Dynamics of horizontal-like maps in higher dimension

We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in C^k, under a natural assumption on the dynamical degrees. We estimate the speed of convergence towards the Green currents, the decay of correlations for the equilibrium measure and the Lyapounov exponents. We show in particular that the equilibrium measure is hyperbolic. We also show that the Green currents are the unique invariant vertical and horizontal positive closed currents. The results apply, in particular, to Henon-like maps, to regular polynomial automorphisms of C^k and to their small pertubations.Comment: Dedicated to Professor Gennadi Henkin on the occasion of his 65th birthday, 37 pages, to appear in Advances in Mat

On the Design of Secure Full-Duplex Multiuser Systems under User Grouping Method

Consider a full-duplex (FD) multiuser system where an FD base station (BS) is designed to simultaneously serve both downlink users and uplink users in the presence of half-duplex eavesdroppers (Eves). Our problem is to maximize the minimum secrecy rate (SR) among all legitimate users by proposing a novel user grouping method, where information signals at the FD-BS are accompanied with artificial noise to degrade the Eves' channel. The SR problem has a highly nonconcave and nonsmooth objective, subject to nonconvex constraints due to coupling between the optimization variables. Nevertheless, we develop a path-following low-complexity algorithm, which invokes only a simple convex program of moderate dimensions at each iteration. We show that our path-following algorithm guarantees convergence at least to a local optima. The numerical results demonstrate the merit of our proposed approach compared to existing well-known ones, i.e., conventional FD and nonorthogonal multiple access.Comment: 6 pages, 3 figure

Convergence of a series associated with the convexification method for coefficient inverse problems

This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman Weight Function in it. In the previous works the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in $L^{2}$ as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the $H^{1}$-norm for a sequence of $L^{2}$-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.Comment: 15 page

On "the authentic damping mechanism" of the phonon damping model

Some general features of the phonon damping model are presented. It is concluded that the fits performed within this model have no physical content