Properties of Random Direction Models

A number of mobility models have been proposed for the purpose of either analyzing or simulating the movement of users in a mobile wireless network. Two of the more popular are the random waypoint and the random direction models. The random waypoint model is physically appealing but difficult to understand. Although the random direction model is less appealing physically, it is much easier to understand. User speeds are easily calculated, unlike for the waypoint model, and, as we will observe, user positions and directions are uniformly distributed. The contribution of this paper is to establish this last property for a rich class of random direction models that allow future movements to depend on past movements. To this end, we consider finite one- and two-dimensional spaces. We consider two variations, the random direction model with wrap around and with reflection. We establish a simple relationship between these two models and, for both, show that positions and directions are uniformly distributed in steady-state for a class of Markov movement models regardless of initial position. In addition, we establish a sample path property for both models, namely that any piecewise linear movement applied to a user preserves the uniform distribution of position and direction provided that users were initially uniformly throughout the space with equal likelihood of being pointed in any direction

Finite element simulation of low-density thermally bonded nonwoven materials: effects of orientation distribution function and arrangement of bond points

A random and discontinuous microstructure is one of the most characteristic features of a low-density thermally bonded nonwoven material, and it affects their mechanical properties significantly. To understand their effect of microstructure on the overall mechanical properties of the nonwoven material, discontinuous models are developed incorporating random discontinuous structures representing microstructures of a real nonwoven material. Experimentally measured elastic material properties of polypropylene fibres are introduced into the models to simulate the tensile behaviour of the material for its both principle directions: machine direction and cross direction. Additionally, varying arrangements of bond points and schemes of fibres’ orientation distribution are implemented in the models to analyse the respective effects

The Space of Solutions of Coupled XORSAT Formulae

The XOR-satisfiability (XORSAT) problem deals with a system of $n$ Boolean variables and $m$ clauses. Each clause is a linear Boolean equation (XOR) of a subset of the variables. A $K$-clause is a clause involving $K$ distinct variables. In the random $K$-XORSAT problem a formula is created by choosing $m$ $K$-clauses uniformly at random from the set of all possible clauses on $n$ variables. The set of solutions of a random formula exhibits various geometrical transitions as the ratio $\frac{m}{n}$ varies. We consider a {\em coupled} $K$-XORSAT ensemble, consisting of a chain of random XORSAT models that are spatially coupled across a finite window along the chain direction. We observe that the threshold saturation phenomenon takes place for this ensemble and we characterize various properties of the space of solutions of such coupled formulae.Comment: Submitted to ISIT 201

Random-Walk Statistics and the Spherical Harmonic Representation of CMB Maps

We investigate the properties of the (complex) coefficients obtained in a spherical harmonic representation of temperature maps of the cosmic microwave background (CMB). We study the effect of the coefficient phase only, as well as the combined effects of phase and amplitude. The method used to check for anomalies is to construct a ``random walk'' trajectory in the complex plane where the step length and direction are given by the amplitude and phase (respectively) of the harmonic coefficient. If the fluctuations comprise a homogeneous and isotropic Gaussian random field on the sky, the path so obtained should be a classical ``Rayleigh flight'' with very well known statistical properties. We illustrate the use of this random-walk representation by using the net walk length as a test statistic, and apply the method to the coefficients obtained from a Wilkinson Microwave Anisotropy Probe (WMAP) preliminary sky temperature map.Comment: 5 pages, 1 figure, uses mn.sty. Version accepted for publication in MNRAS. Includes toy models of non-random walk