854 research outputs found
Evaluation of effective resistances in pseudo-distance-regular resistor networks
In Refs.[1] and [2], calculation of effective resistances on distance-regular
networks was investigated, where in the first paper, the calculation was based
on the stratification of the network and Stieltjes function associated with the
network, whereas in the latter one a recursive formula for effective
resistances was given based on the Christoffel-Darboux identity. In this paper,
evaluation of effective resistances on more general networks called
pseudo-distance-regular networks [21] or QD type networks \cite{obata} is
investigated, where we use the stratification of these networks and show that
the effective resistances between a given node such as and all of the
nodes belonging to the same stratum with respect to
(, belonging to the -th stratum with respect
to the ) are the same. Then, based on the spectral techniques, an
analytical formula for effective resistances such that
(those nodes , of
the network such that the network is symmetric with respect to them) is given
in terms of the first and second orthogonal polynomials associated with the
network, where is the pseudo-inverse of the Laplacian of the network.
From the fact that in distance-regular networks,
is satisfied for all nodes
of the network, the effective resistances
for ( is diameter of the network which
is the same as the number of strata) are calculated directly, by using the
given formula.Comment: 30 pages, 7 figure
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
A methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies
The aim of this work is to provide asymptotic estimates for the splitting of
separatrices in a perturbed 3-degree-of-freedom Hamiltonian system, associated
to a 2-dimensional whiskered torus (invariant hyperbolic torus) whose frequency
ratio is a quadratic irrational number. We show that the dependence of the
asymptotic estimates on the perturbation parameter is described by some
functions which satisfy a periodicity property, and whose behavior depends
strongly on the arithmetic properties of the frequencies.Comment: 5 pages, 1 figur
Alcove path model for
We construct a model for using the alcove path model of Lenart
and Postnikov. We show that the continuous limit of our model recovers a dual
version of the Littelmann path model for given by Li and Zhang.
Furthermore, we consider the dual version of the alcove path model and obtain
analogous results for the dual model, where the continuous limit gives the Li
and Zhang model.Comment: 19 pages, 7 figures; improvements from comments, added more figure
Inverse spectral problems for Sturm-Liouville operators with singular potentials
The inverse spectral problem is solved for the class of Sturm-Liouville
operators with singular real-valued potentials from the space .
The potential is recovered via the eigenvalues and the corresponding norming
constants. The reconstruction algorithm is presented and its stability proved.
Also, the set of all possible spectral data is explicitly described and the
isospectral sets are characterized.Comment: Submitted to Inverse Problem
The Kodaira dimensions of and
We prove that the moduli spaces of curves of genus 22 and 23 are of general
type. To do this, we calculate certain virtual divisor classes of small slope
associated to linear series of rank 6 with quadric relations. We then develop
new tropical methods for studying linear series and independence of quadrics
and show that these virtual classes are represented by effective divisors.Comment: 94 pages, 27 figures; incorporates and supersedes arXiv:1804.01898
and arXiv:1808.0128
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