Constant memory routing in quasi-planar and quasi-polyhedral graphs

We address the problem of online route discovery for a class of graphs that can be embedded either in two or in three-dimensional space. In two dimensions we propose the class of quasi-planar graphs and in three dimensions the class of quasi-polyhedral graphs. In the former case such graphs are geometrically embedded in R² and have an underlying backbone that is planar with convex faces; however within each face arbitrary edges (with arbitrary crossings) are allowed. In the latter case, these graphs are geometrically embedded in R³ and consist of a backbone of convex polyhedra and arbitrary edges within each polyhedron. In both cases we provide a routing algorithm that guarantees delivery. Our algorithms need only “remember” the source and destination nodes and one (respectively, two) reference nodes used to store information about the underlying face (respectively, polyhedron) currently being traversed. The existence of the backbone is used only in proofs of correctness of the routing algorithm; the particular choice is irrelevant and does not affect the behaviour of the algorithm.     

一类平面多项式微分系统赤道环量的代数递推公式

本文研究一类形式相当一般的平面多项式系统赤道环量(Gauss球面的无穷远点奇点量)的计算,建立了系统赤道环量计算的简明的线性代数递推公式.应用递推公式计算赤道环量,只需用系统系数做四则运算,避免了通常计算赤道环量需要的复杂的积分运算和解方程,极易用计算机代数系统作符号推导并且不含舍入误差.     

用户可编程门阵列

第一代的可编程逻辑器件(PLD)以1片器件代替5至10片TTL器件,最新一代可编程器件FPGA可以代替40至400片小规模集成电路。本文介绍四种可编程门阵列的基本结构和主要性能,并给出采用该类器件设计电路的编程技术。     

Smaller planar triangle-free graphs that are not 3-list-colorable

In 1995, Voigt constructed a planar triangle-free graph that is not 3-list-colorable. It has 166 vertices. Gutner then constructed such a graph with 164 vertices. We present two more graphs with these properties. The first graph has 97 vertices and a failing list assignment using triples from a set of six colors, while the second has 109 vertices and a failing list assignment using triples from a set of five colors.     

Exploiting planarity in separation routines for the symmetric traveling salesman problem

At present, the most successful approach for solving large-scale instances of the Symmetric Traveling Salesman Problem to optimality is branch-and-cut. The success of branch-and-cut is due in large part to the availability of effective separation procedures; that is, routines for identifying violated linear constraints.

For two particular classes of constraints, known as comb and domino-parity constraints, it has been shown that separation becomes easier when the underlying graph is planar. We continue this line of research by showing how to exploit planarity in the separation of three other classes of constraints: subtour elimination, 2-matching and simple domino-parity constraints.     

Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x₀ of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x₀ as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.     

Effects of Ar and O2 additives on photopatternable sol-gel etching in an SF6-based plasma for planar lightwave circuit fabrication

We present a novel study of the interaction of SF₆-based plasmas with sol-gel materials in a parallel plate reactive ion etching (RIE) system. The purpose of these experiments was to obtain quantitative measures and optimisation of the RIE parameters, which can be used in the microfabrication of planar lightwave circuit (PLC) devices. The sulfur hexafluoride chemistry is chosen due to its excellent etching properties of SiO₂, which is one of the components of the photopatternable sol-gel materials and is not present in typical photoresist materials. Fast process etching rate and good selectivity is achieved by varying SF₆ flow and power delivered to the electrodes. The study also reveals a marginal influence of oxygen and argon flow on the character of the sol-gel etching. The experimental data obtained can be used as a reference for any sol-gel devices fabricated using widely available RIE reactors.     

Random graphs on surfaces

Counting labelled planar graphs, and typical properties of random labelled planar graphs, have received much attention recently. We start the process here of extending these investigations to graphs embeddable on any fixed surface S. In particular we show that the labelled graphs embeddable on S have the same growth constant as for planar graphs, and the same holds for unlabelled graphs. Also, if we pick a graph uniformly at random from the graphs embeddable on S which have vertex set {1,…,n}, then with probability tending to 1 as n→∞, this random graph either is connected or consists of one giant component together with a few nodes in small planar components.     

Dendriform equations

We investigate solutions for a particular class of linear equations in dendriform algebras. Motivations as well as several applications are provided. The latter follow naturally from the intimate link between dendriform algebras and Rota–Baxter operators, e.g. the Riemann integral map or Jackson's q-integral.     

Determination of planar chlorobiphenyls in animal fat

The occurrence of the highly toxic non-ortho substituted (planar) chlorobiphenyls (CBs) at low levels in wildlife, (human) fat, and fish generates the need for a routine analytical procedure. The use of a fully automated system for the separation of planar and non-planar CBs, consisting of a gel permeation chromatography system, a sample preparation system employing extraction columns, and a porous graphitic carbon HPLC column is described, and tested with horse fat. Gas chromatography on capillary columns, with electron-capture or high resolution mass spectrometric detection, is used for quantitation of the CBs.