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...

We prove lower bounds for the complexity of deciding several relations in imaginary, normEuclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, which yield lower bounds for gcd computations. In each imaginary, norm-Euclidean quadratic integer rin...

The task of determining the greatest common divisors (GCD) for several polynomials which arises in image compression, computer algebra and speech encoding can be formulated as a low rank approximation problem with Sylvester matrix. This paper demonstrates a method based on structured total least norm (STLN) algorithm for matrices with Sylvester structure. We demonstrate the algorithm to compute...

This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U , V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC ’93: International Symposium on Symbolic and Algebr...

where θ = 1 + √ d 2 if d mod 4 = 1 and θ = √ d if d mod 4 = 2, 3. The purpose of this paper is to compute the extended gcd of to quadratic integers in ring Z[θ]. We assume throughout that Z[θ] is principal ideal ring, but not necessarily an euclidean ring. If [a, b+ cθ] is the module {ax+(b+ cθ)y, x, y ∈ Z}, it can be shown [3] that I is an ideal of Z[θ] if and only if I = [a, b+ cθ]; where a, ...

In this correspondence, a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions. In z-domain, the desired image can be regarded as the greatest common polynomial divisor among the distorted versions. With the assumption that the distortion filters are finite impulse response (FIR) and relativ...

BACKGROUND AND PURPOSE There is recent evidence of various types of morphological changes in the hippocampus of a rodent model of medial temporal lobe epilepsy (mTLE). However, little is known about such changes in humans. We examined the histological changes [i.e., neuronal loss, cell genesis, and granule cell dispersion (GCD)] in surgical hippocampal specimens taken from patients with mTLE. ...

Given two polynomials F and G in R[x1, . . . , xn], we are going to find the nontrivial approximate GCD C and polynomials F , G ∈ R[x1, . . . , xn] such that ||F − CF ′|| < and ||G − CG′|| < , for some and some well defined norm. Many papers 1,2,3,5,8,10,11,13,15 have already discussed the problem in the case n = 1. Few of them 2,10,11 mentioned the case n > 1. Approximate GCD computation of un...