We find the formula for the cardinality of maximal set of integers from [1,. .. , n] which does not contain k + 1 pairwise coprimes and has divisors from a specified set of primes. This formula is defined by the set of multiples of the generating set, which does not depend on n. 1 Formulation of the result Let P = {p 1 < p 2 ,. . .} be the set of primes and N be the set of natural numbers. Writ...

We study discrete Fourier transformations of functions of the greatest common divisor: n ∑ k=1 f((k, n)) · exp( − 2πikm/n). Euler’s totient function: φ(n) = n ∑ k=1 (k, n) · exp(−2πik/n) is an example. The greatest common divisor (k, n) = n ∑ m=1 exp(2πikm/n) · ∑ d|n cd(m) d is another result involving Ramanujan’s sum cd(m). The last equation, interestingly, can be evaluated for k in the comple...

where g(n) is a bounded arithmetical function and a a real number. In this paper we investigate the average order of magnitude of functions of the form fa((m, «)) where (m, n) denotes the greatest common divisor of m and « and a = 1. The principal result is embodied in the theorem of §3. The method of the paper is elementary. It is based upon an identity (Lemma 3.1) which enables us to reduce t...

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension a...

The justified fame of the Euclidean algorithm derives from the fact that it has a much larger realm of applicability than just the integers. In particular, let K be any field and let K[x] be the corresponding integral domain of polynomials over K. Given r polynomials h1(x), h2(x), . . . , hr (x) ∈ K[x] whose greatest common divisor is 1 ∈ K (no common zeros in K), there exist polynomials b1(x),...

Musical chords, harmonies or melodies in Just Intonation have note frequencies which are described by a base frequency multiplied by rational numbers. For any local section, these notes can be converted to some base frequency multiplied by whole positive numbers. The structure of the chord can be analysed mathematically by finding functions which are unchanged upon chord transposition. These fu...

In this thesis, we got interested into the Effective Computation of the Topology of Real Algebraic Curves and Surfaces. One can distinguish three main new algorithms in the field of shape representation. Our first algorithm is a certified symbolic-numerical based on sub-resultants properties and computes the topology of a plane algebraic curve with the best known complexity. The second algorith...

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It is a well known fact that the resultants are invariant under translation. We extend this fact to arbitrary composition (where a translation is a particular composition with a linear monic polynomial), and to arbitrary subresultants (where the resultant is the 0-th subresultant).