Let E/k(T ) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R ∈ E(k(T )), write xR = AR/D2 R with relatively prime polynomials AR(T ), DR(T ) ∈ k[T ]. The sequence {DnR}n≥1 is called the elliptic divisibility sequence of R. Let P, Q ∈ E(k(T )) be independent points. We conjecture that deg ( gcd(DnP , DmQ) ) is boun...

We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF (2)-linear combination of Gold functions Tr(x i+1) is semi-bent over GF (2), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold fu...

We introduce an algorithm to interpolate sparse multivariate polynomials with integer coefficients. Our algorithm modifies Ben-Or and Tiwari’s deterministic algorithm for interpolating over rings of characteristic zero to work modulo p, a smooth prime of our choice. We present benchmarks comparing our algorithm to Zippel’s probabilistic sparse interpolation algorithm, demonstrating that our alg...

where z is an integer then gcd(a, b) = gcd(b, c). Indeed any divisor of a and b will divide c, and conversely any divisor of b and c will divide a. We can compute c by taking the remainder after dividing a by b, i.e. c is a mod b. (We will discuss the mod operation in greater details in the next section, but at this point, we only need the definition of c as the remainder of dividing a by b.) B...

In this paper, we propose and study an Asynchronous parallel Greedy Coordinate Descent (Asy-GCD) algorithm for minimizing a smooth function with bounded constraints. At each iteration, workers asynchronously conduct greedy coordinate descent updates on a block of variables. In the first part of the paper, we analyze the theoretical behavior of Asy-GCD and prove a linear convergence rate. In the...

This paper describes parallel matrix transpose algorithms on distributed memory concurrent processors. We assume that the matrix is distributed over a P Q processor template with a block scattered data distribution. P , Q, and the block size can be arbitrary, so the algorithms have wide applicability. The communication schemes of the algorithms are determined by the greatest common divisor (GCD...

Fine and Wilf’s well-known theorem states that any word having periods p, q and length at least p + q − gcd(p, q) also has gcd(p, q), the greatest common divisor of p and q, as a period. Moreover, the length p + q − gcd(p, q) is critical since counterexamples can be provided for shorter words. This result has since been extended to partial words, or finite sequences that may contain a number of...

OBJECTIVE The aim was to compare the corneal topography and tomography parameters of macular corneal dystrophy (MCD), granular corneal dystrophy (GCD) and lattice corneal dystrophy (LCD) patients obtained by Scheimpflug imaging system. METHODS The charts, photographs and topography images of patients were reviewed retrospectively. This study included 73 eyes of 73 patients (28 MCD, 20 GCG and...

PURPOSE The present study investigated the effect of mitomycin C (MMC) on cell viability, apoptosis, and transforming growth factor beta-induced protein (TGFBIp) expression in cultured normal corneal fibroblasts and heterozygote or homozygote granular corneal dystrophy type II (GCD II) corneal fibroblasts. METHODS Keratocytes were obtained from normal cornea or from heterozygote or homozygote...