We investigate Lorentzian minimal surfaces in Lorentzian complex space forms. First, we prove that for such surfaces the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Next, we classify Lorentzian minimal surfaces in the Lorentzian complex plane ${\bf C}^2_1$. Finally, we classify minimal slant surfaces in the Lorentzian complex projective plane $CP^2_1(4)$ and in the Lorentzian complex hyperbolic plane $CH^2_1(-4)$. In particular, our latter results show that if a minimal slant surface in $CP^2_1(4)$ or in $CH^2_1(-4)$ contains no open subset of constant curvature, then it is of Klein-Gordon type which arises from the solutions of certain nonlinear Klein-Gordon equations.

In this paper we deal with special classes of second order elliptic differential operators on the real line. We show that these operators are generators of positive $C_{0}$-semigroups on weighted spaces of continuous functions and we represent them as limits of iterates of integral-type operators. By means of such representation, some qualitative properties of the semigroups are stated.

The purpose of this paper is to present new sufficient conditions for both the metric regularity and the Lipschitzian stability of implicit multifunctions in Asplund spaces. The basic tools of our analysis involve the Fr´echet normal coderivative and the Mordukhovich normal coderivative of set-valued mappings, the basic subgradient estimate for marginal functions and the Ekeland variational principle. Applications to the pointbased characterizations for the metric regularity and the Lipschitzian stability of solution mapping of parametric generalized equations are given.

Let $G$ be a finite group. Based on the Gruenberg-Kegel graph ${\rm GK}(G)$, the order of $G$ can be divided into a product of coprime positive integers. These integers are called the order components of $G$ and the set of order components is denoted by ${\rm OC}(G)$. In this article we prove that, if $S$ is a non-Abelian finite simple group with a disconnected graph ${\rm GK}(S)$, with an exception of $U_4(2)$ and $U_5(2)$, and $G$ is a finite group with ${\rm OC}(G)={\rm OC}(S)$, then $G$ is neither Frobenius nor $2$-Frobenius. For a group $S$ isomorphic to $U_4(2)$ or $U_5(2)$, we construct examples of $2$-Frobenius groups $G$ such that ${\rm OC}(S)={\rm OC}(G)$. In particular, the simple groups $U_4(2)$ and $U_5(2)$ are not recognizable by their order components.

The authors first give a detailed proof on the coincidence between atomic Hardy spaces of Coifman and Weiss on a space of homogeneous type with those Hardy spaces on the same underlying space with the original distance replaced by the measure distance. Then the authors present some general criteria which guarantee the boundedness of considered linear operators from a Hardy space to some Lebesgue space or Hardy space, provided that it maps all atoms into uniformly bounded elements of that Lebesgue space or Hardy space. Third, the authors obtain the boundedness in Hardy spaces of singular integrals with kernels only having weak regularity by characterizing these Hardy spaces with a new kind of molecules, which is deeply related to the kernels of considered singular integrals. Finally, as an application, the authors obtain the boundedness in Hardy spaces of Monge-Ampère singular integral operators.

Let $A$ be the generator of a nondegenerate $\alpha$-times integrated $C$-semigroup $T(\cdot)$ on a complex Banach space $X$ for some $\alpha\geq 0$, $x\in X$ and $f\in L_{loc}^1([0,\infty),X)\cap C((0,\infty),X)$. We first show that the abstract Cauchy problem $ACP (A,C f,C x)$: $u'(t)=A u(t)+C f(t)$ for $t\gt 0$ and $u(0)=C x$ has a strong solution is equivalent to the function $v(\cdot)=T(\cdot)x+T *f(\cdot)\in C^\alpha ([0,\infty),X)$ and $D^\alpha v(\cdot)\in C^1((0,\infty),X)$, and then use it to prove some new existence and approximation theorems concerning strong solutions of $ACP(A,C y+j_{\alpha-1} *C g,C x)$ in $C^1([0,\infty),X)$ and mild solutions of $ACP(A,C y+j_{\alpha-2} *C g,C x)$ (for $\alpha\geq 1$) in $C([0,\infty),X)$ when vectors $x$ and $y$ both satisfy some suitable regularity assumptions and $T(\cdot)$ is locally Lipschitz continuous.

Using the modulus of smoothness $\omega_{\varphi^{\lambda}}^{2}(f,t)_{\omega}$, direct theorem with Jacobi weights of Baskakov operators is established in this paper; In addition, a weak-type inverse theorem of Baskakov operators is obtained in the weighted norm.

Let $q\gt 8$ be an odd integer and $p$ a prime with $p$ a prime with $p\lt q$ and $p\nmid q$. The main purpose of this paper is to study the mean value properties of the character sums over interval $\left[1, \frac{q}{8}\right)$ by using the mean value theorems of the Dirichlet L-functions,and give some interesting mean value formulae.

The 2D $g$-Navier-Stokes equations have the following form, $$ \frac{\partial \mathbf u}{\partial t}-\nu\Delta { \mathbf u} + ( \mathbf u \cdot\nabla)\mathbf u +\nabla p = {\bf f}, \ \ \mbox{in} \ \Omega $$ with the continuity equation $$ \nabla\cdot (g {\mathbf u})= 0, \ \ \mbox{in} \ \Omega, $$ where $g$ is a smooth real valued function. We get the Navier-Stokes equations, for $g$ = $1$. In this paper, we investigate solutions $\{\mathbf u_g, p_g\}$ of the $g$-Navier-Stokes equations, as $g \to 1$ in some suitable spaces.

This paper first introduces a new graph parameter. Let $t$ be a positive integer. A $t$-star-matching of a graph $G$ is a collection of mutually vertex disjoint subgraphs $K_{1,i}$ of $G$ with $1 \leq i \leq t$. The $t$-star-matching number, denoted by $SM_t(G)$, is the maximum number of vertices covered by a $t$-star-matching of $G$. Clearly $SM_1(G)/2$ is the edge independence number of $G$.

An $L(2,1)$-labelling of a graph $G$ is an assignment of nonnegative integers to the vertices of $G$ such that vertices at distance at most two get different numbers and adjacent vertices get numbers which are at least two apart. The $L(2,1)$-labelling number of a graph $G$ is the minimum range of labels over all $L(2,1)$-labellings. If we require the assignment to be one-to-one, then similarly as above we can define the $L'(2,1)$-labelling and the $L'(2,1)$-labelling number of a graph $G$. Given a graph $G$, the path covering number of $G$, denoted by $p_v(G)$, is the smallest number of vertex-disjoint paths covering $V(G)$. By $G^c$ we denote the complement graph of $G$.

In this paper, we design a polynomial time algorithm to compute $SM_t(G)$ for any graph $G$ and any integer $t\geq 2$ and studies the properties of $t$-star-matchings of a graph $G$. For any graph $G$, we determine the path covering numbers of $(\mu(G))^c$ and $(G\times \hat{K}_2)^c$ in terms of $SM_4(G^c)$, and the $L'(2,1)$-labelling umbers of $\mu(G)$ and $G\times \hat{K}_2$ in terms of $SM_4(G^c)$, where $\mu(G)$ is the Mycielskian of $G$ and $G\times \hat{K}_2$ is the direct product of $G$ and $\hat{K}_2$ ($\hat{K}_2$ is a graph obtained from $K_2$ by adding a loop on one of its vertices). Our results imply that the path covering numbers of $(\mu(G))^c$ and $(G\times \hat{K}_2)^c$, the $L'(2,1)$-labelling umbers of $\mu(G)$ and $G\times \hat{K}_2$ can be computed in polynomial time for any graph $G$. So, for any graph $G$, it is polynomial-time solvable to determine whether $(\mu(G))^c$ and $(G\times \hat{K}_2)^c$ has a Hamiltonian path. And consequently, for any graph $G=(V,E)$, it is polynomially solvable to determine whether $\lambda(\mu(G))\leq s$ for each $s\geq |V(\mu(G))|$ and $\lambda(G\times \hat{K}_2)\leq s$ for each $s\geq |V(G\times \hat{K}_2)|$. Using these results, we easily determine $L(2,1)$-labelling numbers and $L'(2,1)$-labelling numbers of several classes of graphs..

In this paper, we introduce a modified extragradient method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone and Lipschitz continuous mapping. Our modified extragradient method is a variant of the so-called extragradient method. We obtain a weak convergent theorem for two sequences generated by this modified extragradient method.

In this paper, we generalize Ky Fan’s minimax inequality to vectorvalued function with values in a topological vector space acting on the product of two other topological vector spaces which are connected by another function. In these results, the concavity or convexity on a function is transferred to another function. And a sufficient condition for the existence of solution for a variational inclusion is given.

The aim of this paper is to define $Y(h,k)$ sum which is related to the Hardy's sums $s_{5}(h,k)$. On the semi-group $G$, matrix operation of this sum is defined. Substituting mediants of Farey fractions into the matrix operation, $Y(h,k)$ sum is generalized. By using contour integration, the reciprocity theorem of the $Y(h,k)$ sum is proved. Moreover, by using $% L(1,\chi )$ function and Gauss sums, generalized character analogs of the Hardy sums are found.

Let $\bf X=\{X_k\}$ be an IID random sequence and $\bf Y=\{Y_k\}$ be an independent random sequence also independent of $\bf X$. Denote by $\mu_{\bf X}$ and $\mu_{\bf X+\bf Y}$ the probability measures on the sequence space induced by $\bf X$ and $\bf X+\bf Y =\{X_k+Y_k\}$, respectively. The problem is to characterize $\mu_{\bf X+\bf Y}\sim\mu_{\bf X}$ in terms of $\mu_{\bf Y}$ in the case where $\bf X$ is non-negative. Sato and Tamashiro [6] first discussed this problem assuming the existence of ${f_{\bf X}(x)=\frac{d\mu_{X_1}}{dx}(x)}$. They gave several necessary or sufficient conditions for $\mu_{\bf X+\bf Y}\sim\mu_{\bf X}$ under some additional assumptions on $f_{\bf X}$ or on $\bf Y$. The authors precisely improve these results. First they rationalize the assumption of the existence of $f_{\bf X}$. Then they prove that the condition (C.6) is necessary for wider classes of $f_\bf X$ with local regularities. They also prove if the $p$-integral $I_p^0(\bf X)\lt \infty$ and $\bf Y\in \ell_p^+$ a.s., then (C.6) is necessary and sufficient. Furthermore, in the typical case where $\bf X$ is exponentially distributed, they prove an explicit necessary and sufficient condition for $\mu_{\bf X+\bf Y}\sim \mu_\bf X$.

In this paper, some new explicit bounds on solutions to a class of new nonlinear retarded integral inequalities are established, which can be used as effective tools in the study of certain nonlinear retarded integral equations. Applications examples are also indicated.

Let $\Omega$ is a bounded domain in $R^N$ with a smooth boundary $\partial\Omega$, $m,n\gt 1$ and $p,q,r,s,a,b$ are positive constants. For the initial and boundary value problem $$ \begin{array}{rl} u_t =\bigtriangleup u^m+v^p-au^r,& \quad x\in\Omega,\quad t\gt 0,\\ v_t =\bigtriangleup v^n+u^q-bv^s, & \quad x\in\Omega,\quad t\gt 0,\\ u=v=0, & \quad x\in\partial\Omega,\quad t\gt 0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),& \quad x\in\Omega, \end{array} $$ we prove that all solutions are globally bounded if $pq \lt \max\{n, r\} \max\{n, s\}$; while there are finite time blowing up solutions if $pq \lt \max\{n, r\} \max\{n, s\}$ and the initial data are sufficiently large. For the critical case $pq=\max\{m,r\}$ $\max\{n,s\}$, the existence or nonexistence of global solutions depends on the relation between the exponents $m$, $n$, $r$, $s$, and also the range of the parameters $a$, $b$.

We study the rational surfaces of revolution in Minkowski 3-space and characterize them with pointwise 1-type Gauss map. In this article, we give a complete classification of rational surfaces of revolution in Minkowski 3-space with pointwise 1-type Gauss map and provide new examples of cones in Minkowski 3-space.

In this paper we study the stability and asymptotic behavior of solutions of a nonlinear iterated Volterra-Fredholm integral equation by using the well known Krasnoselskii’s fixed point theorem.

By making use of the method of differential subordination, we investigate some interesting properties of certain analytic functions associated with the Dziok-Srivastava linear operator.

In this paper, a weighted and exponential generalization of Rado’s inequality is established. As applications, the result is used to obtain a refinement of weighted power means inequality.