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Marcel Abas

Publications and source records attributed to Marcel Abas.

4 recordsLinked to original sources
arXiv2017

Frequency control of singularly perturbed forced Duffing's oscillator

We analyze the dynamics of the forced singularly perturbed differential equation of Duffing's type. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative. We give some generalizations of results obtained recently by B.S. Wu, W.P. Sun and C.W. Lim, Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations, Journal of Sound and Vibration, Volume 307, Issues 3-5, (2007), pp. 953-960. The new method for an analysis of the nonlinear oscillations which is based on the dynamic change of coordinates is proposed.

math.DS↗
arXiv2017

Large Networks of Diameter Two Based on Cayley Graphs

In this contribution we present a construction of large networks of diameter two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley graphs with surprisingly simple underlying groups. For several small degrees we construct Cayley graphs of diameter two and of order greater than $\frac23$ of Moore bound and we show that Cayley graphs of degrees $d\in\{16,17,18,23,24,31,\dots,35\}$ constructed in this paper are the largest currently known vertex-transitive graphs of diameter two.

math.CO↗
arXiv2017

Large Cayley digraphs and bipartite Cayley digraphs of odd diameters

Let $C_{d,k}$ be the largest number of vertices in a Cayley digraph of degree $d$ and diameter $k$, and let $BC_{d,k}$ be the largest order of a bipartite Cayley digraph for given $d$ and $k$. For every degree $d\geq2$ and for every odd $k$ we construct Cayley digraphs of order $2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ and diameter at most $k$, where $k\ge 3$, and bipartite Cayley digraphs of order $2(k-1)\left(\lfloor\frac{d}{2}\rfloor\right)^{k-1}$ and diameter at most $k$, where $k\ge 5$. These constructions yield the bounds $C_{d,k} \ge 2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ for odd $k\ge 3$ and $d\ge \frac{3^{k}}{2k}+1$, and $BC_{d,k} \ge 2(k-1)\left(\lfloor\frac{d}{2}\rfloor\right)^{k-1}$ for odd $k\ge 5$ and $d\ge \frac{3^{k-1}}{k-1}+1$. Our constructions give the best currently known bounds on the orders of large Cayley digraphs and bipartite Cayley digraphs of given degree and odd diameter $k\ge 5$. In our proofs we use new techniques based on properties of group automorphisms of direct products of abelian groups.

math.CO↗