5,579 research outputs found
Inelastic light, neutron, and X-ray scatterings related to the heterogeneous elasticity of glasses
The effects of plasticization of poly(methyl methacrylate) glass on the boson
peaks observed by Raman and neutron scattering are compared. In plasticized
glass the cohesion heterogeneities are responsible for the neutron boson peak
and partially for the Raman one, which is enhanced by the composition
heterogeneities. Because the composition heterogeneities have a size similar to
that of the cohesion ones and form quasiperiodic clusters, as observed by small
angle X-ray scattering, it is inferred that the cohesion heterogeneities in a
normal glass form nearly periodic arrangements too. Such structure at the
nanometric scale explains the linear dispersion of the vibrational frequency
versus the transfer momentum observed by inelastic X-ray scattering.Comment: 9 pages, 2 figures, to be published in J. Non-Cryst. Solids
(Proceedings of the 4th IDMRCS
Enlarged Galilean symmetry of anyons and the Hall effect
Enlarged planar Galilean symmetry, built of both space-time and field
variables and also incorporating the ``exotic'' central extension is
introduced. It is used to describe non-relativistic anyons coupled to an
electromagnetic field. Our theory exhibits an anomalous velocity relation of
the type used to explain the Anomalous Hall Effect. The Hall motions,
characterized by a Casimir of the enlarged algebra, become mandatory for some
critical value(s) of the magnetic field. The extension of our scheme yields the
semiclassical effective model of the Bloch electron.Comment: LaTeX, 7 pages. No figures. One more reference adde
Cellular spanning trees and Laplacians of cubical complexes
We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell
complex in terms of the eigenvalues of its cellular Laplacian operators,
generalizing a previous result for simplicial complexes. As an application, we
obtain explicit formulas for spanning tree enumerators and Laplacian
eigenvalues of cubes; the latter are integers. We prove a weighted version of
the eigenvalue formula, providing evidence for a conjecture on weighted
enumeration of cubical spanning trees. We introduce a cubical analogue of
shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of
shifted cubical complexes, in particular, these eigenvalues are also integers.
Finally, we recover Adin's enumeration of spanning trees of a complete colorful
simplicial complex from the cellular Matrix-Tree Theorem together with a result
of Kook, Reiner and Stanton.Comment: 24 pages, revised version, to appear in Advances in Applied
Mathematic
Simplicial matrix-tree theorems
We generalize the definition and enumeration of spanning trees from the
setting of graphs to that of arbitrary-dimensional simplicial complexes
, extending an idea due to G. Kalai. We prove a simplicial version of
the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the
squares of the orders of their top-dimensional integral homology groups, in
terms of the Laplacian matrix of . As in the graphic case, one can
obtain a more finely weighted generating function for simplicial spanning trees
by assigning an indeterminate to each vertex of and replacing the
entries of the Laplacian with Laurent monomials. When is a shifted
complex, we give a combinatorial interpretation of the eigenvalues of its
weighted Laplacian and prove that they determine its set of faces uniquely,
generalizing known results about threshold graphs and unweighted Laplacian
eigenvalues of shifted complexes.Comment: 36 pages, 2 figures. Final version, to appear in Trans. Amer. Math.
So
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
On Maximal Unbordered Factors
Given a string of length , its maximal unbordered factor is the
longest factor which does not have a border. In this work we investigate the
relationship between and the length of the maximal unbordered factor of
. We prove that for the alphabet of size the expected length
of the maximal unbordered factor of a string of length~ is at least
(for sufficiently large values of ). As an application of this result, we
propose a new algorithm for computing the maximal unbordered factor of a
string.Comment: Accepted to the 26th Annual Symposium on Combinatorial Pattern
Matching (CPM 2015
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