Let $\{T_{t}:t\in{\bf R}\}$ be a measure preserving flow in a probability measure space $(\Omega, {\cal A},\mu)$, and $\{F_{t}:t\in {\bf R}\}$ be a family of real-valued measurable functions on $(\Omega,{\cal A},\mu)$ such that $F_{t+s}=F_{t}+F_{s}\circ T_{t}$ (mod $\mu$) for all $t,\, s \in {\bf R}$. In this paper we deduce necessary and sufficient conditions for the existence of a real-valued measurable function $f$ on $\Omega$, with $f\in L_{p}(\Omega,\mu)$ where $0\leq p\leq \infty$, such that $F_{t}=f\circ T_{t}-f$ (mod $\mu$) for all $t\in {\bf R}$. Related results are also obtained. These may be considered to be continuous parameter refinements of the recent discrete parameter results of Alonso, Hong and Obaya concerning additive real coboundary cocycles.

We consider a two component reaction-diffusion system with a small parameter $\epsilon$ $$ \left \{ \begin{array}{l} u_t = d_u\Delta u + u^{\epsilon_n}(u^mv - au^n),\\ v_t = d_v\Delta v - \displaystyle{\frac{1}{\epsilon}} u^m v, \end{array} \right. $$ where $m$ and $n$ are positive integers, together with zero-flux boundary conditions. It is known that any nonnegative solution becomes spatially homogeneous for large time. In particular, when $n\gt m \geq 1$, $(u^{\epsilon},v^{\epsilon})(t)\to(0,0)$ as $t\to \infty$, while when $m\geq n \geq 1$, there exists some positive constant $v^\epsilon_\infty$ such that $(u^{\epsilon},v^{\epsilon})(t)\to(0,v^\epsilon_\infty)$ as $t\to \infty$. In order to find the value of $v^{\epsilon}_\infty$, we derive a limiting problem when $\epsilon\to 0$ under some conditions on the values of $m$, $n$ and on the initial functions $(u_0,v_0)$, by which an approximate value of $v^\epsilon_\infty$ can be obtained.

In this paper, we shall study the family of operators of the form $$ T_{\vec g}(f)(z)=\int_0^{z_1}\cdots\int_0^{z_n}f(\zeta_1,\cdots,\zeta_n)\prod_{j=1}^ng'_j(\zeta_j)d\zeta_j $$ on Hardy $H^p(D_n)$, the generalized weighted Bergman ${\cal A}_\mu^{p,q}(D_n),$ $p\in (0,\infty),$ and $\alpha$-Bloch ${\cal B}^\alpha (D_n)$ spaces on the polydisk $D_n=\{(z_1,\dots, z_n)\in {\bf C}^n:\,|z_j|\lt 1,\,\,j=1,\dots, n\}.$

We prove the generalized Hyers-Ulam-Rassias stability of the invertible mapping in a Banach module over a unital Banach algebra in the spirit of G\u avruta, and prove the generalized Hyers-Ulam-Rassias stability of linear functional equations in a Hilbert module over a unital $C^*$-algebra in the spirit of G\u avruta.

We consider the commutator of Calder\'{o}n-Zygmund operator and multiplication operator by $b$, that is, the commutator of Coifman, Rochberg and Weiss. We study the boundedness of this operator on the Hardy spaces associated with Herz spaces. We show this commutator is bounded from $H\dot{K}^{\alpha, p}_q$ (Hardy pace) to $\dot{K}^{\alpha, p, \infty}_q$ (weak Herz space).

We present new interval oscillation criteria for certain classes of second order nonlinear differential equations with damping that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of $[t_0,\infty)$, rather than on the whole half-line. Our results are sharper than some previous results and handle the cases which are not covered by known criteria. Finally, several examples that dwell upon the sharp conditions of our results are also included.

Let $X_{i}\sim W_{p}(n_{i},\Sigma,\Theta)$ where $\Theta= diag(\theta^{2}_{i},0,\ldots,0)$, $i=1,\ldots,r+1$. In this article the authors have derived the joint distribution of $U_i=C^{-1}X_{i}C'{}^{-1}$, $i=1,\ldots,r$ where $\sum_{i=1}^{r+1}X_{i}=CC'$ and $C$ is a lower triangular matrix. The joint distribution of $U_{1},\ldots,U_{r}$ is a non-central matrix-variate Dirichlet distribution. Several properties of this distribution such as marginal and conditional distributions, distribution of partial sums, moments and asymptotic results have also been studied.

In this paper we define a normed space $A_p^q \left( {G,A} \right)$ and prove some properties of this space. In particular, we show that the space $ L^p \left( {G,A} \right) \otimes _{L^1 \left( {G,A} \right)} L^q \left( {G,A} \right)$ is isometrically isomorphic to the space $A_q^p \left( {G,A} \right)$ and the space of multipliers from $L_{}^p \left( {G,A} \right)$ to $ L_{}^{q'} \left( {G,A^* } \right)$ is isometrically isomorphic to the dual of the space $A_p^q \left( {G,A} \right)$ if $G$ satisfies a property $P_p^q$.

In this note we show that the uniform convergence theorem for the H1-integral is false.

In this paper, we give a nonabsolute dominated convergence theorem for the K-H integral on the real line. Furthermore, as the converse part, we also give a corresponding Riesz type definition of the K-H integral.