1 S. Sykora, On some number densities related to coprimes, DOI: 10.3247/SL5Math14.005 On some number densities related to coprimes Stanislav Sykora, Extra Byte, www.ebyte.it First published in November 15, 2014 This brief essay explores the limit mean densities of subsets of natural numbers m such that the pair (m, m\b) is either coprime or not coprime. Here m\b = floor(m/b) denotes the operati...

This paper presents an application of-synthesis to the robust control of a large exible space structure experimental testbed called Daisy. A perturbed left-coprime factorization of Daisy's dynamics is used to capture the uncertainty in the modes.-synthesis controller designs based on left-coprime factorizations of 46 th-order collocated and noncollocated models of Daisy are presented together w...

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For x real, let { } $ \lbrace \rbrace$ be the fractional part of (that is, = − ⌊ ⌋ $\lbrace x\rbrace - \lfloor \rfloor$ ). The lonely runner conjecture can stated as follows: for any n positive integers v 1 < 2 ⋯ v_1 v_2 \dots v_n$ , there exists a real number t such that / ( + ) ⩽ i 1/(n+1) \leqslant v_i t\rbrace n/(n+1)$ 1, n$ . In this paper, we prove if ε > 0 \epsilon >0$ and is sufficientl...

In this paper, we show how to use the theory of fractional ideals in order to study the fractional representation approach to analysis and synthesis problems for SISO systems. Within this mathematical framework, we give necessary and su8cient conditions so that a plant is internally/strongly/bistably stabilizable or admits a (weak) coprime factorization. Moreover, we show how to generalize the ...

New proof. Let n be an arbitrary positive integer greater than 1. Since n and n + 1 are consecutive integers, they must be coprime. Hence the number N2 = n(n + 1) must have at least two different prime factors. Similarly, since the integers n(n+1) and n(n+1)+1 are consecutive, and therefore coprime, the number N3 = n(n + 1)[n(n + 1) + 1] must have at least three different prime factors. This pr...