Greatest common divisors and least common multiples of quotients of elements of integral domains have been investigated by Lüneburg and further by Jäger. In this paper we extend these results to invertible fractional ideals. We also lift our earlier study of the greatest common divisor and least common multiple of finitely generated faithful multiplication ideals to finitely generated projectiv...

The gcd-sum is an arithmetic function defined as the sum of the gcd’s of the first n integers with n : g(n) = ∑n i=1(i, n). The function arises in deriving asymptotic estimates for a lattice point counting problem. The function is multiplicative, and has polynomial growth. Its Dirichlet series has a compact representation in terms of the Riemann zeta function. Asymptotic forms for values of par...

A probabilistic algorithm is exhibited that calculates the gcd of many integers using gcds of pairs of integers; the expected number of pairwise gcds required is less than two.

The GCN4 gene encodes a transcriptional activator in yeast whose expression is regulated at the translational level in response to amino acid availability. gcn3 mutations block derepression of GCN4 expression in starvation conditions. gcd1 and gcd12 mutations restore derepression of GCN4 expression in gcn3 deletion mutants, suggesting that GCN3 positively regulates GCN4 indirectly by antagonism...

We consider the problem of computing the monic gcd of two polynomials over a number field L = Q(α1, . . . , αn). Langemyr and McCallum have already shown how Brown’s modular GCD algorithm for polynomials over Q can be modified to work for Q(α) and subsequently, Langemyr extended the algorithm to L[x]. Encarnacion also showed how to use rational number to make the algorithm for Q(α) output sensi...

We have developed a new galactic chemo-dynamical evolution code, called GCD+, for studies of galaxy formation and evolution. This code is based on our original threedimensional tree N-body/smoothed particle hydrodynamics code which includes selfgravity, hydrodynamics, radiative cooling, star formation, supernova feedback, and metal enrichment. GCD+ includes a new Type II (SNe II) and Ia (SNe Ia...

We study the approximate GCD of two univariate polynomials given with limited accuracy or, equivalently, the exact GCD of the perturbed polynomials within some prescribed tolerance. A perturbed polynomial is regarded as a family of polynomials in a clas-siication space, which leads to an accurate analysis of the computation. Considering only the Sylvester matrix singular values, as is frequentl...

GCD computations and variants of the Euclidean algorithm enjoy broad uses in both classical and quantum algorithms. In this paper, we propose quantum circuits for GCD computation with O(n log n) depth with O(n) ancillae. Prior circuit construction needs O(n) running time with O(n) ancillae. The proposed construction is based on the binary GCD algorithm and it benefits from log-depth circuits fo...

We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]/(f). In particular, the structure of the null space of the multiplicat...