## Abstract

We summarize the known results on the integrability of the complex Hamiltonian systems with two degrees of freedom defined by the Hamiltonian functions of the form

where *V*(*q*
_{1},*q*
_{2}) are homogeneous polynomial potentials of degree *k*.

## Abstract

The aim of this paper was to study an oscillatory flow of a Casson fluid through an elastic tube of variable cross section. The radial displacement of tube wall is taken into consideration. The problem is modelled under the assumption that the variation of the cross section of the tube is slow in the axial direction. Cylindrical coordinate system is chosen to study the problem. The analytical expressions for axial velocity and mass flux as a function of pressure gradient are obtained. The change in pressure distribution for various pressure****radius relationships is analyzed by considering different geometries. The effects of elastic parameter, Womersley parameter and Casson parameter on excess pressure and pressure gradient along axial direction are discussed through graphs. The results reveal that the elastic parameter plays a key role in the variation of pressure along the tube. Womersley parameter has significant effect on pressure distribution. Another important observation is that the amplitude of pressure increases for growing values of Casson parameter for both tapered and constricted tubes. In addition, the pressure oscillates more for the case of locally constricted tube when compared to other geometries.

## Abstract

In this work, we deal with the predecessors existence problems in sequential dynamical systems over directed graphs. The results given in this paper extend those existing for such systems over undirected graphs. In particular, we solve the problems on the existence, uniqueness and coexistence of predecessors of any given state vector, characterizing the Garden-of-Eden states at the same time. We are also able to provide a bound for the number of predecessors and Garden-of-Eden state vectors of any of these systems.

## Abstract

In this article, we have presented a parametric finite difference method, a numerical technique for the solution of two point boundary value problems in ordinary differential equations with mixed boundary conditions. We have tested proposed method for the numerical solution of a model problem. The numerical results obtained for the model problem with constructed exact solution depends on the choice of parameters. The computed result of a model problem suggests that proposed method is efficient.

## Abstract

Herpes simplex virus (HSV-2) triples the risk of acquiring human immunodeficiency virus (HIV) and contributes to more than 50% of HIV infections in other parts of the world. A deterministic mathematical model for the co-interaction of HIV and HSV-2 in a community, with all the relevant biological detail and poor HSV-2 treatment adherence is proposed. The threshold parameters of the model are determined and stabilities are analysed. Further, we applied optimal control theory. We proved the existence of the optimal control and characterized the controls using Pontryagin’s maximum principle. The controls represent monitoring and counselling of individuals infected with HSV-2 only and the other represent monitoring and counselling of individuals dually infected with HIV and HSV-2. Numerical results suggests that more effort should be devoted to monitoring and counselling of individuals dually infected with HIV and HSV-2 as compared to those infected with HSV-2 only. Overall, the study demonstrate that, though time dependent controls will be effective on controlling HIV cases, they may not be sustainable for certain time intervals.

## Abstract

The product graph *G _{m} *G_{p}* of two given graphs

*G*and

_{m}*G*, defined by J.C. Bermond et al.[J Combin Theory, Series B 36(1984) 32-48] in the context of the so-called (

_{p}*Δ,D*)-problem, is one interesting model in the design of large reliable networks. This work deals with sufficient conditions that guarantee these product graphs to be hamiltonian-connected. Moreover, we state product graphs for which provide panconnectivity of interconnection networks modeled by a product of graphs with faulty elements.

## Abstract

The present study aims to explore the effects of an adapted classical dance intervention on the psychological and functional status of institutionalized elder people using a Bayesian network. All participants were assessed at baseline and after the 9 weeks period of the intervention. Measures included balance and gait, psychological well-being, depression, and emotional distress.

According to the Bayesian network obtained, the dance intervention increased the likelihood of presenting better psychological well-being, balance, and gait. Besides, it also decreased the probabilities of presenting emotional distress and depression. These findings demonstrate that dancing has functional and psychological benefits for institutionalized elder people. Moreover it highlights the importance of promoting serious leisure variety in the daily living of institutionalized elder adults.

## Abstract

Chaotic behaviour of dynamical systems, their routes to chaos, and the intermittency in particular are interesting and investigated subjects in nonlinear dynamics. The studying of these phenomena in non-smooth dynamical systems is of the special scientists’ interest. In this paper we study the type-III intermittency route to chaos in strongly nonlinear non-smooth discontinuous 2-DOF vibroimpact system. We apply relatively new mathematical tool – continuous wavelet transform CWT – for investigation this phenomenon. We show that CWT applying allows to detect and determine the chaotic motion and the intermittency with great confidence and reliability, gives the possibility to demonstrate intermittency route to chaos, to distinguish and analyze the laminar and turbulent phases.

## Abstract

This paper propose Noether symmetries and the conserved quantities of the relative motion systems on time scales. The Lagrange equations with delta derivatives on time scales are presented for the system. Based upon the invariance of Hamilton action on time scales, under the infinitesimal transformations with respect to the time and generalized coordinates, the Hamilton’s principle, the Noether theorems and conservation quantities are given for the systems on time scales. Lastly, an example is given to show the application the conclusion.

## Abstract

In the paper we present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform. The selected algorithms represent diverse lines of approach to this problem and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for the Fourier series convergence acceleration. We use C++ and Python languages with arbitrary precision mathematical libraries to address some crucial issues of numerical implementation. The test set includes Laplace transforms considered as difficult to compute as well as some others commonly applied in fractional calculus. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods using Fourier series expansion with accelerated convergence can be assumed as general purpose high-accuracy algorithms. They can be applied to a wide variety of inversion problems.