510 Mathematik
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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.
Ergodicity of PCA
(2004)
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.
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.
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.
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
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.
We study a natural Dirac operator on a Lagrangian submanifold of a Kähler 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.
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 Käuhler 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 Kähler surface is a Kähler metric.
This thesis discusses theoretical and practical aspects of modelling of light propagation in non-aged and aged step-index polymer optical fibres (POFs). Special attention has been paid in describing optical characteristics of non-ideal fibres, scattering and attenuation, and in combining application-oriented and theoretical approaches. The precedence has been given to practical issues, but much effort has been also spent on the theoretical analysis of basic mechanisms governing light propagation in cylindrical waveguides.As a result a practically usable general POF model based on the raytracing approach has been developed and implemented. A systematic numerical optimisation of its parameters has been performed to obtain the best fit between simulated and measured optical characteristics of numerous non-aged and aged fibre samples. The model was verified by providing good agreement, especially for the non-aged fibres. The relations found between aging time and optimal values of model parameters contribute to a better understanding of the aging mechanisms of POFs.
Modelling and simulation of light propagation in non-aged and aged step-index polymer optical fibres
(2004)
This thesis discusses theoretical and practical aspects of modelling of light propagation in non-aged and aged step-index polymer optical fibres (POFs). Special attention has been paid in describing optical characteristics of non-ideal fibres, scattering and attenuation, and in combining application-oriented and theoretical approaches. The precedence has been given to practical issues, but much effort has been also spent on the theoretical analysis of basic mechanisms governing light propagation in cylindrical waveguides. As a result a practically usable general POF model based on the raytracing approach has been developed and implemented. A systematic numerical optimisation of its parameters has been performed to obtain the best fit between simulated and measured optical characteristics of numerous non-aged and aged fibre samples. The model was verified by providing good agreement, especially for the non-aged fibres. The relations found between aging time and optimal values of model parameters contribute to a better understanding of the aging mechanisms of POFs.
Was misst TIMSS?
(2001)
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 Fähigkeiten herausgearbeitet werden können. Dabei werden fachliche Anforderungen, Irritationsmomente und das durch die Aufgabe transportierte Bild vom jeweiligen Fach ebenso erfasst wie Momente, die man eher als Testfähigkeit bezeichnen würde.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.
The objective of this thesis is to provide new space compaction techniques for testing or concurrent checking of digital circuits. In particular, the work focuses on the design of space compactors that achieve high compaction ratio and minimal loss of testability of the circuits. In the first part, the compactors are designed for combinational circuits based on the knowledge of the circuit structure. Several algorithms for analyzing circuit structures are introduced and discussed for the first time. The complexity of each design procedure is linear with respect to the number of gates of the circuit. Thus, the procedures are applicable to large circuits. In the second part, the first structural approach for output compaction for sequential circuits is introduced. Essentially, it enhances the first part. For the approach introduced in the third part it is assumed that the structure of the circuit and the underlying fault model are unknown. The space compaction approach requires only the knowledge of the fault-free test responses for a precomputed test set. The proposed compactor design guarantees zero-aliasing with respect to the precomputed test set.
Als Grundlage vieler statistischer Verfahren wird der Prozess der Entstehung von Daten modelliert, um dann weitere Schätz- und Testverfahren anzuwenden. Diese Arbeit befasst sich mit der Frage, wie diese Spezifikation für parametrische Modelle selbst getestet werden kann. In Erweiterung bestehender Verfahren werden Tests mit festem Kern eingeführt und ihre asymptotischen Eigenschaften werden analysiert. Es wird gezeigt, dass die Bestimmung der kritischen Werte mit mehreren Stichprobenwiederholungsverfahren möglich ist. Von diesen ist eine neue Monte-Carlo-Approximation besonders wichtig, da sie die Komplexität der Berechnung deutlich verringern kann. Ein bedingter Kleinste-Quadrate-Schätzer für nichtlineare parametrische Modelle wird definiert und seine wesentlichen asymptotischen Eigenschaften werden hergeleitet. Sämtliche Versionen der Tests und alle neuen Konzepte wurden in Simulationsstudien untersucht, deren wichtigste Resultate präsentiert werden. Die praktische Anwendbarkeit der Testverfahren wird an einem Datensatz zur Produktwahl dargelegt, der mit multinomialen Logit-Modellen analysiert werden soll.