Quasiconcavity, the condition that the level sets of a positive graph are convex, is known to hold for solutions of certain semilinear equations. We survey some techniques that can be used to show quasiconcavity for solutions of quasilinear elliptic equations with form similar to the equation of constant mean curvature.

The paper gives counterexamples in abstract ergodic theory of an equicontinuous semigroup $\mathcal{S}$ of linear operators on a locally convex space $X$. In particular, it is shown that the orbit of an element $x\in X$ may contain a unique fixed point of $\cal{S}$ without $x$ being necessarily ergodic.

In this paper, we first show that for any Banach space $X$, $\ell_1[X]\subseteq\ell_\infty\langle X\rangle$ by Khinchin's inequality. Then by the relationship between Banach-valued sequence spaces and tensor products, we show that for any Banach space $X$, $\ell_1\stackrel{\vee}{\otimes}X\subseteq c_0\stackrel{\wedge}{\otimes}X$.

We consider the following quasilinear elliptic problems in a bounded smooth domain $Z$ of ${\Bbb R}^N$, $N\ge 3$: $$\left\{\begin{array}{lr} Lu= \sum_{i,j=1}^{N}a_{ij}(x,u) \frac{\partial^2u} {\partial x_i\partial x_j}+ \sum_{i=1}^{N}b_{i}(x,u) \frac{\partial u} {\partial x_i}+c(x,u)u=f(x) &\mbox{in } Z, \\ u=0 &\mbox{on } \partial Z,\end{array}\right . $$ where $f(x)\in L^{p}(Z)$ and all the coefficients $a_{ij}, b_{i}, c$ are Carath\'{e}dory functions. Suppose that $a_{ij}\in C^{0,1}(\bar{Z}\times \Bbb R)$, $a_{ij}$, $\partial a_{ij}/\partial x_i$, $\partial a_{ij}/\partial r$, $b_{i}$, $c\in L^{\infty}(Z\times \Bbb R)$, $c\le 0$ for $i,j=1,...N$ and the oscillations of $a_{ij}=a_{ij}(x,r)$ with respect to $r$ are sufficiently small. A global estimate for a solution $u\in W^{2,p}(Z)\cap W^{1,p}_{0}(Z)$ is established and the existence of a strong solution $u\in W^{2,p}(Z)\cap W^{1,p}_{0}(Z)$ is proved for $p\gt N$. Furthermore, we replace $f(x)$ by $f(x,r,\xi)$ which is defined on $Z\times {\Bbb R}\times {\Bbb R}^N$ and is a $Carath\acute{e}dory$ function. Assume that $$ |f(x,r,\xi)|\le C_{0}+h(|r|)|\xi|^{\theta},\qquad 0\le\theta \lt 2,$$ where $C_0$ is a nonnegative constant, $h(|r|)$ is a locally bounded function, and $-c\ge \alpha_{0}\gt 0$ for some constant $\alpha_{0}$. We prove the existence of solution $u\in W^{2,p}(Z)\cap W^{1,p}_{0}(Z)$ for the equation $Lu=f(x,u,\nabla u)$.

Two nonlinear oscillators of relaxation type are studied for representing periodic orbits in terms of two inverse functions of x exp(x). The limit of the limit cycle of singularly perturbed van der Pol differential equation is approximated analytically; while the periodic orbit of singularly perturbed Lotka-Volterra system is represented in exact manner. These results are in an excellent agreement with numerical results computed via a stiff/nonstiff Maple ODE solver “NODES package” authored by Lawrence F. Shampine and Robert M. Corless. Some remarks are provided for the relaxation period.

[3] proves that if $T$ is a strong $c$-partite tournament $(c\geq 3)$, then there is a $(k-3)$-reducible $k$-cycle in $T$, for all $k=3,4,\cdots, c$. In this paper we investigate the smallest number of $(k-3)$-reducible $k$-cycles in strong $c$-partite tournaments for $3\leq k\leq c$ and give some related problems.

In the classical Hahn-Banach-Kantorovich theorem, the range space $Y$ is Dedekind complete. In this paper, by extending the arguments of the original Hahn-Banach-Kantorovich theorem and using an idea of Y. A. Abramovich and A. W. Wickstead, we can weaken the order theoretic assumption on $Y$\ and obtain more general results in the settings of Banach lattices as well as ordered linear spaces.

Let $R$ be a commutative artinian ring, and $f(x)\in R[x]$ be a non-constant monic polynomial. The main purpose of this paper is to determine the socle series of $R[x]/\langle f(x)\rangle$ in terms of the socle series of $R$. As an application of the results proved, it is proved that $R$ is a $QF$-ring if and only if $R[x]/\langle f(x)\rangle$ is a $QF$-ring. As another application , a necessary and sufficient condition for a local artinian ring $R$ having a semisimple ideal $B$, with $R/B$ a $PIR$, to be a split extension of a $PIR$ by a semisimple module, is given.

Consider $n$ disjoint triangles and a cycle on the 3$n$ vertices of the $n$ triangles. In 1986, Du, Hsu, and Hwang conjectured that the union of the $n$ triangles and the cycle has independent number $n$. Soon later, Paul Erd\"os improved it to a stronger version that every cycle-plus-triangle graph is 3-colorable. This conjecture was proved by H. Fleischner and M. Stiebitz. In this note, we want to give an extension of the above conjecture with an application in switching networks.

We prove the variational principle for dimensions for Poincar´e recurrences, in the case of invariant sets of dynamical systems with continuous time. To achieve this goal we show that these dimensions can be expressed as roots of a non–homogeneous Bowen equation.