Let S = {x1, x2, . . . , xn} be a set of distinct positive integers such that gcd(xi, xj) ∈ S for 1 ≤ i, j ≤ n. Such a set is called GCD-closed. In 1875/1876, H.J.S. Smith showed that, if the set S is “factor-closed”, then the determinant of the matrix eij = gcd(xi, xj) is det(E) = ∏n m=1 φ(xm), where φ denotes Euler’s Phi-function. Since the early 1990’s there has been a rebirth of interest in...

Purpose. The purpose of this study was to elucidate the pathophysiological process in primary cultured corneal fibroblasts (PCFs) from normal subjects and granular corneal dystrophy (GCD) II patients, by using cDNA microarrays. Methods. PCFs were isolated from the corneas of normal subjects and GCD II patients who were heterozygous and homozygous for the TGFBI R124H mutation. RNA was isolated f...

As a consequence of Algorithm 1 below, there always exist integers x, y ∈ Z such that ax+by = gcd(a, b). We say that a and b are co-prime (or relatively prime) if gcd(a, b) = 1, i.e., ax = 1 mod b. From this, x is the multiplicative inverse of a modulo b, and likewise y is the multiplicative inverse of b modulo a. The following deterministic algorithm shows that gcd(a, b) (and additionally, the...

The computation of the Greatest Common Divisor (GCD) of a set of polynomials is an important issue in computational mathematics and it is linked to Control Theory very strong. In this paper we present different matrix-based methods, which are developed for the efficient computation of the GCD of several polynomials. Some of these methods are naturally developed for dealing with numerical inaccu...

Let φ stand for the Euler function. Given a positive integer n, let σ(n) stand for the sum of the positive divisors of n and let τ(n) be the number of divisors of n. We obtain an asymptotic estimate for the counting function of the set {n : gcd(φ(n), τ(n)) = gcd(σ(n), τ(n)) = 1}. Moreover, setting l(n) := gcd(τ(n), τ(n+ 1)), we provide an asymptotic estimate for the size of #{n 6 x : l(n) = 1}.

groups are the groups Dk,l,e = {( α 0 0 δ ) ∈ D | α = 1, δ = 1, ααδδ = 1 } with l, k, e ∈ Z≥1 and gcd(k, l, e) = 1, where gα = k gcd(k,l) and gδ = l gcd(k,l) . If G is a one-dimensional algebraic subgroup of D then G is equal to a group Dm,n = {( α 0 0 δ ) ∈ D | αδ = 1 } with m,n ∈ Z. For all m,n ∈ Z we have that Dm,n/D m,n

If S is a numerical semigroup with embedding dimension equal to three whose minimal generators are pairwise relatively prime numbers, then S = 〈a, b, cb − da〉 with a, b, c, d positive integers such that gcd(a, b) = gcd(a, c) = gcd(b, d) = 1, c ∈ {2, . . . , a− 1}, and a < b < cb− da. In this paper we give formulas, in terms of a, b, c, d, for the genus, the Frobenius number, and the set of pseu...

We present two new parallel algorithms which compute the GCD of n integers of O(n) bits in O(n/ logn) time with O(n) processors in the worst case, for any ε > 0 in CRCW PRAM model. More generally, we prove that computing the GCD of m integers of O(n) bits can be achieved in O(n / logn) parallel time with O(mn ) processors, for any 2 ≤ m ≤ n/ logn, i.e. the parallel time does not depend on the n...