The purpose of this paper is to construct solutions for the problems when capillary-gravity waves are generated by a vertical plane wavemaker in consideration of surface tension and the edge condition proposed by L. M. Hocking [4]. The uniqueness of these solutions will also be proven here.

Let I be an ideal of an exchange ring R. We say that I is weakly stable provided that every regular element in 1+I is one-sided unit-regular. If I is weakly stable, then so is Mn(I) as an ideal of Mn(R). Also every square regular matrix over such an ideal admits a diagonal reduction by right or left invertible matrices. These extend the corresponding results of [1],[6-8],[10] and [12-13].

In this paper we establish extension theorems for additive mappings ϕ : A+ × B+ → C+, where A, B are Riesz spaces (lattice ordered spaces or vector lattices) and C is an order complete Riesz space, to the whole of A × B, thereby extending well-known results for additive mappings between Riesz spaces. We prove, in particular, that when A, B and C are order complete Riesz spaces, the ordered vector space Bb(A × B,C) of all order bounded bilinear mappings has the structure of a lattice space.

The purpose of this paper is to construct a random Ishikawa iterative sequence with errors for random strongly pseudo-contractive operator T in separable Banach spaces and to study that under suitable conditions this random iterative sequence with errors converges to a random fixed point of T. As applications, we utilize our results to study the existence problems of solutions for some kinds of nonlinear random operator equations in uniformly smooth separable Banach spaces.

We prove existence-uniqueness theorems for some kinds of nonlinear parabolic equations (cf. [2, 3, 15]) with singular initial data and non- Lipschitz’s nonlinearities in a framework of Colombeau’s algebras using different kinds of regularization for singularities appearing in the equations. We establish the convergence of a family of regularized solutions to the classical solutions (if they exist), when nonlinear term g(u) is of Lipschitz’s class and ε → 0. Moreover, we find solutions not available in classical approach.

A kind of spherical zonal translation network operator is constructed with the help of the de la Vallée poussin means of the spherical harmonic polynomial and the Riesz means of the Jacobi orthogonal polynomials, and, moreover, its degree of approximation in $L^p(S^q)$ spaces is deduced. The method presented in the present paper is actually a way of constructing spherical zonal translation network operators with spherical linear bounded operators.

Let $\Bbb{T}$ be a time scale such that $0,T\in \Bbb{T},$ $a_i\geq 0$ for $% i=1,\ldots ,m-2.$ Let $\xi _i$ satisfy $0\lt \xi _1\lt \xi _2\lt \ldots \lt \xi _{m-2}\lt \rho (T)$ and $\sum_{i=1}^{m-2}a_i\lt 1.$ We consider the following $p$% -Laplacian $m$-point boundary value problem on time scales \begin{eqnarray*} \left( \varphi _p\left( u^\Delta (t)\right) \right) ^\nabla +a(t)f(t,u(t)) &\hspace{-0.2cm}=\hspace{-0.2cm}&0,t\in (0,T), \\ u^\Delta (0) &\hspace{-0.2cm}=\hspace{-0.2cm}&0,\text{ }u(T)=\sum_{i=1}^{m-2}a_iu(\xi _i), \end{eqnarray*} where $a\in C_{ld}[(0,T),\Bbb{[}0,\infty )]$ and $f\in C\left( (0,T)\times \Bbb{[}0,\infty ),\Bbb{[}0,\infty )\right) .$ Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by applying Krasnosel'skii's fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett-Williams fixed point theorem. In particular, our criteria extend and improve some known results.

A ring $R$ is called strongly reversible, if whenever polynomials $f(x),$ $g(x)$ in $R[x]$ satisfy $f(x)g(x)=0$, then $g(x)f(x)=0$. It is proved that a ring $R$ is strongly reversible if and only if its polynomial ring $R[x]$ is strongly reversible if and only if its Laurent polynomial ring $R[x,x^{-1}]$ is strongly reversible. We also show that for a right Ore ring $R$ with $Q$ its classical right quotient ring, $R$ is strongly reversible if and only if $Q$ is strongly reversible.

An $(m_{1},\ldots,m_{r})$-cycle is the union of edge-disjoint $m_{i}$-cycles for $1\le i\le r$. An $(m_{1},\ldots,m_{r})$-cycle system of the complete graph $K_{v}$, $(\pmb{V},\pmb{C})$, is said to be $k$-cyclic if $\pmb{V}=Z_{v}$ and for $k\in Z_{v}$, $C+k\in \pmb{C}$ whenever $C\in \pmb{C}$. Let $m_{i}$ ($1\le i\le r$ ) be even integers ($\gt 2$) and let $\sum_{i=1}^{r}m_{i}=m=ks$ with $\gcd(k,s)=1$ and $k$ odd. Suppose $v$ is the least positive integer such that $v(v-1)\equiv0\pmod{2m}$ and $\gcd(v,m)=k.$ In this paper, it is proved that if there is a $k$-cyclic $(m_{1},\ldots,m_{r})$-cycle system of order $v$, then for any positive integer $p$, a $k$-cyclic $(m_{1},\ldots,m_{r})$ cycle system of order $2pm+v$ exists. As the main consequence of this paper, the necessary and sufficient conditions for the existence of a $k$-cyclic $(m_{1},\ldots,m_{r})$-cycle system of order $v$ with $m_{i}$ even and $\sum_{i=1}^{r}m_{i}\le20$ are given.

The purpose of this paper is to study the solvability for a class of generalized vector variational inequalities with pseudomonotonicity in reflexive Banach spaces. Utilizing the KKM-Fan lemma and the Nadler’s result, we derive the solvability results for this class of generalized vector variational inequalities with pseudomonotonicity. Utilizing these results, we also establish some existence theorems for zero points of pseudomonotone multifunctions including the characterization of the existence of zero points.

We prove that each nonzero algebra homomorphism π : C(X) −→ R is countably evaluated. This is applied to give a simple and direct proof (from the algebraic view) of the fact that each Lindel¨of space is realcompact.

New necessary and sufficient conditions for Banach spaces to have bounded approximation properties are established, which are easier to check than known ones. Also using these it is shown that for a Banach space $X$, the dual $X^{*}$ has the bounded approximation property if and only if $X$ has the bounded approximation property approximated by the weak adjoint operator topology, and if $X^{*}$ has the bounded weak approximation property, then $X$ has the bounded weak approximation property approximated by the weak adjoint operator topology.

Three new Bessel function distributions are introduced by taking products of a Bessel function pdf of the first kind and a Bessel function pdf of the second kind. Various particular cases and expressions for moments are derived for each distribution. The work is motivated by recent developments in the electrical and electronic engineering literature.

Assume we have a set of $k$ colors and to each vertex of a graph $G$ we assign an arbitrary subset of these colors. If we require that each vertex to which an empty set is assigned has in its neighborhood all $k$ colors, then this is called the $k$-rainbow dominating function of a graph $G$. The corresponding invariant $\gamma_{{\rm r}k}(G)$, which is the minimum sum of numbers of assigned colors over all vertices of $G$, is called the $k$-rainbow domination number of $G$. In this paper we connect this new concept to usual domination in (products of) graphs, and present its application to paired-domination of Cartesian products of graphs. Finally, we present a linear algorithm for determining a minimum $2$-rainbow dominating set of a tree.

Assume that $F$ is a nonlinear operator on a real Hilbert space $H$ which is strongly monotone and Lipschitzian with constants $\eta \gt 0$ and $\kappa \gt 0$, respectively on a nonempty closed convex subset $C$ of $H$. Assume also that $C$ is the intersection of the fixed point sets of a finite number of nonexpansive mappings on $H$. We develop an implicit hybrid steepest-descent method which generates an iterative sequence $\{u_n\}$ from an arbitrary initial point $u_0\in H$. We characterize the weak convergence of $\{u_n\}$ to the unique solution $u^*$ of the variational inequality: $$\langle F(u^*),v-u^*\rangle\geq0\quad\forall v\in C.$$ Applications to constrained generalized pseudoinverse are included.

We study polynomials over finite fields with a given value set. By constructing relations among the coefficients of Lagrange Interpolation Formula of these polynomials, we obtain its new kind of expression. Using this we find some characterizations for the set and the number of such polynomials.

From the viewpoint of space-time geometry and the trick of space-time track, the author would like to investigate the $\mathcal{L}$-geodesics, Perelman's reduced volume and Huisken's monotonicity formula.