We study the topology of moduli spaces of polygons with xed side lengths in the Euclidean plane. We establish a duality between the spaces of marked Euclidean polygons with xed side lengths and marked convex Euclidean polygons with prescribed angles. 1. We consider the space P n of all polygons with n distinguished vertices in the Euclidean plane E 2 whose sides have nonnegative length allowing...

The complete multipartite graph Kn(2t) having n partite sets of size 2t, with n ≥ 6 and t ≥ 1, is shown to have a decomposition into gregarious 6-cycles, that is, the cycles which have at most one vertex from any particular partite set. Complete sets of differences of numbers in Zn are used to produce starter cycles and obtain other cycles by rotating the cycles around the n-gon of the partite ...

Define a graph GT (n) with one node for each triangulation of a convex n-gon. Place an edge between each pair of nodes that differ by a single flip: two triangles forming a quadrilateral are exchanged for the other pair of triangles forming the same quadrilateral. In this paper we introduce a tree of all triangulations of polygons with any number of vertices which gives a unified framework in w...

We study the space Nm d of clouds in Rd (ordered sets of m points modulo the action of the group of affine isometries). We show that Nm d is a smooth space, stratified over a certain hyperplane arrangement in Rm. We give an algorithm to list all the chambers and other strata (this is independent of d). With the help of a computer, we obtain the list of all the chambers for m ≤ 9 and all the str...

| In this paper, we study various algorithmic questions regarding the computation of an optimal three n-ger planar grasp. We present a novel O(n 2 logn)-time algorithm to compute such an optimal grasp for an arbitrary simple n-gon. This algorithm can be used for nding \good" immobilizing sets. We also discuss several variations on the problem and many intriguing open questions in the area that ...

What is the minimum perimeter of a convex lattice n-gon? This question was answered by Jarník in 1926. We solve the same question, and prove a limit shape result, in the case when perimeter is measured by a (not necessarily symmetric) norm.

By a triangulation of the ∞-gon, we mean a maximal set of non-intersecting arcs connecting non-neighbouring integers: We adopt the philosophy that the integers can be viewed as the vertices of the ∞-gon, and that the arcs can be viewed as diagonals. There are two obvious ways to achieve such maximal sets; they are shown in the following two sketches where the arcs must be continued ad infinitum...

PURPOSE To examine the relationship between intraocular pressure (IOP), anthropomorphic, demographic, socioeconomic, systemic, and ocular factors and glaucomatous optic neuropathy (GON) in Chinese people. METHODS Chinese people (n = 2000), aged 40 to 79 years, were selected from the Singapore electoral register. Of the 1717 considered eligible for examination, 1232 participated, representing ...

Recently, the importance of biosimulation softwares in systems biology has been emphasized and received considerable attentions. Since the information conventionally created with most of biosimulation softwares does not have a common format for modeling, it has been very difficult to exchange pathway models among them. At present, the Systems Biology group of ERATO Kitano Symbiosis System Proje...

Synapses are surrounded by a layer of extracellular matrix (ECM), which is instrumental for their development and maintenance. ECM composition is dynamically controlled by proteases, but how the precise composition of the ECM affects synaptic morphology is largely unknown. Through an unbiased forward genetic screen, we found that Caenorhabditis elegans gon-1, a conserved extracellular ADAMTS pr...