Consider lattice paths in Z taking unit steps north (N) and east (E). Fix positive integers r, s and put an equivalence relation on points of Z by letting v, w be equivalent if v − w = `(r, s) for some ` ∈ Z. Call a lattice path valid if whenever it enters a point v with an E-step, then any further points of the path in the equivalence class of v are also entered with an E-step. Loehr and Warri...

The two-point cluster function C2(rl, r2) provides a measure of clustering in continuum models of disordered many-particle systems and thus is a useful signature of the microstructure. For a two-phase disordered medium, C2(rt, r2) is defined to be the probability of finding two points at positions r t and r_, in the same cluster o f one o f the phases. An exact ana/ytical expression is found fo...

I n this paper, we present a new snake-based method to delineate contours where the m e r has to specify only the two distant end points without having to supply an almost complete polygonal approximation. W e achieve much better convergence properties than those of traditional snakes by propagating the image information along the curve from both end points towards its center. W e use our new m...

Let G = (V,E) be a weighted undirected graph with vertex set V , edge set E and a weight function d. Thus, d(u, v) denotes the weight of any edge (u, v) ∈ E. A matching M ⊆ E is a collection of edges such that every node in V is incident to at most one edge in M . The matching is perfect if every node in V is incident to exactly one edge in M . The cost of the matching is given by ∑ (u,v)∈M d(u...

We construct oriented matroids of rank 3 on 13 points whose realization spaces are disconnected. They are defined on smaller points than the known examples with this property. Moreover, we construct the one on 13 points whose realization space is a connected and non-irreducible semialgebraic variety. 1 Oriented Matroids and Matrices Throughout this section, we fix positive integers r and n. Let...

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...

We improve a number of existing algorithms for determining the location of one or more undesirable facilities amidst a set P of n demand points, under various constraints and distance functions. We assume that the demand points reside within some given bounded region R. Applying concepts and techniques from Computational Geometry, we provide e cient algorithms for the following problems: 1. Bri...