where T (A′) denotes the space spanned by the elements of the form uky and xv> k , for 1 ≤ k ≤ r, Ω(A′) denotes the space of matrices that have the same support toA′, ‖ ·‖ denotes the spectral norm and ‖ · ‖∞ denotes the largest entry in magnitude. Lemma 1. Let A∗ ∈ RN×N be a similarity matrix of rank r obeying the incoherence properties (A1) and (A2), with μ = max(μ0, μ1). Suppose we observe m...

The max-plus (or tropical) algebra is obtained by replacing the addition by the maximisation (or the minimisation) and the multiplication by the addition. It arises in the dynamic programming approach to deterministic optimal control. In particular , the evolution semigroup of a first order Hamilton-Jacobi equation is linear in the max-plus sense if the Hamiltonian is convex in the adjoint vari...

We (analytically) calculate the energy spectrum corresponding to various experimental and numerical turbulence data analyzed by Benzi et al. 1]. We nd two bottleneck phenomena: While the local scaling exponent r (r) of the structure function decreases monotonically, the local scaling exponent p (p) of the corresponding spectrum has a minimum of p (p min) 0:45 at p min (10) ?1 and a maximum of p...

Recently we generalized the max algebra system to the class of nonnegative  tensors. In this paper we give some basic properties for the left (right) inverse, under the new system. The existence of order 2 left (right) inverse of tensors is characterized. Also we generalize the direct product of matrices to the direct product of tensors (of the same order, but may be different dimensions) and i...

Free probability analogues of the basics of extreme value theory are obtained, based on Ando’s spectral order. This includes classification of freely max-stable laws and their domains of attraction, using “free extremal convolutions” on the distributions. These laws coincide with the limit laws in the classical peaks-over-threshold approach. A free extremal projection-valued process over a meas...

We develop classification results for max–stable processes, based on their spectral representations. The structure of max–linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable max–stable processes based on the notion of co–spectral functions. In particular, we discuss the spectrally continuous–discrete, the conse...