@unpublished{RoellyFradon2006, author = {Roelly, Sylvie and Fradon, Myriam}, title = {Infinite system of Brownian balls : equilibrium measures are canonical Gibbs}, url =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} } @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{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{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{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} } @phdthesis{Demircioglu2007, author = {Demircioglu, Aydin}, title = {Reconstruction of deligne classes and cocycles}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13755}, school = {Universit{\"a}t Potsdam}, year = {2007}, abstract = {In der vorliegenden Arbeit verallgemeinern wir im Wesentlichen zwei Theoreme von Mackaay-Picken und Picken (2002, 2004). Im ihrem Artikel zeigen Mackaay und Picken,dass es eine bijektive Korrespodenz zwischen Deligne 2-Klassen \$\xi \in \check{H}^2(M, \mathcal{D}^2)\$ und Holonomie Abbildungen von der zweiten d{\"u}nnen Homotopiegruppe \$\pi_2^2(M)\$ in die abelsche Gruppe \$U(1)\$ gibt. Im zweiten Artikel wird eine Verallgemeinerung dieses Theorems bewiesen: Picken zeigt, dass es eine Bijektion gibt zwischen Deligne 2-Kozykeln und gewissen 2-dimensionalen topologischen Quantenfeldtheorien. In dieser Arbeit zeigen wir, dass diese beiden Theoreme in allen Dimensionen gelten.Wir betrachten zun{\"a}chst den Holonomie Fall und k{\"o}nnen mittels simplizialen Methoden nachweisen, dass die Gruppe der glatten Deligne \$d\$-Klassen isomorph ist zu der Gruppe der glatten Holonomie Abbildungen von der \$d\$-ten d{\"u}nnen Homotopiegruppe \$\pi_d^d(M)\$ nach \$U(1)\$, sofern \$M\$ eine \$(d-1)\$-zusammenh{\"a}ngende Mannigfaltigkeit ist. Wir vergleichen dieses Resultat mit einem Satz von Gajer (1999). Gajer zeigte, dass jede Deligne \$d\$-Klasse durch eine andere Klasse von Holonomie-Abbildungen rekonstruiert werden kann, die aber nicht nur Holonomien entlang von Sph{\"a}ren, sondern auch entlang von allgemeinen \$d\$-Mannigfaltigkeiten in \$M\$ enth{\"a}lt. Dieser Zugang ben{\"o}tigt dann aber nicht, dass \$M\$ hoch-zusammenh{\"a}ngend ist. Wir zeigen, dass im Falle von flachen Deligne \$d\$-Klassen unser Rekonstruktionstheorem sich von Gajers unterscheidet, sofern \$M\$ nicht als \$(d-1)\$, sondern nur als \$(d-2)\$-zusammenh{\"a}ngend angenommen wird. Stiefel Mannigfaltigkeiten besitzen genau diese Eigenschaft, und wendet man unser Theorem auf diese an und vergleicht das Resultat mit dem von Gajer, so zeigt sich, dass es zuviele Deligne Klassen rekonstruiert. Dies bedeutet, dass unser Rekonstruktionsthreorem ohne die Zusatzbedingungen an die Mannigfaltigkeit M nicht auskommt, d.h. unsere Rekonstruktion ben{\"o}tigt zwar weniger Informationen {\"u}ber die Holonomie entlang von d-dimensionalen Mannigfaltigkeiten, aber daf{\"u}r muss M auch \$(d-1)\$-zusammenh{\"a}ngend angenommen werden. Wir zeigen dann, dass auch das zweite Theorem verallgemeinert werden kann: Indem wir das Konzept einer Picken topologischen Quantenfeldtheorie in beliebigen Dimensionen einf{\"u}hren, k{\"o}nnen wir nachweisen, dass jeder Deligne \$d\$-Kozykel eine solche \$d\$-dimensionale Feldtheorie mit zwei besonderen Eigenschaften, der d{\"u}nnen Invarianz und der Glattheit, induziert. Wir beweisen, dass jede \$d\$-dimensionale topologische Quantenfeldtheorie nach Picken mit diesen zwei Eigenschaften auch eine Deligne \$d\$-Klasse definiert und pr{\"u}fen nach, dass diese Konstruktion sowohl surjektiv als auch injektiv ist. Demzufolge sind beide Gruppen isomorph.}, language = {en} } @article{EvansHyde2022, author = {Evans, Myfanwy E. and Hyde, Stephen T.}, title = {Symmetric Tangling of Honeycomb Networks}, series = {Symmetry}, volume = {14}, journal = {Symmetry}, edition = {9}, publisher = {MDPI}, address = {Basel, Schweiz}, issn = {2073-8994}, doi = {10.3390/sym14091805}, pages = {1 -- 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{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{ImkellerRoelly2007, author = {Imkeller, Peter and Roelly, Sylvie}, title = {Die Wiederentdeckung eines Mathematikers: Wolfgang D{\"o}blin}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-16397}, year = {2007}, abstract = {"Considerons une particule mobile se mouvant aleatoirement sur la droite (ou sur un segment de droite). Supposons qu'il existe une probabilite F(x,y;s,t) bien definie pour que la particule se trouvant a l'instant s dans la position x se trouve a l'instant t (> s) a gauche de y, probabilite independante du mouvement anterieur de la particule...." Mit diesen Worten beginnt eines der ber{\"u}hmtesten mathematischen Manuskripte des letzten Jahrhunderts. Es stammt vom Soldaten Wolfgang D{\"o}blin, Sohn des deutschen Schriftstellers Alfred D{\"o}blin, und tr{\"a}gt den Titel "Sur l'equation de Kolmogoroff". Seine Ver{\"o}ffentlichung verbindet sich mit einer unglaublichen Geschichte. Wolfgang D{\"o}blin, stationiert mit seiner Einheit in den Ardennen im Winter 1939/1940, arbeitete an diesem Manuskript. Er entschloss sich, es als versiegeltes Manuskript an die Academie des Sciences in Paris zu schicken. Aber er kehrte nie aus diesem Krieg zur{\"u}ck. Sein Manuskript blieb 60 Jahre unter Verschluss im Archiv, und wurde erst im Jahre 2000 ge{\"o}ffnet. Wie weit D{\"o}blin damit seiner Zeit voraus war, wurde erkannt, nachdem es von Bernard Bru und Marc Yor ausgewertet worden war. Im ersten Satz umschreibt W. D{\"o}blin gleichzeitig das Programm des Manuskripts: "Wir betrachten ein bewegliches Teilchen, das sich zuf{\"a}llig auf der Geraden (oder einem Teil davon) bewegt." Er widmet sich damit der Aufgabe, die Fundamente eines Gebiets zu legen, das wir heute als stochastische Analysis bezeichnen.}, language = {de} } @unpublished{PraLouisMinelli2008, author = {Pra, Paolo Dai and Louis, Pierre-Yves and Minelli, Ida G.}, title = {Complete monotone coupling for Markov processes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-18286}, year = {2008}, abstract = {We formalize and 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 continuoustime but not in discrete-time.}, language = {de} } @phdthesis{Fischer2022, author = {Fischer, Jens Walter}, title = {Random dynamics in collective behavior - consensus, clustering \& extinction of populations}, doi = {10.25932/publishup-55372}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-553725}, school = {Universit{\"a}t Potsdam}, pages = {242}, year = {2022}, abstract = {The echo chamber model describes the development of groups in heterogeneous social networks. By heterogeneous social network we mean a set of individuals, each of whom represents exactly one opinion. The existing relationships between individuals can then be represented by a graph. The echo chamber model is a time-discrete model which, like a board game, is played in rounds. In each round, an existing relationship is randomly and uniformly selected from the network and the two connected individuals interact. If the opinions of the individuals involved are sufficiently similar, they continue to move closer together in their opinions, whereas in the case of opinions that are too far apart, they break off their relationship and one of the individuals seeks a new relationship. In this paper we examine the building blocks of this model. We start from the observation that changes in the structure of relationships in the network can be described by a system of interacting particles in a more abstract space. These reflections lead to the definition of a new abstract graph that encompasses all possible relational configurations of the social network. This provides us with the geometric understanding necessary to analyse the dynamic components of the echo chamber model in Part III. As a first step, in Part 7, we leave aside the opinions of the inidividuals and assume that the position of the edges changes with each move as described above, in order to obtain a basic understanding of the underlying dynamics. Using Markov chain theory, we find upper bounds on the speed of convergence of an associated Markov chain to its unique stationary distribution and show that there are mutually identifiable networks that are not apparent in the dynamics under analysis, in the sense that the stationary distribution of the associated Markov chain gives equal weight to these networks. In the reversible cases, we focus in particular on the explicit form of the stationary distribution as well as on the lower bounds of the Cheeger constant to describe the convergence speed. The final result of Section 8, based on absorbing Markov chains, shows that in a reduced version of the echo chamber model, a hierarchical structure of the number of conflicting relations can be identified. We can use this structure to determine an upper bound on the expected absorption time, using a quasi-stationary distribution. This hierarchy of structure also provides a bridge to classical theories of pure death processes. We conclude by showing how future research can exploit this link and by discussing the importance of the results as building blocks for a full theoretical understanding of the echo chamber model. Finally, Part IV presents a published paper on the birth-death process with partial catastrophe. The paper is based on the explicit calculation of the first moment of a catastrophe. This first part is entirely based on an analytical approach to second degree recurrences with linear coefficients. The convergence to 0 of the resulting sequence as well as the speed of convergence are proved. On the other hand, the determination of the upper bounds of the expected value of the population size as well as its variance and the difference between the determined upper bound and the actual value of the expected value. For these results we use almost exclusively the theory of ordinary nonlinear differential equations.}, language = {en} } @article{PornsawadSapsakulBoeckmann2019, author = {Pornsawad, Pornsarp and Sapsakul, Nantawan and B{\"o}ckmann, Christine}, title = {A modified asymptotical regularization of nonlinear ill-posed problems}, series = {Mathematics}, volume = {7}, journal = {Mathematics}, edition = {5}, publisher = {MDPI}, address = {Basel, Schweiz}, issn = {2227-7390}, doi = {10.3390/math7050419}, pages = {19}, year = {2019}, abstract = {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.}, language = {en} } @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{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} }