@article{WangHainzlZoelleretal.2012, author = {Wang, Lifeng and Hainzl, Sebastian and Z{\"o}ller, Gert and Holschneider, Matthias}, title = {Stress- and aftershock-constrained joint inversions for coseismic and postseismic slip applied to the 2004 M6.0 Parkfield earthquake}, series = {Journal of geophysical research : Solid earth}, volume = {117}, journal = {Journal of geophysical research : Solid earth}, publisher = {American Geophysical Union}, address = {Washington}, issn = {2169-9313}, doi = {10.1029/2011JB009017}, pages = {18}, year = {2012}, abstract = {Both aftershocks and geodetically measured postseismic displacements are important markers of the stress relaxation process following large earthquakes. Postseismic displacements can be related to creep-like relaxation in the vicinity of the coseismic rupture by means of inversion methods. However, the results of slip inversions are typically non-unique and subject to large uncertainties. Therefore, we explore the possibility to improve inversions by mechanical constraints. In particular, we take into account the physical understanding that postseismic deformation is stress-driven, and occurs in the coseismically stressed zone. We do joint inversions for coseismic and postseismic slip in a Bayesian framework in the case of the 2004 M6.0 Parkfield earthquake. We perform a number of inversions with different constraints, and calculate their statistical significance. According to information criteria, the best result is preferably related to a physically reasonable model constrained by the stress-condition (namely postseismic creep is driven by coseismic stress) and the condition that coseismic slip and large aftershocks are disjunct. This model explains 97\% of the coseismic displacements and 91\% of the postseismic displacements during day 1-5 following the Parkfield event, respectively. It indicates that the major postseismic deformation can be generally explained by a stress relaxation process for the Parkfield case. This result also indicates that the data to constrain the coseismic slip model could be enriched postseismically. For the 2004 Parkfield event, we additionally observe asymmetric relaxation process at the two sides of the fault, which can be explained by material contrast ratio across the fault of similar to 1.15 in seismic velocity.}, language = {en} } @article{Wallenta2012, author = {Wallenta, D.}, title = {Elliptic quasicomplexes on compact closed manifolds}, series = {Integral equations and operator theor}, volume = {73}, journal = {Integral equations and operator theor}, number = {4}, publisher = {Springer}, address = {Basel}, issn = {0378-620X}, doi = {10.1007/s00020-012-1983-7}, pages = {517 -- 536}, year = {2012}, abstract = {We consider quasicomplexes of pseudodifferential operators on a smooth compact manifold without boundary. To each quasicomplex we associate a complex of symbols. The quasicomplex is elliptic if this symbol complex is exact away from the zero section. We prove that elliptic quasicomplexes are Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes and prove a generalisation of the Atiyah-Singer index theorem.}, language = {en} } @article{KellerValleriani2012, author = {Keller, Peter and Valleriani, Angelo}, title = {Single-molecule stochastic times in a reversible bimolecular reaction}, series = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, volume = {137}, journal = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, number = {8}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0021-9606}, doi = {10.1063/1.4747337}, pages = {7}, year = {2012}, abstract = {In this work, we consider the reversible reaction between reactants of species A and B to form the product C. We consider this reaction as a prototype of many pseudobiomolecular reactions in biology, such as for instance molecular motors. We derive the exact probability density for the stochastic waiting time that a molecule of species A needs until the reaction with a molecule of species B takes place. We perform this computation taking fully into account the stochastic fluctuations in the number of molecules of species B. We show that at low numbers of participating molecules, the exact probability density differs from the exponential density derived by assuming the law of mass action. Finally, we discuss the condition of detailed balance in the exact stochastic and in the approximate treatment.}, language = {en} } @article{KurtenbachEickerMayerGuerretal.2012, author = {Kurtenbach, E. and Eicker, A. and Mayer-Guerr, T. and Holschneider, Matthias and Hayn, M. and Fuhrmann, M. and Kusche, J.}, title = {Improved daily GRACE gravity field solutions using a Kalman smoother}, series = {Journal of geodynamics}, volume = {59}, journal = {Journal of geodynamics}, number = {3}, publisher = {Elsevier}, address = {Oxford}, issn = {0264-3707}, doi = {10.1016/j.jog.2012.02.006}, pages = {39 -- 48}, year = {2012}, abstract = {Different GRACE data analysis centers provide temporal variations of the Earth's gravity field as monthly, 10-daily or weekly solutions. These temporal mean fields cannot model the variations occurring during the respective time span. The aim of our approach is to extract as much temporal information as possible out of the given GRACE data. Therefore the temporal resolution shall be increased with the goal to derive daily snapshots. Yet, such an increase in temporal resolution is accompanied by a loss of redundancy and therefore in a reduced accuracy if the daily solutions are calculated individually. The approach presented here therefore introduces spatial and temporal correlations of the expected gravity field signal derived from geophysical models in addition to the daily observations, thus effectively constraining the spatial and temporal evolution of the GRACE solution. The GRACE data processing is then performed within the framework of a Kalman filter and smoother estimation procedure. The approach is at first investigated in a closed-loop simulation scenario and then applied to the original GRACE observations (level-1B data) to calculate daily solutions as part of the gravity field model ITG-Grace2010. Finally, the daily models are compared to vertical GPS station displacements and ocean bottom pressure observations. From these comparisons it can be concluded that particular in higher latitudes the daily solutions contain high-frequent temporal gravity field information and represent an improvement to existing geophysical models.}, language = {en} } @article{BettenbuehlRusconiEngbertetal.2012, author = {Bettenb{\"u}hl, Mario and Rusconi, Marco and Engbert, Ralf and Holschneider, Matthias}, title = {Bayesian selection of Markov Models for symbol sequences application to microsaccadic eye movements}, series = {PLoS one}, volume = {7}, journal = {PLoS one}, number = {9}, publisher = {PLoS}, address = {San Fransisco}, issn = {1932-6203}, doi = {10.1371/journal.pone.0043388}, pages = {10}, year = {2012}, abstract = {Complex biological dynamics often generate sequences of discrete events which can be described as a Markov process. The order of the underlying Markovian stochastic process is fundamental for characterizing statistical dependencies within sequences. As an example for this class of biological systems, we investigate the Markov order of sequences of microsaccadic eye movements from human observers. We calculate the integrated likelihood of a given sequence for various orders of the Markov process and use this in a Bayesian framework for statistical inference on the Markov order. Our analysis shows that data from most participants are best explained by a first-order Markov process. This is compatible with recent findings of a statistical coupling of subsequent microsaccade orientations. Our method might prove to be useful for a broad class of biological systems.}, language = {en} } @article{IochumLevyVassilevich2012, author = {Iochum, B. and Levy, C. and Vassilevich, D. V.}, title = {Global and local aspects of spectral actions}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {45}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {37}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/45/37/374020}, pages = {19}, year = {2012}, abstract = {The principal object in noncommutative geometry is the spectral triple consisting of an algebra A, a Hilbert space H and a Dirac operator D. Field theories are incorporated in this approach by the spectral action principle, which sets the field theory action to Tr f (D-2/Lambda(2)), where f is a real function such that the trace exists and Lambda is a cutoff scale. In the low-energy (weak-field) limit, the spectral action reproduces reasonably well the known physics including the standard model. However, not much is known about the spectral action beyond the low-energy approximation. In this paper, after an extensive introduction to spectral triples and spectral actions, we study various expansions of the spectral actions (exemplified by the heat kernel). We derive the convergence criteria. For a commutative spectral triple, we compute the heat kernel on the torus up to the second order in gauge connection and consider limiting cases. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.}, language = {en} } @article{BlanchardMathe2012, author = {Blanchard, Gilles and Mathe, Peter}, title = {Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration}, series = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data}, volume = {28}, journal = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data}, number = {11}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {0266-5611}, doi = {10.1088/0266-5611/28/11/115011}, pages = {23}, year = {2012}, abstract = {The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which corrects both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration it is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.}, language = {en} } @article{SchachtschneiderHolschneiderMandea2012, author = {Schachtschneider, R. and Holschneider, Matthias and Mandea, M.}, title = {Error distribution in regional modelling of the geomagnetic field}, series = {Geophysical journal international}, volume = {191}, journal = {Geophysical journal international}, number = {3}, publisher = {Wiley-Blackwell}, address = {Hoboken}, issn = {0956-540X}, doi = {10.1111/j.1365-246X.2012.05675.x}, pages = {1015 -- 1024}, year = {2012}, abstract = {In this study we analyse the error distribution in regional models of the geomagnetic field. Our main focus is to investigate the distribution of errors when combining two regional patches to obtain a global field from regional ones. To simulate errors in overlapping patches we choose two different data region shapes that resemble that scenario. First, we investigate the errors in elliptical regions and secondly we choose a region obtained from two overlapping circular spherical caps. We conduct a Monte-Carlo simulation using synthetic data to obtain the expected mean errors. For the elliptical regions the results are similar to the ones obtained for circular spherical caps: the maximum error at the boundary decreases towards the centre of the region. A new result emerges as errors at the boundary vary with azimuth, being largest in the major axis direction and minimal in the minor axis direction. Inside the region there is an error decay towards a minimum at the centre at a rate similar to the one in circular regions. In the case of two combined circular regions there is also an error decay from the boundary towards the centre. The minimum error occurs at the centre of the combined regions. The maximum error at the boundary occurs on the line containing the two cap centres, the minimum in the perpendicular direction where the two circular cap boundaries meet. The large errors at the boundary are eliminated by combining regional patches. We propose an algorithm for finding the boundary region that is applicable to irregularly shaped model regions.}, language = {en} } @article{Baumgaertel2012, author = {Baumg{\"a}rtel, Hellmut}, title = {On a critical radiation density in the Friedmann equation}, series = {Journal of mathematical physics}, volume = {53}, journal = {Journal of mathematical physics}, number = {12}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0022-2488}, doi = {10.1063/1.4771668}, pages = {9}, year = {2012}, abstract = {The paper presents a classification of the basic types of admissible solutions of the general Friedmann equation with non-vanishing cosmological constant and for the case that radiation and matter do not couple. There are four distinct types. The classification uses first the discriminant of a polynomial of the third degree, closely related to the right hand side of the Friedmann equation. The decisive term is then a critical radiation density which can be calculated explicitly.}, language = {en} } @phdthesis{Keller2012, author = {Keller, Peter}, title = {Mathematical modeling of molecular motors}, address = {Potsdam}, pages = {116 S.}, year = {2012}, language = {en} } @phdthesis{DiGesu2012, author = {Di Ges{\`u}, Giacomo}, title = {Semiclassical spectral analysis of discrete Witten Laplacians}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-65286}, school = {Universit{\"a}t Potsdam}, year = {2012}, abstract = {A discrete analogue of the Witten Laplacian on the n-dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low-lying spectrum. Our proof, inspired by work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of metastable Markov processes on the lattice.}, language = {en} } @phdthesis{Branding2012, author = {Branding, Volker}, title = {The evolution equations for Dirac-harmonic Maps}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64204}, school = {Universit{\"a}t Potsdam}, year = {2012}, abstract = {This thesis investigates the gradient flow of Dirac-harmonic maps. Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points of this energy functional couple the equation for harmonic maps with spinor fields. At present, many analytical properties of Dirac-harmonic maps are known, but a general existence result is still missing. In this thesis the existence question is studied using the evolution equations for a regularized version of Dirac-harmonic maps. Since the energy functional for Dirac-harmonic maps is unbounded from below the method of the gradient flow cannot be applied directly. Thus, we first of all consider a regularization prescription for Dirac-harmonic maps and then study the gradient flow. Chapter 1 gives some background material on harmonic maps/harmonic spinors and summarizes the current known results about Dirac-harmonic maps. Chapter 2 introduces the notion of Dirac-harmonic maps in detail and presents a regularization prescription for Dirac-harmonic maps. In Chapter 3 the evolution equations for regularized Dirac-harmonic maps are introduced. In addition, the evolution of certain energies is discussed. Moreover, the existence of a short-time solution to the evolution equations is established. Chapter 4 analyzes the evolution equations in the case that the domain manifold is a closed curve. Here, the existence of a smooth long-time solution is proven. Moreover, for the regularization being large enough, it is shown that the evolution equations converge to a regularized Dirac-harmonic map. Finally, it is discussed in which sense the regularization can be removed. In Chapter 5 the evolution equations are studied when the domain manifold is a closed Riemmannian spin surface. For the regularization being large enough, the existence of a global weak solution, which is smooth away from finitely many singularities is proven. It is shown that the evolution equations converge weakly to a regularized Dirac-harmonic map. In addition, it is discussed if the regularization can be removed in this case.}, language = {en} } @unpublished{Murr2012, author = {Murr, R{\"u}diger}, title = {Reciprocal classes of Markov processes : an approach with duality formulae}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63018}, year = {2012}, abstract = {In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.}, language = {en} } @phdthesis{Nehring2012, author = {Nehring, Benjamin}, title = {Point processes in statistical mechanics : a cluster expansion approach}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-62682}, school = {Universit{\"a}t Potsdam}, year = {2012}, abstract = {A point process is a mechanism, which realizes randomly locally finite point measures. One of the main results of this thesis is an existence theorem for a new class of point processes with a so called signed Levy pseudo measure L, which is an extension of the class of infinitely divisible point processes. The construction approach is a combination of the classical point process theory, as developed by Kerstan, Matthes and Mecke, with the method of cluster expansions from statistical mechanics. Here the starting point is a family of signed Radon measures, which defines on the one hand the Levy pseudo measure L, and on the other hand locally the point process. The relation between L and the process is the following: this point process solves the integral cluster equation determined by L. We show that the results from the classical theory of infinitely divisible point processes carry over in a natural way to the larger class of point processes with a signed Levy pseudo measure. In this way we obtain e.g. a criterium for simplicity and a characterization through the cluster equation, interpreted as an integration by parts formula, for such point processes. Our main result in chapter 3 is a representation theorem for the factorial moment measures of the above point processes. With its help we will identify the permanental respective determinantal point processes, which belong to the classes of Boson respective Fermion processes. As a by-product we obtain a representation of the (reduced) Palm kernels of infinitely divisible point processes. In chapter 4 we see how the existence theorem enables us to construct (infinitely extended) Gibbs, quantum-Bose and polymer processes. The so called polymer processes seem to be constructed here for the first time. In the last part of this thesis we prove that the family of cluster equations has certain stability properties with respect to the transformation of its solutions. At first this will be used to show how large the class of solutions of such equations is, and secondly to establish the cluster theorem of Kerstan, Matthes and Mecke in our setting. With its help we are able to enlarge the class of Polya processes to the so called branching Polya processes. The last sections of this work are about thinning and splitting of point processes. One main result is that the classes of Boson and Fermion processes remain closed under thinning. We use the results on thinning to identify a subclass of point processes with a signed Levy pseudo measure as doubly stochastic Poisson processes. We also pose the following question: Assume you observe a realization of a thinned point process. What is the distribution of deleted points? Surprisingly, the Papangelou kernel of the thinning, besides a constant factor, is given by the intensity measure of this conditional probability, called splitting kernel.}, language = {en} } @phdthesis{Murr2012, author = {Murr, R{\"u}diger}, title = {Reciprocal classes of Markov processes : an approach with duality formulae}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-62091}, school = {Universit{\"a}t Potsdam}, year = {2012}, abstract = {This work is concerned with the characterization of certain classes of stochastic processes via duality formulae. In particular we consider reciprocal processes with jumps, a subject up to now neglected in the literature. In the first part we introduce a new formulation of a characterization of processes with independent increments. This characterization is based on a duality formula satisfied by processes with infinitely divisible increments, in particular L{\´e}vy processes, which is well known in Malliavin calculus. We obtain two new methods to prove this duality formula, which are not based on the chaos decomposition of the space of square-integrable function- als. One of these methods uses a formula of partial integration that characterizes infinitely divisible random vectors. In this context, our characterization is a generalization of Stein's lemma for Gaussian random variables and Chen's lemma for Poisson random variables. The generality of our approach permits us to derive a characterization of infinitely divisible random measures. The second part of this work focuses on the study of the reciprocal classes of Markov processes with and without jumps and their characterization. We start with a resume of already existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. Thus we are able to connect the results of characterizations via duality formulae with the theory of stochastic mechanics by our interpretation, and to stochastic optimal control theory by the mathematical approach. As an application we are able to prove an invariance property of the reciprocal class of a Brownian diffusion under time reversal. In the context of pure jump processes we derive the following new results. We describe the reciprocal classes of Markov counting processes, also called unit jump processes, and obtain a characterization of the associated reciprocal class via a duality formula. This formula contains as key terms a stochastic derivative, a compensated stochastic integral and an invariant of the reciprocal class. Moreover we present an interpretation of the characterization of a reciprocal class in the context of stochastic optimal control of unit jump processes. As a further application we show that the reciprocal class of a Markov counting process has an invariance property under time reversal. Some of these results are extendable to the setting of pure jump processes, that is, we admit different jump-sizes. In particular, we show that the reciprocal classes of Markov jump processes can be compared using reciprocal invariants. A characterization of the reciprocal class of compound Poisson processes via a duality formula is possible under the assumption that the jump-sizes of the process are incommensurable.}, language = {en} } @unpublished{AntonioukKiselevStepanenkoetal.2012, author = {Antoniouk, Alexandra Viktorivna and Kiselev, Oleg and Stepanenko, Vitaly and Tarkhanov, Nikolai Nikolaevich}, title = {Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61987}, year = {2012}, abstract = {The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character.}, language = {en} } @unpublished{AlsaedyTarkhanov2012, author = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich}, title = {The method of Fischer-Riesz equations for elliptic boundary value problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61792}, year = {2012}, abstract = {We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.}, language = {en} } @unpublished{DyachenkoTarkhanov2012, author = {Dyachenko, Evgueniya and Tarkhanov, Nikolai Nikolaevich}, title = {Degeneration of boundary layer at singular points}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60135}, year = {2012}, abstract = {We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.}, language = {en} } @unpublished{Baer2012, author = {B{\"a}r, Christian}, title = {Some properties of solutions to weakly hypoelliptic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60064}, year = {2012}, abstract = {A linear differential operator L is called weakly hypoelliptic if any local solution u of Lu = 0 is smooth. We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which covers all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p solution must vanish.}, language = {en} } @unpublished{Baer2012, author = {B{\"a}r, Christian}, title = {Renormalized integrals and a path integral formula for the heat kernel on a manifold}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60052}, year = {2012}, abstract = {We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.}, language = {en} } @unpublished{PfaeffleStephan2012, author = {Pf{\"a}ffle, Frank and Stephan, Christoph A.}, title = {Chiral asymmetry and the spectral action}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60046}, year = {2012}, abstract = {We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.}, language = {en} } @unpublished{PfaeffleStephan2012, author = {Pf{\"a}ffle, Frank and Stephan, Christoph A.}, title = {The Holst action by the spectral action principle}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60032}, year = {2012}, abstract = {We investigate the Holst action for closed Riemannian 4-manifolds with orthogonal connections. For connections whose torsion has zero Cartan type component we show that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on left-handed spinor fields.}, language = {en} } @unpublished{BaerBallmann2012, author = {B{\"a}r, Christian and Ballmann, Werner}, title = {Boundary value problems for elliptic differential operators of first order}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60023}, year = {2012}, abstract = {We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.}, language = {en} } @unpublished{BaerPfaeffle2012, author = {B{\"a}r, Christian and Pf{\"a}ffle, Frank}, title = {Wiener measures on Riemannian manifolds and the Feynman-Kac formula}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59998}, year = {2012}, abstract = {This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schr{\"o}dinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.}, language = {en} } @unpublished{PfaeffleStephan2012, author = {Pf{\"a}ffle, Frank and Stephan, Christoph A.}, title = {On gravity, torsion and the spectral action principle}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59989}, year = {2012}, abstract = {We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine-Connes spectral action, deduce the equations of motions and discuss critical points.}, language = {en} } @unpublished{BaerGinoux2012, author = {B{\"a}r, Christian and Ginoux, Nicolas}, title = {Classical and quantum fields on Lorentzian manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59973}, year = {2012}, abstract = {We construct bosonic and fermionic locally covariant quantum fields theories on curved backgrounds for large classes of fields. We investigate the quantum field and n-point functions induced by suitable states.}, language = {en} } @unpublished{Nehring2012, author = {Nehring, Benjamin}, title = {Construction of point processes for classical and quantum gases}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59648}, year = {2012}, abstract = {We propose a new construction of point processes, which generalizes the class of infinitely divisible point processes. Examples are the quantum Boson and Fermion gases as well as the classical Gibbs point processes, where the interaction is given by a stable and regular pair potential.}, language = {en} } @unpublished{RattanaBoeckmann2012, author = {Rattana, Amornrat and B{\"o}ckmann, Christine}, title = {Matrix methods for computing Eigenvalues of Sturm-Liouville problems of order four}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59279}, year = {2012}, abstract = {This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.}, language = {en} } @unpublished{AlsaedyTarkhanov2012, author = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich}, title = {Spectral projection for the dbar-Neumann problem}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-58616}, year = {2012}, abstract = {We show that the spectral kernel function of the dbar-Neumann problem on a non-compact strongly pseudoconvex manifold is smooth up to the boundary.}, language = {en} } @unpublished{Tarkhanov2012, author = {Tarkhanov, Nikolai Nikolaevich}, title = {A simple numerical approach to the Riemann hypothesis}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57645}, year = {2012}, abstract = {The Riemann hypothesis is equivalent to the fact the the reciprocal function 1/zeta (s) extends from the interval (1/2,1) to an analytic function in the quarter-strip 1/2 < Re s < 1 and Im s > 0. Function theory allows one to rewrite the condition of analytic continuability in an elegant form amenable to numerical experiments.}, language = {en} } @unpublished{ShlapunovTarkhanov2012, author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57759}, year = {2012}, abstract = {We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.}, language = {en} } @unpublished{GrudskyTarkhanov2012, author = {Grudsky, Serguey and Tarkhanov, Nikolai Nikolaevich}, title = {Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57745}, year = {2012}, abstract = {We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions.}, language = {en} } @unpublished{KoppitzMusunthia2012, author = {Koppitz, J{\"o}rg and Musunthia, Tiwadee}, title = {Maximal subsemigroups containing a particular semigroup}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57465}, year = {2012}, abstract = {We study maximal subsemigroups of the monoid T(X) of all full transformations on the set X = N of natural numbers containing a given subsemigroup W of T(X), where each element of a given set U is a generator of T(X) modulo W. This note continues the study of maximal subsemigroups of the monoid of all full transformations on an infinite set.}, language = {en} } @unpublished{BlanchardMathe2012, author = {Blanchard, Gilles and Math{\´e}, Peter}, title = {Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57117}, year = {2012}, abstract = {The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which takes into account both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration this modification is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.}, language = {en} } @unpublished{KleinLeonardRosenberger2012, author = {Klein, Markus and L{\´e}onard, Christian and Rosenberger, Elke}, title = {Agmon-type estimates for a class of jump processes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56995}, year = {2012}, abstract = {In the limit we analyze the generators of families of reversible jump processes in the n-dimensional space associated with a class of symmetric non-local Dirichlet forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of certain eikonal equation. Fine results are sensitive to the rate functions being twice differentiable or just Lipschitz. Our estimates are similar to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice.}, language = {en} } @unpublished{KleinRosenberger2012, author = {Klein, Markus and Rosenberger, Elke}, title = {Tunneling for a class of difference operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56989}, year = {2012}, abstract = {We analyze a general class of difference operators containing a multi-well potential and a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we treat the eigenvalue problem as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix similar to the analysis for the Schr{\"o}dinger operator, and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.}, language = {en} } @unpublished{KellerRoellyValleriani2012, author = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo}, title = {On time duality for quasi-birth-and-death processes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56973}, year = {2012}, abstract = {We say that (weak/strong) time duality holds for continuous time quasi-birth-and-death-processes if, starting from a fixed level, the first hitting time of the next upper level and the first hitting time of the next lower level have the same distribution. We present here a criterion for time duality in the case where transitions from one level to another have to pass through a given single state, the so-called bottleneck property. We also prove that a weaker form of reversibility called balanced under permutation is sufficient for the time duality to hold. We then discuss the general case.}, language = {en} } @unpublished{TarkhanovWallenta2012, author = {Tarkhanov, Nikolai Nikolaevich and Wallenta, Daniel}, title = {The Lefschetz number of sequences of trace class curvature}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56969}, year = {2012}, abstract = {For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number.}, language = {en} } @unpublished{KiselevTarkhanov2012, author = {Kiselev, Oleg M. and Tarkhanov, Nikolai Nikolaevich}, title = {Scattering of autoresonance trajectories upon a separatrix}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56880}, year = {2012}, abstract = {We study asymptotic properties of solutions to the primary resonance equation with large amplitude on a long time interval.}, language = {en} } @unpublished{BlanchardDelattreRoquain2012, author = {Blanchard, Gilles and Delattre, Sylvain and Roquain, {\´E}tienne}, title = {Testing over a continuum of null hypotheses}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56877}, year = {2012}, abstract = {We introduce a theoretical framework for performing statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses. This extends the standard statistical setting for multiple hypotheses testing, which is restricted to a finite set. This work is motivated by numerous modern applications where the observed signal is modeled by a stochastic process over a continuum. As a measure of type I error, we extend the concept of false discovery rate (FDR) to this setting. The FDR is defined as the average ratio of the measure of two random sets, so that its study presents some challenge and is of some intrinsic mathematical interest. Our main result shows how to use the p-value process to control the FDR at a nominal level, either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting, the latter one leading to a less conservative procedure. The interest of this approach is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables. Conceptually, an interesting feature of the setting advocated here is that it focuses directly on the intrinsic hypothesis space associated with a testing model on a random process, without referring to an arbitrary discretization.}, language = {en} }