29 research outputs found
From self-similar groups to self-similar sets and spectra
The survey presents developments in the theory of self-similar groups leading
to applications to the study of fractal sets and graphs, and their associated
spectra
``Smoke Rings'' in Ferromagnets
It is shown that bulk ferromagnets support propagating non-linear modes that
are analogous to the vortex rings, or ``smoke rings'', of fluid dynamics. These
are circular loops of {\it magnetic} vorticity which travel at constant
velocity parallel to their axis of symmetry. The topological structure of the
continuum theory has important consequences for the properties of these
magnetic vortex rings. One finds that there exists a sequence of magnetic
vortex rings that are distinguished by a topological invariant (the Hopf
invariant). We present analytical and numerical results for the energies,
velocities and structures of propagating magnetic vortex rings in ferromagnetic
materials.Comment: 4 pages, 3 eps-figures, revtex with epsf.tex and multicol.sty. To
appear in Physical Review Letters. (Postscript problem fixed.
Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals
We investigate the existence of the meromorphic extension of the spectral
zeta function of the Laplacian on self-similar fractals using the classical
results of Kigami and Lapidus (based on the renewal theory) and new results of
Hambly and Kajino based on the heat kernel estimates and other probabilistic
techniques. We also formulate conjectures which hold true in the examples that
have been analyzed in the existing literature
Spectral analysis on infinite Sierpinski fractafolds
A fractafold, a space that is locally modeled on a specified fractal, is the
fractal equivalent of a manifold. For compact fractafolds based on the
Sierpinski gasket, it was shown by the first author how to compute the discrete
spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian.
A similar problem was solved by the second author for the case of infinite
blowups of a Sierpinski gasket, where spectrum is pure point of infinite
multiplicity. Both works used the method of spectral decimations to obtain
explicit description of the eigenvalues and eigenfunctions. In this paper we
combine the ideas from these earlier works to obtain a description of the
spectral resolution of the Laplacian for noncompact fractafolds. Our main
abstract results enable us to obtain a completely explicit description of the
spectral resolution of the fractafold Laplacian. For some specific examples we
turn the spectral resolution into a "Plancherel formula". We also present such
a formula for the graph Laplacian on the 3-regular tree, which appears to be a
new result of independent interest. In the end we discuss periodic fractafolds
and fractal fields
Geometric Inverse Problem of Radiative Heat Transfer as Applied to the Development of Alternate Spacecraft Orientation Systems
Studying the psycho-physical status of students of higher education institutions of Yekaterinburg
The aim of the study – to determine the possibility of prompt examination of the psychophysiological state of students of 1-2 courses of five universities in Yekaterinburg.Цель исследования является определение возможности оперативного обследования психофизиологического состояния студентов 1-2 курсов пяти вузов Екатеринбурга