Refine
Has Fulltext
- yes (30) (remove)
Year of publication
- 2012 (30) (remove)
Document Type
- Preprint (26)
- Doctoral Thesis (4)
Language
- English (30)
Is part of the Bibliography
- yes (30)
Keywords
- Gibbs point processes (2)
- Heat equation (2)
- Riemannian manifold (2)
- counting process (2)
- infinite divisibility (2)
- infinitely divisible point processes (2)
- reciprocal class (2)
- Boundary value methods (1)
- Boundary value problems for first order systems (1)
- Brownian bridge (1)
- Brownian motion (1)
- CCR-algebra (1)
- Cluster Entwicklung (1)
- Cluster expansion (1)
- Determinantal point processes (1)
- Dirac-harmonic maps (1)
- Dirac-harmonische Abbildungen (1)
- Dirac-type operator (1)
- Dirichlet form (1)
- Dirichlet problem (1)
- Duality formula (1)
- Dualitätsformeln (1)
- Eigenvalues (1)
- Elliptic operators (1)
- Eyring-Kramers Formel (1)
- Feynman-Kac formula (1)
- Finite difference method (1)
- Fischer-Riesz equations (1)
- Fourth order Sturm-Liouville problem (1)
- Gibbssche Punktprozesse (1)
- Gradient flow (1)
- Gradientenfluss (1)
- Green formula (1)
- Green's operator (1)
- Heat Flow (1)
- Hypoelliptic operators (1)
- Lefschetz number (1)
- Levy Maß (1)
- Levy measure (1)
- Levy process (1)
- Liouville theorem (1)
- Lipschitz domains (1)
- Lévy measure (1)
- Malliavin calculus (1)
- Metastabilität (1)
- Montel theorem (1)
- Numerov's method (1)
- Perturbed complexes (1)
- Renormalized integral (1)
- Semiklassische Spektralasymptotik (1)
- Spin Geometrie (1)
- Spin Geometry (1)
- Sturm-Liouville problems (1)
- Tunneleffekt (1)
- Vitali theorem (1)
- Wave operator (1)
- Wiener measure (1)
- Wärmefluss (1)
- Zählprozesse (1)
- absorbing boundary (1)
- boundary layer (1)
- boundary regularity (1)
- boundary value problems (1)
- characteristic boundary point (1)
- characteristic points (1)
- cluster expansion (1)
- coercivity (1)
- completeness (1)
- conditional Wiener measure (1)
- continuous time Markov chain (1)
- curvature (1)
- cusp (1)
- dbar-Neumann problem (1)
- decay of eigenfunctions (1)
- determinantal point processes (1)
- determinantische Punktprozesse (1)
- discontinuous Robin condition (1)
- discrete Witten complex (1)
- diskreter Witten-Laplace-Operator (1)
- duality formulae (1)
- elliptic boundary conditions (1)
- finsler distance (1)
- generalized Laplace operator (1)
- globally hyperbolic spacetime (1)
- hitting times (1)
- hypoelliptic estimate (1)
- interaction matrix (1)
- jump process (1)
- low-lying eignvalues (1)
- metastability (1)
- nichtlineare partielle Differentialgleichung (1)
- nonlinear partial differential equations (1)
- nonsmooth curves (1)
- path integral (1)
- permanental- (1)
- regularisation (1)
- rescaled lattice (1)
- reziproke Klassen (1)
- root functions (1)
- semi-classical difference operator (1)
- semiclassical spectral asymptotics (1)
- singular integral equations (1)
- spectral kernel function (1)
- stochastic mechanics (1)
- stochastische Mechanik (1)
- strongly pseudoconvex domains (1)
- the first boundary value problem (1)
- time duality (1)
- tunneling (1)
- unendlich teilbare Punktprozesse (1)
- unendliche Teilbarkeit (1)
Institute
- Institut für Mathematik (30) (remove)
In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.
A point process is a mechanism, which realizes randomly locally finite point measures. One of the main results of this thesis is an existence theorem for a new class of point processes with a so called signed Levy pseudo measure L, which is an extension of the class of infinitely divisible point processes. The construction approach is a combination of the classical point process theory, as developed by Kerstan, Matthes and Mecke, with the method of cluster expansions from statistical mechanics. Here the starting point is a family of signed Radon measures, which defines on the one hand the Levy pseudo measure L, and on the other hand locally the point process. The relation between L and the process is the following: this point process solves the integral cluster equation determined by L. We show that the results from the classical theory of infinitely divisible point processes carry over in a natural way to the larger class of point processes with a signed Levy pseudo measure. In this way we obtain e.g. a criterium for simplicity and a characterization through the cluster equation, interpreted as an integration by parts formula, for such point processes. Our main result in chapter 3 is a representation theorem for the factorial moment measures of the above point processes. With its help we will identify the permanental respective determinantal point processes, which belong to the classes of Boson respective Fermion processes. As a by-product we obtain a representation of the (reduced) Palm kernels of infinitely divisible point processes. In chapter 4 we see how the existence theorem enables us to construct (infinitely extended) Gibbs, quantum-Bose and polymer processes. The so called polymer processes seem to be constructed here for the first time. In the last part of this thesis we prove that the family of cluster equations has certain stability properties with respect to the transformation of its solutions. At first this will be used to show how large the class of solutions of such equations is, and secondly to establish the cluster theorem of Kerstan, Matthes and Mecke in our setting. With its help we are able to enlarge the class of Polya processes to the so called branching Polya processes. The last sections of this work are about thinning and splitting of point processes. One main result is that the classes of Boson and Fermion processes remain closed under thinning. We use the results on thinning to identify a subclass of point processes with a signed Levy pseudo measure as doubly stochastic Poisson processes. We also pose the following question: Assume you observe a realization of a thinned point process. What is the distribution of deleted points? Surprisingly, the Papangelou kernel of the thinning, besides a constant factor, is given by the intensity measure of this conditional probability, called splitting kernel.
We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine-Connes spectral action, deduce the equations of motions and discuss critical points.
We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.
This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.
On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators
(2012)
We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.
The Riemann hypothesis is equivalent to the fact the the reciprocal function 1/zeta (s) extends from the interval (1/2,1) to an analytic function in the quarter-strip 1/2 < Re s < 1 and Im s > 0. Function theory allows one to rewrite the condition of analytic continuability in an elegant form amenable to numerical experiments.
For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number.