Vol. 3, No. 2 (2014): Regular Issue

In this paper, we solve generalized pantograph equation by changing the problem to a system of ordinary equations and using the variational iteration method. We discuss convergence of the proposed method to the exact solution. Finally, illustrative examples are given to demonstrate the efficiency of the method.

The Celtic stone, sometimes also called wobbles tone or rattleback usually is a semi-ellipsoidal solid with a special mass distribution. Most celts lied on aflat and horizontal base, set in rotational motion around a vertical axis can rotate in only one direction. In this work the dynamics of the celt is simulated numerically, but the solid is forced untypically, i.e. it is situated on a harmonically vibrating base. Essential part of the model are approximate functions describing the contact forces, i.e. dry friction forces and rolling resistance. They are based on previous works of the authors, but some modifications of friction model are made, which can be described as a generalization of the earlier used Padé approximants. Periodic, quasiperiodic and chaotic dynamics of a harmonically forced rattleback is found and presented by the use of Poincaré maps and bifurcation diagrams.

This paper is concerned with the existence and uniqueness of mild solution of some fractional impulsive equations withfinite delay. Firstly,we introduce the fractional calculus,Gronwall inequality, leray-schauder’sfixed point theorem,Secondly with the help of them, the sufficient condition for the existence and uniqueness of solutions is presented. Finally we give an example to illustrate our main results.

By a successive reduction of complexity in the appearance data collected in situ, we have shown that the way people maintain their appearance constitutes a multi-level dynamical process evolving in several incomparable time scales. The slowest component of this process represents a gradual diffusion of features (with a measurable pace of growing variance) from a proto-costume, which is likely has the 19th century military uniform as a prototype for men, and a robe de soirée - for women. The rapidly varying component of the process is statistically reminiscent of adiabatic processes in thermodynamics which evolve rapidly, without exchange of heat of a system with its environment. The appearance of other people can be considered as being in a state of "thermal equilibrium" between the almost imperceptible, subtle modifications of a traditional costume and the one day gusts of unpredictable fashion.

We discuss the relation between the self-similarity reduction of the generalized mixed mKdV-sinh-Gordon model and the fourth-order equation obtained by Kudryashov inJ. Phys. A: Math. Theor.35(2002) 93-99. Also, it is shown two particular solutions for this equation. Then, we extend the mixed AKNS-Lund-Regge model and study its self-similarity reduction. We obtain thefifth Painlevé equation as a particular case of this reduction and a fourth-order second-degree equation otherwise. The relation between a integrable model and a fourth-order second-degree equation is interesting because the general solution of this equation must have the Painlevé property due the Ablowitz, Ramani and Segur conjecture.

Structure of a complex network provides important information about its performance and may be used to predict changes in network performance. Degree distributions are used to model the network structure. Four degree distributions, including the power law, Weibull, Poisson, and negative binomial, are applied in this research to three complex networks, the Krebs, HIV, and Power Grid networks. To improve accuracy of network modeling, the maximum likelihood estimation method and expectation-maximization algorithm are used to estimate parameters of the four degree distributions. Several statistical analyses and a simulation study are conducted to determine which degree distribution best describes the network structure. The results show that the degree distributions with two descriptive parameters, Weibull and negative binomial, provide better estimations than the onedescriptive-parameter degree distributions, power law and Poisson.