In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schr¨odinger type equations. Different formulations are used to study these singular limits, e.g., the kinetic formulation of the linear Schr¨odinger equation based on the Wigner transform is well suited for global-in-time analysis without using WKB-(expansion) techniques, while the modified Madelung transformation reformulating Schr¨odinger equations in terms of a dispersive perturbation of a quasilinear symmetric hyperbolic system usually only gives local-in-time results due to the hyperbolic nature of the limit equations. Deterministic analogues of turbulence are also discussed. There, turbulent diffusion appears naturally in the zero dispersion limit.

This is an expository article concerning complex analysis, in particular, several complex variables. Several subjects are discussed here to demonstrate the development and the diversity of several complex variables. Hopefully, the brief introduction to complex analysis in several variables would motivate the reader’s interests to this subject.

Many interesting families of rapidly convergent series representations for the Riemann Zeta function $\zeta (2n+1)$ $(n\in {\Bbb N})$ were considered recently by various authors. In this survey-cum-expository paper, the author presents a systematic (and historical) investigation of these series representations. Relevant connections of the results presented here with several other known series representations for $\zeta (2n+1)$ $(n\in {\Bbb N})$ are also pointed out. In one of many computationally useful special cases presented here, it is observed that $\zeta (3)$ can be represented by means of a series which converges much faster than that in Euler's celebrated formula as well as the series used recently by Ap\'{e}ry in his proof of the irrationality of $\zeta (3)$. Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of this series are capable of producing an accuracy of seven decimal places.

A renewed algorithm is presented to calculate the mixed volume of the support ${\cal A}=({\cal A}_1,\dots,{\cal A}_n)$ of a polynomial system $P({\bf x})=(p_1({\bf x}),\dots, p_n({\bf x}))$ in $\Bbb C^n$. The key ingredient is a specially tailored application of LP feasibility tests, which allows us to calculate the mixed cells, their volumes constituting the mixed volume, in a mixed subdivision of ${\cal A}$ more efficiently. The problem of finding mixed cells plays a crucial role in polyhedral homotopy methods for finding all isolated zeros of $P({\bf x})$. Our new algorithm advances the speed of mixed volume computation by a considerable margin, illustrated by numerical examples.

We study the third-order ordinary differential equations (ODEs) which are relevant to the steady state of the Kuramoto-Sivashinsky equation, and/or to a model of dendritic growth of needle crystals. We show that there is no monotonic solution for certain range of parameter.

Two kinds of Palais-Smale condition, $(PS)_c$ and $(PS)^*_c$, for nondifferentiable functionals are studied. It is shown that $(PS)_c$ implies $(PS)^*_c$ and that they are equivalent for convex functionals. This points out a gap in the proof of Costa and Goncalves [5, Proposition 3]. Some other nonsmooth versions of known smooth results are also obtained.

An inductive limit $(E,t)=\text{ind}\,(E_n,t_n)$ is said to have property (P) if every closed absolutely convex neighborhood in $(E_n,t_n)$ is closed in $(E_{n+1},t_{n+1})$. This property was introduced and investigated by J. Kucera. In this paper we give some equivalent descriptions of property (P) and prove that property (P) implies ultraregularity. Particularly, if all $(E_n,t_n)$ are metrizable locally convex spaces, we have: $(E,t)$ is ultraregular if and only if $(E,t)$ is a strict inductive limit and for each $n\in {\Bbb N}$, there is $m=m(n)\in {\Bbb N}$ such that $\overline E_n^{E}\subset E_m;$ $(E,t)$ has property (P) if and only if $(E,t)$ is a strict inductive limit and each $E_n$ is closed in $(E_{n+1},t_{n+1}).$

This paper proves that for any integer $n\geq 4$ and any rational number $r$, $2\leq r\leq n-2$, there exists a graph $G$ which has circular chromatic number $r$ and which does not contain $K_n$ as a minor.

Hwang and Xu defined equivalent double-loop networks and gave one such result showing that the L-shapes of the two equivalent networks are recombinations of three rectangles. Recently, R¨odseth gave an elegant algebraic theorem for equivalent multi-loop networks. We show that its double-loop version yields equivalent networks of the 3- rectangle version. We also show that other seemingly different geometric recombinations also all turn out to be special cases of the 3-rectangle version.

Adaptive group testing in the presence of a large percentage of defectives is best done by individual testing rather than by pooling. The fraction of items which must be defective to make individual testing optimal remains unknown, and is conjectured to be 1/3. In this paper it is shown that when the number of items is sufficiently large, and the fraction of defective items is at least $1/\log_{3/2}3$, individual testing is optimal.

The radio channel assignment problem can be cast as a graph coloring problem. Vertices correspond to transmitter locations and their labels (colors) to radio channels. The assignment of frequencies to each transmitter (vertex) must avoid interference which depends on the seperation each pair of vertices has. Two levels of interference are assumed in the problem we are concerned. Based on this channel assignment problem, we proposed a graph labelling problem which has two constraints instead of one. We consider the question of finding the minimum edge of this labelling. Several classes of graphs including one that is important to a telecommunication problem have been studied.

In this paper, we study the problem of determining the largest number of maximum independent sets of a graph of order n. Solutions to this problem are given for various classes of graphs, including general graphs, trees, forests, (connected) graphs with at most one cycle, connected graphs and triangle-free graphs. Extremal graphs achieving the maximum values are also given.