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The classical Lefschetz fixed point formula expresses the number of fixed points of a continuous map f : M-->M in terms of the transformation induced by f on the cohomology of M. In 1966 Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they presented a holomorphic Lefschetz formula for compact complex manifolds without boundary, a result, in the framework of algebraic geometry due to Eichler (1957) for holomorphic curves. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, hence the Atiyah- Bott theory is no longer applicable. To get rid of the difficulties related to the boundary behaviour of the Dolbeault cohomology, Donelli and Fefferman (1986) derived a fixed point formula for the Bergman metric. The purpose of this paper is to present a holomorphic Lefschetz formula on a strictly convex domain in C-n, n>1
Consider the perturbed harmonic oscillator Ty=-y''+x(2)y+q(x)y in L-2(R), where the real potential q belongs to the Hilbert space H={q', xq is an element of L-2(R)}. The spectrum of T is an increasing sequence of simple eigenvalues lambda(n)(q)=1+2n+mu(n), ngreater than or equal to0, such that mu(n)-->0 as n-->infinity. Let psi(n)(x,q) be the corresponding eigenfunctions. Define the norming constants nu(n)(q)=lim(xup arrowinfinity)log |psi(n) (x,q)/psi(n) (-x,q)|. We show that {mu(n)}(0)(infinity) is an element of H {nu(n)}(0)(infinity) is an element of H-0 for some real Hilbert space and some subspace H-0 subset of H. Furthermore, the mapping Psi:q-- >Psi(q)=({lambda(n)(q)}(0)(infinity), {nu(n)(q)}(0)(infinity)) is a real analytic isomorphism between H and S x H-0, where S is the set of all strictly increasing sequences s={s(n)}(0)(infinity) such that s(n)=1+2n+h(n), {h(n)}(0)(infinity) is an element of H. The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator -y"py, p is an element of L-2(0,1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces H,H-0. We obtain their basic properties, using their representation as spaces of analytic functions in the disk
Local asymptotic types
(2004)
We discuss the role of gravitational excitons/radions in different cosmological scenarios. Gravitational excitons are massive moduli fields which describe conformal excitations of the internal spaces and which, due to their Planck-scale suppressed coupling to matter fields, are WIMPs. It is demonstrated that, depending on the concrete scenario, observational cosmological data set strong restrictions on the allowed masses and initial oscillation amplitudes of these particles
Metastability in reversible diffusion processes : I. Sharp asymptotics for capacities and exit times
(2004)
We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form -epsilonDelta+ delF(.) del on R-d or subsets of R-d, where F is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one as epsilon down arrow 0, to the capacities of suitably constructed sets. We show that these capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of F at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring - Kramers formula in dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes
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.
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.
Multidimensional cosmological models : Cosmological and astrophysical implications and constraints
(2004)
We investigate four-dimensional effective theories which are obtained by dimensional reduction of multidimensional cosmological models with factorizable geometry and we consider the interaction between conformal excitations of the internal space (geometrical moduli excitations) and Abelian gauge fields. It is assumed that the internal space background can be stabilized by minima of an effective potential. The conformal excitations over such a background have the form of massive scalar fields (gravitational excitons) propagating in the external spacetime. We discuss cosmological and astrophysical implications of the interaction between gravexcitons and four-dimensional photons as well as constraints arising on multidimensional models of the type considered in our paper. In particular, we show that due to the experimental bounds on the variation of the fine-structure constant, gravexcitons should decay before nucleosynthesis starts. For a successful nucleosynthesis, the masses of the decaying gravexcitons should be mgreater than or similar to10(4) GeV. Furthermore, we discuss the possible contribution of gravexcitons to ultrahigh-energy cosmic rays. It is shown that, at energies Esimilar to10(20) eV, the decay length of gravexcitons with masses mgreater than or similar to10(4) GeV is very small, but that for mless than or similar to10(2) GeV it becomes much larger than the Greisen-Zatsepin-Kuzmin cutoff distance. Finally, we investigate the possibility for gravexciton-photon oscillations in strong magnetic fields of astrophysical objects. The corresponding estimates indicate that even the high-magnetic- field strengths B of magnetars (special types of pulsars with B>B(critical)similar to4.4x10(13) G) are not sufficient for an efficient and copious production of gravexcitons
The classical Lefschetz formula expresses the number of fixed points of a continuous map f: M -> M in terms of the transformation induced by f on the cohomology of M. In 1966, Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they obtained a holomorphic Lefschetz formula on compact complex manifolds without boundary. Brenner and Shubin (1981, 1991) extended the Atiyah-Bott theory to compact manifolds with boundary. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, therefore the Atiyah- Bott theory is not applicable. Bypassing difficulties related to the boundary behaviour of Dolbeault cohomology, Donnelly and Fefferman (1986) obtained a formula for the number of fixed points in terms of the Bergman metric. The aim of this paper is to obtain a Lefschetz formula on relatively compact strictly pseudoconvex subdomains of complex manifolds X with smooth boundary, that is, to find the total Lefschetz number for a holomorphic endomorphism f(*) of the Dolbeault complex and to express it in terms of local invariants of the fixed points of f.
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