@article{Roos2019, author = {Roos, Saskia}, title = {The Dirac operator under collapse to a smooth limit space}, series = {Annals of global analysis and geometry}, volume = {57}, journal = {Annals of global analysis and geometry}, number = {1}, publisher = {Springer}, address = {Dordrecht}, issn = {0232-704X}, doi = {10.1007/s10455-019-09691-8}, pages = {121 -- 151}, year = {2019}, abstract = {Let (M-i, g(i))(i is an element of N) be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov-Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator D-B on B. We give an explicit description of D-B and characterize the special case where D-B equals the Dirac operator on B.}, language = {en} } @article{GueneysuKeller2018, author = {G{\"u}neysu, Batu and Keller, Matthias}, title = {Scattering the Geometry of Weighted Graphs}, series = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics}, volume = {21}, journal = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics}, number = {3}, publisher = {Springer}, address = {Dordrecht}, issn = {1385-0172}, doi = {10.1007/s11040-018-9285-1}, pages = {15}, year = {2018}, abstract = {Given two weighted graphs (X, b(k), m(k)), k = 1, 2 with b(1) similar to b(2) and m(1) similar to m(2), we prove a weighted L-1-criterion for the existence and completeness of the wave operators W-+/- (H-2, H-1, I-1,I-2), where H-k denotes the natural Laplacian in l(2)(X, m(k)) w.r.t. (X, b(k), m(k)) and I-1,I-2 the trivial identification of l(2)(X, m(1)) with l(2) (X, m(2)). In particular, this entails a general criterion for the absolutely continuous spectra of H-1 and H-2 to be equal.}, language = {en} } @article{FedchenkoTarkhanov2017, author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich}, title = {A Rado theorem for the porous medium equation}, series = {Boletin de la Sociedad Matem{\´a}tica Mexicana}, volume = {24}, journal = {Boletin de la Sociedad Matem{\´a}tica Mexicana}, number = {2}, publisher = {Springer}, address = {Cham}, issn = {1405-213X}, doi = {10.1007/s40590-017-0169-3}, pages = {427 -- 437}, year = {2017}, abstract = {We prove that if u is a locally Lipschitz continuous function on an open set chi subset of Rn + 1 satisfying the nonlinear heat equation partial derivative(t)u = Delta(vertical bar u vertical bar(p-1) u), p > 1, weakly away from the zero set u(-1) (0) in chi, then u is a weak solution to this equation in all of chi.}, language = {en} } @article{DimitrovaKoppitz2017, author = {Dimitrova, Ilinka and Koppitz, J{\"o}rg}, title = {On the semigroup of all partial fence-preserving injections on a finite set}, series = {Journal of Algebra and Its Applications}, volume = {16}, journal = {Journal of Algebra and Its Applications}, number = {12}, publisher = {World Scientific}, address = {Singapore}, issn = {0219-4988}, doi = {10.1142/S0219498817502231}, pages = {14}, year = {2017}, abstract = {For n∈N , let Xn={a1,a2,…,an} be an n-element set and let F=(Xn;R1 generated from identity by taking powers is isomorphic to P and can be embedded into a coppice by taking inverses.}, language = {en} } @article{FischerKeller2021, author = {Fischer, Florian and Keller, Matthias}, title = {Riesz decompositions for Schr{\"o}dinger operators on graphs}, series = {Journal of mathematical analysis and applications}, volume = {495}, journal = {Journal of mathematical analysis and applications}, number = {1}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0022-247X}, doi = {10.1016/j.jmaa.2020.124674}, pages = {22}, year = {2021}, abstract = {We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.}, language = {en} } @article{Denecke2020, author = {Denecke, Klaus-Dieter}, title = {Partial clones}, series = {Asian-European journal of mathematics : AEJM}, volume = {13}, journal = {Asian-European journal of mathematics : AEJM}, number = {8}, publisher = {World Scientific}, address = {Singapore}, issn = {1793-5571}, doi = {10.1142/S1793557120501612}, pages = {19}, year = {2020}, abstract = {A set C of operations defined on a nonempty set A is said to be a clone if C is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the n-ary operations defined on set A for all natural numbers n >= 1 and the operations are the so-called superposition operations S-m(n) for natural numbers m, n >= 1 and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set A and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.}, language = {en} } @article{Baer2021, author = {B{\"a}r, Christian}, title = {The Faddeev-LeVerrier algorithm and the Pfaffian}, series = {Linear algebra and its applications}, volume = {630}, journal = {Linear algebra and its applications}, publisher = {Elsevier}, address = {New York}, issn = {0024-3795}, doi = {10.1016/j.laa.2021.07.023}, pages = {39 -- 55}, year = {2021}, abstract = {We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n(beta+1)) where nis the size of the matrix and O(n(beta)) is the cost of multiplying n x n-matrices, beta is an element of [2, 2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.}, language = {en} } @misc{KleinRosenberger2018, author = {Klein, Markus and Rosenberger, Elke}, title = {The tunneling effect for a class of difference operators}, series = {Reviews in Mathematical Physics}, volume = {30}, journal = {Reviews in Mathematical Physics}, number = {4}, publisher = {World Scientific}, address = {Singapore}, issn = {0129-055X}, doi = {10.1142/S0129055X18300029}, pages = {42}, year = {2018}, abstract = {We analyze a general class of self-adjoint difference operators H-epsilon = T-epsilon + V-epsilon on l(2)((epsilon Z)(d)), where V-epsilon is a multi-well potential and v(epsilon) is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30-35]). Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs. The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian H-epsilon is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance d induced by H-epsilon, we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first n eigenvalues of H-epsilon converge to the first n eigenvalues of the direct sum of harmonic oscillators on R-d located at several wells. In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low-lying eigenvalues of H-epsilon. These are obtained from eigenfunctions or quasimodes for the operator H-epsilon acting on L-2(R-d), via restriction to the lattice (epsilon Z)(d). Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrodinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted l(2)-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two "wells" (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrodinger operator in [22].}, language = {en} } @misc{ZoellerHolschneider2018, author = {Z{\"o}ller, Gert and Holschneider, Matthias}, title = {Reply to "Comment on 'The Maximum Possible and the Maximum Expected Earthquake Magnitude for Production-Induced Earthquakes at the Gas Field in Groningen, The Netherlands' by Gert Z{\"o}ller and Matthias Holschneider" by Mathias Raschke}, series = {Bulletin of the Seismological Society of America}, volume = {108}, journal = {Bulletin of the Seismological Society of America}, number = {2}, publisher = {Seismological Society of America}, address = {Albany}, issn = {0037-1106}, doi = {10.1785/0120170131}, pages = {1029 -- 1030}, year = {2018}, language = {en} } @article{LiuMuenchPeyerimhoff2018, author = {Liu, Shiping and M{\"u}nch, Florentin and Peyerimhoff, Norbert}, title = {Bakry-Emery curvature and diameter bounds on graphs}, series = {Calculus of variations and partial differential equations}, volume = {57}, journal = {Calculus of variations and partial differential equations}, number = {2}, publisher = {Springer}, address = {Heidelberg}, issn = {0944-2669}, doi = {10.1007/s00526-018-1334-x}, pages = {9}, year = {2018}, abstract = {We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry-{\´E}mery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet-Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from Fathi and Shu (Bernoulli 24(1):672-698, 2018) and Horn et al. (J f{\"u}r die reine und angewandte Mathematik (Crelle's J), 2017, https://doi.org/10.1515/crelle-2017-0038) and solve a conjecture from Cushing et al. (Bakry-{\´E}mery curvature functions of graphs, 2016).}, language = {en} } @article{KellerSchwarz2018, author = {Keller, Matthias and Schwarz, Michael}, title = {The Kazdan-Warner equation on canonically compactifiable graphs}, series = {Calculus of variations and partial differential equations}, volume = {57}, journal = {Calculus of variations and partial differential equations}, number = {2}, publisher = {Springer}, address = {Heidelberg}, issn = {0944-2669}, doi = {10.1007/s00526-018-1329-7}, pages = {18}, year = {2018}, abstract = {We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.}, language = {en} } @article{LesurWardinskiBaerenzungetal.2017, author = {Lesur, Vincent and Wardinski, Ingo and B{\"a}renzung, Julien and Holschneider, Matthias}, title = {On the frequency spectra of the core magnetic field Gauss coefficients}, series = {Physics of the earth and planetary interiors}, volume = {276}, journal = {Physics of the earth and planetary interiors}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0031-9201}, doi = {10.1016/j.pepi.2017.05.017}, pages = {145 -- 158}, year = {2017}, abstract = {From monthly mean observatory data spanning 1957-2014, geomagnetic field secular variation values were calculated by annual differences. Estimates of the spherical harmonic Gauss coefficients of the core field secular variation were then derived by applying a correlation based modelling. Finally, a Fourier transform was applied to the time series of the Gauss coefficients. This process led to reliable temporal spectra of the Gauss coefficients up to spherical harmonic degree 5 or 6, and down to periods as short as 1 or 2 years depending on the coefficient. We observed that a k(-2) slope, where k is the frequency, is an acceptable approximation for these spectra, with possibly an exception for the dipole field. The monthly estimates of the core field secular variation at the observatory sites also show that large and rapid variations of the latter happen. This is an indication that geomagnetic jerks are frequent phenomena and that significant secular variation signals at short time scales - i.e. less than 2 years, could still be extracted from data to reveal an unexplored part of the core dynamics.}, language = {en} } @article{BrungsGraeter2017, author = {Brungs, Hans H. and Gr{\"a}ter, Joachim}, title = {On central extensions of SL(2, F) admitting left-orderings}, series = {Journal of Algebra}, volume = {486}, journal = {Journal of Algebra}, publisher = {Elsevier}, address = {San Diego}, issn = {0021-8693}, doi = {10.1016/j.jalgebra.2017.05.025}, pages = {288 -- 327}, year = {2017}, abstract = {For an arbitrary euclidean field F we introduce a central extension (G(F), Phi) of SL(2, F) admitting a left-ordering and study its algebraic properties. The elements of G(F) are order preserving bijections of the convex hull of Q in F. If F = R then G(F) is isomorphic to the classical universal covering group of the Lie group SL(2, R). Among other results we show that G(F) is a perfect group which possesses a rank 1 cone of exceptional type. We also prove that its centre is an infinite cyclic group and investigate its normal subgroups.}, language = {en} } @article{SomogyvariReich2020, author = {Somogyv{\´a}ri, M{\´a}rk and Reich, Sebastian}, title = {Convergence tests for transdimensional Markov chains in geoscience imaging}, series = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences}, volume = {52}, journal = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences}, number = {5}, publisher = {Springer}, address = {Heidelberg}, issn = {1874-8961}, doi = {10.1007/s11004-019-09811-x}, pages = {651 -- 668}, year = {2020}, abstract = {Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations, which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and must be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles that are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization.}, language = {en} } @article{Ly2020, author = {Ly, Ibrahim}, title = {A Cauchy problem for the Cauchy-Riemann operator}, series = {Afrika Matematika}, volume = {32}, journal = {Afrika Matematika}, number = {1-2}, publisher = {Springer}, address = {Heidelberg}, issn = {1012-9405}, doi = {10.1007/s13370-020-00810-4}, pages = {69 -- 76}, year = {2020}, abstract = {We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution.}, language = {en} } @unpublished{LyTarkhanov2015, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {A Rad{\´o} theorem for p-harmonic functions}, volume = {4}, number = {3}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-71492}, pages = {10}, year = {2015}, abstract = {Let A be a nonlinear differential operator on an open set X in R^n and S a closed subset of X. Given a class F of functions in X, the set S is said to be removable for F relative to A if any weak solution of A (u) = 0 in the complement of S of class F satisfies this equation weakly in all of X. For the most extensively studied classes F we show conditions on S which guarantee that S is removable for F relative to A.}, language = {en} } @unpublished{LyTarkhanov2015, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {Asymptotic expansions at nonsymmetric cuspidal points}, volume = {4}, number = {7}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-78199}, pages = {11}, year = {2015}, abstract = {We study asymptotics of solutions to the Dirichlet problem in a domain whose boundary contains a nonsymmetric conical point. We establish a complete asymptotic expansion of solutions near the singular point.}, language = {en} } @article{MalassTarkhanov2020, author = {Malass, Ihsane and Tarkhanov, Nikolaj Nikolaevič}, title = {A perturbation of the de Rham complex}, series = {Journal of Siberian Federal University : Mathematics \& Physics}, volume = {13}, journal = {Journal of Siberian Federal University : Mathematics \& Physics}, number = {5}, publisher = {Siberian Federal University}, address = {Krasnojarsk}, issn = {1997-1397}, doi = {10.17516/1997-1397-2020-13-5-519-532}, pages = {519 -- 532}, year = {2020}, abstract = {We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings.}, language = {en} } @phdthesis{Jakobs2019, author = {Jakobs, Friedrich}, title = {Dubrovin-rings and their connection to Hughes-free skew fields of fractions}, doi = {10.25932/publishup-43556}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-435561}, school = {Universit{\"a}t Potsdam}, pages = {ix, 62}, year = {2019}, abstract = {One method of embedding groups into skew fields was introduced by A. I. Mal'tsev and B. H. Neumann (cf. [18, 19]). If G is an ordered group and F is a skew field, the set F((G)) of formal power series over F in G with well-ordered support forms a skew field into which the group ring F[G] can be embedded. Unfortunately it is not suficient that G is left-ordered since F((G)) is only an F-vector space in this case as there is no natural way to define a multiplication on F((G)). One way to extend the original idea onto left-ordered groups is to examine the endomorphism ring of F((G)) as explored by N. I. Dubrovin (cf. [5, 6]). It is possible to embed any crossed product ring F[G; η, σ] into the endomorphism ring of F((G)) such that each non-zero element of F[G; η, σ] defines an automorphism of F((G)) (cf. [5, 10]). Thus, the rational closure of F[G; η, σ] in the endomorphism ring of F((G)), which we will call the Dubrovin-ring of F[G; η, σ], is a potential candidate for a skew field of fractions of F[G; η, σ]. The methods of N. I. Dubrovin allowed to show that specific classes of groups can be embedded into a skew field. For example, N. I. Dubrovin contrived some special criteria, which are applicable on the universal covering group of SL(2, R). These methods have also been explored by J. Gr{\"a}ter and R. P. Sperner (cf. [10]) as well as N.H. Halimi and T. Ito (cf. [11]). Furthermore, it is of interest to know if skew fields of fractions are unique. For example, left and right Ore domains have unique skew fields of fractions (cf. [2]). This is not the general case as for example the free group with 2 generators can be embedded into non-isomorphic skew fields of fractions (cf. [12]). It seems likely that Ore domains are the most general case for which unique skew fields of fractions exist. One approach to gain uniqueness is to restrict the search to skew fields of fractions with additional properties. I. Hughes has defined skew fields of fractions of crossed product rings F[G; η, σ] with locally indicable G which fulfill a special condition. These are called Hughes-free skew fields of fractions and I. Hughes has proven that they are unique if they exist [13, 14]. This thesis will connect the ideas of N. I. Dubrovin and I. Hughes. The first chapter contains the basic terminology and concepts used in this thesis. We present methods provided by N. I. Dubrovin such as the complexity of elements in rational closures and special properties of endomorphisms of the vector space of formal power series F((G)). To combine the ideas of N.I. Dubrovin and I. Hughes we introduce Conradian left-ordered groups of maximal rank and examine their connection to locally indicable groups. Furthermore we provide notations for crossed product rings, skew fields of fractions as well as Dubrovin-rings and prove some technical statements which are used in later parts. The second chapter focuses on Hughes-free skew fields of fractions and their connection to Dubrovin-rings. For that purpose we introduce series representations to interpret elements of Hughes-free skew fields of fractions as skew formal Laurent series. This 1 Introduction allows us to prove that for Conradian left-ordered groups G of maximal rank the statement "F[G; η, σ] has a Hughes-free skew field of fractions" implies "The Dubrovin ring of F [G; η, σ] is a skew field". We will also prove the reverse and apply the results to give a new prove of Theorem 1 in [13]. Furthermore we will show how to extend injective ring homomorphisms of some crossed product rings onto their Hughes-free skew fields of fractions. At last we will be able to answer the open question whether Hughes--free skew fields are strongly Hughes-free (cf. [17, page 53]).}, language = {en} } @misc{BeniniSchenkel2017, author = {Benini, Marco and Schenkel, Alexander}, title = {Quantum field theories on categories fibered in groupoids}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {895}, issn = {1866-8372}, doi = {10.25932/publishup-43154}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-431541}, pages = {48}, year = {2017}, abstract = {We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories.}, language = {en} } @misc{KarpuzCevikKoppitzetal.2013, author = {Karpuz, Eylem Guzel and {\c{C}}evik, Ahmet Sinan and Koppitz, J{\"o}rg and Cangul, Ismail Naci}, title = {Some fixed-point results on (generalized) Bruck-Reilly ∗-extensions of monoids}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {942}, issn = {1866-8372}, doi = {10.25932/publishup-43270}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-432701}, pages = {11}, year = {2013}, abstract = {In this paper, we determine necessary and sufficient conditions for Bruck-Reilly and generalized Bruck-Reilly ∗-extensions of arbitrary monoids to be regular, coregular and strongly π-inverse. These semigroup classes have applications in various field of mathematics, such as matrix theory, discrete mathematics and p-adic analysis (especially in operator theory). In addition, while regularity and coregularity have so many applications in the meaning of boundaries (again in operator theory), inverse monoids and Bruck-Reilly extensions contain a mixture fixed-point results of algebra, topology and geometry within the purposes of this journal.}, language = {en} } @article{AlSaedyTarchanov2020, author = {Al-Saedy, Ammar Jaffar Muhesin and Tarchanov, Nikolaj Nikolaevič}, title = {A degree theory for Lagrangian boundary value problems}, series = {Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University; mathematics \& physics}, volume = {13}, journal = {Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University; mathematics \& physics}, number = {1}, publisher = {Sibirskij Federalʹnyj Universitet}, address = {Krasnojarsk}, issn = {1997-1397}, doi = {10.17516/1997-1397-2020-13-1-5-25}, pages = {5 -- 25}, year = {2020}, abstract = {We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems.}, language = {en} } @article{ClavierGuoPaychaetal.2019, author = {Clavier, Pierre J. and Guo, Li and Paycha, Sylvie and Zhang, Bin}, title = {An algebraic formulation of the locality principle in renormalisation}, series = {European Journal of Mathematics}, volume = {5}, journal = {European Journal of Mathematics}, number = {2}, publisher = {Springer}, address = {Cham}, issn = {2199-675X}, doi = {10.1007/s40879-018-0255-8}, pages = {356 -- 394}, year = {2019}, abstract = {We study the mathematical structure underlying the concept of locality which lies at the heart of classical and quantum field theory, and develop a machinery used to preserve locality during the renormalisation procedure. Viewing renormalisation in the framework of Connes and Kreimer as the algebraic Birkhoff factorisation of characters on a Hopf algebra with values in a Rota-Baxter algebra, we build locality variants of these algebraic structures, leading to a locality variant of the algebraic Birkhoff factorisation. This provides an algebraic formulation of the conservation of locality while renormalising. As an application in the context of the Euler-Maclaurin formula on lattice cones, we renormalise the exponential generating function which sums over the lattice points in a lattice cone. As a consequence, for a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate germs.}, language = {en} } @incollection{ClavierGuoPaychaetal.2020, author = {Clavier, Pierre J. and Guo, Li and Paycha, Sylvie and Zhang, Bin}, title = {Renormalisation and locality}, series = {Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2}, booktitle = {Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2}, publisher = {European Mathematical Society Publishing House}, address = {Z{\"u}rich}, isbn = {978-3-03719-205-4 print}, doi = {10.4171/205}, pages = {85 -- 132}, year = {2020}, language = {en} } @inproceedings{AudinDucourtiouxOuedraogoetal.2017, author = {Audin, Mich{\`e}le and Ducourtioux, Catherine and Ou{\´e}draogo, Fran{\c{c}}oise and Schulz, Ren{\´e} and Delgado, Julio and Ruzhansky, Michael and Lebeau, Gilles}, title = {Integral Fourier operators}, editor = {Paycha, Sylvie}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-413-5}, issn = {2199-4951}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-402657}, pages = {xxi, 229}, year = {2017}, abstract = {This volume of contributions based on lectures delivered at a school on Fourier Integral Operators held in Ouagadougou, Burkina Faso, 14-26 September 2015, provides an introduction to Fourier Integral Operators (FIO) for a readership of Master and PhD students as well as any interested layperson. Considering the wide spectrum of their applications and the richness of the mathematical tools they involve, FIOs lie the cross-road of many a field. This volume offers the necessary background, whether analytic or geometric, to get acquainted with FIOs, complemented by more advanced material presenting various aspects of active research in that area.}, language = {en} } @article{ChelkhLyTarkhanov2020, author = {Chelkh, W. and Ly, Ibrahim and Tarkhanov, Nikolai}, title = {A remark on the Laplace transform}, series = {Siberian Mathematical Journal}, volume = {61}, journal = {Siberian Mathematical Journal}, number = {4}, publisher = {Consultants Bureau, Springer}, address = {New York}, issn = {0037-4466}, doi = {10.1134/S0037446620040151}, pages = {755 -- 762}, year = {2020}, abstract = {The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function cos root z. This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of cos root z using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces.}, language = {en} } @article{KayaFreitag2022, author = {Kaya, Adem and Freitag, Melina A.}, title = {Conditioning analysis for discrete Helmholtz problems}, series = {Computers and mathematics with applications : an international journal}, volume = {118}, journal = {Computers and mathematics with applications : an international journal}, publisher = {Elsevier Science}, address = {Amsterdam}, issn = {0898-1221}, doi = {10.1016/j.camwa.2022.05.016}, pages = {171 -- 182}, year = {2022}, abstract = {In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.}, language = {en} } @article{KellerSchwarz2020, author = {Keller, Matthias and Schwarz, Michael}, title = {Courant's nodal domain theorem for positivity preserving forms}, series = {Journal of spectral theory}, volume = {10}, journal = {Journal of spectral theory}, number = {1}, publisher = {EMS Publishing House}, address = {Z{\"u}rich}, issn = {1664-039X}, doi = {10.4171/JST/292}, pages = {271 -- 309}, year = {2020}, abstract = {We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods.}, language = {en} } @article{KolasinskiMenne2017, author = {Kolasinski, Slawomir and Menne, Ulrich}, title = {Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds}, series = {Nonlinear Differential Equations and Applications NoDEA}, volume = {24}, journal = {Nonlinear Differential Equations and Applications NoDEA}, publisher = {Springer}, address = {Basel}, issn = {1021-9722}, doi = {10.1007/s00030-017-0436-z}, pages = {56}, year = {2017}, abstract = {This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds.}, language = {en} } @article{LyTarkhanov2020, author = {Ly, Ibrahim and Tarkhanov, Nikolaj Nikolaevič}, title = {Asymptotic expansions at nonsymmetric cuspidal points}, series = {Mathematical notes}, volume = {108}, journal = {Mathematical notes}, number = {1-2}, publisher = {Springer Science}, address = {New York}, issn = {0001-4346}, doi = {10.1134/S0001434620070238}, pages = {219 -- 228}, year = {2020}, abstract = {We study the asymptotics of solutions to the Dirichlet problem in a domain X subset of R3 whose boundary contains a singular point O. In a small neighborhood of this point, the domain has the form {z > root x(2) + y(4)}, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat'ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point.}, language = {en} } @article{Clavier2020, author = {Clavier, Pierre J.}, title = {Double shuffle relations for arborified zeta values}, series = {Journal of algebra}, volume = {543}, journal = {Journal of algebra}, publisher = {Elsevier}, address = {San Diego}, issn = {0021-8693}, doi = {10.1016/j.jalgebra.2019.10.015}, pages = {111 -- 155}, year = {2020}, abstract = {Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multiple zeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborified zeta values are algebra morphisms for these new products on trees.}, language = {en} } @misc{SerthPodlesnyBornsteinetal.2017, author = {Serth, Sebastian and Podlesny, Nikolai and Bornstein, Marvin and Lindemann, Jan and Latt, Johanna and Selke, Jan and Schlosser, Rainer and Boissier, Martin and Uflacker, Matthias}, title = {An interactive platform to simulate dynamic pricing competition on online marketplaces}, series = {2017 IEEE 21st International Enterprise Distributed Object Computing Conference (EDOC)}, journal = {2017 IEEE 21st International Enterprise Distributed Object Computing Conference (EDOC)}, publisher = {Institute of Electrical and Electronics Engineers}, address = {New York}, isbn = {978-1-5090-3045-3}, issn = {2325-6354}, doi = {10.1109/EDOC.2017.17}, pages = {61 -- 66}, year = {2017}, abstract = {E-commerce marketplaces are highly dynamic with constant competition. While this competition is challenging for many merchants, it also provides plenty of opportunities, e.g., by allowing them to automatically adjust prices in order to react to changing market situations. For practitioners however, testing automated pricing strategies is time-consuming and potentially hazardously when done in production. Researchers, on the other side, struggle to study how pricing strategies interact under heavy competition. As a consequence, we built an open continuous time framework to simulate dynamic pricing competition called Price Wars. The microservice-based architecture provides a scalable platform for large competitions with dozens of merchants and a large random stream of consumers. Our platform stores each event in a distributed log. This allows to provide different performance measures enabling users to compare profit and revenue of various repricing strategies in real-time. For researchers, price trajectories are shown which ease evaluating mutual price reactions of competing strategies. Furthermore, merchants can access historical marketplace data and apply machine learning. By providing a set of customizable, artificial merchants, users can easily simulate both simple rule-based strategies as well as sophisticated data-driven strategies using demand learning to optimize their pricing strategies.}, language = {en} } @article{TaghvaeideWiljesMehtaetal.2017, author = {Taghvaei, Amirhossein and de Wiljes, Jana and Mehta, Prashant G. and Reich, Sebastian}, title = {Kalman filter and its modern extensions for the continuous-time nonlinear filtering problem}, series = {Journal of dynamic systems measurement and control}, volume = {140}, journal = {Journal of dynamic systems measurement and control}, number = {3}, publisher = {ASME}, address = {New York}, issn = {0022-0434}, doi = {10.1115/1.4037780}, pages = {11}, year = {2017}, abstract = {This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman-Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.}, language = {en} } @article{deWiljesReichStannat2018, author = {de Wiljes, Jana and Reich, Sebastian and Stannat, Wilhelm}, title = {Long-Time stability and accuracy of the ensemble Kalman-Bucy Filter for fully observed processes and small measurement noise}, series = {SIAM Journal on Applied Dynamical Systems}, volume = {17}, journal = {SIAM Journal on Applied Dynamical Systems}, number = {2}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {1536-0040}, doi = {10.1137/17M1119056}, pages = {1152 -- 1181}, year = {2018}, abstract = {The ensemble Kalman filter has become a popular data assimilation technique in the geosciences. However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman-Bucy filter is consistent with the classic Kalman-Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system.}, language = {en} } @article{HammPelivanGrottetal.2020, author = {Hamm, Maximilian and Pelivan, Ivanka and Grott, Matthias and de Wiljes, Jana}, title = {Thermophysical modelling and parameter estimation of small solar system bodies via data assimilation}, series = {Monthly notices of the Royal Astronomical Society}, volume = {496}, journal = {Monthly notices of the Royal Astronomical Society}, number = {3}, publisher = {Oxford Univ. Press}, address = {Oxford}, issn = {0035-8711}, doi = {10.1093/mnras/staa1755}, pages = {2776 -- 2785}, year = {2020}, abstract = {Deriving thermophysical properties such as thermal inertia from thermal infrared observations provides useful insights into the structure of the surface material on planetary bodies. The estimation of these properties is usually done by fitting temperature variations calculated by thermophysical models to infrared observations. For multiple free model parameters, traditional methods such as least-squares fitting or Markov chain Monte Carlo methods become computationally too expensive. Consequently, the simultaneous estimation of several thermophysical parameters, together with their corresponding uncertainties and correlations, is often not computationally feasible and the analysis is usually reduced to fitting one or two parameters. Data assimilation (DA) methods have been shown to be robust while sufficiently accurate and computationally affordable even for a large number of parameters. This paper will introduce a standard sequential DA method, the ensemble square root filter, for thermophysical modelling of asteroid surfaces. This method is used to re-analyse infrared observations of the MARA instrument, which measured the diurnal temperature variation of a single boulder on the surface of near-Earth asteroid (162173) Ryugu. The thermal inertia is estimated to be 295 +/- 18 Jm(-2) K-1 s(-1/2), while all five free parameters of the initial analysis are varied and estimated simultaneously. Based on this thermal inertia estimate the thermal conductivity of the boulder is estimated to be between 0.07 and 0.12,Wm(-1) K-1 and the porosity to be between 0.30 and 0.52. For the first time in thermophysical parameter derivation, correlations and uncertainties of all free model parameters are incorporated in the estimation procedure that is more than 5000 times more efficient than a comparable parameter sweep.}, language = {en} } @article{AcevedoDeWiljesReich2017, author = {Acevedo, Walter and De Wiljes, Jana and Reich, Sebastian}, title = {Second-order accurate ensemble transform particle filters}, series = {SIAM journal on scientific computing}, volume = {39}, journal = {SIAM journal on scientific computing}, number = {5}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {1064-8275}, doi = {10.1137/16M1095184}, pages = {A1834 -- A1850}, year = {2017}, abstract = {Particle filters (also called sequential Monte Carlo methods) are widely used for state and parameter estimation problems in the context of nonlinear evolution equations. The recently proposed ensemble transform particle filter (ETPF) [S. Reich, SIAM T. Sci. Comput., 35, (2013), pp. A2013-A2014[ replaces the resampling step of a standard particle filter by a linear transformation which allows for a hybridization of particle filters with ensemble Kalman filters and renders the resulting hybrid filters applicable to spatially extended systems. However, the linear transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes. Here we address both of these shortcomings by developing second order accurate extensions of the ETPF. These extensions allow one in particular to replace the exact solution of a linear transport problem by its Sinkhorn approximation. It is also demonstrated that the nonlinear ensemble transform filter arises as a special case of our general framework. We illustrate the performance of the second-order accurate filters for the chaotic Lorenz-63 and Lorenz-96 models and a dynamic scene-viewing model. The numerical results for the Lorenz-63 and Lorenz-96 models demonstrate that significant accuracy improvements can be achieved in comparison to a standard ensemble Kalman filter and the ETPF for small to moderate ensemble sizes. The numerical results for the scene-viewing model reveal, on the other hand, that second-order corrections can lead to statistically inconsistent samples from the posterior parameter distribution.}, language = {en} } @phdthesis{Rafler2009, author = {Rafler, Mathias}, title = {Gaussian loop- and P{\´o}lya processes : a point process approach}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-029-8}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-38706}, school = {Universit{\"a}t Potsdam}, pages = {ix, 162}, year = {2009}, abstract = {This thesis considers on the one hand the construction of point processes via conditional intensities, motivated by the partial Integration of the Campbell measure of a point process. Under certain assumptions on the intensity the existence of such a point process is shown. A fundamental example turns out to be the P{\´o}lya sum process, whose conditional intensity is a generalisation of the P{\´o}lya urn dynamics. A Cox process representation for that point process is shown. A further process considered is a Poisson process of Gaussian loops, which represents a noninteracting particle system derived from the discussion of indistinguishable particles. Both processes are used to define particle systems locally, for which thermodynamic limits are determined.}, language = {en} } @unpublished{MeleardRoelly2013, author = {M{\´e}l{\´e}ard, Sylvie and Roelly, Sylvie}, title = {Evolutive two-level population process and large population approximations}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64604}, year = {2013}, abstract = {We are interested in modeling the Darwinian evolution of a population described by two levels of biological parameters: individuals characterized by an heritable phenotypic trait submitted to mutation and natural selection and cells in these individuals influencing their ability to consume resources and to reproduce. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We are looking for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses.}, language = {en} } @phdthesis{Ludewig2016, author = {Ludewig, Matthias}, title = {Path integrals on manifolds with boundary and their asymptotic expansions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-94387}, school = {Universit{\"a}t Potsdam}, pages = {146}, year = {2016}, abstract = {It is "scientific folklore" coming from physical heuristics that solutions to the heat equation on a Riemannian manifold can be represented by a path integral. However, the problem with such path integrals is that they are notoriously ill-defined. One way to make them rigorous (which is often applied in physics) is finite-dimensional approximation, or time-slicing approximation: Given a fine partition of the time interval into small subintervals, one restricts the integration domain to paths that are geodesic on each subinterval of the partition. These finite-dimensional integrals are well-defined, and the (infinite-dimensional) path integral then is defined as the limit of these (suitably normalized) integrals, as the mesh of the partition tends to zero. In this thesis, we show that indeed, solutions to the heat equation on a general compact Riemannian manifold with boundary are given by such time-slicing path integrals. Here we consider the heat equation for general Laplace type operators, acting on sections of a vector bundle. We also obtain similar results for the heat kernel, although in this case, one has to restrict to metrics satisfying a certain smoothness condition at the boundary. One of the most important manipulations one would like to do with path integrals is taking their asymptotic expansions; in the case of the heat kernel, this is the short time asymptotic expansion. In order to use time-slicing approximation here, one needs the approximation to be uniform in the time parameter. We show that this is possible by giving strong error estimates. Finally, we apply these results to obtain short time asymptotic expansions of the heat kernel also in degenerate cases (i.e. at the cut locus). Furthermore, our results allow to relate the asymptotic expansion of the heat kernel to a formal asymptotic expansion of the infinite-dimensional path integral, which gives relations between geometric quantities on the manifold and on the loop space. In particular, we show that the lowest order term in the asymptotic expansion of the heat kernel is essentially given by the Fredholm determinant of the Hessian of the energy functional. We also investigate how this relates to the zeta-regularized determinant of the Jacobi operator along minimizing geodesics.}, language = {en} } @misc{Scharrer2016, type = {Master Thesis}, author = {Scharrer, Christian}, title = {Relating diameter and mean curvature for varifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-97013}, school = {Universit{\"a}t Potsdam}, pages = {42}, year = {2016}, abstract = {The main results of this thesis are formulated in a class of surfaces (varifolds) generalizing closed and connected smooth submanifolds of Euclidean space which allows singularities. Given an indecomposable varifold with dimension at least two in some Euclidean space such that the first variation is locally bounded, the total variation is absolutely continuous with respect to the weight measure, the density of the weight measure is at least one outside a set of weight measure zero and the generalized mean curvature is locally summable to a natural power (dimension of the varifold minus one) with respect to the weight measure. The thesis presents an improved estimate of the set where the lower density is small in terms of the one dimensional Hausdorff measure. Moreover, if the support of the weight measure is compact, then the intrinsic diameter with respect to the support of the weight measure is estimated in terms of the generalized mean curvature. This estimate is in analogy to the diameter control for closed connected manifolds smoothly immersed in some Euclidean space of Peter Topping. Previously, it was not known whether the hypothesis in this thesis implies that two points in the support of the weight measure have finite geodesic distance.}, language = {en} } @unpublished{ShlapunovTarkhanov2016, author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {An open mapping theorem for the Navier-Stokes equations}, volume = {5}, number = {10}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-98687}, pages = {80}, year = {2016}, abstract = {We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed H{\"o}lder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fr{\´e}chet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of H{\"o}lder spaces.}, language = {en} } @unpublished{BlanchardKraemer2016, author = {Blanchard, Gilles and Kr{\"a}mer, Nicole}, title = {Convergence rates of kernel conjugate gradient for random design regression}, volume = {5}, number = {8}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-94195}, pages = {31}, year = {2016}, abstract = {We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.}, language = {en} } @phdthesis{Beinrucker2015, author = {Beinrucker, Andre}, title = {Variable selection in high dimensional data analysis with applications}, school = {Universit{\"a}t Potsdam}, pages = {VII, 107}, year = {2015}, language = {en} } @unpublished{FedosovTarkhanov2015, author = {Fedosov, Boris and Tarkhanov, Nikolai Nikolaevich}, title = {Deformation quantisation and boundary value problems}, volume = {4}, number = {5}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-77150}, pages = {27}, year = {2015}, abstract = {We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.}, language = {en} } @unpublished{ElinShoikhetTarkhanov2015, author = {Elin, Mark and Shoikhet, David and Tarkhanov, Nikolai Nikolaevich}, title = {Analytic semigroups of holomorphic mappings and composition operators}, volume = {4}, number = {6}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-77914}, pages = {30}, year = {2015}, abstract = {In this paper we study the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators. We also provide a brief review around this topic.}, language = {en} } @incollection{Petsche2015, author = {Petsche, Hans-Joachim}, title = {Einf{\"u}hrung}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, booktitle = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {9 -- 14}, year = {2015}, language = {de} } @article{Petsche2015, author = {Petsche, Hans-Joachim}, title = {Raum und Zahl - philosophische Kontexte}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, journal = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {15 -- 33}, year = {2015}, language = {de} } @article{SixtusFischer2015, author = {Sixtus, Elena and Fischer, Martin H.}, title = {Eine kognitionswissenschaftliche Betrachtung der Konzepte "Raum" und "Zahl"}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, journal = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {35 -- 62}, year = {2015}, language = {de} } @article{Proeve2015, author = {Pr{\"o}ve, Ralf}, title = {Numerische Zeichen und die Repr{\"a}sentation von Sinn}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, journal = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {63 -- 76}, year = {2015}, language = {de} } @article{Heimann2015, author = {Heimann, Heinz-Dieter}, title = {Kreise - Punkte - Linien}, series = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, journal = {Raum und Zahl im Fokus der Wissenschaften : eine multidisziplin{\"a}re Vorlesungsreihe}, publisher = {Trafo}, address = {Berlin}, isbn = {978-3-86464-082-7}, pages = {111 -- 127}, year = {2015}, language = {de} } @phdthesis{Chutsagulprom2016, author = {Chutsagulprom, Nawinda}, title = {Ensemble-based filters dealing with non-Gaussianity and nonlinearity}, school = {Universit{\"a}t Potsdam}, pages = {95}, year = {2016}, language = {en} } @unpublished{RoellyVallois2016, author = {Roelly, Sylvie and Vallois, Pierre}, title = {Convoluted Brownian motion}, volume = {5}, number = {9}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-96339}, pages = {37}, year = {2016}, abstract = {In this paper we analyse semimartingale properties of a class of Gaussian periodic processes, called convoluted Brownian motions, obtained by convolution between a deterministic function and a Brownian motion. A classical example in this class is the periodic Ornstein-Uhlenbeck process. We compute their characteristics and show that in general, they are neither Markovian nor satisfy a time-Markov field property. Nevertheless, by enlargement of filtration and/or addition of a one-dimensional component, one can in some case recover the Markovianity. We treat exhaustively the case of the bidimensional trigonometric convoluted Brownian motion and the higher-dimensional monomial convoluted Brownian motion.}, language = {en} } @unpublished{FedchenkoTarkhanov2017, author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich}, title = {A Rad{\´o} Theorem for the Porous Medium Equation}, series = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam}, volume = {6}, journal = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam}, number = {1}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-102735}, pages = {12}, year = {2017}, abstract = {We prove that each locally Lipschitz continuous function satisfying the porous medium equation away from the set of its zeroes is actually a weak solution of this equation in the whole domain.}, language = {en} } @unpublished{ShlapunovTarkhanov2017, author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {Golusin-Krylov Formulas in Complex Analysis}, series = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam}, volume = {6}, journal = {Preprints des Instituts f{\"u}r Mathematik der Universit{\"a}t Potsdam}, number = {2}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-102774}, pages = {25}, year = {2017}, abstract = {This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.}, language = {en} } @unpublished{VasilievTarkhanov2016, author = {Vasiliev, Serguei and Tarkhanov, Nikolai Nikolaevich}, title = {Construction of series of perfect lattices by layer superposition}, volume = {5}, number = {11}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-100591}, pages = {11}, year = {2016}, abstract = {We construct a new series of perfect lattices in n dimensions by the layer superposition method of Delaunay-Barnes.}, language = {en} } @phdthesis{Bartels1999, author = {Bartels, Knut}, title = {Tests zur Modellspezifikation in der nichtlinearen Regression}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0000171}, school = {Universit{\"a}t Potsdam}, year = {1999}, abstract = {Als Grundlage vieler statistischer Verfahren wird der Prozess der Entstehung von Daten modelliert, um dann weitere Sch{\"a}tz- und Testverfahren anzuwenden. Diese Arbeit befasst sich mit der Frage, wie diese Spezifikation f{\"u}r parametrische Modelle selbst getestet werden kann. In Erweiterung bestehender Verfahren werden Tests mit festem Kern eingef{\"u}hrt und ihre asymptotischen Eigenschaften werden analysiert. Es wird gezeigt, dass die Bestimmung der kritischen Werte mit mehreren Stichprobenwiederholungsverfahren m{\"o}glich ist. Von diesen ist eine neue Monte-Carlo-Approximation besonders wichtig, da sie die Komplexit{\"a}t der Berechnung deutlich verringern kann. Ein bedingter Kleinste-Quadrate-Sch{\"a}tzer f{\"u}r nichtlineare parametrische Modelle wird definiert und seine wesentlichen asymptotischen Eigenschaften werden hergeleitet. S{\"a}mtliche Versionen der Tests und alle neuen Konzepte wurden in Simulationsstudien untersucht, deren wichtigste Resultate pr{\"a}sentiert werden. Die praktische Anwendbarkeit der Testverfahren wird an einem Datensatz zur Produktwahl dargelegt, der mit multinomialen Logit-Modellen analysiert werden soll.}, language = {de} } @article{Meyerhoefer2001, author = {Meyerh{\"o}fer, Wolfram}, title = {Was misst TIMSS?}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-224}, pages = {1 -- 11}, year = {2001}, abstract = {Bei der Erstellung und Interpretation mathematischer Leistungstests steht die Frage, was eine Aufgabe mißt. Der Artikel stellt mit der strukturalen oder objektiven Hermeneutik eine Methode vor, mit der die verschiedenen Dimensionen der von einer Aufgabe erfassten F{\"a}higkeiten herausgearbeitet werden k{\"o}nnen. Dabei werden fachliche Anforderungen, Irritationsmomente und das durch die Aufgabe transportierte Bild vom jeweiligen Fach ebenso erfasst wie Momente, die man eher als Testf{\"a}higkeit bezeichnen w{\"u}rde.Am Beispiel einer TIMSS-Aufgabe wird diskutiert, dass das von den Testerstellern benutzte theoretische Konstrukt kaum geeignet ist, nachhaltig zu beschreiben, was eine Aufgabe misst.}, language = {de} } @unpublished{RoellyFradon2006, author = {Roelly, Sylvie and Fradon, Myriam}, title = {Infinite system of Brownian balls : equilibrium measures are canonical Gibbs}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6720}, year = {2006}, abstract = {We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.}, language = {en} } @phdthesis{Rosenberger2006, author = {Rosenberger, Elke}, title = {Asymptotic spectral analysis and tunnelling for a class of difference operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-7393}, school = {Universit{\"a}t Potsdam}, year = {2006}, abstract = {We analyze the asymptotic behavior in the limit epsilon to zero for a wide class of difference operators H_epsilon = T_epsilon + V_epsilon with underlying multi-well potential. They act on the square summable functions on the lattice (epsilon Z)^d. We start showing the validity of an harmonic approximation and construct WKB-solutions at the wells. Then we construct a Finslerian distance d induced by H and show that short integral curves are geodesics and d gives the rate for the exponential decay of Dirichlet eigenfunctions. In terms of this distance, we give sharp estimates for the interaction between the wells and construct the interaction matrix.}, subject = {Mathematische Physik}, language = {en} } @unpublished{MeklerBoeckmannSokolovskaia2000, author = {Mekler, A. A. and B{\"o}ckmann, Christine and Sokolovskaia, N.}, title = {Particle distribution from spectral Mie-scattering: kernel representation and singular-value spectrum}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14948}, year = {2000}, abstract = {This paper deals with the Mie scattering kernels for multi-spectral data. The kernels may be represented in form of power series. Furthermore, the singular-value spectrum and the degree of ill-posedness in dependence on the refractive index of the particles are numerically approximated. A special hybrid regularization technique allows us to determine via inversion the particle distributions of different types.}, language = {en} } @phdthesis{Ong2007, author = {Ong, James Kwan Yau}, title = {The predictability problem}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15025}, school = {Universit{\"a}t Potsdam}, year = {2007}, abstract = {Wir versuchen herauszufinden, ob das subjektive Maß der Cloze-Vorhersagbarkeit mit der Kombination objektiver Maße (semantische und n-gram-Maße) gesch{\"a}tzt werden kann, die auf den statistischen Eigenschaften von Textkorpora beruhen. Die semantischen Maße werden entweder durch Abfragen von Internet-Suchmaschinen oder durch die Anwendung der Latent Semantic Analysis gebildet, w{\"a}hrend die n-gram-Wortmaße allein auf den Ergebnissen von Internet-Suchmaschinen basieren. Weiterhin untersuchen wir die Rolle der Cloze-Vorhersagbarkeit in SWIFT, einem Modell der Blickkontrolle, und w{\"a}gen ab, ob andere Parameter den der Vorhersagbarkeit ersetzen k{\"o}nnen. Unsere Ergebnisse legen nahe, dass ein computationales Modell, welches Vorhersagbarkeitswerte berechnet, nicht nur Maße beachten muss, die die Relatiertheit eines Wortes zum Kontext darstellen; das Vorhandensein eines Maßes bez{\"u}glich der Nicht-Relatiertheit ist von ebenso großer Bedeutung. Obwohl hier jedoch nur Relatiertheits-Maße zur Verf{\"u}gung stehen, sollte SWIFT ebensogute Ergebnisse liefern, wenn wir Cloze-Vorhersagbarkeit mit unseren Maßen ersetzen.}, language = {en} } @phdthesis{Koh2008, author = {Koh, Dennis}, title = {The evolution equation for closed magnetic geodesics}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-940793-24-9}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-16647}, school = {Universit{\"a}t Potsdam}, pages = {60}, year = {2008}, abstract = {Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations whose solutions describe the orbits of closed p-branes under the effect of a "generalized physical force". For the corresponding evolution equation, which is a system of parabolic nonlinear partial differential equations associated to the elliptic PDE, we can establish existence of short time solutions.}, language = {en} } @unpublished{Murr2008, author = {Murr, R{\"u}diger}, title = {Dualit{\"a}tsformeln f{\"u}r Brownsche Bewegung und f{\"u}r eine Irrfahrt mit Anwendung am Konvergenzergebnis von Donsker}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49476}, year = {2008}, abstract = {Aus dem Inhalt: 0.1 Danksagung 0.2 Einleitung 1 Allgemeines und Grundlagen 1.1 Die Brownsche Bewegung 2 Die Dualit{\"a}tsformel des Wienermaßes 2.1 Wienermaß erf{\"u}llt Dualit{\"a}tsformel 2.2 Dualit{\"a}tsformel charakterisiert Wienermaß 3 Die diskrete Dualit{\"a}tsformel der Irrfahrt 3.1 Verallgemeinerte symmetrische Irrfahrt erf{\"u}llt diskrete Dualit{\"a}tsformel 3.2 Diskrete Dualit{\"a}tsformel charakterisiert verallgemeinerte symmetrische Irrfahrt 4 Donskers Theorem und die Dualit{\"a}tsformeln 4.1 Straffheit der renormierten stetigen Irrfahrt 4.2 Konvergenz der Irrfahrt 5 Anhang}, language = {de} } @unpublished{FedchenkoTarkhanov2014, author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich}, title = {An index formula for Toeplitz operators}, volume = {3}, number = {12}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72499}, pages = {24}, year = {2014}, abstract = {We prove a Fedosov index formula for the index of Toeplitz operators connected with the Hardy space of solutions to an elliptic system of first order partial differential equations in a bounded domain of Euclidean space with infinitely differentiable boundary.}, language = {en} } @phdthesis{Dyachenko2014, author = {Dyachenko, Evgeniya}, title = {Elliptic problems with small parameter}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72056}, school = {Universit{\"a}t Potsdam}, year = {2014}, abstract = {In this thesis we consider diverse aspects of existence and correctness of asymptotic solutions to elliptic differential and pseudodifferential equations. We begin our studies with the case of a general elliptic boundary value problem in partial derivatives. A small parameter enters the coefficients of the main equation as well as into the boundary conditions. Such equations have already been investigated satisfactory, but there still exist certain theoretical deficiencies. Our aim is to present the general theory of elliptic problems with a small parameter. For this purpose we examine in detail the case of a bounded domain with a smooth boundary. First of all, we construct formal solutions as power series in the small parameter. Then we examine their asymptotic properties. It suffices to carry out sharp two-sided \emph{a priori} estimates for the operators of boundary value problems which are uniform in the small parameter. Such estimates failed to hold in functional spaces used in classical elliptic theory. To circumvent this limitation we exploit norms depending on the small parameter for the functions defined on a bounded domain. Similar norms are widely used in literature, but their properties have not been investigated extensively. Our theoretical investigation shows that the usual elliptic technique can be correctly carried out in these norms. The obtained results also allow one to extend the norms to compact manifolds with boundaries. We complete our investigation by formulating algebraic conditions on the operators and showing their equivalence to the existence of a priori estimates. In the second step, we extend the concept of ellipticity with a small parameter to more general classes of operators. Firstly, we want to compare the difference in asymptotic patterns between the obtained series and expansions for similar differential problems. Therefore we investigate the heat equation in a bounded domain with a small parameter near the time derivative. In this case the characteristics touch the boundary at a finite number of points. It is known that the solutions are not regular in a neighbourhood of such points in advance. We suppose moreover that the boundary at such points can be non-smooth but have cuspidal singularities. We find a formal asymptotic expansion and show that when a set of parameters comes through a threshold value, the expansions fail to be asymptotic. The last part of the work is devoted to general concept of ellipticity with a small parameter. Several theoretical extensions to pseudodifferential operators have already been suggested in previous studies. As a new contribution we involve the analysis on manifolds with edge singularities which allows us to consider wider classes of perturbed elliptic operators. We examine that introduced classes possess a priori estimates of elliptic type. As a further application we demonstrate how developed tools can be used to reduce singularly perturbed problems to regular ones.}, 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} } @phdthesis{Kroencke2013, author = {Kr{\"o}ncke, Klaus}, title = {Stability of Einstein Manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69639}, school = {Universit{\"a}t Potsdam}, year = {2013}, abstract = {This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum.}, language = {en} } @unpublished{GairingHoegeleKosenkovaetal.2013, author = {Gairing, Jan and H{\"o}gele, Michael and Kosenkova, Tetiana and Kulik, Alexei Michajlovič}, title = {Coupling distances between L{\´e}vy measures and applications to noise sensitivity of SDE}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68886}, year = {2013}, abstract = {We introduce the notion of coupling distances on the space of L{\´e}vy measures in order to quantify rates of convergence towards a limiting L{\´e}vy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the L{\´e}vy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two L{\´e}vy diffusions in terms of the couping distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data.}, language = {en} } @unpublished{FedchenkoTarkhanov2013, author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich}, title = {A Class of Toeplitz Operators in Several Variables}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68932}, year = {2013}, abstract = {We introduce the concept of Toeplitz operator associated with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We characterise those Toeplitz operators which are Fredholm, thus initiating the index theory.}, 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} } @unpublished{ConfortiRoelly2015, author = {Conforti, Giovanni and Roelly, Sylvie}, title = {Reciprocal class of random walks on an Abelian group}, volume = {4}, number = {1}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72604}, pages = {22}, year = {2015}, abstract = {Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.}, language = {en} } @phdthesis{Bettenbuehl2015, author = {Bettenb{\"u}hl, Mario}, title = {Microsaccades}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-122-6}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72622}, school = {Universit{\"a}t Potsdam}, pages = {iv, 126}, year = {2015}, abstract = {The first thing we do upon waking is open our eyes. Rotating them in our eye sockets, we scan our surroundings and collect the information into a picture in our head. Eye movements can be split into saccades and fixational eye movements, which occur when we attempt to fixate our gaze. The latter consists of microsaccades, drift and tremor. Before we even lift our eye lids, eye movements - such as saccades and microsaccades that let the eyes jump from one to another position - have partially been prepared in the brain stem. Saccades and microsaccades are often assumed to be generated by the same mechanisms. But how saccades and microsaccades can be classified according to shape has not yet been reported in a statistical manner. Research has put more effort into the investigations of microsaccades' properties and generation only since the last decade. Consequently, we are only beginning to understand the dynamic processes governing microsaccadic eye movements. Within this thesis, the dynamics governing the generation of microsaccades is assessed and the development of a model for the underlying processes. Eye movement trajectories from different experiments are used, recorded with a video-based eye tracking technique, and a novel method is proposed for the scale-invariant detection of saccades (events of large amplitude) and microsaccades (events of small amplitude). Using a time-frequency approach, the method is examined with different experiments and validated against simulated data. A shape model is suggested that allows for a simple estimation of saccade- and microsaccade related properties. For sequences of microsaccades, in this thesis a time-dynamic Markov model is proposed, with a memory horizon that changes over time and which can best describe sequences of microsaccades.}, language = {en} } @unpublished{AlsaedyTarkhanov2015, author = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich}, title = {Weak boundary values of solutions of Lagrangian problems}, volume = {4}, number = {2}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72617}, pages = {24}, year = {2015}, abstract = {We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems.}, language = {en} } @unpublished{Tarkhanov2015, author = {Tarkhanov, Nikolai Nikolaevich}, title = {A spectral theorem for deformation quantisation}, volume = {4}, number = {4}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-72425}, pages = {8}, year = {2015}, abstract = {We present a construction of the eigenstate at a noncritical level of the Hamiltonian function. Moreover, we evaluate the contributions of Morse critical points to the spectral decomposition.}, language = {en} } @misc{Schorsch2008, type = {Master Thesis}, author = {Schorsch, Andrea}, title = {Statistische Eigenschaften von Clusterverfahren}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29026}, school = {Universit{\"a}t Potsdam}, year = {2008}, abstract = {Die vorliegende Diplomarbeit besch{\"a}ftigt sich mit zwei Aspekten der statistischen Eigenschaften von Clusterverfahren. Zum einen geht die Arbeit auf die Frage der Existenz von unterschiedlichen Clusteranalysemethoden zur Strukturfindung und deren unterschiedlichen Vorgehensweisen ein. Die Methode des Abstandes zwischen Mannigfaltigkeiten und die K-means Methode liefern ausgehend von gleichen Daten unterschiedliche Endclusterungen. Der zweite Teil dieser Arbeit besch{\"a}ftigt sich n{\"a}her mit den asymptotischen Eigenschaften des K-means Verfahrens. Hierbei ist die Menge der optimalen Clusterzentren konsistent. Bei Vergr{\"o}ßerung des Stichprobenumfangs gegen Unendlich konvergiert diese in Wahrscheinlichkeit gegen die Menge der Clusterzentren, die das Varianzkriterium minimiert. Ebenfalls konvergiert die Menge der optimalen Clusterzentren f{\"u}r n gegen Unendlich gegen eine Normalverteilung. Es hat sich dabei ergeben, dass die einzelnen Clusterzentren voneinander abh{\"a}ngen.}, language = {de} } @phdthesis{Penisson2010, author = {P{\´e}nisson, Sophie}, title = {Conditional limit theorems for multitype branching processes and illustration in epidemiological risk analysis}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-45307}, school = {Universit{\"a}t Potsdam}, year = {2010}, abstract = {This thesis is concerned with the issue of extinction of populations composed of different types of individuals, and their behavior before extinction and in case of a very late extinction. We approach this question firstly from a strictly probabilistic viewpoint, and secondly from the standpoint of risk analysis related to the extinction of a particular model of population dynamics. In this context we propose several statistical tools. The population size is modeled by a branching process, which is either a continuous-time multitype Bienaym{\´e}-Galton-Watson process (BGWc), or its continuous-state counterpart, the multitype Feller diffusion process. We are interested in different kinds of conditioning on non-extinction, and in the associated equilibrium states. These ways of conditioning have been widely studied in the monotype case. However the literature on multitype processes is much less extensive, and there is no systematic work establishing connections between the results for BGWc processes and those for Feller diffusion processes. In the first part of this thesis, we investigate the behavior of the population before its extinction by conditioning the associated branching process X_t on non-extinction (X_t≠0), or more generally on non-extinction in a near future 0≤θ<∞ (X_{t+θ}≠0), and by letting t tend to infinity. We prove the result, new in the multitype framework and for θ>0, that this limit exists and is non-degenerate. This reflects a stationary behavior for the dynamics of the population conditioned on non-extinction, and provides a generalization of the so-called Yaglom limit, corresponding to the case θ=0. In a second step we study the behavior of the population in case of a very late extinction, obtained as the limit when θ tends to infinity of the process conditioned by X_{t+θ}≠0. The resulting conditioned process is a known object in the monotype case (sometimes referred to as Q-process), and has also been studied when X_t is a multitype Feller diffusion process. We investigate the not yet considered case where X_t is a multitype BGWc process and prove the existence of the associated Q-process. In addition, we examine its properties, including the asymptotic ones, and propose several interpretations of the process. Finally, we are interested in interchanging the limits in t and θ, as well as in the not yet studied commutativity of these limits with respect to the high-density-type relationship between BGWc processes and Feller processes. We prove an original and exhaustive list of all possible exchanges of limit (long-time limit in t, increasing delay of extinction θ, diffusion limit). The second part of this work is devoted to the risk analysis related both to the extinction of a population and to its very late extinction. We consider a branching population model (arising notably in the epidemiological context) for which a parameter related to the first moments of the offspring distribution is unknown. We build several estimators adapted to different stages of evolution of the population (phase growth, decay phase, and decay phase when extinction is expected very late), and prove moreover their asymptotic properties (consistency, normality). In particular, we build a least squares estimator adapted to the Q-process, allowing a prediction of the population development in the case of a very late extinction. This would correspond to the best or to the worst-case scenario, depending on whether the population is threatened or invasive. These tools enable us to study the extinction phase of the Bovine Spongiform Encephalopathy epidemic in Great Britain, for which we estimate the infection parameter corresponding to a possible source of horizontal infection persisting after the removal in 1988 of the major route of infection (meat and bone meal). This allows us to predict the evolution of the spread of the disease, including the year of extinction, the number of future cases and the number of infected animals. In particular, we produce a very fine analysis of the evolution of the epidemic in the unlikely event of a very late extinction.}, language = {en} } @phdthesis{Jankowski2004, author = {Jankowski, \poundsukasz}, title = {Modelling and simulation of light propagation in non-aged and aged step-index polymer optical fibres}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0001649}, school = {Universit{\"a}t Potsdam}, year = {2004}, abstract = {Kunststofflichtwellenleiter (POFs) stellen ein verh{\"a}ltnism{\"a}ßig neues Medium zur optische Datenkommunikation {\"u}ber kurzen Strecken dar. W{\"a}hrend ihrer Einsatzdauer unterliegen POFs unterschiedlichen Arten von Umweltbeanspruchungen, haupts{\"a}chlich durch hohe Temperatur, hohe Feuchtigkeit und mechanischen Belastungen. Zahlreiche experimentelle Forschungen besch{\"a}ftigten sich mit der standardisierten Pr{\"u}fung der Zuverl{\"a}ssigkeit von im Handel erh{\"a}ltlichen Fasern. Jedoch gab es bisher wenig Erfolg bei der Bem{\"u}hung, zwei grundlegende optische Erscheinungen, Absorption und Streuung, die die Lichtausbreitung in Fasern stark beeinflussen, zu verstehen und praktisch zu modellieren: Diese beiden Effekte beschreiben nicht nur die Qualit{\"a}t neuer Fasern, sondern sie werden auch stark durch die Alterungsprozess beeinflusst. Der Hauptzweck dieser Doktorarbeit war es, ein praktisch verwendbares und theoretisch gut fundiertes Modell der Lichtausbreitung in nicht gealterten und gealterten POFs zu entwickeln und es durch optische Experimente zu verifizieren. Dabei wurden anwendungsorientierte Aspekte mit theoretischer POF-Modellierung kombiniert. Die Arbeit enth{\"a}lt die erste bekannte Anwendung der Wellenanalyse zur Untersuchung der winkelabh{\"a}ngigen Eigenschaften der Streuung in Lichtwellenleitern. F{\"u}r die praktischen Experimente wurden mehrere POF-Proben unterschiedlicher Hersteller k{\"u}nstlich gealtert, indem sie bis 4500 Stunden bei 100 °C gelagert wurden (ohne Feuchtekontrolle). Die Parameter der jeweiligen Simulationen wurden mittels einer systematischen Optimierung an die gemessen optischen Eigenschaften der gealterten Proben angeglichen. Die Resultate deuten an, dass der {\"U}bertragungsverlust der gealterten Fasern in den ersten Tagen und Wochen der Alterung am st{\"a}rksten durch eine wesentliche physikalische Verschlechterung der Kern-Mantel-Grenzfl{\"a}che verursacht wird. Chemische Effekte des Alterungsprozesses scheinen im Faserkernmaterial zuerst nach einigen Monaten aufzutreten.}, language = {en} } @book{Jankowski2004, author = {Jankowski, \poundsukasz}, title = {Modelling and simulation of light propagation in non-aged and aged step-index polymer optical fibres. - [{\"u}berarb. Diss.]}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0001659}, publisher = {Universit{\"a}t Potsdam}, year = {2004}, abstract = {Kunststofflichtwellenleiter (POFs) stellen ein verh{\"a}ltnism{\"a}ßig neues Medium zur optische Datenkommunikation {\"u}ber kurzen Strecken dar. W{\"a}hrend ihrer Einsatzdauer unterliegen POFs unterschiedlichen Arten von Umweltbeanspruchungen, haupts{\"a}chlich durch hohe Temperatur, hohe Feuchtigkeit und mechanischen Belastungen. Zahlreiche experimentelle Forschungen besch{\"a}ftigten sich mit der standardisierten Pr{\"u}fung der Zuverl{\"a}ssigkeit von im Handel erh{\"a}ltlichen Fasern. Jedoch gab es bisher wenig Erfolg bei der Bem{\"u}hung, zwei grundlegende optische Erscheinungen, Absorption und Streuung, die die Lichtausbreitung in Fasern stark beeinflussen, zu verstehen und praktisch zu modellieren: Diese beiden Effekte beschreiben nicht nur die Qualit{\"a}t neuer Fasern, sondern sie werden auch stark durch die Alterungsprozess beeinflusst. Der Hauptzweck dieser Doktorarbeit war es, ein praktisch verwendbares und theoretisch gut fundiertes Modell der Lichtausbreitung in nicht gealterten und gealterten POFs zu entwickeln und es durch optische Experimente zu verifizieren. Dabei wurden anwendungsorientierte Aspekte mit theoretischer POF-Modellierung kombiniert. Die Arbeit enth{\"a}lt die erste bekannte Anwendung der Wellenanalyse zur Untersuchung der winkelabh{\"a}ngigen Eigenschaften der Streuung in Lichtwellenleitern. F{\"u}r die praktischen Experimente wurden mehrere POF-Proben unterschiedlicher Hersteller k{\"u}nstlich gealtert, indem sie bis 4500 Stunden bei 100 °C gelagert wurden (ohne Feuchtekontrolle). Die Parameter der jeweiligen Simulationen wurden mittels einer systematischen Optimierung an die gemessen optischen Eigenschaften der gealterten Proben angeglichen. Die Resultate deuten an, dass der {\"U}bertragungsverlust der gealterten Fasern in den ersten Tagen und Wochen der Alterung am st{\"a}rksten durch eine wesentliche physikalische Verschlechterung der Kern-Mantel-Grenzfl{\"a}che verursacht wird. Chemische Effekte des Alterungsprozesses scheinen im Faserkernmaterial zuerst nach einigen Monaten aufzutreten.}, language = {en} } @phdthesis{Becker2005, author = {Becker, Christian}, title = {On the Riemannian geometry of Seiberg-Witten moduli spaces}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-5425}, school = {Universit{\"a}t Potsdam}, year = {2005}, abstract = {In this thesis, we give two constructions for Riemannian metrics on Seiberg-Witten moduli spaces. Both these constructions are naturally induced from the L2-metric on the configuration space. The construction of the so called quotient L2-metric is very similar to the one construction of an L2-metric on Yang-Mills moduli spaces as given by Groisser and Parker. To construct a Riemannian metric on the total space of the Seiberg-Witten bundle in a similar way, we define the reduced gauge group as a subgroup of the gauge group. We show, that the quotient of the premoduli space by the reduced gauge group is isomorphic as a U(1)-bundle to the quotient of the premoduli space by the based gauge group. The total space of this new representation of the Seiberg-Witten bundle carries a natural quotient L2-metric, and the bundle projection is a Riemannian submersion with respect to these metrics. We compute explicit formulae for the sectional curvature of the moduli space in terms of Green operators of the elliptic complex associated with a monopole. Further, we construct a Riemannian metric on the cobordism between moduli spaces for different perturbations. The second construction of a Riemannian metric on the moduli space uses a canonical global gauge fixing, which represents the total space of the Seiberg-Witten bundle as a finite dimensional submanifold of the configuration space. We consider the Seiberg-Witten moduli space on a simply connected K\äuhler surface. We show that the moduli space (when nonempty) is a complex projective space, if the perturbation does not admit reducible monpoles, and that the moduli space consists of a single point otherwise. The Seiberg-Witten bundle can then be identified with the Hopf fibration. On the complex projective plane with a special Spin-C structure, our Riemannian metrics on the moduli space are Fubini-Study metrics. Correspondingly, the metrics on the total space of the Seiberg-Witten bundle are Berger metrics. We show that the diameter of the moduli space shrinks to 0 when the perturbation approaches the wall of reducible perturbations. Finally we show, that the quotient L2-metric on the Seiberg-Witten moduli space on a K\ähler surface is a K\ähler metric.}, subject = {Eichtheorie}, language = {en} } @misc{Ginoux2003, author = {Ginoux, Nicolas}, title = {Une nouvelle estimation extrins{\`e}que du spectre de l'op{\´e}rateur de Dirac}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-5644}, year = {2003}, abstract = {Nous {\´e}tablissons une nouvelle majoration optimale pour les plus petites valeurs propres de l'op{\´e}rateur de Dirac sur une hypersurface compacte de l'espace hyperbolique.}, language = {fr} } @misc{RoellySortais2004, author = {Roelly, Sylvie and Sortais, Michel}, title = {Space-time asymptotics of an infinite-dimensional diffusion having a long- range memory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6700}, year = {2004}, abstract = {We develop a cluster expansion in space-time for an infinite-dimensional system of interacting diffusions where the drift term of each diffusion depends on the whole past of the trajectory; these interacting diffusions arise when considering the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.}, language = {en} } @misc{RoellyThieullen2005, author = {Roelly, Sylvie and Thieullen, Mich{\`e}le}, title = {Duality formula for the bridges of a Brownian diffusion : application to gradient drifts}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6710}, year = {2005}, abstract = {In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C[0; 1]; R-d) Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.}, language = {en} } @misc{Louis2005, author = {Louis, Pierre-Yves}, title = {Increasing coupling for probabilistic cellular automata}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6593}, year = {2005}, abstract = {We give a necessary and sufficient condition for the existence of an increasing coupling of N (N >= 2) synchronous dynamics on S-Zd (PCA). Increasing means the coupling preserves stochastic ordering. We first present our main construction theorem in the case where S is totally ordered; applications to attractive PCAs are given. When S is only partially ordered, we show on two examples that a coupling of more than two synchronous dynamics may not exist. We also prove an extension of our main result for a particular class of partially ordered spaces.}, subject = {Wahrscheinlichkeitstheorie}, language = {en} }