Fréchet differentiability of solution mappings for semilinear second order evolution equations

In this paper we establish the strong Fréchet differentiability of maps from the set of initial values and forcing terms into the set of solutions for semilinear second order evolution equations. Also, under the weaker conditions of nonlinear terms we establish the strong Gâteaux differentiability of the solution maps. An application of results to semilinear strongly damped wave equations is given.     

Almost automorphic solutions for stochastic differential equations driven by Lévy noise

We introduce the concepts of Poisson square-mean almost automorphy and almost automorphy in distribution. Under suitable conditions on the coefficients, we establish the existence of solutions which are almost automorphic in distribution for some semilinear stochastic differential equations with infinite dimensional Lévy noise. We further discuss the global asymptotic stability of these solutions. Finally, to illustrate the theoretical results obtained in this paper, we give several examples.     

Extensions of Fréchet ?-Subdifferential Calculus and Applications

In this paper, we establish some calculus rules for the limiting Fréchet ?-subdifferentials of marginal functions and composite functions. Necessary conditions for approximate solutions of a constrained optimization problem are derived.     

Sur l'existence, l'unicité, la stabilité et les propriétés de grandes déviations des solutions d'équations différentielles stochastiques rétrogrades à horizon aléatoire. Application à des problèmes de perturbations singulières

We study the existence, uniqueness and stability of solutions of backward stochastic differential equations with random terminal time under new assumptions; then we establish a large deviation principle for the solutions of such equations, related to a family of Markov processes, the diffusion coefficient of which tends to zero. Finally we apply these results to the analysis of some singular perturbation problems for a class of nonlinear partial differential equations.     

Variational theory and domain decomposition for nonlocal problems

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.     

Poincaré type theorems for non-autonomous systems

In this paper we establish analytic equivalence theorems of Poincaré and Poincaré-Dulac type for analytic non-autonomous differential systems based on the dichotomy spectrum of their linear part. As applications of the theorem, normal forms linearize for two illustrative examples.     

Completeness of the System of Values of a Holomorphic Vector Function in Fréchet Spaces

We establish necessary and sufficient conditions for the system of values of a holomorphic vector function to be complete in a Fréchet space. The results obtained are illustrated by several examples.     

Large deviation principles for 2-D stochastic Navier-Stokes equations driven by Lévy processes

In this paper, we establish a large deviation principle for the two-dimensional stochastic Navier-Stokes equations driven by Lévy processes, which involves the study of the Lévy noise and the investigation of the effect of the highly nonlinear, unbounded drifts.     

Image des opérateurs d?entrelacements normalisés et pôles des séries d?Eisenstein

This paper is about the pole of some Eisenstein series for classical groups over a number field. In a previous paper, we have shown how to normalize intertwining operators in such a way that they are holomorphic for positive parameters. Here we show that the image of such operators is (in the interesting cases) either 0 or an irreducible representation. This enables us to compute explicitly the residue of the Eisenstein series obtained from square integrable cohomological representations. At the end of the paper we give necessary and sufficient conditions in terms of Arthur?s data in order that a square integrable cohomological representation is cuspidal; the conditions are not totally satisfactory and we explain what we expect when Arthur?s results will be fully available.     

Continuous Frechet differentiability with respect to a Lipschitz domain and a stability estimate for direct acoustic scattering problems

We consider direct acoustic scattering problems with eithera sound-soft or sound-hard obstacle, or lossy boundary conditions,and establish continuous Fréchet differentiability withrespect to the shape of the scatterer of the scattered fieldand its corresponding far-field pattern. Our proof is basedon the implicit function theorem, and assumes that the boundaryof the scatterer as well as the deformation are only Lipschitzcontinuous. From continuous Fréchet differentiability,we deduce a stability estimate governing the variation of thefar-field pattern with respect to the shape of the scatterer.We illustrate this estimate with numerical results obtainedfor a two-dimensional high-frequency acoustic scattering problem.     

Optimal control and dependence modeling of insurance portfolios with Lévy dynamics

In this paper we are interested in optimizing proportional reinsurance and investment policies in a multidimensional Lévy-driven insurance model. The criterion is that of maximizing exponential utility. Solving the classical Hamilton-Jacobi-Bellman equation yields that the optimal retention level keeps a constant amount of claims regardless of time and the company’s wealth level.A special feature of our construction is to allow for dependencies of the risk reserves in different business lines. Dependence is modeled via an Archimedean Lévy copula. We derive a sufficient and necessary condition for an Archimedean Lévy generator to create a multidimensional positive Lévy copula in arbitrary dimension.Based on these results we identify structure conditions for the generator and the Lévy measure of an Archimedean Lévy copula under which an insurance company reinsures a larger fraction of claims from one business line than from another.     

A Berry-Esséen bound for student's statistic in the non-I.I.D. case

We establish a Berry-Esséen bound for Student's statistic for independent (nonidentically) distributed random variables. In particular, the bound implies a sharp estimate similar to the classical Berry-Esséen bound. In the i.i.d. case it yields sufficient conditions for the Central Limit Theorem for studentized sums. For non-i.i.d. random variables the bound shows that the Lindeberg condition is sufficient for the Central Limit Theorem for studentized sums.Research supported by the SFB 343 in Bielefeld.     

Paley-Wiener spaces with vanishing conditions and Painlevé VI transcendents

We modify the classical Paley-Wiener spaces PWx of entire functions of finite exponential type at most x>0, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points z₁,…,zn. We compute the reproducing kernels and relate their variations with respect to x to a Krein differential system, whose coefficient (which we call the μ-function) and solutions have determinantal expressions. Arguments specific to the case where the “trivial zeros” z₁,…,zn are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non-linear first order differential equations. A computation, having this non-linear system at his start, obtains quasi-algebraic and among them rational Painlevé transcendents of the sixth kind as certain quotients of such μ-functions.     

Functoriality of the Bimodule Associated to a Hilsum–Skandalis Map

We extend the functoriality of the Connes convolution algebra to the category of Hilsum–Skandalis maps between separated smooth étale groupoids. Thereby we establish that Morita equivalent separated smooth étale groupoids have Morita equivalent convolution algebras, analogously to the results of Muhly, Renault and Williams, and others, on the C^*algebras of Morita equivalent groupoids.     

Holomorphic transforms with application to affine processes

In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multi-dimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

On the number of limit cycles in general planar piecewise linear systems of node–node types

We investigate the existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with node–node dynamics. Using the Liénard-like canonical form with seven parameters, some sufficient and necessary conditions for the existence of limit cycles are given by studying the fixed points of proper Poincaré maps. In particular, we prove the existence of at least two nested limit cycles and describe some parameter regions where two limit cycles exist. The main results are applied to the PWL Morris–Lecar neural model to determine the existence and stability of the limit cycles.     

Linear regularity and ?-regularity of nonconvex sets

In this paper, we discuss some sufficient conditions for the linear regularity and bounded linear regularity (and their variations) of finitely many closed (not necessarily convex) sets in a normed vector space. The accompanying necessary conditions are also given in the setting of Asplund spaces.     

The center conditions and local bifurcation of critical periods for a Liénard system

In this paper we study the problems of centers and isochronous centers and the local bifurcation of critical periods for a Liénard system with forth damping. Calculating the singular point values and period constants, we find all center conditions and isochronous center conditions. Moreover, the numbers of local critical periods bifurcating from centers and isochronous centers is obtained by computing the orders of weak centers.     

Réflexivité d'une extension d'un opérateur sous-normal par un opérateur algébrique

A bounded linear operator on a Hilbert space is said to be reflexive if the operators which leave invariant the invariant subspaces of T are wot-limits of polynomials in T. In this paper we give a necessary and sufficient condition for an extension of a subnormal operator by an algebraic one to be reflexive.We also give a formula for the reflexivity defect of such extensions.