Subspaces and quotients of Banach spaces with shrinking unconditional bases

The main result is that a separable Banach space with the weak* unconditional tree property is isomorphic to a subspace as well as a quotient of a Banach space with a shrinking unconditional basis. A consequence of this is that a Banach space is isomorphic to a subspace of a space with a shrinking unconditional basis if and only if it is isomorphic to a quotient of a space with a shrinking unconditional basis, which solves a problem dating to the 1970s. The proof of the main result also yields that a uniformly convex space with the unconditional tree property is isomorphic to a subspace as well as a quotient of a uniformly convex space with an unconditional finite dimensional decomposition.     

On the properties of orthorecursive expansions with respect to subspaces

In a Hilbert space, for orthorecursive expansions with respect to closed subspaces, we establish a criterion for expansions of elements of a certain finite-dimensional subspace with respect to a finite sequence of subspaces to coincide with the expanded elements. This implies a criterion for an element to be equal to its orthorecursive expansion with respect to a finite sequence of subspaces. We also obtain a number of results related to the best approximations of elements by partial sums of their orthorecursive expansions with respect to a sequence of finite-dimensional subspaces.     

On Martingale Approximation of Adapted Processes

We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to which it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.     

Homogenization and propagation of chaos to a nonlinear diffusion with sticky reflection

Summary We study a process reflecting in a domain. The process follows Wentzell non-sticky boundary conditions while being adsorbed at the boundary at a certain rate with respect to local time and desorbed at a rate with respect to natural time. We show that when the rates go to infinity with a converging ratio, the process converges to a process with sticky reflection having the limit ratio as the sojourn coefficient. We then study a mean-field interacting system of such particles. We show propagation of chaos to a nonlinear diffusion with sticky reflection when we perform this homogenization simultaneously as the number of particles goes to infinity.     

Gaussian moving averages,semimartingales and option pricing

We provide a characterization of the Gaussian processes with stationary increments that can be represented as a moving average with respect to a two-sided Brownian motion. For such a process we give a necessary and sufficient condition to be a semimartingale with respect to the filtration generated by the two-sided Brownian motion. Furthermore, we show that this condition implies that the process is either of finite variation or a multiple of a Brownian motion with respect to an equivalent probability measure. As an application we discuss the problem of option pricing in financial models driven by Gaussian moving averages with stationary increments. In particular, we derive option prices in a regularized fractional version of the Black–Scholes model.     

Hitting Times in Markov Chains with Restart and their Application to Network Centrality

Motivated by applications in telecommunications, computer science and physics, we consider a discrete-time Markov process with restart. At each step the process either with a positive probability restarts from a given distribution, or with the complementary probability continues according to a Markov transition kernel. The main contribution of the present work is that we obtain an explicit expression for the expectation of the hitting time (to a given target set) of the process with restart. The formula is convenient when considering the problem of optimization of the expected hitting time with respect to the restart probability. We illustrate our results with two examples in uncountable and countable state spaces and with an application to network centrality.     

Zur Kohomologie von Gruppen mit abelschen Sylowuntergruppen

In 1962 Atiyah showed, with, the help of topological methods, the existence of a spectral sequence joining the cohomology ring of a finite group (with values in the rational integers) and the ring of generalized complex characters of the group. This paper undertakes to get that connection (at least in part) in a purely algebraic way. To that effect such ring filtrations of the character ring of a group are described that are functorially compatible with group morphisms and compatible with the induction of representations belonging to subgroups. The graded ring formed canonically by a filtration of that kind is to be compared with the cohomology ring of the same group, and it is shown, the group assumed to have abelian Sylow subgroups or to be a periodic group in a cohomological sense, that by taking a certain filtration its graded ring can be identified with a homomorphic part of the cohomology ring. (That corresponds to the infinite term of Atiyah's spectral sequence). The generalization of this result with respect to all finite groups has not yet been achieved; there is, however, a reduction theorem saying that it is sufficient to deal with groups of prime power order only. - As an application one gets lower bounds for the orders of the cohomology groups belonging to a finite group, especially, as a corollary, the theorem of its infinite cohomological dimension.     

Solving nonlinear equations by differentiation with respect to a parameter

Differentiation with respect to a parameter is proposed as a method for solving a system of nonlinear equations with a poorly conditioned Jacobian matrix. Singular and polar decompositions of the Jacobian matrix lead to a system of ordinary differential equations with small parameters multiplying the derivatives. This system is solved by introducing auxiliary linear differential equations. The numerical solution of a poorly conditioned system is reduced in this case to the approximation of an exponential with a large negative argument.     

Random homogenization and convergence to integrals with respect to the Rosenblatt process

This paper concerns the random fluctuation theory of a one dimensional elliptic equation with highly oscillatory random coefficient. Theoretical studies show that the rescaled random corrector converges in distribution to a stochastic integral with respect to Brownian motion when the random coefficient has short-range correlation. When the random coefficient has long-range correlation, it was shown for a large class of random processes that the random corrector converged to a stochastic integral with respect to fractional Brownian motion. In this paper, we construct a class of random coefficients for which the random corrector converges to a non-Gaussian limit. More precisely, for this class of random coefficients with long-range correlation, the properly rescaled corrector converges in distribution to a stochastic integral with respect to a Rosenblatt process.     

Differential games with a given value function

The set of solutions of a differential game with a terminal payoff functional is investigated. A method is obtained that allows us to establish whether a given function is a value of some differential game with a terminal payoff functional. The condition obtained is in fact the condition for the given function to be a minimax (viscosity) solution of some Hamilton-Jacobi equation with Hamiltonian homogeneous in the third variable. We also obtain a sufficient condition for a function to belong to the set of values of differential games with a terminal payoff function.     

关于非完整力学系统相对部分变量的稳定性*

本文给出研究非完整系统相对部分变量稳定性的一种方法,并得到非完整系统相对部分变量的一些稳定性定理:同时,本文还得到一类非完整系统相对全部变量稳定性与相对部分变量稳定性的关系。     

Parabolic variational inequality with parameter and gradient constraints

This paper concerns a singular control problem whose value function is governed by a time-dependent HJB equation with gradient constraints. The method is to transform a two-dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies. Moreover we analyze the behaviors of the free boundary surfaces. The main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter.     

Coexistence in host–pathogen systems

Lanchier and Neuhauser have initiated the study of host–symbiont systems but have concentrated on the case in which the birth rates for unassociated hosts are equal. Here we allow the birth rates to be different and identify cases in which a host with a specialist pathogen can coexist with a second species. Our calculations suggest that it is possible for two hosts with specialist pathogens to coexist but it is not possible for a host with a specialist mutualist to coexist with a second species.     

Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points

It was established in [1] that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is nonsimple.     

Stochastic integration,trace and the skeleton of wiener functionals

It is shown that the notion of trace induced by a given complete orthonormal system relates the Skorohod integral with a corresponding Ogawa‐type integral evaluated with respect to the same orthonormal systems. Similarly the multiple Wiener‐Ito integral is shown to be related to a multiple Ogawa‐type integral induced by a complete orthonormal system via the Hu‐Meyer formula with suitably defined multiple traces. The notion of skeleton of a Wiener functional relative to a given orthonormal system is defined and yields what seems to be a “natural” extension of Wiener functionals to the Cameron Martin space and the Wiener processes with a different scale.     

Approximating connected facility location with buy-at-bulk edge costs via random sampling

In the connected facility location problem with buy-at-bulk edge costs we are given a set of clients with positive demands and a set of potential facilities with opening costs in an undirected graph with edge lengths obeying the triangle inequality. Moreover, we are given a set of access cable types, each with a cost per unit length and a capacity such that the cost per capacity decreases from small to large cables, and a core cable type of infinite capacity. The task is to open some facilities and to connect them by a Steiner tree using core cables, and to build a forest network using access cables such that the edge capacities suffice to simultaneously route all client demands unsplit to the open facilities. The objective is to minimize the total cost of opening facilities, building the core Steiner tree, and installing the access cables. In this paper, we devise a constant-factor approximation algorithm for this problem based on a random sampling technique.     

A collocation method for high-frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.     

Sparse Linear Algebra and Geophysical Migration: A Review of Direct and Iterative Methods

The pre-stack depth migration of reflection seismic data can be expressed, in the framework of waveform inversion, as a linear least squares problem. Together with the precise definition of this operator, we detail additional main characteristics of the forward model, like its huge size, its sparsity and the composition with convolution. It ends up with a so-called discrete ill-posed problem, whose acceptable solutions have to undergo a regularization procedure. Both direct and iterative methods have been implemented with specific attention to the convolution, and then applied to a given data set: a synthetic 2-dimensional profile of revealing size with some added noise. The efficiency with regard to computational effort and storage requirements is evaluated. The needed regularization of the solution is thoroughly studied in both cases. From the point of the global inverse problem, the extra feature of providing a solution that can be differentiated with respect to a parameter such as background velocity is also discussed.     

Strong competition model with non-uniform dispersal in a heterogeneous environment

In this study, a strong competition model was considered between two species in a heterogeneous environment. For a system with two different constant diffusion rates for each competitor, the fast diffuser can be selected evolutionally under suitable assumptions if the competing interaction between the species is strong. We also claim that a strongly interacting competition leads to a more evolutionary selection than that with the same population dynamics if a species moves with a certain non-uniform dispersal. Furthermore, species with a certain non-uniform dispersal have a competitive advantage over linear random diffusers. In addition, a species with highly sensitive dispersal response to the environment may survive. These strongly competitive advantages were demonstrated by investigating the stability of semi-trivial solutions of the system with non-uniform dispersal and comparing it to the conditions of the model with constant diffusion.