Moduli Spaces of Curves with Quasi-Symmetric Weierstrass GAp Seqnences

In this paper we give an explicit construction of the moduli space of the pointed complete Gorenstein curves of arithmetic genus g with a given quasi-symmetric Weierstrass semigroup, that is, a Weierstrass semigroup whose last gap is equal to 2g – 2. We identify such a curve with its image under the canonical embedding in the (g – 1)-dimensional projective space. By normalizing the coefficients of the quadratic relations and by constructing Gröbner bases of the canonical ideal, we obtain the equations of the moduli space in terms of Buchberger's criterion. Moreover, by analyzing syzygies of the canonical ideal we establish criteria that make the computations less expensive.     

Universal Hashing and Geometric Codes

We describe a new application of algebraic coding theory to universal hashing and authentication without secrecy. This permits to make use of the hitherto sharpest weapon of coding theory, the construction of codes from algebraic curves. We show in particular how codes derived from Artin-Schreier curves, Hermitian curves and Suzuki curves yield classes of universal hash functions which are substantially better than those known before.     

Some Elementary Ramanujan Graphs

We give elementary constructions of two infinite families of Ramanujan graphs of unbounded degree. The first uses the geometry of buildings over finite fields, and the second uses triangulations of modular curves.Mathematics Subject Classiffications (2000). Primary: 05C25; secondary: 05C50, 51E24     

The fundamental group of the complement of the branch curve of ℂℙ1 × T

Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.     

Remarks on lines and minimal rational curves

We determine all of lines in the moduli space M of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section. This work was supported by the Competitive Earmarked Research Grant (Grant No. HKU7025/03P) of the Research Grant Council, Hong Kong     

Local heat transfer from a hot plate to a water jet

Jet impingement boiling is very efficient in cooling of hot surfaces as a part of the impinging liquid evaporates. Because of its importance to many cooling procedures, investigations on basic mechanisms of jet impingement boiling heat transfer are needed. Until now, most of the experimental studies, carried out under steady-state conditions, used a heat flux controlled system and were limited by the critical heat flux (CHF). The present study focuses on steady-state experiments along the entire boiling curve for hot plate temperatures of up to 700°C. A test section has been built up simulating a hot plate. It is divided into 8 independently heated modules of 10 mm length to enable local heat transfer measurements. By means of temperature controlled systems for each module local steady-state experiments in the whole range between single phase heat transfer and film boiling are possible. By solving the two dimensional inverse heat conduction problem, the local heat flux and the corresponding wall temperature on the surface of each module can be computed. The measurements show important differences between boiling curves measured at the stagnation line and those obtained in the parallel flow region. At the stagnation line, the transition boiling regime is characterised by very high heat fluxes, extended to large wall superheats. Inversely, boiling curves in the parallel flow region are very near to classical ones obtained for forced convection boiling. The analysis of temperature fluctuations measured at a depth of 0.8 mm from the boiling surface enables some conclusions on the boiling mechanism in the different boiling regimes.     

Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time

The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (“trees”). We describe a polynomial-time algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.     

A Combinatorial Algorithm Related to the Geometry of the Moduli Space of Pointed Curves

As pointed out in Arbarello and Cornalba (J. Alg. Geom. 5 (1996), 705–749), a theorem due to Di Francesco, Itzykson, and Zuber (see Di Francesco, Itzykson, and Zuber, Commun. Math. Phys. 151 (1993), 193–219) should yield new relations among cohomology classes of the moduli space of pointed curves. The coefficients appearing in these new relations can be determined by the algorithm we introduce in this paper.     

Using Conventional Edge Detectors and Postsmoothing for Segmentation of Spotted Microarray Images

Segmentation of spotted microarray images is important in generating gene expression data. It aims to distinguish foreground pixels from background pixels for a given spot of a microarray image. Edge detection in the image processing literature is a closely related research area, because spot boundary curves separating foregrounds from backgrounds in a microarray image can be treated as edges. However, for generating gene expression data, segmentation methods for handling spotted microarray images are required to classify each pixel as either a foreground or a background pixel; most conventional edge detectors in the image processing literature do not have this classification property, because their detected edge pixels are often scattered in the whole design space and consequently the foreground or background pixels are not defined. In this article, we propose a general postsmoothing procedure for estimating spot boundary curves from the detected edge pixels of conventional edge detectors, such that these conventional edge detectors together with the proposed postsmoothing procedure can be used for segmentation of spotted microarray images. Numerical studies show that this proposal works well in applications.

Datasets and computer code are available in the online supplements.