Computation of the invariant measure for a Lévy driven SDE: Rate of convergence

We study the rate of convergence of some recursive procedures based on some “exact” or “approximate” Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by “exact” and “approximate” Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.     

Counting perfect matchings in n-extendable graphs

The structural theory of matchings is used to establish lower bounds on the number of perfect matchings in n-extendable graphs. It is shown that any such graph on p vertices and q edges contains at least ⌈(n+1)!/4[q-p-(n-1)(2Δ-3)+4]⌉ different perfect matchings, where Δ is the maximum degree of a vertex in G.     

Topological decoupling, linearization and perturbation on inhomogeneous time scales

We derive a linearization theorem in the framework of dynamic equations on time scales. This extends a recent result from [Y. Xia, J. Cao, M. Han, A new analytical method for the linearization of dynamic equation on measure chains, J. Differential Equations 235 (2007) 527-543] in various directions: Firstly, in our setting the linear part need not to be hyperbolic and due to the existence of a center manifold this leads to a generalized global Hartman-Grobman theorem for nonautonomous problems. Secondly, we investigate the behavior of the topological conjugacy under parameter variation.These perturbation results are tailor-made for future applications in analytical discretization theory, i.e., to study the relationship between ODEs and numerical schemes applied to them.     

A renewal jump-diffusion process with threshold dividend strategy

In this paper, we consider a jump-diffusion risk process with the threshold dividend strategy. Both the distributions of the inter-arrival times and the claims are assumed to be in the class of phase-type distributions. The expected discounted dividend function and the Laplace transform of the ruin time are discussed. Motivated by Asmussen [S. Asmussen, Stationary distributions for fluid flow models with or without Brownian noise, Stochastic Models 11 (1) (1995) 21–49], instead of studying the original process, we study the constructed fluid flow process and their closed-form formulas are obtained in terms of matrix expression. Finally, numerical results are provided to illustrate the computation.     

Pfaffian labelings and signs of edge colorings

We relate signs of edge-colorings (as in classical Penrose’s result) with “Pfaffian labelings”, a generalization of Pfaffian orientations, whereby edges are labeled by elements of an Abelian group with an element of order two. In particular, we prove a conjecture of Goddyn that all k-edge-colorings of a k-regular Pfaffian graph G have the same sign. We characterize graphs that admit a Pfaffian labeling in terms of bricks and braces in their matching decomposition and in terms of their drawings in the projective plane. Partially supported by NSF grants 0200595 and 0354742.     

Superconducting screen printed thick films of YBa2Cu3O7 and Bi1.6Pb0.4Sr1.6Ca2.4Cu3O10 on polycrystalline substrates

Thick films of YBa₂Cu₃O₇ and Bi_1.6Pb_0.4Sr_1.6Ca_2.4Cu₂O₁₀ have been prepared using a screen printing method on polycrystalline substrates. The films have been characterized by X-ray diffraction, electrical measurements and SEM observations. YBa₂Cu₃O₇ films exhibited T_c ranging from 80 to 87 K on Y₂BaCuO₅ substrates while Bi_1.6Pb_0.5Sr_1.6Ca_2.4Cu₃O₁₀ films on polycrystalline MgO reached the zero resistance state at 104 K with critical current densities of up to 10 A/cm² at 77 K.     

Kinetic interface roughening and magneto resistance of sputter-deposited Fe/Ni75B25 multilayers

The roughening of interfaces as a function of layer thickness and magneto transport properties have been investigated on sputter-deposited Fe/Ni₇₅B₂₅ multilayer films. X-ray reflectivity data were recorded for Ni₇₅B₂₅(72 nm) film and for [Fe(2 nm)/Ni₇₅B₂₅(2 nm)]₁₆ and [Fe(4 nm)/Ni₇₅B₂₅(4 nm)]₈ multilayer films. A power law dependence of the interfacial width of growing Fe/Ni₇₅B₂₅ interfaces was observed. The resulting growth exponents β were found to be in the range of 0.55–0.58 in the initial growth stage of the multilayer with lower Fe/Ni₇₅B₂₅ repetition thickness and at approximately 0.34 for multilayer with higher repetition thickness. The growth exponents were compared with theoretical calculations. High resolution electron microscopy revealed the columnar growth of the Fe/Ni₇₅B₂₅ multilayer. Additionally, an increase of magnetoresistance was observed by the multilayering of Ni₇₅B₂₅ films with Fe interlayers.     

Perturbations of Pseudodifferential Operators with Negative Definite Symbol

Pseudodifferential operators with negative definite symbols appear as generators of jump-type Markov processes. The purpose of this paper is to treat the large jumps of the process by a perturbation approach for the generator. This is of particular interest since in this way the generators are made accessible to a symbolic calculus of pseudodifferential operators. The main auxiliary result consists of a characterization of tightness of the jump measures in terms of the symbol.     

Lower bounds for numbers of ABC-hits

By an ABC-hit, we mean a triple (a,b,c) of relatively prime positive integers such that a+b=c and rad(abc)<c. Denote by N(X) the number of ABC-hits (a,b,c) with c?X. In this paper we discuss lower bounds for N(X). In particular we prove that for every ?>0 and X large enough N(X)?exp((logX)^1/2−?).     

Evidence for shape coexistence at medium spin in Rb

Four previously known rotational bands in ⁷⁶Rb have been extended to moderate spins using the Gammasphere and Microball γ ray and charged particle detector arrays and the ⁴⁰Ca(⁴⁰Ca, 3pn) reaction at a beam energy of 165 MeV. The properties of two of the negative-parity bands can only readily be interpreted in terms of the highly successful Cranked Nilsson-Strutinsky model calculations if they have the same configuration in terms of the number of g9/2 particles, but they result from different nuclear shapes (one near-oblate and the other near-prolate). These data appear to constitute a unique example of shape coexisting structures at medium spins.