Physical Layer Network Coding for Two-Way Relaying with QAM

The design of modulation schemes for the physical layer network-coded two way relaying scenario was studied in [1], [3], [4] and [5]. In [7] it was shown that every network coding map that satisfies the exclusive law is representable by a Latin Square and conversely, and this relationship can be used to get the network coding maps satisfying the exclusive law. But, only the scenario in which the end nodes use $M$-PSK signal sets is addressed in [7] and [8]. In this paper, we address the case in which the end nodes use $M$-QAM signal sets. In a fading scenario, for certain channel conditions $\gamma e^{j \theta}$, termed singular fade states, the MA phase performance is greatly reduced. By formulating a procedure for finding the exact number of singular fade states for QAM, we show that square QAM signal sets give lesser number of singular fade states compared to PSK signal sets. This results in superior performance of $M$-QAM over $M$-PSK. It is shown that the criterion for partitioning the complex plane, for the purpose of using a particular network code for a particular fade state, is different from that used for $M$-PSK. Using a modified criterion, we describe a procedure to analytically partition the complex plane representing the channel condition. We show that when $M$-QAM ($M >4$) signal set is used, the conventional XOR network mapping fails to remove the ill effects of $\gamma e^{j \theta}=1$, which is a singular fade state for all signal sets of arbitrary size. We show that a doubly block circulant Latin Square removes this singular fade state for $M$-QAM.Comment: 13 pages, 14 figures, submitted to IEEE Trans. Wireless Communications. arXiv admin note: substantial text overlap with arXiv:1203.326

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

A Robust and Efficient Method for Solving Point Distance Problems by Homotopy

The goal of Point Distance Solving Problems is to find 2D or 3D placements of points knowing distances between some pairs of points. The common guideline is to solve them by a numerical iterative method (\emph{e.g.} Newton-Raphson method). A sole solution is obtained whereas many exist. However the number of solutions can be exponential and methods should provide solutions close to a sketch drawn by the user.Geometric reasoning can help to simplify the underlying system of equations by changing a few equations and triangularizing it.This triangularization is a geometric construction of solutions, called construction plan. We aim at finding several solutions close to the sketch on a one-dimensional path defined by a global parameter-homotopy using a construction plan. Some numerical instabilities may be encountered due to specific geometric configurations. We address this problem by changing on-the-fly the construction plan.Numerical results show that this hybrid method is efficient and robust

Discrete models of force chain networks

A fundamental property of any material is its response to a localized stress applied at a boundary. For granular materials consisting of hard, cohesionless particles, not even the general form of the stress response is known. Directed force chain networks (DFCNs) provide a theoretical framework for addressing this issue, and analysis of simplified DFCN models reveal both rich mathematical structure and surprising properties. We review some basic elements of DFCN models and present a class of homogeneous solutions for cases in which force chains are restricted to lie on a discrete set of directions.Comment: 17 pages, 6 figures, dcds-B.cls; Minor corrections to version 2, but including an important factor of 2; Submitted to Discrete and Continuous Dynamical Systems B for special issue honoring David Schaeffe