magnetic fields from satellite magnetometers: KALMAG. The method's advantage is that it allows to study a precisely estimated posterior error covariance matrix. We present the results of a simultaneous estimation of the magnetic signals of 8 major tides from 17 years of Swarm and CHAMP data. For the first time, robustly derived posterior error distributions are reported along with the reported tidal magnetic fields. The results are compared to other estimates that are either based on numerical forward models or on satellite inversions of the same data. For all comparisons, maximal differences and the corresponding globally averaged RMSE are reported. We found that the inter-product differences are comparable with the KALMAG-based errors only in a global mean sense. Here, all approaches give values of the same order, e.g., 0.09 nT-0.14 nT for M2. Locally, the KALMAG posterior errors are up to one order smaller than the inter-product differences, e.g., 0.12 nT vs. 0.96 nT for M2.}, language = {en} } @article{SaposhnikovMorosovSaposhnikovetal.1997, author = {Saposhnikov, V. V. and Morosov, Andrej and Saposhnikov, Vl. V. and G{\"o}ssel, Michael}, title = {Design of self-checking unidirectional combinational circuits with low area overhead}, year = {1997}, language = {en} } @article{LilienkampvonSpechtWeatherilletal.2022, author = {Lilienkamp, Henning and von Specht, Sebastian and Weatherill, Graeme and Caire, Giuseppe and Cotton, Fabrice}, title = {Ground-Motion modeling as an image processing task}, series = {Bulletin of the Seismological Society of America}, volume = {112}, journal = {Bulletin of the Seismological Society of America}, number = {3}, publisher = {Seismological Society of America}, address = {Albany}, issn = {0037-1106}, doi = {10.1785/0120220008}, pages = {1565 -- 1582}, year = {2022}, abstract = {We construct and examine the prototype of a deep learning-based ground-motion model (GMM) that is both fully data driven and nonergodic. We formulate ground-motion modeling as an image processing task, in which a specific type of neural network, the U-Net, relates continuous, horizontal maps of earthquake predictive parameters to sparse observations of a ground-motion intensity measure (IM). The processing of map-shaped data allows the natural incorporation of absolute earthquake source and observation site coordinates, and is, therefore, well suited to include site-, source-, and path-specific amplification effects in a nonergodic GMM. Data-driven interpolation of the IM between observation points is an inherent feature of the U-Net and requires no a priori assumptions. We evaluate our model using both a synthetic dataset and a subset of observations from the KiK-net strong motion network in the Kanto basin in Japan. We find that the U-Net model is capable of learning the magnitude???distance scaling, as well as site-, source-, and path-specific amplification effects from a strong motion dataset. The interpolation scheme is evaluated using a fivefold cross validation and is found to provide on average unbiased predictions. The magnitude???distance scaling as well as the site amplification of response spectral acceleration at a period of 1 s obtained for the Kanto basin are comparable to previous regional studies.}, language = {en} } @article{ZoellerUllahBindietal.2017, author = {Z{\"o}ller, Gert and Ullah, Shahid and Bindi, Dino and Parolai, Stefano and Mikhailova, Natalya}, title = {The largest expected earthquake magnitudes in Central Asia}, series = {Seismicity, fault rupture and earthquake hazards in slowly deforming regions}, volume = {432}, journal = {Seismicity, fault rupture and earthquake hazards in slowly deforming regions}, publisher = {The Geological Society}, address = {London}, isbn = {978-1-86239-745-3}, issn = {0305-8719}, doi = {10.1144/SP432.3}, pages = {29 -- 40}, year = {2017}, abstract = {The knowledge of the largest expected earthquake magnitude in a region is one of the key issues in probabilistic seismic hazard calculations and the estimation of worst-case scenarios. Earthquake catalogues are the most informative source of information for the inference of earthquake magnitudes. We analysed the earthquake catalogue for Central Asia with respect to the largest expected magnitudes m(T) in a pre-defined time horizon T-f using a recently developed statistical methodology, extended by the explicit probabilistic consideration of magnitude errors. For this aim, we assumed broad error distributions for historical events, whereas the magnitudes of recently recorded instrumental earthquakes had smaller errors. The results indicate high probabilities for the occurrence of large events (M >= 8), even in short time intervals of a few decades. The expected magnitudes relative to the assumed maximum possible magnitude are generally higher for intermediate-depth earthquakes (51-300 km) than for shallow events (0-50 km). For long future time horizons, for example, a few hundred years, earthquakes with M >= 8.5 have to be taken into account, although, apart from the 1889 Chilik earthquake, it is probable that no such event occurred during the observation period of the catalogue.}, language = {en} } @article{HarutjunjanSchulze2004, author = {Harutjunjan, Gohar and Schulze, Bert-Wolfgang}, title = {Reduction of orders in boundary value problems without transmission property}, issn = {0025-5645}, year = {2004}, abstract = {Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary}, language = {en} } @article{HarutjunjanSchulze2004, author = {Harutjunjan, Gohar and Schulze, Bert-Wolfgang}, title = {Parametrices of mixed elliptic problems}, issn = {0025-584X}, year = {2004}, abstract = {Mixed elliptic problems for differential operators A in a domain Q with smooth boundary Y are studied in the form Au = f in Omega, T+/-u = g+/- on Y+/-, where Y+/- subset of Y are manifolds with a common boundary Z, such that Y- boolean OR Y+ = Y and Y- boolean AND Y+ = z, with boundary conditions T+/- on Y+/- (with smooth coefficients up to Z from the respective side) satisfying the Shapiro-Lopatinskij condition. We consider such problems in standard Sobolev spaces and characterise natural extra conditions on the interface Z with an analogue of Shapiro-Lopatinskij ellipticity for an associated transmission problem on the boundary; then the extended operator is Fredholm. The transmission operators on the boundary with respect to Z belong to a complete pseudo-differential calculus, a modification of the algebra of boundary value problems without the transmission property. We construct parametrices of elliptic elements in that calculus, and we obtain parametrices of the original mixed problems under additional conditions on the interface. We consider the Zaremba problem and other mixed problems for the Laplace operator, determine the number of extra conditions and calculate the index of associated Fredholm operators. (C) 2004 WILEY-VCH Verlag GmbH \& Co. KGaA, Weinheim}, language = {en} } @book{NazajkinskijSavinSchulzeetal.2003, author = {Nazajkinskij, Vladimir E. and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Elliptic theory on manifolds with nonisolated singularities : 5 Index formulas for elliptic problems on manifolds with edges}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {84 S.}, year = {2003}, language = {en} } @book{Schulze2003, author = {Schulze, Bert-Wolfgang}, title = {Toeplitz operators, and ellipticity of boundary value problems with global projection conditions}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {66 S.}, year = {2003}, language = {en} } @book{DinesHarutjunjanSchulze2003, author = {Dines, Nicoleta and Harutjunjan, Gohar and Schulze, Bert-Wolfgang}, title = {The Zaremba problem in edge sobolev spaces}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {38 S.}, year = {2003}, language = {en} } @book{DinesSchulze2003, author = {Dines, Nicoleta and Schulze, Bert-Wolfgang}, title = {Melin-edges representations of elliptic operators}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {43 S.}, year = {2003}, language = {en} } @book{Schulze2003, author = {Schulze, Bert-Wolfgang}, title = {Crack theory with singularities at the boundary}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {36 S.}, year = {2003}, language = {en} } @book{NazajkinskijSavinSchulzeetal.2003, author = {Nazajkinskij, Vladimir E. and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Localization (surgery) in elliptic theory}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {32 S.}, year = {2003}, language = {en} } @book{NazajkinskijSavinSchulzeetal.2003, author = {Nazajkinskij, Vladimir E. and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Eta Invariant and the Spectral Flow}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {31 S.}, year = {2003}, language = {en} } @book{NazajkinskijSavinSchulzeetal.2003, author = {Nazajkinskij, Vladimir E. and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Manifolds with isolated singularities}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {38 S.}, year = {2003}, language = {en} } @book{NazajkinskijSavinSchulzeetal.2003, author = {Nazajkinskij, Vladimir E. and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Index defects in elliptic theory}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {30 S.}, year = {2003}, language = {en} } @book{FedosovSchulzeTarkhanov2003, author = {Fedosov, Boris V. and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {On index theorem for symplectic orbifolds}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {44 S.}, year = {2003}, language = {en} } @book{DeDonnoSchulze2003, author = {De Donno, G. and Schulze, Bert-Wolfgang}, title = {Meromorphic symbolic structures for boundary value problems on manifolds with edges}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {40 S.}, year = {2003}, language = {en} } @book{NazajkinskijSavinSchulzeetal.2003, author = {Nazajkinskij, Vladimir E. and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Pseudodifferential Operators}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {25 S.}, year = {2003}, language = {en} } @book{SchulzeTarkhanov2003, author = {Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {Symbol algebra for manifolds with cuspidal singularities}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {37 S.}, year = {2003}, language = {en} } @book{KapanadzeSchulze2003, author = {Kapanadze, David and Schulze, Bert-Wolfgang}, title = {Asymptotics of potentials in the edge calculus}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {34 S.}, year = {2003}, language = {en} } @book{KrainerSchulze2001, author = {Krainer, Thomas and Schulze, Bert-Wolfgang}, title = {On the inverse of parabolic systems of partial differential equation of general form in an infinite space-time cylinder : Chapters VI - VII}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {220 S.}, year = {2001}, language = {en} } @book{KrainerSchulze2001, author = {Krainer, Thomas and Schulze, Bert-Wolfgang}, title = {On the inverse of parabolic systems of partial differential equation of general form in an infinite space-time cylinder : Chapters I - II}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {220 S.}, year = {2001}, language = {en} } @book{EgorovKondratievSchulze2001, author = {Egorov, Yu and Kondratiev, V. A. and Schulze, Bert-Wolfgang}, title = {On completeness of eigenfunctions of an elliptic operator on a manifold with concial points}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgrupe Partielle Differentialgleichun}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgrupe Partielle Differentialgleichun}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-339X}, pages = {11 S.}, year = {2001}, language = {en} } @book{NazajkinskijSchulzeSternin2001, author = {Nazajkinskij, Vladimir E. and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Localization Problem in Index Theory of Elliptic Operators}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {52 S.}, year = {2001}, language = {en} } @book{HarutjunjanSchulze2002, author = {Harutjunjan, Gohar and Schulze, Bert-Wolfgang}, title = {Asymptotics and relative index on a cylinder with conical cross section}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {45 S.}, year = {2002}, language = {en} } @book{KapanadzeSchulze2001, author = {Kapanadze, David and Schulze, Bert-Wolfgang}, title = {Symbolic calcullus for boundary value problems on manifolds with edges}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {39 S.}, year = {2001}, language = {en} } @book{HarutjunjanSchulze2001, author = {Harutjunjan, Gohar and Schulze, Bert-Wolfgang}, title = {Mixed problems and edge calculus : symbol structures}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {43 S.}, year = {2001}, language = {en} } @book{SchulzeSeiler2001, author = {Schulze, Bert-Wolfgang and Seiler, J{\"o}rg}, title = {The edge algebra structure of boundary value problems}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {52 S.}, year = {2001}, language = {en} } @book{Schulze2001, author = {Schulze, Bert-Wolfgang}, title = {Operators with symbol hierarchies and iterated asymptotics}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, publisher = {Univ.}, address = {Potsdam}, issn = {1437-739X}, pages = {53 S.}, year = {2001}, language = {en} } @article{SchulzeSeiler2004, author = {Schulze, Bert-Wolfgang and Seiler, J{\"o}rg}, title = {Boundary value problems with global projection conditions}, issn = {0022-1236}, year = {2004}, abstract = {Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus. (C) 2003 Elsevier Inc. All rights reserved}, language = {en} } @article{FedosovSchulzeTarkhanov2004, author = {Fedosov, Boris and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {On the index theorem for symplectic orbifolds}, issn = {0373-0956}, year = {2004}, abstract = {We give an explicit construction of the trace on the algebra of quantum observables on a symplectiv orbifold and propose an index formula}, language = {en} } @article{NazajkinskijSavinSterninetal.2004, author = {Nazajkinskij, Vladimir E. and Savin, Anton and Sternin, Boris Ju. and Schulze, Bert-Wolfgang}, title = {On the index of differential operators on manifolds with edges}, issn = {1064-5624}, year = {2004}, language = {en} } @article{NazajkinskijSavinSterninetal.2004, author = {Nazajkinskij, Vladimir E. and Savin, Anton and Sternin, Boris Ju. and Schulze, Bert-Wolfgang}, title = {On the existence of elliptic problems on manifolds with edges}, issn = {1064-5624}, year = {2004}, language = {en} } @article{NazajkinskijSavinSterninetal.2004, author = {Nazajkinskij, Vladimir E. and Savin, Anton and Sternin, Boris Ju. and Schulze, Bert-Wolfgang}, title = {Pseudodifferential operators on manifolds with edges}, issn = {1064-5624}, year = {2004}, language = {en} } @article{ChangSchulze2018, author = {Chang, Der-Chen and Schulze, Bert-Wolfgang}, title = {Corner spaces and Mellin quantization}, series = {Journal of nonlinear and convex analysis : an international journal}, volume = {19}, journal = {Journal of nonlinear and convex analysis : an international journal}, number = {2}, publisher = {Yokohama Publishers}, address = {Yokohama}, issn = {1345-4773}, pages = {179 -- 195}, year = {2018}, abstract = {Manifolds with corners in the present investigation are non-smooth configurations - specific stratified spaces - with an incomplete metric such as cones, manifolds with edges, or corners of piecewise smooth domains in Euclidean space. We focus here on operators on such "corner manifolds" of singularity order <= 2, acting in weighted corner Sobolev spaces. The corresponding corner degenerate pseudo-differential operators are formulated via Mellin quantizations, and they also make sense on infinite singular cones.}, language = {en} } @article{ChangSchulze2017, author = {Chang, Der-Chen and Schulze, Bert-Wolfgang}, title = {Ellipticity on spaces with higher singularities}, series = {Science China Mathematics}, volume = {60}, journal = {Science China Mathematics}, number = {11}, publisher = {Science China Press}, address = {Beijing}, issn = {1674-7283}, doi = {10.1007/s11425-016-0519-9}, pages = {2053 -- 2076}, year = {2017}, abstract = {We study corner-degenerate pseudo-differential operators of any singularity order and develop ellipticity based on the principal symbolic hierarchy, associated with the stratification of the underlying space. We construct parametrices within the calculus and discuss the aspect of additional trace and potential conditions along lower-dimensional strata.}, language = {en} } @article{KhalilSchulze2017, author = {Khalil, Sara and Schulze, Bert-Wolfgang}, title = {Boundary problems on a manifold with edge}, series = {Asian-European Journal of Mathematics}, volume = {10}, journal = {Asian-European Journal of Mathematics}, number = {2}, publisher = {World Scientific}, address = {Singapore}, issn = {1793-5571}, doi = {10.1142/S1793557117500875}, pages = {43}, year = {2017}, abstract = {We establish a calculus of boundary value problems (BVPs) on a manifold N with boundary and edge, based on Boutet de Monvel's theory of BVPs in the case of a smooth boundary and on the edge calculus, where in the present case the model cone has a base which is a compact manifold with boundary. The corresponding calculus with boundary and edge is a unification of both structures and controls different operator-valued symbolic structures, in order to obtain ellipticity and parametrices.}, language = {en} } @article{ChangHedayatMahmoudiSchulze2017, author = {Chang, Der-Chen and Hedayat Mahmoudi, Mahdi and Schulze, Bert-Wolfgang}, title = {Singular degenerate operators}, series = {Applicable analysis : an international journal}, volume = {96}, journal = {Applicable analysis : an international journal}, number = {14}, publisher = {Routledge, Taylor \& Francis Group}, address = {Abingdon}, issn = {0003-6811}, doi = {10.1080/00036811.2017.1336546}, pages = {2434 -- 2456}, year = {2017}, abstract = {We outline some simplified and more general method for constructing parametrices on higher singular spaces. We also outline basic ideas on operators on manifolds with conical or edge singularities.}, language = {en} } @article{HarutyunyanSchulze2006, author = {Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {The Zaremba problem with singular interfaces as a corner boundary value problem}, series = {Potential analysis : an international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis}, volume = {25}, journal = {Potential analysis : an international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis}, publisher = {Springer}, address = {Dordrecht}, issn = {0926-2601}, doi = {10.1007/s11118-006-9020-6}, pages = {327 -- 369}, year = {2006}, abstract = {We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y, i.e., Au = f in int X, T (+/-) u = g(+/-) on int Y+/-, where Y is subdivided into subsets Y+/- with an interface Z and boundary conditions T+/- on Y+/- that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z subset of Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T- Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in Bull. Sci. Math. ( to appear). With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.}, language = {en} } @article{FladHarutyunyanSchulze2015, author = {Flad, Heinz-J{\"u}rgen and Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {Singular analysis and coupled cluster theory}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {17}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, number = {47}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c5cp01183c}, pages = {31530 -- 31541}, year = {2015}, abstract = {The primary motivation for systematic bases in first principles electronic structure simulations is to derive physical and chemical properties of molecules and solids with predetermined accuracy. This requires a detailed understanding of the asymptotic behaviour of many-particle Coulomb systems near coalescence points of particles. Singular analysis provides a convenient framework to study the asymptotic behaviour of wavefunctions near these singularities. In the present work, we want to introduce the mathematical framework of singular analysis and discuss a novel asymptotic parametrix construction for Hamiltonians of many-particle Coulomb systems. This corresponds to the construction of an approximate inverse of a Hamiltonian operator with remainder given by a so-called Green operator. The Green operator encodes essential asymptotic information and we present as our main result an explicit asymptotic formula for this operator. First applications to many-particle models in quantum chemistry are presented in order to demonstrate the feasibility of our approach. The focus is on the asymptotic behaviour of ladder diagrams, which provide the dominant contribution to short-range correlation in coupled cluster theory. Furthermore, we discuss possible consequences of our asymptotic analysis with respect to adaptive wavelet approximation.}, language = {en} } @article{FladFladHarutyunyanSchulze2018, author = {Flad, Heinz-J{\"u}rgen and Flad-Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {Ellipticity of the quantum mechanical Hamiltonians}, series = {Journal of pseudo-differential operators and applications}, volume = {9}, journal = {Journal of pseudo-differential operators and applications}, number = {3}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-017-0201-4}, pages = {451 -- 467}, year = {2018}, abstract = {In paper (Flad and Harutyunyan in Discrete Contin Dyn Syst 420-429, 2011) is shown that the Hamiltonian of the helium atom in the Born-Oppenheimer approximation, in the case if two particles coincide, is an edge-degenerate operator, which is elliptic in the corresponding edge calculus. The aim of this paper is an analogous investigation in the case if all three particles coincide. More precisely, we show that the Hamiltonian in the mentioned case is a corner-degenerate operator, which is elliptic as an operator in the corner analysis.}, language = {en} } @article{HedayatMahmoudiSchulze2018, author = {Hedayat Mahmoudi, Mahdi and Schulze, Bert-Wolfgang}, title = {A new approach to the second order edge calculus}, series = {Journal of pseudo-differential operators and applications}, volume = {9}, journal = {Journal of pseudo-differential operators and applications}, number = {2}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-017-0191-2}, pages = {265 -- 300}, year = {2018}, abstract = {We establish essential steps of an iterative approach to operator algebras, ellipticity and Fredholm property on stratified spaces with singularities of second order. We cover, in particular, corner-degenerate differential operators. Our constructions are focused on the case where no additional conditions of trace and potential type are posed, but this case works well and will be considered in a forthcoming paper as a conclusion of the present calculus.}, language = {en} } @article{KhalilSchulze2019, author = {Khalil, Sara and Schulze, Bert-Wolfgang}, title = {Calculus on a Manifold with Edge and Boundary}, series = {Complex analysis and operator theory}, volume = {13}, journal = {Complex analysis and operator theory}, number = {6}, publisher = {Springer}, address = {Basel}, issn = {1661-8254}, doi = {10.1007/s11785-018-0800-y}, pages = {2627 -- 2670}, year = {2019}, abstract = {We study elements of the calculus of boundary value problems in a variant of Boutet de Monvel's algebra (Acta Math 126:11-51, 1971) on a manifold N with edge and boundary. If the boundary is empty then the approach corresponds to Schulze (Symposium on partial differential equations (Holzhau, 1988), BSB Teubner, Leipzig, 1989) and other papers from the subsequent development. For non-trivial boundary we study Mellin-edge quantizations and compositions within the structure in terms a new Mellin-edge quantization, compared with a more traditional technique. Similar structures in the closed case have been studied in Gil et al.}, language = {en} } @article{FladHarutyunyanSchulze2016, author = {Flad, H. -J. and Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {Asymptotic parametrices of elliptic edge operators}, series = {Journal of pseudo-differential operators and applications}, volume = {7}, journal = {Journal of pseudo-differential operators and applications}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-016-0159-7}, pages = {321 -- 363}, year = {2016}, abstract = {We study operators on singular manifolds, here of conical or edge type, and develop a new general approach of representing asymptotics of solutions to elliptic equations close to the singularities. We introduce asymptotic parametrices, using tools from cone and edge pseudo-differential algebras. Our structures are motivated by models of many-particle physics with singular Coulomb potentials that contribute higher order singularities in Euclidean space, determined by the number of particles.}, language = {en} } @article{SchulzeSeiler2019, author = {Schulze, Bert-Wolfgang and Seiler, J{\"o}rg}, title = {Elliptic complexes on manifolds with boundary}, series = {The journal of geometric analysis}, volume = {29}, journal = {The journal of geometric analysis}, number = {1}, publisher = {Springer}, address = {New York}, issn = {1050-6926}, doi = {10.1007/s12220-018-0014-6}, pages = {656 -- 706}, year = {2019}, abstract = {We show that elliptic complexes of (pseudo) differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi, and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah-Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.}, language = {en} } @article{ChangViahmoudiSchulze2016, author = {Chang, D. -C. and Viahmoudi, M. Hedayat and Schulze, Bert-Wolfgang}, title = {PSEUDO-DIFFERENTIAL ANALYSIS WITH TWISTED SYMBOLIC STRUCTURE}, series = {Journal of nonlinear and convex analysis : an international journal}, volume = {17}, journal = {Journal of nonlinear and convex analysis : an international journal}, publisher = {Yokohama Publishers}, address = {Yokohama}, issn = {1345-4773}, pages = {1889 -- 1937}, year = {2016}, abstract = {This paper is devoted to pseudo-differential operators and new applications. We establish necessary extensions of the standard calculus to specific classes of operator-valued symbols occurring in principal symbolic hierarchies of operators on manifolds with singularities or stratified spaces.}, language = {en} } @article{ChangSchulze2017, author = {Chang, D. -C. and Schulze, Bert-Wolfgang}, title = {Calculus on spaces with higher singularities}, series = {Journal of pseudo-differential operators and applications}, volume = {8}, journal = {Journal of pseudo-differential operators and applications}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-016-0180-x}, pages = {585 -- 622}, year = {2017}, abstract = {We establish extensions of the standard pseudo-differential calculus to specific classes of operators with operator-valued symbols occurring in symbolic hierarchies motivated by manifolds with higher singularities or stratified spaces.}, language = {en} } @article{ChangQianSchulze2015, author = {Chang, Der-Chen and Qian, Tao and Schulze, Bert-Wolfgang}, title = {Corner Boundary Value Problems}, series = {Complex analysis and operator theory}, volume = {9}, journal = {Complex analysis and operator theory}, number = {5}, publisher = {Springer}, address = {Basel}, issn = {1661-8254}, doi = {10.1007/s11785-014-0424-9}, pages = {1157 -- 1210}, year = {2015}, abstract = {Boundary value problems on a manifold with smooth boundary are closely related to the edge calculus where the boundary plays the role of an edge. The problem of expressing parametrices of Shapiro-Lopatinskij elliptic boundary value problems for differential operators gives rise to pseudo-differential operators with the transmission property at the boundary. However, there are interesting pseudo-differential operators without the transmission property, for instance, the Dirichlet-to-Neumann operator. In this case the symbols become edge-degenerate under a suitable quantisation, cf. Chang et al. (J Pseudo-Differ Oper Appl 5(2014):69-155, 2014). If the boundary itself has singularities, e.g., conical points or edges, then the symbols are corner-degenerate. In the present paper we study elements of the corresponding corner pseudo-differential calculus.}, language = {en} } @article{ChangHabalSchulze2014, author = {Chang, Der-Chen and Habal, Nadia and Schulze, Bert-Wolfgang}, title = {The edge algebra structure of the Zaremba problem}, series = {Journal of pseudo-differential operators and applications}, volume = {5}, journal = {Journal of pseudo-differential operators and applications}, number = {1}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-013-0088-7}, pages = {69 -- 155}, year = {2014}, abstract = {We study mixed boundary value problems, here mainly of Zaremba type for the Laplacian within an edge algebra of boundary value problems. The edge here is the interface of the jump from the Dirichlet to the Neumann condition. In contrast to earlier descriptions of mixed problems within such an edge calculus, cf. (Harutjunjan and Schulze, Elliptic mixed, transmission and singular crack problems, 2008), we focus on new Mellin edge quantisations of the Dirichlet-to-Neumann operator on the Neumann side of the boundary and employ a pseudo-differential calculus of corresponding boundary value problems without the transmission property at the interface. This allows us to construct parametrices for the original mixed problem in a new and transparent way.}, language = {en} } @article{LyuQianSchulze2015, author = {Lyu, Xiaojing and Qian, Tao and Schulze, Bert-Wolfgang}, title = {Order filtrations of the edge algebra}, series = {Journal of pseudo-differential operators and applications}, volume = {6}, journal = {Journal of pseudo-differential operators and applications}, number = {3}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-015-0126-8}, pages = {279 -- 305}, year = {2015}, abstract = {By edge algebra we understand a pseudo-differential calculus on a manifold with edge. The operators have a two-component principal symbolic hierarchy which determines operators up to lower order terms. Those belong to a filtration of the corresponding operator spaces. We give a new characterisation of this structure, based on an alternative representation of edge amplitude functions only containing holomorphic edge-degenerate Mellin symbols.}, language = {en} } @article{SchroheSchulze1997, author = {Schrohe, Elmar and Schulze, Bert-Wolfgang}, title = {A symbol algebra for pseudodifferential boundary value problems on manifolds with edges}, year = {1997}, language = {en} } @book{EgorovSchulze1997, author = {Egorov, Jurij V. and Schulze, Bert-Wolfgang}, title = {Pseudo-differential operators, singularities, applicatons}, series = {Operator theory}, volume = {93}, journal = {Operator theory}, publisher = {Birkh{\"a}user}, address = {Basel}, isbn = {3-7643-5484-4}, pages = {XII; 349 S.}, year = {1997}, language = {en} } @article{SchulzeWei2009, author = {Schulze, Bert-Wolfgang and Wei, Ya-wei}, title = {Edge-boundary problems with singular trace conditions}, issn = {0232-704X}, doi = {10.1007/s10455-008-9143-7}, year = {2009}, abstract = {The ellipticity of boundary value problems on a smooth manifold with boundary relies on a two-component principal symbolic structure (sigma(psi), sigma(partial derivative)), consisting of interior and boundary symbols. In the case of a smooth edge on manifolds with boundary, we have a third symbolic component, namely, the edge symbol sigma(boolean AND), referring to extra conditions on the edge, analogously as boundary conditions. Apart from such conditions 'in integral form' there may exist singular trace conditions, investigated in Kapanadze et al., Internal Equations and Operator Theory, 61, 241-279, 2008 on 'closed' manifolds with edge. Here, we concentrate on the phenomena in combination with boundary conditions and edge problem.}, language = {en} } @article{Schulze2009, author = {Schulze, Bert-Wolfgang}, title = {On a paper of Krupchyk, Tarkhanov, and Tuomela}, issn = {0022-1236}, doi = {10.1016/j.jfa.2008.07.024}, year = {2009}, abstract = {We compare the above-mentioned article with the content of a previous publication}, language = {en} } @article{SchulzeWei2014, author = {Schulze, Bert-Wolfgang and Wei, Y.}, title = {The Mellin-edge quantisation for corner operators}, series = {Complex analysis and operator theory}, volume = {8}, journal = {Complex analysis and operator theory}, number = {4}, publisher = {Springer}, address = {Basel}, issn = {1661-8254}, doi = {10.1007/s11785-013-0289-3}, pages = {803 -- 841}, year = {2014}, abstract = {We establish a quantisation of corner-degenerate symbols, here called Mellin-edge quantisation, on a manifold with second order singularities. The typical ingredients come from the "most singular" stratum of which is a second order edge where the infinite transversal cone has a base that is itself a manifold with smooth edge. The resulting operator-valued amplitude functions on the second order edge are formulated purely in terms of Mellin symbols taking values in the edge algebra over . In this respect our result is formally analogous to a quantisation rule of (Osaka J. Math. 37:221-260, 2000) for the simpler case of edge-degenerate symbols that corresponds to the singularity order 1. However, from the singularity order 2 on there appear new substantial difficulties for the first time, partly caused by the edge singularities of the cone over that tend to infinity.}, language = {en} } @article{KuehnSorgenfreiGiangrisostomietal.2018, author = {K{\"u}hn, Danilo and Sorgenfrei, Nomi and Giangrisostomi, Erika and Jay, Raphael and Musazay, Abdurrahman and Ovsyannikov, Ruslan and Strahlman, Christian and Svensson, Svante and M{\aa}rtensson, Nils and F{\"o}hlisch, Alexander}, title = {Capabilities of angle resolved time of flight electron spectroscopy with the 60 degrees wide angle acceptance lens}, series = {Journal of electron spectroscopy and related phenomena : the international journal on theoretical and experimental aspects of electron spectroscopy}, volume = {224}, journal = {Journal of electron spectroscopy and related phenomena : the international journal on theoretical and experimental aspects of electron spectroscopy}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0368-2048}, doi = {10.1016/j.elspec.2017.06.008}, pages = {45 -- 50}, year = {2018}, abstract = {The simultaneous detection of energy, momentum and temporal information in electron spectroscopy is the key aspect to enhance the detection efficiency in order to broaden the range of scientific applications. Employing a novel 60 degrees wide angle acceptance lens system, based on an additional accelerating electron optical element, leads to a significant enhancement in transmission over the previously employed 30 degrees electron lenses. Due to the performance gain, optimized capabilities for time resolved electron spectroscopy and other high transmission applications with pulsed ionizing radiation have been obtained. The energy resolution and transmission have been determined experimentally utilizing BESSY II as a photon source. Four different and complementary lens modes have been characterized. (C) 2017 The Authors. Published by Elsevier B.V.}, language = {en} } @article{ZoellerHainzlTilmannetal.2020, author = {Z{\"o}ller, Gert and Hainzl, Sebastian and Tilmann, Frederik and Woith, Heiko and Dahm, Torsten}, title = {Comment on: Wikelski, Martin; M{\"u}ller, Uschi; Scocco, Paola; Catorci, Andrea; Desinov, Lev V.; Belyaev, Mikhail Y.; Keim, Daniel A.; Pohlmeier, Winfried; Fechteler, Gerhard; Mai, Martin P. : Potential short-term earthquake forecasting by farm animal monitoring. - Ethology. - 126 (2020), 9. - S. 931 - 941. -ISSN 0179-1613. - eISSN 1439-0310. - doi 10.1111/eth.13078}, series = {Ethology}, volume = {127}, journal = {Ethology}, number = {3}, publisher = {Wiley}, address = {Hoboken}, issn = {0179-1613}, doi = {10.1111/eth.13105}, pages = {302 -- 306}, year = {2020}, abstract = {Based on an analysis of continuous monitoring of farm animal behavior in the region of the 2016 M6.6 Norcia earthquake in Italy, Wikelski et al., 2020; (Seismol Res Lett, 89, 2020, 1238) conclude that animal activity can be anticipated with subsequent seismic activity and that this finding might help to design a "short-term earthquake forecasting method." We show that this result is based on an incomplete analysis and misleading interpretations. Applying state-of-the-art methods of statistics, we demonstrate that the proposed anticipatory patterns cannot be distinguished from random patterns, and consequently, the observed anomalies in animal activity do not have any forecasting power.}, language = {en} } @unpublished{Schulze2003, author = {Schulze, Bert-Wolfgang}, title = {Toeplitz operators, and ellipticity of boundary value problems with global projection conditions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26510}, year = {2003}, abstract = {Ellipticity of (pseudo-) differential operators A on a compact manifold X with boundary (or with edges) Y is connected with boundary (or edge) conditions of trace and potential type, formulated in terms of global projections on Y together with an additional symbolic structure. This gives rise to operator block matrices A with A in the upper left corner. We study an algebra of such operators, where ellipticity is equivalent to the Fredhom property in suitable scales of spaces: Sobolev spaces on X plus closed subspaces of Sobolev spaces on Y which are the range of corresponding pseudo-differential projections. Moreover, we express parametrices of elliptic elements within our algebra and discuss spectral boundary value problems for differential operators.}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2003, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Elliptic theory on manifolds with nonisolated singularities : V. Index formulas for elliptic problems on manifolds with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26500}, year = {2003}, abstract = {For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.}, language = {en} } @unpublished{FedosovSchulzeTarkhanov2003, author = {Fedosov, Boris and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {On index theorem for symplectic orbifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26550}, year = {2003}, abstract = {We give an explicit construction of the trace on the algebra of quantum observables on a symplectic orbifold and propose an index formula.}, language = {en} } @unpublished{HarutjunjanSchulze2002, author = {Harutjunjan, Gohar and Schulze, Bert-Wolfgang}, title = {Asymptotics and relative index on a cylinder with conical cross section}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26446}, year = {2002}, abstract = {We study pseudodifferential operators on a cylinder IR x B with cross section B that conical singularities. Configurations of that kind are the local model of cornere singularities with base spaces B. Operators A in our calculus are assumed to have symbols ฮฑ which are meromorphic in the complex covariable with values in the space of all cone operators on B. In case ฮฑ is dependent of the axial variable t โˆˆ IR, we show an explicit formula for solutions of the homogeneous equation. Each non-bjectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula.}, language = {en} } @unpublished{NazaikinskiiSavinSchulzeetal.2003, author = {Nazaikinskii, Vladimir and Savin, Anton and Schulze, Bert-Wolfgang and Sternin, Boris}, title = {Differential operators on manifolds with singularities : analysis and topology : Chapter 1: Localization (surgery) in elliptic theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26546}, year = {2003}, abstract = {Contents: Chapter 1: Localization (Surgery) in Elliptic Theory 1.1. The Index Locality Principle 1.1.1. What is locality? 1.1.2. A pilot example 1.1.3. Collar spaces 1.1.4. Elliptic operators 1.1.5. Surgery and the relative index theorem 1.2. Surgery in Index Theory on Smooth Manifolds 1.2.1. The BooรŸ-Wojciechowski theorem 1.2.2. The Gromov-Lawson theorem 1.3. Surgery for Boundary Value Problems 1.3.1. Notation 1.3.2. General boundary value problems 1.3.3. A model boundary value problem on a cylinder 1.3.4. The Agranovich-Dynin theorem 1.3.5. The Agranovich theorem 1.3.6. Bojarski's theorem and its generalizations 1.4. (Micro)localization in Lefschetz theory 1.4.1. The Lefschetz number 1.4.2. Localization and the contributions of singular points 1.4.3. The semiclassical method and microlocalization 1.4.4. The classical Atiyah-Bott-Lefschetz theorem}, language = {en} } @unpublished{KapanadzeSchulze2003, author = {Kapanadze, David and Schulze, Bert-Wolfgang}, title = {Asymptotics of potentials in the edge calculus}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26530}, year = {2003}, abstract = {Boundary value problems on manifolds with conical singularities or edges contain potential operators as well as trace and Green operators which play a similar role as the corresponding operators in (pseudo-differential) boundary value problems on a smooth manifold. There is then a specific asymptotic behaviour of these operators close to the singularities. We characterise potential operators in terms of actions of cone or edge pseudo-differential operators (in the neighbouring space) on densities supported by sbmanifolds which also have conical or edge singularities. As a byproduct we show the continuity of such potentials as continuous perators between cone or edge Sobolev spaces and subspaces with asymptotics.}, language = {en} } @unpublished{DeDonnoSchulze2003, author = {De Donno, G. and Schulze, Bert-Wolfgang}, title = {Meromorphic symbolic structures for boundary value problems on manifolds with edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26570}, year = {2003}, abstract = {We investigate the ideal of Green and Mellin operators with asymtotics for a manifold with edge-corner singularities and boundary which belongs to the structure of parametrices of elliptic boundary value problems on a configuration with corners whose base manifolds have edges.}, language = {en} }