The recent discovery that the exponent of matrix multiplication is determined by the rank of the symmetrized matrix multiplication tensor has invigorated interest in better understanding symmetrized matrix multiplication. I present an explicit rank 18 Waring decomposition of $sM_{\langle 3\rangle}$ and describe its symmetry group.

The problem of evaluation of Lyapunov exponent in queueing network analysis is considered based on models and methods of idempotent algebra. General existence conditions for Lyapunov exponent to exist in generalized linear stochastic dynamic systems are given, and examples of evaluation of the exponent for systems with matrices of particular types are presented. A method which allow one to get ...

This review paper describes certain elementary methods of group theory for studying the algebraic complexity of matrix multiplication, as measured by the exponent 2 ≤ ω ≤ 3 of matrix multiplication. which is conjectured to be 2. The seed of these methods lies in two ideas of H. Cohn, C. Umans et. al., firstly, that it is possible to " realize " a matrix product via the regular algebra of a (fin...

Control theory has long provided a rich source of motivation for developments in matrix theory. Accordingly, we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control. The problems discussed include robust stability, matrix exponent& induced norms, stabilizability and pole assignability, and nonstandard matrix equa...

We calculate analytically the largest Lyapunov exponent of the so-called αXY Hamiltonian in the high energy regime. This system consists of a d-dimensional lattice of classical spins with interactions that decay with distance following a powerlaw, the range being adjustable. In disordered regimes the Lyapunov exponent can be easily estimated by means of the “stochastic approach”, a theoretical ...

We consider the static and dynamic phases in a Rosenzweig-Porter (RP) random matrix ensemble with tailed distribution of off-diagonal elements form large-deviation ansatz. present general theory survival probability such random-matrix model show that {\it averaged} may decay time as simple exponent, stretch-exponent power-law or slower. Correspondingly, we identify exponential, stretch-exponent...

We study detection of random signals corrupted by noise that over time switch their values (states) from a finite set of possible values, where the switchings occur at unknown points in time. We model such signals by means of a random duration model that to each possible state assigns a probability mass function which controls the statistics of durations of that state occurrences. Assuming two ...

We prove injectivity and relative asphericity for "layered" systems of equations over torsion-free groups, when the exponent matrix is invertible over Z.