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.