Periodic Trajectories and Topology of the Integrable Boltzmann System

We consider the Boltzmann system corresponding to the motion of a billiard with a linear boundary under the influence of a gravitational field. We derive analytic conditions of Cayley's type for periodicity of its trajectories and provide geometric descriptions of caustics. The topology of the phase space is discussed using Fomenko graphs.Comment: 18 pages, 14 figure

Marketing Research on Passenger Satisfaction With Public Transport Service in the City of Belgrade

The aim of this paper is to determine, based on conducted marketing research, the level of passenger satisfaction with public transport services for the purpose of making better marketing decisions in the example of the City of Belgrade. The main task is to test the hypothesis on the existence of significant influence of factors, such as quality service, attitude and behaviour of employees (e.g. driver), adequate informing, quality of vehicles, line routes and timetable, on passenger satisfaction. Correlation coefficient and regression analysis were used for interpreting the obtained results and examining the formulated hypothesis. Empirical research has shown that there is a significant correlation between the aforementioned factors and passenger satisfaction with public transport services. The obtained results provided recommendations and guidelines for improving and increasing the quality of public transport services. The research results also provide the basis for future research that could examine the relationship between passenger satisfaction with services and sub-groups within the analyzed factors.</p

On Elliptical Billiards in the Lobachevsky Space and associated Geodesic Hierarchies

We derive Cayley's type conditions for periodical trajectories for the billiard within an ellipsoid in the Lobachevsky space. It appears that these new conditions are of the same form as those obtained before for the Euclidean case. We explain this coincidence by using theory of geodesically equivalent metrics and show that Lobachevsky and Euclidean elliptic billiards can be naturally considered as a part of a hierarchy of integrable elliptical billiards.Comment: 14 pages, to appear in Journal of Geometry and Physic

Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics

We study geometry of confocal quadrics in pseudo-Euclidean spaces of an arbitrary dimension $d$ and any signature, and related billiard dynamics. The goal is to give a complete description of periodic billiard trajectories within ellipsoids. The novelty of our approach is based on introduction of a new discrete combinatorial-geometric structure associated to a confocal pencil of quadrics, a colouring in $d$ colours, by which we decompose quadrics of $d+1$ geometric types of a pencil into new relativistic quadrics of $d$ relativistic types. Deep insight of related geometry and combinatorics comes from our study of what we call discriminat sets of tropical lines $\Sigma^+$ and $\Sigma^-$ and their singularities. All of that enable usto get an analytic criterion describing all periodic billiard trajectories, including the light-like ones as those of a special interest.Comment: 29 pages, 7 figure