This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with existing approaches, this method is based on convex optimization, and thus has polynomial-time complexity. As an application, we use this method to study powers of the construction given by Coppersmith and Winograd [Journal of Symbolic Computation, 1990] and obtain the upper bound ω < 2.3728639 on the exponent of square matrix multiplication, which slightly improves the best known upper bound.

Let α be the maximal value such that the product of an n × n α matrix by an n α × n matrix can be computed with n 2+o(1) arithmetic operations. In this paper we show that α > 0.30298, which improves the previous record α > 0.29462 by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an n × n k matrix by an n k × n matrix, for any value k = 1. The complexity of this algorithm is better than all known algorithms for rectangular matrix multiplication. In the case of square matrix multiplication (i.e., for k = 1), we recover exactly the complexity of the algorithm by Coppersmith and Winograd (Journal of Symbolic Computation, 1990).These new upper bounds can be used to improve the time complexity of several known algorithms that rely on rectangular matrix multiplication. For example, we directly obtain a O(n 2.5302 )-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, improving over the O(n 2.575 )-time algorithm by Zwick (JACM 2002), and also improve the time complexity of sparse square matrix multiplication. Short description of the approach by Coppersmith and Winograd. The results [9,13,16,24,27] mentioned above are all obtained by extending the approach by Coppersmith and Winograd [10]. This approach is an illustration of a general methodology initiated in the 1970's based on the theory of bilinear and trilinear forms, through which most of the improvements for matrix multiplication have been obtained. Informally, the idea is to start with a basic construction (some small trilinear form), and then exploit general properties of matrix multiplication (in particular Schönhage's asymptotic sum inequality [23]) to derive an upper bound on the exponent ω from this construction. The main contributions of [10] consist of two parts: the discovery of new basic constructions and the introduction of strong techniques to analyze them. In their paper, Coppersmith and Winograd actually present three algorithms, based on three different basic constructions. The first basic construction (Section 6 in [10]) is the simplest of the three and leads to the upper bound ω < 2.40364. The second basic construction (Section 7 in [10]), that we will refer in this paper as F q (here q ∈ N is a parameter), leads to the upper bound

All types of sarcomas may be observed in the heart with a predominance of undifferentiated sarcomas. Histological grading, unlike histological type, seems to correlate with survival which remains extremely poor.

This paper considers the problem of efficiently transmitting quantum states through a network. It has been known for some time that without additional assumptions it is impossible to achieve this task perfectly in general -indeed, it is impossible even for the simple butterfly network. As additional resource we allow free classical communication between any pair of network nodes. It is shown that perfect quantum network coding is achievable in this model whenever classical network coding is possible over the same network when replacing all quantum capacities by classical capacities. More precisely, it is proved that perfect quantum network coding using free classical communication is possible over a network with k source-target pairs if there exists a classical linear (or even vector-linear) coding scheme over a finite ring. Our proof is constructive in that we give explicit quantum coding operations for each network node. This paper also gives an upper bound on the number of classical communication required in terms of k, the maximal fan-in of any network node, and the size of the network.