A convex polygon is a polygon whose vertices are points on the integer lattice with interior angles all convex. Let $a(v)$ be the least possible area of a convex lattice polygon with $v$ vertices. It is known that $cv^{2.5}\leq a(v)\leq (15/784)v^3 + o(v^3)$. In this paper, we prove that $a(v)\geq (1/1152)v^3 + O(v^2)$.

In this paper, we study the nonoscillatory property of the second order nonlinear differential equation of advanced type: $$ (r(t)\phi(x(t))x'(t))' + p(t)f(x(g(t))) = 0. $$ We prove an existence theorem for nonoscillatory solutions and compare the oscillation of nonlinear equations.

An analysis of connectivity reliability in daisy chain computer networks led to a combinatorial problem that is most easily described in terms of necklaces. Basically n beads, some black and some white, are to be arranged into a necklace. The problem is to nd the arrangements that satisfy some constraints on the separations between black beads. Thus, no black beads may be allowed to be adjacent and the number of black beads with s beads between them may be specied. Hwang and Wright gave a matrix method of counting those necklaces. Here we use generating functions for similar counts, but with an added condition that all s separating beads must be white. To reduce the set of arrangements to more manageable size we also count necklaces when two that dier only by a rotation are considered the same.

We rst show the existence of a random innitesimal generator of a given random positive semigroup under some conditions. Then we represent it as a random second order integro-dierential operator.

We consider a semi-preinvex programming as follows: $$ (\mbox{P})~~\left\{\begin{array}{l} \inf~f(x)\\ \mbox{ subject to}~x\in K\subseteq X,~g(x)\in -D,\end{array}\right . $$ where $K$ is a semi-connected subset; $f:K\to (Y,C)$ and $g:K\to (Z,D)$ are semi-preinvex maps; while $(Y,C)$ and $(Z,D)$ are ordered vector spaces with order cones $C$ and $D$, respectively. If $f$ and $g$ are arc-directionally differentiable semi-preinvex maps with respect to a continuous map: $\gamma :[0,1]\to K\subseteq X$ with $\gamma (0)=0$ and $\gamma '(0^+)=u$, then the necessary and sufficient conditions for optimality of (P) is established. It is also established that a solution of an unconstrained semi-preinvex optimization problem is related to a solution of a semi-prevariational inequality .

Various ball separation properties in Banach spaces and their relationships with the extremal structures of the unit ball of dual space are presented. The relationships of these properties with the ball convexity, ball topology in Banach spaces and Banach spaces with the Radon- Nikodyn property are discussed.

Gram-Schmidt orthonormalization in Banach spaces is considered. Using this orthonormalization process we can prove that if $P$ is a projection on a reflexive Banach space $X$ with a basis $\{e_n;f_n\}$, then there exists a basis $\{u_n;g_n\}$ of $X$ such that $\{g_n\}\approx\{f_n\}$ and the matrix of $P$ with respect to $\{u_n;g_n\}$ has the property that all but a finite number of entries of each column and each row are zero.

Let $C$ be a nonempty subset (not necessarily closed and convex) of a Hilbert space, and $T: C \rightarrow C$ be a nonlinear mapping (not necessarily asymptotically nonexpansive). In this paper, we study the convergence of $(1/n)\sum^{n-1}_{i=0} T^i x(x\in C)$ as $n \rightarrow \infty $.

A classical Lotka-Volterra system of two rst-order nonlinear dierential equations modeling predator prey competition in population biology has been known to have an algebraic relation between two dependent variables for its periodic behavior in the phase plane since pioneering works by Lotka [12] on chemical reaction, Lotka [13] on parasitology, and Volterra [24] on shing activity in the upper Adriatic Sea. The techniques of Volterra [24], Hsu [10], Waldvogel [25, 26], Rothe [19], and Shih [22] in obtaining an integral representation of the period of Lotka-Volterra system are surveyed. These integrals are then shown to be equivalent.

This paper is concerned with a functional differential equation $x'(z) = x(az + bx(z))$, where $a \neq 1$ and $b \neq 0$. By constructing a convergent power series solution $y(z)$ of a companion equation of the form $\beta y'(\beta z) = y'(z) [y (\beta ^2z) - ay(\beta z) + a]$, analytic solutions of the form $(y (\beta y^{-1} (z)) - az) /b$ for the original differential equation are obtained.

We prove a uniform CLT for Markov chains for functions dominated by a function in a $L^{2}$-space. Using empirical process CLT's for stationary sequences satisfying a variety of mixing conditions one can obtain similar results. However, our conditions are less restrictive than those required for a mixing process application to these problems, and an example is given to these differences.

Solutions of the n-th order linear ordinary differential equations \begin{eqnarray*} \begin{array}{l} (z+b)^l \prod^{n-l}_{k=1}(z+a_k)\varphi_n+ \sum^n_{k=1}\varphi_{n-k} \{C^\lambda_{k}\{Q(z)\}_{k}+C^\lambda_{k-1}\{G(z)\}_{k-1}\}=f \\[] (z\neq -a_{k}\ (k=1,2,\ldots , n-l)\ z\neq -b;~a_i\neq a_j\neq b\ {\rm if}\ i\neq j;\ n\gt l,\ l\geq 2) \end{array} \end{eqnarray*} and the partial differential equations \begin{eqnarray*} \begin{array}{ll} (z+b)^l \prod^{n-l}_{k=1}(z+a_k)\cdot \frac{\partial^{n}\mu}{\partial z^{n}} & +\sum^{n-1}_{k=1} \frac{\partial^{n-k}\mu}{\partial z^{n-k}} \{C^\lambda_{k}\{Q(z)\}_{k}+C^\lambda_{k-1}\{G(z)\}_{k-1}\} \\[] &\hspace{-0.3cm} +\alpha\mu(z,t)=M \frac{\partial^2\mu}{\partial t^2}+N \frac{\partial\mu}{\partial t} \end{array} \end{eqnarray*} $$ (z\neq -a_{k}\ (k=1,2,\ldots , n-l)\ z\neq -b;\ a_i\neq a_j\neq b\ {\rm if}\ i\neq j;\ n\gt l,\ l\geq 2) $$ are discussed.

We introduce an innite dimension completely stationary ergodic Markov chain; investigate its properties; establish a sucient condition that the Markov property of a Markov chain is preserved by its induced multi-state sequence and nally derive the limiting distribution of the sum of the Markov chain. It is the simplest, nontrivial, innite dimension Markov chain which possesses all the nice properties - stationarity, irreducibility, positive recurrency, aperiodicity, : : :, etc.

This paper is concerned with two cosine-function-related functions which are called cosine step response and cosine cumulative output. We study some of their properties, such as measurability, continuity, innitesimal operator, compactness, positivity, almost periodicity, and asymptotic behavior.

Let $\Omega$ be a bounded open set in ${\Bbb R}^N$ and $1