Integer-Forcing MIMO Linear Receivers Based on Lattice Reduction

A new architecture called integer-forcing (IF) linear receiver has been recently proposed for multiple-input multiple-output (MIMO) fading channels, wherein an appropriate integer linear combination of the received symbols has to be computed as a part of the decoding process. In this paper, we propose a method based on Hermite-Korkine-Zolotareff (HKZ) and Minkowski lattice basis reduction algorithms to obtain the integer coefficients for the IF receiver. We show that the proposed method provides a lower bound on the ergodic rate, and achieves the full receive diversity. Suitability of complex Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm (CLLL) to solve the problem is also investigated. Furthermore, we establish the connection between the proposed IF linear receivers and lattice reduction-aided MIMO detectors (with equivalent complexity), and point out the advantages of the former class of receivers over the latter. For the $2 \times 2$ and $4\times 4$ MIMO channels, we compare the coded-block error rate and bit error rate of the proposed approach with that of other linear receivers. Simulation results show that the proposed approach outperforms the zero-forcing (ZF) receiver, minimum mean square error (MMSE) receiver, and the lattice reduction-aided MIMO detectors.Comment: 9 figures and 11 pages. Modified the title, abstract and some parts of the paper. Major change from v1: Added new results on applicability of the CLLL reductio

A Potential Foundation for Emergent Space-Time

We present a novel derivation of both the Minkowski metric and Lorentz transformations from the consistent quantification of a causally ordered set of events with respect to an embedded observer. Unlike past derivations, which have relied on assumptions such as the existence of a 4-dimensional manifold, symmetries of space-time, or the constant speed of light, we demonstrate that these now familiar mathematics can be derived as the unique means to consistently quantify a network of events. This suggests that space-time need not be physical, but instead the mathematics of space and time emerges as the unique way in which an observer can consistently quantify events and their relationships to one another. The result is a potential foundation for emergent space-time.Comment: The paper was originally titled "The Physics of Events: A Potential Foundation for Emergent Space-Time". We changed the title (and abstract) to be more direct when the paper was accepted for publication at the Journal of Mathematical Physics. 24 pages, 15 figure

Tropical Theta Functions and Log Calabi-Yau Surfaces

We generalize the standard combinatorial techniques of toric geometry to the study of log Calabi-Yau surfaces. The character and cocharacter lattices are replaced by certain integral linear manifolds described by Gross, Hacking, and Keel, and monomials on toric varieties are replaced with the canonical theta functions which GHK defined using ideas from mirror symmetry. We describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher-rank cluster varieties.Comment: 40 pages, 2 figures. The final publication is available at Springer via http://dx.doi.org/10.1007/s00029-015-0221-y, Selecta Math. (2016

Weyl's law and quantum ergodicity for maps with divided phase space

For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the distribution of eigenphases. If the map has one ergodic component, and is periodic on the remaining domains, we prove the Schnirelman-Zelditch-Colin de Verdiere Theorem on the equidistribution of eigenfunctions with respect to the ergodic component of the classical map (quantum ergodicity). We apply our main theorems to quantised linked twist maps on the torus. In the Appendix, S. Zelditch connects these studies to some earlier results on `pimpled spheres' in the setting of Riemannian manifolds. The common feature is a divided phase space with a periodic component.Comment: Colour figures. Black & white figures available at http://www2.maths.bris.ac.uk/~majm. Appendix by Steve Zelditc

Clifford algebra and the projective model of homogeneous metric spaces: Foundations

This paper is to serve as a key to the projective (homogeneous) model developed by Charles Gunn (arXiv:1101.4542 [math.MG]). The goal is to explain the underlying concepts in a simple language and give plenty of examples. It is targeted to physicists and engineers and the emphasis is on explanation rather than rigorous proof. The projective model is based on projective geometry and Clifford algebra. It supplements and enhances vector and matrix algebras. It also subsumes complex numbers and quaternions. Projective geometry augmented with Clifford algebra provides a unified algebraic framework for describing points, lines, planes, etc, and their transformations, such as rotations, reflections, projections, and translations. The model is relevant not only to Euclidean space but to a variety of homogeneous metric spaces.Comment: 89 pages, 140 figures (many include 3D PRC vector graphics

Honeycomb tessellations and canonical bases for permutohedral blades

This paper studies two families of piecewise constant functions which are determined by the $(n-2)$-skeleta of collections of honeycomb tessellations of $\mathbb{R}^{n-1}$ with standard permutohedra. The union of the codimension $1$ cones obtained by extending the facets which are incident to a vertex of such a tessellation is called a blade. We prove ring-theoretically that such a honeycomb, with 1-skeleton built from a cyclic sequence of segments in the root directions $e_i-e_{i+1}$, decomposes locally as a Minkowski sum of isometrically embedded components of hexagonal honeycombs: tripods and one-dimensional subspaces. For each triangulation of a cyclically oriented polygon there exists such a factorization. This consequently gives resolution to an issue proposed and developed by A. Ocneanu, to find a structure theory for an object he discovered during his investigations into higher Lie theories: permutohedral blades. We introduce a certain canonical basis for a vector space spanned by piecewise constant functions of blades which is compatible with various quotient spaces appearing in algebra, topology and scattering amplitudes. Various connections to scattering amplitudes are discussed, giving new geometric interpretations for certain combinatorial identities for one-loop Parke-Taylor factors. We give a closed formula for the graded dimension of the canonical blade basis. We conjecture that the coefficients of the generating function numerators for the diagonals are symmetric and unimodal.Comment: Added references; new section on configuration space

Virtual polytopes

Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as the Grothendick group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. A selection of applications demonstrates their versatility