An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds

For a given self-map f of M, a closed smooth connected and simply-connected manifold of dimension m ≥ 4, we provide an algorithm for estimating the values of the topological invariant Dm r [f], which equals the minimal number of r-periodic points in the smooth homotopy class of f. Our results are based on the combinatorial scheme for computing Dm r [f] introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013), 63–84]. An open-source implementation of the algorithm programmed in C++ is publicly available at http://www.pawelpilarczyk.com/combtop/.

On the impact of overlapping access points in detecting node encounters

Data traces, consisting of logs about the use of mobile and wireless networks, have been used to study the statistics of encounters between mobile nodes, in an attempt to predict the performance of opportunistic networks. Understanding the role and potential of mobile devices as relaying nodes in message dissemination and delivery depends on the knowledge about patterns and number of encounters among nodes. Data traces about the use of WiFi networks are widely available and can be used to extract large datasets of encounters between nodes. However, these logs only capture indirect encounters between nodes, and the resulting encounters datasets might not realistically represent the spatial and temporal behaviour of nodes. This paper addresses the impact of overlapping between the coverage areas of different Access Points of WiFi networks in extracting encounters datasets from the usage logs. Simulation and real-world experimental results show that indirect encounter traces extracted directly from these logs strongly underestimate the opportunities for direct node-to- node message exchange in opportunistic networks.

Global and local coordinates

This chapter described the global and local coordinate systems utilized in the formulation of spatial multibody systems. Global coordinate system is considered in the present work to denote the inertia frame. Additionally, body-fixed coordinate systems, also called local coordinate systems, are utilized to describe local properties of points that belong to a particular body. Furthermore, the process of transforming local coordinates into global coordinates is characterized by considering a transformation matrix. In the present work, Cartesian coordinates are utilized to locate the center of mass of each rigid body, as well as the location of any point that belongs to a body.