Distributing Points on the Torus via Modular Inverses

Distributing Points on the Sphere, I

The problem of “evenly” distributing points on a sphere has a long history. Albeit its intuitive meaning, it is necessary to define “even distribution” mathematically. Various metrics, θs, 0 ≤ s ≤ ∞ (see below for definitions), whose extrema may correspond to even distributions have been proposed. The starting point of this study is Problem 7 in [Smale 00] where the implications of the distribu...

Distributing points on the sphere: partitions, separation, quadrature and energy

Torsion Points on Modular Curves

Let N ≥ 23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the Q-valued points of the modular curve X0(N) which map to torsion points on J0(N) via the cuspidal embedding. We give some generalizations to other modular curves, and to noncuspidal embeddings of X0(N) into J0(N).

GCD-Free Algorithms for Computing Modular Inverses

This paper describes new algorithms for computing a modular inverse e−1 mod f given coprime integers e and f . Contrary to previously reported methods, we neither rely on the extended Euclidean algorithm, nor impose conditions on e or f . The main application of our gcd-free technique is the computation of an RSA private key in both standard and CRT modes based on simple modular arithmetic oper...

OWL/ZIP: Distributing Large and Modular Ontologies

Ontologies published on the web can be quite large, from a couple of megabytes to more than a gigabyte. Deploying, importing and using such ontologies can be a problem, both in terms of bandwidth and load time over the web, and in terms of physically storing them. Some ontologies in BioPortal for example are shipped in their compressed form (via their web services), which allows filesize reduct...