TY - GEN A1 - Reich, Sebastian T1 - Algebrodifferentialgleichungen und Vektorfelder auf Mannigfaltigkeiten N2 - In diesem Beitrag wird der Zusammenhang zwischen Algebrodifferentialgleichungen (ADGL) und Vektorfeldern auf Mannigfaltigkeiten untersucht. Dazu wird zunächst der Begriff der regulären ADGL eingeführt, wobei unter eirter regulären ADGL eine ADGL verstanden wird, deren Lösungsmenge identisch mit der Lösungsmenge eines Vektorfeldes ist. Ausgehend von bekannten Aussagen über die Lösungsmenge eines Vektorfeldes werden analoge Aussagen für die Lösungsmenge einer regulären ADGL abgeleitet. Es wird eine Reduktionsmethode angegeben, die auf ein Kriterium für die Begularität einer ADGL und auf die Definition des Index einer nichtlinearen ADGL führt. Außerdem wird gezeigt, daß beliebige Vektorfelder durch reguläre ADGL so realisiert werden können, daß die Lösungsmenge des Vektorfeldes mit der der realisierenden ADGL identisch ist. Abschließend werden die für autonome ADGL gewonnenen Aussagen auf den Fall der nichtautonomen ADGL übertragen. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 160 Y1 - 1980 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-47290 ER - TY - GEN A1 - Reich, Sebastian T1 - Differential-algebraic equations and applications in circuit theory N2 - Technical and physical systems, especially electronic circuits, are frequently modeled as a system of differential and nonlinear implicit equations. In the literature such systems of equations are called differentialalgebraic equations (DAEs). It turns out that the numerical and analytical properties of a DAE depend on an integer called the index of the problem. For example, the well-known BDF method of Gear can be applied, in general, to a DAE only if the index does not exceed one. In this paper we give a geometric interpretation of higherindex DAEs and indicate problems arising in connection with such DAEs by means of several examples. N2 - Die mathematische Modellierung technisch physikalischer Systeme wie elektrische Netzwerke, führt häufig auf ein System von Differentialgleichungen und nichtlinearen impliziten Gleichungen sogenannten Algebrodifferentialgleichungen (ADGL). Es zeigt sich, daß die numerischen und analytischen Eigenschaften von ADGL durch den Index des Problems charakterisiert werden können. Insbesondere können die bekannten Integrationsformeln von Gear im allgemeinen nur auf ADGL mit dem Index eins angewendet werden. In diesem Beitrag wird eine geometrische Interpretation von ADGL mit einem höheren Index gegeben sowie auf Probleme im Zusammenhang mit derartigen ADGL an Hand verschiedener Beispiele hingewiesen. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 156 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-46646 ER - TY - GEN A1 - Reich, Sebastian T1 - On a geometrical interpretation of differential-algebraic equations N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 157 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-46683 ER - TY - GEN A1 - Reich, Sebastian T1 - Momentum conserving symplectic integrators N2 - 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). T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 044 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-16824 ER - TY - GEN A1 - Reich, Sebastian T1 - On the local qualitative behavior of differential-algebraic equations N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 159 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-46739 ER - TY - GEN A1 - Reich, Sebastian T1 - On an existence and uniqueness theory for nonlinear differential-algebraic equations N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 158 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-46706 ER - TY - GEN A1 - Pornsawad, Pornsarp A1 - Sungcharoen, Parada A1 - Böckmann, Christine T1 - Convergence rate of the modified Landweber method for solving inverse potential problems T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1034 KW - nonlinear operator KW - regularization KW - modified Landweber method KW - discrepancy principle KW - logarithmic source condition Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471942 SN - 1866-8372 IS - 1034 ER - TY - GEN A1 - Pornsawad, Pornsarp A1 - Sapsakul, Nantawan A1 - Böckmann, Christine T1 - A modified asymptotical regularization of nonlinear ill-posed problems T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))−𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1335 KW - nonlinear operator KW - regularization KW - discrepancy principle KW - asymptotic method KW - optimal rate Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-473433 SN - 1866-8372 IS - 1335 ER - TY - GEN A1 - Perera, Upeksha A1 - Böckmann, Christine T1 - Solutions of direct and inverse even-order Sturm-Liouville problems using Magnus expansion T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1336 KW - higher-order Sturm–Liouville problems KW - inverse Sturm–Liouville problems KW - Magnus expansion Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-473414 SN - 1866-8372 IS - 1336 ER - TY - GEN A1 - Omel'chenko, Oleh T1 - Travelling chimera states in systems of phase oscillators with asymmetric nonlocal coupling T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We study travelling chimera states in a ring of nonlocally coupled heterogeneous (with Lorentzian distribution of natural frequencies) phase oscillators. These states are coherence-incoherence patterns moving in the lateral direction because of the broken reflection symmetry of the coupling topology. To explain the results of direct numerical simulations we consider the continuum limit of the system. In this case travelling chimera states correspond to smooth travelling wave solutions of some integro-differential equation, called the Ott–Antonsen equation, which describes the long time coarse-grained dynamics of the oscillators. Using the Lyapunov–Schmidt reduction technique we suggest a numerical approach for the continuation of these travelling waves. Moreover, we perform their linear stability analysis and show that travelling chimera states can lose their stability via fold and Hopf bifurcations. Some of the Hopf bifurcations turn out to be supercritical resulting in the observation of modulated travelling chimera states. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1169 KW - chimera states KW - nonlocally coupled phase oscillators KW - Ott–Antonsen equation KW - forced symmetry breaking KW - travelling waves KW - continuation KW - stability Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-518141 SN - 1866-8372 IS - 2 SP - 611 EP - 642 ER - TY - GEN A1 - Nüsken, Nikolas A1 - Reich, Sebastian A1 - Rozdeba, Paul J. T1 - State and parameter estimation from observed signal increments T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean–Vlasov equations as the starting point to derive ensemble Kalman–Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 916 KW - parameter estimation KW - continuous-time data assimilation KW - ensemble Kalman filter KW - correlated noise KW - multi-scale diffusion processes Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-442609 SN - 1866-8372 IS - 916 ER - TY - GEN A1 - Molahajloo, Shahla A1 - Pirhayati, Mohammad T1 - Traces of pseudo-differential operators on compact and Hausdorff groups T2 - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe N2 - We give a characterization of and a trace formula for trace class pseudo-differential operators on compact Hausdorff groups. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 893 KW - pseudo-differential operators KW - compact groups Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436371 SN - 1866-8372 IS - 893 SP - 361 EP - 369 ER - TY - GEN A1 - Mickelsson, Jouko A1 - Paycha, Sylvie T1 - The logarithmic residue density of a generalized Laplacian T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold defines an invariant polynomial-valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas provide a pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either a Campbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 649 KW - residue KW - index KW - Dirac operators Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-413680 SN - 1866-8372 IS - 649 ER - TY - GEN A1 - Mazzonetto, Sara A1 - Salimova, Diyora T1 - Existence, uniqueness, and numerical approximations for stochastic burgers equations T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1393 KW - stochastic Burgers equations KW - SPDEs KW - mild solution KW - existence KW - numerical approximation Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-515796 SN - 1866-8372 IS - 4 ER - TY - GEN A1 - Louis, Pierre-Yves T1 - Ergodicity of PCA BT - equivalence between spatial and temporal mixing conditions N2 - For a general attractive Probabilistic Cellular Automata on S-Zd, we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A). This condition means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on {1,+1}(Zd), wit a naturally associated Gibbsian potential rho, we prove that a (spatial-) weak mixing condition (WM) for rho implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to rho hods. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition. KW - Wahrscheinlichkeitstheorie KW - Wechselwirkende Teilchensysteme KW - Stochastische Zellulare Automaten KW - Interacting particle systems KW - Probabilistic Cellular Automata KW - ERgodicity of Markov Chains KW - Gibbs measures Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6589 ER - TY - GEN A1 - Louis, Pierre-Yves T1 - Increasing coupling for probabilistic cellular automata N2 - 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. KW - Wahrscheinlichkeitstheorie KW - stochastische Anordnung KW - stochastische Zellulare Automaten KW - Kopplung KW - stochastic ordering KW - Probabilistic Cellular Automata KW - monotone coupling Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6593 ER - TY - GEN A1 - Leimkuhler, Benedict A1 - Reich, Sebastian T1 - Symplectic integration of constrained Hamiltonian systems N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 032 KW - differential-algebraic equations KW - constrained Hamiltonian systems KW - canonical discretization schemes KW - symplectic methods Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15653 ER - TY - GEN A1 - Kolbe, Benedikt Maximilian A1 - Evans, Myfanwy E. T1 - Isotopic tiling theory for hyperbolic surfaces T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1347 KW - isotopic tiling theory KW - Delaney–Dress tiling theory KW - mapping class groups KW - Orbifolds KW - maps on surfaces KW - hyperbolic tilings Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-544285 SN - 1866-8372 IS - 1 ER - TY - GEN A1 - Keller, Matthias A1 - Pinchover, Yehuda A1 - Pogorzelski, Felix T1 - From hardy to rellich inequalities on graphs T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1379 KW - 35R02 KW - 39A12 (primary) KW - 26D15 KW - 31C20 KW - 35B09 KW - 58E35 (secondary) Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-542140 SN - 1866-8372 IS - 3 ER - TY - GEN A1 - Karpuz, Eylem Guzel A1 - Çevik, Ahmet Sinan A1 - Koppitz, Jörg A1 - Cangul, Ismail Naci T1 - Some fixed-point results on (generalized) Bruck-Reilly ∗-extensions of monoids T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 942 KW - Bruck-Reilly extension KW - generalized Bruck-Reilly ∗-extension KW - π -inverse monoid KW - regular monoid Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-432701 SN - 1866-8372 IS - 942 ER -