Positive solutions to generalized Dickman equation

The paper investigates large-time behaviour of positive solutions to a generalized Dickman equation. The asymptotic behaviour of dominant and subdominant positive solutions is analysed and a structure formula describing behaviour of all solutions is proved. A criterion is also given for sufficient conditions on initial functions to generate positive solutions with prescribed asymptotic behaviour with values of their weighted limits computed.     

Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems

This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein's formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift-Hohenberg equation and reaction-diffusion systems of activator-inhibitor type.     

Quantum differential systems and construction of rational structures

We consider mirror symmetry (A-side vs B-side, namely singularity side) in the framework of quantum differential systems. We focuse on the logarithmic non-resonant case, which describes the geometric situation and we show that such systems provide a good framework in order to generalize the construction of the rational structure given by Katzarkov, Kontsevich and Pantev for the complex projective space. As an application, we give a closed formula for the rational structure defined by the Lefschetz thimbles on the flat sections of the Gauss-Manin connection associated with the Landau–Ginzburg models of weighted projective spaces (a class of Laurent polynomials). As a by-product, using a mirror theorem, we get a rational structure on the orbifold cohomology of weighted projective spaces. The formula on the B-side is more complicated than the one on the A-side (the latter agrees with one of Iritani’s results), depending on numerous combinatorial data which are rearranged after the mirror transformation.     

Scattering Matrix,Phase Shift,Spectral Shift and Trace Formula for One-dimensional Dissipative Schrödinger-type Operators

The paper is devoted to Schr?dinger operators with dissipative boundary conditions on bounded intervals. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed. In particular, the trace formula and the Birman-Krein formula are verified directly. The results are exploited for dissipative Schr?dinger-Poisson systems. In friendship dedicated to P. Exner on the occasion of his 60^th birthday This work was supported by DFG, Grant 1480/2.     

On the structure of additive systems of integers

A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an analogous concept where only sums of elements are considered. We establish a bijection between sum systems and sum-and-distance systems of corresponding size, and show that sum systems are equivalent to principal reversible cuboids, which are tensors with integer entries and a symmetry of ‘reversible square’ type. We prove a structure theorem for principal reversible cuboids, which gives rise to an explicit construction formula for all sum systems in terms of joint ordered factorisations of their component set cardinalities.

     

Operatorenkalkül über freien Monoiden III: Lagrangeinversion und Sheffersysteme

In this part the relations between the theory of binomial systems for noncommuting variables (of part II) and the results ofRota, Cigler et al. for the commutative case are studied in some detail. Afterwards for binomial systems of diagonal type there are given generalizations of the Rodrigues formula and the theorem ofLagrange-Good. A short development of Sheffer sequences follows. Finally the results are extended to binomial systems the structure of which is determined by any partial order of linear type.     

Dynamik von verankerten meerestechnischen Konstruktionen

The dynamical behaviour of moored structures in ocean engineering in an irregular sea is usually investigated by means of simplified mathematical models. Nonlinear phenomena like the influence of the motion on the mooring geometry or the fluid‐structure interaction are not considered. In this paper a method is described for analyzing the dynamical behaviour of mooring systems including the hydrodynamic forces due to the fluid surrounding the structure. The model for the dynamic analysis is generated using the multibody system method.     

Optical Tunnelling from One-Dimensional Square-Well Potentials

The authors study a one-dimensional model for optical tunnellingwith a refractive index in the shape of a square well. The relevanceof the model and its limiting cases are discussed. The mainresult is the leading behaviour of the exponentially small imaginarypart of the eigenvalue which determines the radiation loss.The leading behaviour of the imaginary part is calculated usingBerry's formula which controls the asymptotic expansion of theAiry function Bi(z) to better-than-exponential accuracy.     

On Stability of Symplectic Algorithms

1.FundamentalDeflnitionsLemma1.Thesolutionofalinearoofinarydtherentialequationwithcon8tantcoeffcientY=AYissta6leifalleigenvalue8ofAhaven0nP6sitivercalpartsandtheeigenvalueswithnullrealpartaresingleroots0ftheminimalp0lynomial.,/P\ThelinearHamiltoniansystemcanbeden0tedasZ=JSZwhereZ=(q),J=(ELs),andtheHamiltonianfuncti0nH(z)=ty.Lemma2.Thesolution80flinearHamiltoniansy8temsarecmticallysta6leifalleigenvaluesofJShavenullrsalpartandaresinglerootsojtheminitnalp0lyno?nial.Definiti0n1.Whenthemo…     

A formula for the lyapunov numbers of a stochastic flow. application to a perturbation theorem

We present a formula for the Lyapunov exponents of the flow of a nonlinear stochastic system. (These exponents characterise the asymptotic behaviour of the derivative flow, and negative exponents are associated with clustering of the flow). This formula is analogous to that of Khas'minskii, who deals with a linear system. We use this fojoruila to show that if we have an ordinary dynamical system which is Lyapunov stable (i.e. all the exponents are negative) then so are certain stochastic perturbations of it.     

Outputs and time lags for linear bioreactors

Linear systems of convection reaction-diffusion equations for bioreactors are shown to have a structure which allows a geometric factorization of steady state problems giving a significant reduction in their dimensionality. Moreover, convection dominated linear systems with quasisymmetric reaction terms may be further simplified by matrix transformations, which uncouple the differential equations. The boundary conditions are also uncoupled when the diagonal diffusivity matrix D governing diffusion in the bioparticle is a scalar multiple of the corresponding matrix H describing the diffusivity characteristic of the fluid boundary layers around the bioparticles. The dominant transient behaviour of the systems may be handled by establishing an analogous system of time independent equations for mean action time variables and higher moments. These equations have the same amenable structure. Outputs, time lags and various mean residence and first passage times, associated with establishing steady outputs from a concentration free initial state, can be expressed in terms of the steady state solutions and mean action time variables.     

Jumps of the fundamental solution matrix in discontinuous systems and applications

This paper deals with differential equations with discontinuous right-hand side. The concept of a solution for a discontinuous system is defined on the basis of differential inclusions using Filippov’s method. We study in particular the behaviour of solutions crossing a discontinuity surface transversally. A formula characterizing jumps of the fundamental solution matrix is derived. As an application of it, the concept of Poincaré mapping is defined for such systems.     

Paths and cycles of hypergraphs

Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs are constructed. The systems suit the structure properties of relational databases. The concepts of pseudo cycles and essential cycles of hypergraphs are introduced. They are relative to each other. Whether a family of cycles of a hypergraph is dependent or independent is defined. An enumeration formula for the maximum number of independent essential cycles of a hypergraph is given.     

The eigen-structure of the Jacobian in multi-class Lighthill-Whitham-Richards traffic flow models

Characteristic-based High Resolution Shock Capturing schemes for hyperbolic systems of conservation laws require, in their basic design structure, knowledge on the complete eigen-decomposition of the Jacobian matrix of the system. For the Multi-Class Lighthill-Witham-Richards (MCLWR) Traffic flow model considered in [4], there is no explicit formula for the eigenvalues of the Jacobian matrix, which can only be determined numerically. However, once they are determined, the eigen-vectors are easily computed and straightforward formulas can be obtained by exploiting the specific structure of the Jacobian matrix in these models. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Methode de laplace: etude variationnelle des fluctuations de diffusions de type

This article concerns the asymptotic behaviour of diffusions consisting of various systems of mean-field type interacting particles. We specifically study the following properties: propagation of chaos; critical and non critical fluctuations, using the Laplace method which has been developed by Bolthausen in the general Banach valued random variables context; and finally the large deviations of laws of trajectorial empirical measures.

We also study Gibbs variational formula associated with this problem and show that it reduces to a finite dimensional problem     

Approximate Equilibrium Results for the Multi-Server Queue (GI/M/r)

Approximate formulae for the equilibrium queue-size and queueing-time distribution in the system GI/M/r are derived. The degree of approximation achieved depends on the particular formula used and on the values of the various parameters involved; the relative percentage errors incurred seem, however, to be well below 5 per cent for most practical purposes. It is believed that the underlying idea of "similarity" in the behaviour of queueing processes may also lead, if extended, to useful approximate formulae for more complex systems.     

Geometric integration of non-autonomous linear Hamiltonian problems

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.     

碟形扁壳在均布载荷下的非线性稳定分析*

本文对碟形扁壳的非线性稳定问题进行了分析.用修正迭代法得到了弹性特征的二次近似表达式,给出了一组算例,并讨论了参数β和k对稳定特性的影响.     

Damage detection in plates by wave propagation analyses

At locations with defects the propagation of elastic waves in solids is disturbed. This behaviour can be used for the development of new structural health monitoring and quality assurance systems. In the sense of adaptive structure systems actuators of piezo–electric materials [1] are implemented to generate high frequency waves (Lamb waves) in plate like structures, [2]. At flaws, imperfections of the structure and surface defects these waves show reflections, refractions and mode conversions. The observation of the wave propagation and the wave behaviour at possible flaws and surface defects is performed by an optical measurement system. Measurement data of the thin walled metallic plates are compared with numerical and analytical solutions. First results of the detection of the wave propagation in metallic plates are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive solutions for Lotka–Volterra competition system with large cross-diffusion in a spatially heterogeneous environment

In this paper, we study the stationary problem for the Lotka–Volterra competition system with cross-diffusion in a spatially heterogeneous environment. Although some sufficient conditions for the existence of positive solutions are obtained by using global bifurcation theory, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we focus on the asymptotic behaviour of positive solutions and derive two shadow systems as the cross-diffusion coefficient tends to infinity, moreover, the structure of positive solutions of the limiting system is analysed. The result of asymptotic behaviour also reveals different phenomena from that studied in Wang and Li (2013).