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Infinite families of noncototients

A. FlammenkampF. Luca — 2000

Colloquium Mathematicae

For any positive integer n let ϕ(n) be the Euler function of n. A positive integer n is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression ( 2 m k ) m 1 consists entirely of noncototients. We then use computations to detect seven such positive integers k.

On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ

A. GrytczukF. LucaM. Wójtowicz — 2000

Colloquium Mathematicae

For any positive integer n let ϕ(n) and σ(n) be the Euler function of n and the sum of divisors of n, respectively. In [5], Mąkowski and Schinzel conjectured that the inequality σ(ϕ(n)) ≥ n/2 holds for all positive integers n. We show that the lower density of the set of positive integers satisfying the above inequality is at least 0.74.

Domain decomposition methods and scientific computing applications

Luca F. Pavarino — 2005

Bollettino dell'Unione Matematica Italiana

This paper reviews the basic mathematical ideas and convergence analysis of domain decomposition methods. These are parallel and scalable iterative methods for the efficient numerical solution of partial differential equations. Two examples are then presented showing the application of domain decomposition methods to large-scale numerical simulations in computational mechanics and electrocardiology.

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