@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}
}
@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{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{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{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{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{Reich1980,
  author    = {Reich, Sebastian},
  title     = {Algebrodifferentialgleichungen und Vektorfelder auf Mannigfaltigkeiten},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-47290},
  year      = {1980},
  abstract  = {In diesem Beitrag wird der Zusammenhang zwischen Algebrodifferentialgleichungen (ADGL) und Vektorfeldern auf Mannigfaltigkeiten untersucht. Dazu wird zun{\"a}chst der Begriff der regul{\"a}ren ADGL eingef{\"u}hrt, wobei unter eirter regul{\"a}ren ADGL eine ADGL verstanden wird, deren L{\"o}sungsmenge identisch mit der L{\"o}sungsmenge eines Vektorfeldes ist. Ausgehend von bekannten Aussagen {\"u}ber die L{\"o}sungsmenge eines Vektorfeldes werden analoge Aussagen f{\"u}r die L{\"o}sungsmenge einer regul{\"a}ren ADGL abgeleitet. Es wird eine Reduktionsmethode angegeben, die auf ein Kriterium f{\"u}r die Begularit{\"a}t einer ADGL und auf die Definition des Index einer nichtlinearen ADGL f{\"u}hrt. Außerdem wird gezeigt, daß beliebige Vektorfelder durch regul{\"a}re ADGL so realisiert werden k{\"o}nnen, daß die L{\"o}sungsmenge des Vektorfeldes mit der der realisierenden ADGL identisch ist. Abschließend werden die f{\"u}r autonome ADGL gewonnenen Aussagen auf den Fall der nichtautonomen ADGL {\"u}bertragen.},
  language  = {de}
}
@misc{Reich1992,
  author    = {Reich, Sebastian},
  title     = {Differential-algebraic equations and applications in circuit theory},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46646},
  year      = {1992},
  abstract  = {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.},
  language  = {en}
}
@misc{Ginoux2004,
  author    = {Ginoux, Nicolas},
  title     = {Dirac operators on Lagrangian submanifolds},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-5627},
  year      = {2004},
  abstract  = {We study a natural Dirac operator on a Lagrangian submanifold of a K{\"a}hler manifold. We first show that its square coincides with the Hodge - de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.},
  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{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{Louis2004,
  author    = {Louis, Pierre-Yves},
  title     = {Ergodicity of PCA},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6589},
  year      = {2004},
  abstract  = {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.},
  subject      = {Wahrscheinlichkeitstheorie},
  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}
}
@misc{RoellyDaiPra2004,
  author    = {Roelly, Sylvie and Dai Pra, Paolo},
  title     = {An existence result for infinite-dimensional Brownian diffusions with non- regular and non Markovian drift},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6684},
  year      = {2004},
  abstract  = {We prove in this paper an existence result for infinite-dimensional stationary interactive Brownian diffusions. The interaction is supposed to be small in the norm ||.||∞ but otherwise is very general, being possibly non-regular and non-Markovian. Our method consists in using the characterization of such diffusions as space-time Gibbs fields so that we construct them by space-time cluster expansions in the small coupling parameter.},
  language  = {en}
}
@misc{NueskenReichRozdeba2019,
  author    = {N{\"u}sken, Nikolas and Reich, Sebastian and Rozdeba, Paul J.},
  title     = {State and parameter estimation from observed signal increments},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {916},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-44260},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-442609},
  pages     = {25},
  year      = {2019},
  abstract  = {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.},
  language  = {en}
}
@misc{PornsawadSungcharoenBoeckmann2020,
  author    = {Pornsawad, Pornsarp and Sungcharoen, Parada and B{\"o}ckmann, Christine},
  title     = {Convergence rate of the modified Landweber method for solving inverse potential problems},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {1034},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-47194},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-471942},
  pages     = {24},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}
@misc{BandaraRosen2019,
  author    = {Bandara, Menaka Lashitha and Ros{\´e}n, Andreas},
  title     = {Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions},
  series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  number    = {758},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-43407},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-434078},
  pages     = {1253 -- 1284},
  year      = {2019},
  abstract  = {On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator in depends Riesz continuously on perturbations of local boundary conditions The Lipschitz bound for the map depends on Lipschitz smoothness and ellipticity of and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.},
  language  = {en}
}