@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{LeimkuhlerReich1994, author = {Leimkuhler, Benedict and Reich, Sebastian}, title = {Symplectic integration of constrained Hamiltonian systems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15653}, year = {1994}, abstract = {A Hamiltonian system in potential form (formula in the original abstract) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in Rn. In this paper methods which reduce "Hamiltonian differential algebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically.}, language = {en} } @misc{EvansHyde2022, author = {Evans, Myfanwy E. and Hyde, Stephen T.}, title = {Symmetric Tangling of Honeycomb Networks}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1282}, issn = {1866-8372}, doi = {10.25932/publishup-57084}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-570842}, pages = {13}, year = {2022}, abstract = {Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.}, language = {en} } @misc{AscherChinReich1994, author = {Ascher, Uri M. and Chin, Hongsheng and Reich, Sebastian}, title = {Stabilization of DAEs and invariant manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15625}, year = {1994}, abstract = {Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble. We then show how to improve it. We further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a simple way, generally different from the ODE discretization, and (iii) use orthogonal projections whenever possible. The best methods thus obtained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions.}, language = {en} } @misc{AscherChinPetzoldetal.1994, author = {Ascher, Uri M. and Chin, Hongsheng and Petzold, Linda R. and Reich, Sebastian}, title = {Stabilization of constrained mechanical systems with DAEs and invariant manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15698}, year = {1994}, abstract = {Many methods have been proposed for the simulation of constrained mechanical systems. The most obvious of these have mild instabilities and drift problems. Consequently, stabilization techniques have been proposed A popular stabilization method is Baumgarte's technique, but the choice of parameters to make it robust has been unclear in practice. Some of the simulation methods that have been proposed and used in computations are reviewed here, from a stability point of view. This involves concepts of differential-algebraic equation (DAE) and ordinary differential equation (ODE) invariants. An explanation of the difficulties that may be encountered using Baumgarte's method is given, and a discussion of why a further quest for better parameter values for this method will always remain frustrating is presented. It is then shown how Baumgarte's method can be improved. An efficient stabilization technique is proposed, which may employ explicit ODE solvers in case of nonstiff or highly oscillatory problems and which relates to coordinate projection methods. Examples of a two-link planar robotic arm and a squeezing mechanism illustrate the effectiveness of this new stabilization method.}, language = {en} } @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{PereraBoeckmann2019, author = {Perera, Upeksha and B{\"o}ckmann, Christine}, title = {Solutions of direct and inverse even-order Sturm-Liouville problems using Magnus expansion}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1336}, issn = {1866-8372}, doi = {10.25932/publishup-47341}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-473414}, pages = {24}, year = {2019}, abstract = {In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm-Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.}, language = {en} } @misc{Reich1995, author = {Reich, Sebastian}, title = {Smoothed dynamics of highly oscillatory Hamiltonian systems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15639}, year = {1995}, abstract = {We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method.}, language = {en} } @misc{FladHarutyunyanSchulze2015, author = {Flad, Heinz-J{\"u}rgen and Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {Singular analysis and coupled cluster theory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-102306}, pages = {31530 -- 31541}, year = {2015}, abstract = {The primary motivation for systematic bases in first principles electronic structure simulations is to derive physical and chemical properties of molecules and solids with predetermined accuracy. This requires a detailed understanding of the asymptotic behaviour of many-particle Coulomb systems near coalescence points of particles. Singular analysis provides a convenient framework to study the asymptotic behaviour of wavefunctions near these singularities. In the present work, we want to introduce the mathematical framework of singular analysis and discuss a novel asymptotic parametrix construction for Hamiltonians of many-particle Coulomb systems. This corresponds to the construction of an approximate inverse of a Hamiltonian operator with remainder given by a so-called Green operator. The Green operator encodes essential asymptotic information and we present as our main result an explicit asymptotic formula for this operator. First applications to many-particle models in quantum chemistry are presented in order to demonstrate the feasibility of our approach. The focus is on the asymptotic behaviour of ladder diagrams, which provide the dominant contribution to shortrange correlation in coupled cluster theory. Furthermore, we discuss possible consequences of our asymptotic analysis with respect to adaptive wavelet approximation.}, language = {en} } @misc{Ginoux2003, author = {Ginoux, Nicolas}, title = {Remarques sur le spectre de l'op{\´e}rateur de Dirac}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-5630}, year = {2003}, abstract = {Nous d{\´e}crivons un nouvelle famille d'exemples d'hypersurfaces de la sph{\`e}re satisfaisant le cas d'{\´e}galit{\´e} de la majoration extrins{\`e}que de C. B{\"a}r de la plus petite valeur propre de l'op{\´e}rateur de Dirac.}, language = {fr} } @misc{RoellyDereudre2004, author = {Roelly, Sylvie and Dereudre, David}, title = {Propagation of Gibbsiannes for infinite-dimensional gradient Brownian diffusions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6918}, year = {2004}, abstract = {We study the (strong-)Gibbsian character on R Z d of the law at time t of an infinitedimensional gradient Brownian diffusion , when the initial distribution is Gibbsian.}, language = {en} } @misc{Reich1995, author = {Reich, Sebastian}, title = {On the local qualitative behavior of differential-algebraic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46739}, year = {1995}, abstract = {A theoretical famework for the investigation of the qualitative behavior of differential-algebraic equations (DAEs) near an equilibrium point is established. The key notion of our approach is the notion of regularity. A DAE is called regular locally around an equilibrium point if there is a unique vector field such that the solutions of the DAE and the vector field are in one-to-one correspondence in a neighborhood of this equili Drium point. Sufficient conditions for the regularity of an equilibrium point are stated. This in turn allows us to translate several local results, as formulated for vector fields, to DAEs that are regular locally around a g: ven equilibrium point (e.g. Local Stable and Unstable Manifold Theorem, Hopf theorem). It is important that ihese theorems are stated in terms of the given problem and not in terms of the corresponding vector field.}, language = {en} } @misc{RoellyDereudre2004, author = {Roelly, Sylvie and Dereudre, David}, title = {On Gibbsianness of infinite-dimensional diffusions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6692}, year = {2004}, abstract = {The authors analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the d-dimensional lattice. In the first part of the paper, these processes are characterized as Gibbs states on path spaces. In the second part of the paper, they study the Gibbsian character on R^{Z^d} of the law at time t of the infinite-dimensional diffusion X(t), when the initial law is Gibbsian. AMS Classifications: 60G15 , 60G60 , 60H10 , 60J60}, language = {en} } @misc{Reich1991, author = {Reich, Sebastian}, title = {On an existence and uniqueness theory for nonlinear differential-algebraic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46706}, year = {1991}, abstract = {An existence and uniqueness theory is developed for general nonlinear and nonautonomous differential-algebraic equations (DAEs) by exploiting their underlying differential-geometric structure. A DAE is called regular if there is a unique nonautonomous vector field such that the solutions of the DAE and the solutions of the vector field are in one-to-one correspondence. Sufficient conditions for regularity of a DAE are derived in terms of constrained manifolds. Based on this differential-geometric characterization, existence and uniqueness results are stated for regular DAEs. Furthermore, our not ons are compared with techniques frequently used in the literature such as index and solvability. The results are illustrated in detail by means of a simple circuit example.}, language = {en} } @misc{Reich1990, author = {Reich, Sebastian}, title = {On a geometrical interpretation of differential-algebraic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46683}, year = {1990}, abstract = {The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE. and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lowsr-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena.}, language = {en} } @misc{DaiPraLouisMinelli2006, author = {Dai Pra, Paolo and Louis, Pierre-Yves and Minelli, Ida}, title = {Monotonicity and complete monotonicity for continuous-time Markov chains}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-7665}, year = {2006}, abstract = {We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time.}, subject = {Stochastik}, language = {en} } @misc{Reich1994, author = {Reich, Sebastian}, title = {Momentum conserving symplectic integrators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-16824}, year = {1994}, abstract = {In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).}, language = {en} } @misc{ChampagnatRoelly2008, author = {Champagnat, Nicolas and Roelly, Sylvie}, title = {Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-18610}, year = {2008}, abstract = {A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process - the conditioned multitype Feller branching diffusion - are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too .}, language = {en} } @misc{KolbeEvans2020, author = {Kolbe, Benedikt Maximilian and Evans, Myfanwy E.}, title = {Isotopic tiling theory for hyperbolic surfaces}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1}, issn = {1866-8372}, doi = {10.25932/publishup-54428}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-544285}, pages = {30}, year = {2020}, abstract = {In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.}, 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} }