Description
We consider the adjacency matrix associated with a graph that describes transitions between $2^N$ states of the discrete Preisach memory model. This matrix can also be associated with the "last-in-first-out" inventory management rule. We present an explicit solution for the spectrum by showing that the characteristic polynomial is the product of Chebyshev polynomials. The eigenvalue distribution (density of states) is explicitly calculated and is shown to approach a scaled Devil's staircase. The eigenvectors of the adjacency matrix are also expressed analytically. This is joint work with Andreas Amann, Daniel Kim, and Dmitrii Rachinski. We also examine a mechanistic model of turbulence, a binary tree of masses connected by springs. We analyze the behavior of this linear model: a formula is presented for the analytical calculation of the eigenvalues and the optimal damping - at which the decay of the total mechanical energy is maximized. The discrete energy spectrum of the mechanistic model (defined as the total mechanical energy stored in each level) can be tuned to display the features of the Kolmogorov-spectrum. This is joint work with Bendegúz Dezső Bak.