This paper studies infinite-dimensional bilinear control systems described by $y'(t)=Ay(t)+u(t)By(t)$ where $A$ generates a semigroup $(e^{tA})_{t\geq 0}$ on a Banach space $Y$ (state space), $B:D(B)(\subset Y)\rightarrow Y$ is an unbounded linear operator and $u\in L^{p}_{loc}(0,\infty)$ is a scalar control. Sufficient conditions are given for $B$ to be admissible, i.e for any $t$, the integral $\int^{t}_{0}u(s)e^{(t-s)A}By(s)ds$ should be in $Y$ and depends continuously on $u\in L^{p}(0,\infty)$, $y\in L^{q}(0,\infty;Y)$ for some appropriate positive numbers $p$, $q$. This approach enables us to obtain, through an integrated form, a unique solution for the bilinear system. The results are applied to a heat equation.

We study in this paper the space $L^\infty_0({\cal S},M_a({\cal S}))$ of a locally compact semigroup ${\cal S}$. That space consists of all $\mu$-measurable ($\mu\in M_a({\cal S})$) functions vanishing at infinity, where $M_a({\cal S})$ denotes the algebra of all measures with continuous translations. We introduce an Arens multiplication on the dual $L^\infty_0({\cal S},M_a({\cal S}))^*$ of $L^\infty_0({\cal S},M_a({\cal S}))$ under which $M_a({\cal S})$ is an ideal. We then give some characterizations for Arens regularity of $M_a({\cal S})$ and $L^\infty_0({\cal S},M_a({\cal S}))^*$. As the main result, we show that $M_a({\cal S})$ or $L^\infty_0({\cal S},M_a({\cal S}))^*$ is Arens regular if and only if ${\cal S}$ is finite.

We introduce and study a new class of lattice ordered algebras.

The order of the weight of orthogonal polynomials is analyzed, when this weight function shows singularities on the boundary of a region in the complex plane.

Necessary and sufficient coefficient conditions are established for certain classes of analytic functions to be univalent in the unit disk. In addition, characterizations and convolution results are established for these classes of functions. As a consequence univalency preservation type operators are discussed.

For arbitrary compact set $K\subset C$ we consider the uniform polynomial approximation error $E_{n} (f, K)$ of an entire function $f$ and relate it to the generalized order, generalized lower order and generalized type of $f$.

The notion of pair-wise tangential mappings, which is a generalization of mappings satisfying (E.A) property, is introduced and used to prove a common fixed point theorem of Gregus type for a quadruple of self mappings of a metric space satisfying a strict general contractive condition of integral type. Our main result generalizes a recent result of A. Djoudi, A. Alioche [Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type, J. Math. Anal. Appl. 329 (2007), 31-45].

This paper concerns the study of the numerical approximation for the following boundary value problem $$ \left\{ \begin{array}{ll} \hbox{$(u^{m})_{t}=u_{xx}$, $<x<1$,\; $t>0$,} \\ \hbox{$u_{x}(0,t)=0$,\quad $u_{x}(1,t)=-u^{-\beta}(1,t)$,\quad $t>0$,} \\ \hbox{$u(x,0)=u_{0}(x)>0$,\quad $0\leq x\leq 1$,} \\ \end{array} \right.$$ where $m\geq1$, $\beta>0$. We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis.

In this note we show that, for an arbitrary Hausdorff locally m-convex topology on a subalgebra $A$ of the algebra $C(X)$, the boundedness radius $\beta$ is nothing but the uniform norm, whenever $A$ is a $C_b(X)$-module and closed under the complex conjugation. We then deduce a Theorem of Kaplansky-Meyer type for subalgebras.

We prove that a topological vector space $E$ is Fréchet-Urysohn if and only if it has a bounded tightness, i.e. for any subset $A$ of $E$ and each point $x$ in the closure of $A$ there exists a bounded subset of $A$ whose closure contains $x$. This answers a question of Nyikos on $C_p(X)$ (personal communication). We also raise two related questions for topological groups.

The existence of common fixed point results for a Banach operator pair under certain generalized $\varphi$-contractions is established. As applications, the corresponding invariant best approximation results are proved. Our results unify and generalize various known results to the more general class of noncommuting mappings.

A parabolic surface in hyperbolic space $\mathbb H^3$ is a surface invariant by a group of parabolic isometries. In this paper we describe all parabolic surfaces with constant Gaussian curvature. We study the qualitative properties such as completeness and embeddedness.

We give a characterization of quasinormable weighted Fr\'echet spaces of entire functions under the assumption that the system of weights belongs to the class $(E)_{a,A}$ of Bierstedt-Bonet-Taskinen.

In this paper we characterize the derivations from $\frak E_p(\widehat{G})$ into $\frak E_q(\widehat{G})$, $1\leq p,q\leq\infty$. We find necessary and sufficient conditions for weak amenability of Banach algebras $\frak E_p(\widehat{G})$, $1\leq p\leq\infty$. Moreover we find a necessary condition for the weak amenability of the convolution Banach algebra $A(G)$ on a compact group $G$, where its sufficiency was proved earlier by Ghahramani and Lau.

Given an infinite-dimensional Banach space $X$ and a Banach space $Y$ with no finite cotype, we determine whether or not every continuous linear operator from $X$ to $Y$ is absolutely $(q;p)$-summing for various choices of $p$ and $q$, including the case $p=q$. If $X$ assumes its cotype, the problem is solved for all choices of $p$ and $q$. Applications to the theory of dominated multilinear mappings are also provided.