10 research outputs found
Some Applications of the Percolation Theory. Brief Review of the Century Beginning
The review is a brief description of the state of problems in percolation
theory and their numerous applications, which are analyzed on base of
interesting papers published in the last 15-20 years. At the submitted papers
are studied both the cluster system of the physical body and its impact on the
object in general, and adequate mathematical tools for description of critical
phenomena too. Of special interest are the data, first, the point of phase
transition of certain of percolation system is not really a point, but it is a
critical interval, and second, in vicinity of percolation threshold observed
many different infinite clusters instead of one infinite cluster that appears
in traditional consideration.Comment: arXiv admin note: text overlap with arXiv:1104.5376, arXiv:1205.0691
by other author
Constructive fractals in set theory. Tutorial. (In Russian)
Classical geometric fractals - Cantor set and Sierpinski continua - are
presented in the manual as set-theoretic objects.Comment: in Russia
The selected models of the mesostructure of composites: percolation, clusters, and force fields
This book presents the role of mesostructure on the properties of composite materials. A complex percolation model is developed for the material structure containing percolation clusters of phases and interior boundaries. Modeling of technological cracks and the percolation in the Sierpinski carpet are described. The interaction of mesoscopic interior boundaries of the material, including the fractal nature of interior boundaries, the oscillatory nature of it interaction and also the stochastic model of the interior boundaries’ interaction, the genesis, structure, and properties are discussed. One of part of the book introduces the percolation model of the long-range effect which is based on the notion on the multifractal clusters with transforming elements, and the theorem on the field interaction of multifractals is described. In addition small clusters, their characteristic properties and the criterion of stability are presented
Spectral Analysis of a Q-difference Operator
For a number q bigger than 1, we consider a q-difference version of a second-order singular differential operator which depends on a real parameter. We give three exact parameter intervals in which the operator is semibounded from above, not semibounded, and semibounded from below, respectively. We also provide two exact parameter sets in which the operator is symmetric and self-adjoint, respectively. Our model exhibits a more complex behavior than in the classical continuous case but reduces to it when q approaches 1