Rank and border rank of Kronecker powers of tensors and Strassen's laser method

We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor Tcw,q is the square of its border rank for q> 2 and that the border rank of its Kronecker cube is the cube of its border rank for q> 4. This answers questions raised implicitly by Coppersmith & Winograd (1990, §11)and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range. In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith–Winograd tensor, Tskewcw,q. For q= 2 , the Kronecker square of this tensor coincides with the 3 × 3 determinant polynomial, det 3∈ C9⊗ C9⊗ C9, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two. We determine new upper bounds for the (Waring) rank and the (Waring) border rank of det 3, exhibiting a strict submultiplicative behaviour for Tskewcw,2 which is promising for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3⊗ C3⊗ C3

Local equations for equivariant evolutionary models

Phylogenetic varieties related to equivariant substitution models have been studied largely in the last years. One of the main objectives has been finding a set of generators of the ideal of these varieties, but this has not yet been achieved in some cases (for example, for the general Markov model this involves the open “salmon conjecture”, see [2]) and it is not clear how to use all generators in practice. Motivated by applications in biology, we tackle the problem from another point of view. The elements of the ideal that could be useful for applications in phylogenetics only need to describe the variety around certain points of no evolution (see [13]). We produce a collection of explicit equations that describe the variety on a Zariski open neighborhood of these points (see Theorem 5.4). Namely, for any tree T on any number of leaves (and any degrees at the interior nodes) and for any equivariant model on any set of states ¿, we compute the codimension of the corresponding phylogenetic variety. We prove that this variety is smooth at general points of no evolution and, if a mild technical condition is satisfied (“d-claw tree hypothesis”), we provide an algorithm to produce a complete intersection that describes the variety around these points.Peer ReviewedPostprint (author's final draft

An Investigation into the Performance Evaluation of Connected Vehicle Applications: From Real-World Experiment to Parallel Simulation Paradigm

A novel system was developed that provides drivers lane merge advisories, using vehicle trajectories obtained through Dedicated Short Range Communication (DSRC). It was successfully tested on a freeway using three vehicles, then targeted for further testing, via simulation. The failure of contemporary simulators to effectively model large, complex urban transportation networks then motivated further research into distributed and parallel traffic simulation. An architecture for a closed-loop, parallel simulator was devised, using a new algorithm that accounts for boundary nodes, traffic signals, intersections, road lengths, traffic density, and counts of lanes; it partitions a sample, Tennessee road network more efficiently than tools like METIS, which increase interprocess communications (IPC) overhead by partitioning more transportation corridors. The simulator uses logarithmic accumulation to synchronize parallel simulations, further reducing IPC. Analyses suggest this eliminates up to one-third of IPC overhead incurred by a linear accumulation model

Portable high-performance programs

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1999.Includes bibliographical references (p. 159-169).by Matteo Frigo.Ph.D

Renormalisation in discrete elasticity

This thesis deals with the statistical mechanics of lattice models. It has two main contributions. On the one hand we implement a general framework for a rigorous renormalisation group approach to gradient models. This approach relies on work by Bauerschmidt, Brydges, and Slade and extends earlier results for gradient interface models by Adams, Kotecký and Müller. On the other hand we use those results to analyse microscopic models for discrete elasticity at small positive temperature and in particular prove convexity properties of the free energy. The first Chapter is introductory and discusses the necessary mathematical background and the physical motivation for this thesis. Chapters 2 to 4 then contain a complete and almost self contained implementation of the renormalisation group approach for gradient models. Chapter 2 is concerned with a new construction of a finite range decomposition with improved regularity. Finite range decompositions are an important ingredient in the renormalisation group approach but also appear at various other places. The new finite range decomposition helps to avoid a loss of regularity and several technical problems that were present in the earlier applications of the renormalisation group technique to gradient models. In the third Chapter we analyse generalized gradient models and discrete models for elasticity and we state our main results: At low temperatures the surface tension is locally uniformly convex and the scaling limit is Gaussian. Moreover, we show that those statements can be reduced to a general statement about perturbations of massless Gaussian measures using suitable null Lagrangians. This is a first step towards a mathematical understanding of elastic behaviour of crystalline solids at positive temperatures starting from microscopic models. The fourth Chapter contains the renormalisation group analysis of gradient models. The main result is a bound for certain perturbations of Gaussian gradient measures that implies the results of the previous chapters. This generalizes earlier results for scalar nearest neighbour models to vector-valued finite range interactions. We also require a much weaker growth assumption for the perturbation. This is possible because we introduce a new solution to the large field problem based on an alternative construction of the weight functions using Gaussian calculus. The last Chapter has a slightly different focus. We investigate gradient interface models for a specific class of non-convex potentials for which phase transitions occur in dimension two. The analysis of these potentials is based on the relation to a random conductance model. We study properties of this random conductance model and in particular prove correlation inequalities and reprove the phase transition result relying on planar duality instead of reflection positivity