In this paper we will prove the coexistence of unbounded solutions and periodic solutions for a class of planar systems with asymmetric nonlinearities \begin{eqnarray*}\label{abstract} \left \{ \begin{array}{lll} u'=v-\alpha u^{+}+\beta u^{-} \\ v'=-\mu u^{+}+\gamma u^{-}-g(u)+p(t), \end{array} \right. \end{eqnarray*} where $g(u)$ is continuous and bounded, $p(t)$ is a continuous $2\pi$-periodic function and $\alpha, \beta\in \mathbb{R}, \mu, \gamma$ are positive constants.

The main purpose of this paper is to study, under a suitable geometric conditions, the indirect boundary stabilization for coupled Schrödinger equations with variable coefficients and one Dirichlet boundary feedback. The polynomial energy decay rate for smooth solutions is obtained by the combination of the Riemannian geometry method in $\left[ Ya1\right] $ and the ideas of I. Lasiecka and R. Triggiani in $\left[ LT3\right].

This paper deals with some perturbation of the so called prescribed scalar Q-curvature type equations on compact Riemannian manifolds; these equations are fourth order elliptic and of critical Sobolev growth. Sufficient conditions are given to have at least two distinct solutions first without using the concentration-compactness technic but with a suitable range of the parameters and secondly by using the concentration-compactness methods.

Let $p>0$ and let $\Omega\subset\Bbb C^{d}$ be a bounded, strictly pseudoconvex domain with boundary of class $C^{2}$. We consider a family of directions in the form of a continuous function $\gamma:\partial\Omega\times[0,1]\ni(z,t)\rightarrow\gamma(z,t)\in\overline{\Omega}$ satisfying some natural properties. Then for a given lower semicontinuous, strictly positive function $H$ on $\partial\Omega$ we construct a holomorphic function $f\in\Bbb O(\Omega)$ such that $H(z)=\int_{0}^{1}\left|f(\gamma(z,t))\right|^{p}dt$ for $\eta$-almost all $z\in\partial\Omega$ where $\eta$ is a given pro\-ba\-bility measure on $\partial\Omega$.

Given two sequences of monic orthogonal polynomials $\{P_{n}\}_{_{n\geq 0}}$ and $\{B_{n}\}_{_{n\geq 0}}$ such that $B_{2n}(x)=P_{n}(x^{2}),n\geq0,$ we show that the Laguerre-Hahn character of one of them remains valid for the other. Then we give relations between their classes and the coefficients of their structure relations. As an application, with an appropriate choice of the sequence $\{P_{n}\}_{n\geq 0},$ we obtain a new nonsymmetric semi-classical sequence of polynomials $\{B_{n}\}_{_{n\geq 0}}$ of class $s=1$.

For a fixed generalized reflection matrix $P$, i.e., $P^T=P, P^2= I$, and $P\neq \pm I$, then a matrix $A $ is called a symmetric $P$-symmetric matrix if $A=A^T$ and $(PA)^T=PA$. This paper is mainly concerned with finding the least squares symmetric $P$-symmetric solutions to the matrix inverse problem $AX=B$ with a submatrix constraint, where $X$ and $B$ are given matrices of suitable size. By applying the generalized singular value decomposition and the canonical correlation decomposition, an analytical expression of the least squares solutions is derived basing on the Projection Theorem in Hilbert inner products spaces. Moreover, in the corresponding solution set, the analytical expression of the unique minimum-norm solution is described in detail.

In this short note, completing a sequence of studies, we consider the $k$-Grassmannians of a number of polar geometries of finite rank $n$. We classify those subspaces that are isomorphic to the $j$-Grassmannian of a projective $m$-space. In almost all cases, these are parabolic, that is, they are the residues of a flag of the polar geometry. Exceptions only occur when the subspace is isomorphic to the Grassmannian of $2$-spaces in a projective $m$-space and we describe these in some detail. This Witt-type result implies that automorphisms of the Grassmannian are almost always induced by automorphisms of the underlying polar space.

The aim of this paper is twofold. First, we investigate the properties of the composition of harmonic mappings with harmonic mappings, and the composition of biharmonic mappings with harmonic mappings. Second, we consider the Goodman-Saff conjecture for biharmonic mappings in the unit disk. In fact, we show that the answer to the Goodman-Saff conjecture is positive for a special class of univalently biharmonic mappings which contains the set of all harmonic univalent mappings.

A necessary and sufficient condition for the uniform exponential dichotomy is pointed out using a discrete-time argument. Thus are extended known results due to Barbashin, Datko, Lovelady, Pazy[13], Preda.

In this paper we give a numerical method for the study of the decay of the energy of a network of strings. This method is based on a theoretical and numerical construction of some operator say of type {\Large $\tau$} by the D'Alembert formula. Moreover we give some illustrations of this method.

Dans cet article on présente une méthode numérique pour étudier la dé\-crois\-sance de l'énergie d'un réseau des cordes. Une telle méthode est basée sur la construction, théorique et numérique des opérateurs de type {\Large $\tau$}, par la formule de D'Alembert. On donnera à la fin quelques exemples d'applications.

We give a simple and easy proof of a result obtained by W. Rehder who generalized, in his turn, a famous result by Beck and Putnam. We then give a generalization to unbounded operators.

For the generalized-Euler-constant function \[ a\mapsto \gamma(a):=\underset{n\rightarrow\infty}{\lim} \left(\sum_{i=0}^{n-1}\frac{1}{a+i}-\ln\frac{a+n-1}{a}\right) \] defined on $\R^+$, the expansion $\gamma(a)=\sum_{j=2}^{\infty}\frac{(-1)^j}{j}\,\zeta(j,a)$, where $\zeta(j,a)$ is the Hurwitz zeta function, is derived and a formula for its numerical computation is presented.

The Lusternik-Schnirelmann category of a space is a homotopy invariant. Cone-decompositions are used to give an upper bound for Lusternik-Schnirelmann categories of topological spaces. The purpose of this paper is to construct cone-decompositions of the special unitary groups, for which we use a filtration due to Miller. We observe also that Miller's filtration is closely related to a CW-decomposition.