5,821 research outputs found
On a Problem of Erd\H{o}s, Herzog and Sch\"onheim
Let be distinct primes.
In 1970, Erd\H os, Herzog and Sch\"{o}nheim proved that if is a set
of divisors of , , no two members of the set being coprime and if no
additional member may be included in without contradicting this
requirement then .
They asked to determine all sets such that the equality holds. In this
paper we solve this problem. We also pose several open problems for further
research.Comment: 12 page
Fixed energy universality for generalized Wigner matrices
We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the
spectrum for generalized symmetric and Hermitian Wigner matrices. Previous
results concerning the universality of random matrices either require an
averaging in the energy parameter or they hold only for Hermitian matrices if
the energy parameter is fixed. We develop a homogenization theory of the Dyson
Brownian motion and show that microscopic universality follows from mesoscopic
statistics
Asymptotics of resolvent integrals: The suppression of crossings for analytic lattice dispersion relations
We study the so called crossing estimate for analytic dispersion relations of
periodic lattice systems in dimensions three and higher. Under a certain
regularity assumption on the behavior of the dispersion relation near its
critical values, we prove that an analytic dispersion relation suppresses
crossings if and only if it is not a constant on any affine hyperplane. In
particular, this will then be true for any dispersion relation which is a Morse
function. We also provide two examples of simple lattice systems whose
dispersion relations do not suppress crossings in the present sense.Comment: 47 page
Delocalization and Diffusion Profile for Random Band Matrices
We consider Hermitian and symmetric random band matrices in dimensions. The matrix entries , indexed by x,y \in
(\bZ/L\bZ)^d, are independent, centred random variables with variances s_{xy}
= \E |h_{xy}|^2. We assume that is negligible if exceeds the
band width . In one dimension we prove that the eigenvectors of are
delocalized if . We also show that the magnitude of the matrix
entries \abs{G_{xy}}^2 of the resolvent is self-averaging
and we compute \E \abs{G_{xy}}^2. We show that, as and , the behaviour of \E |G_{xy}|^2 is governed by a diffusion operator
whose diffusion constant we compute. Similar results are obtained in higher
dimensions
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
Quantitative Derivation of the Gross-Pitaevskii Equation
Starting from first principle many-body quantum dynamics, we show that the
dynamics of Bose-Einstein condensates can be approximated by the time-dependent
nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the
convergence. Initial data are constructed on the bosonic Fock space applying an
appropriate Bogoliubov transformation on a coherent state with expected number
of particles N. The Bogoliubov transformation plays a crucial role; it produces
the correct microscopic correlations among the particles. Our analysis shows
that, on the level of the one particle reduced density, the form of the initial
data is preserved by the many-body evolution, up to a small error which
vanishes as N^{-1/2} in the limit of large N.Comment: 62 pages. Small improvements, new appendi
On the swap-distances of different realizations of a graphical degree sequence
One of the first graph theoretical problems which got serious attention
(already in the fifties of the last century) was to decide whether a given
integer sequence is equal to the degree sequence of a simple graph (or it is
{\em graphical} for short). One method to solve this problem is the greedy
algorithm of Havel and Hakimi, which is based on the {\em swap} operation.
Another, closely related question is to find a sequence of swap operations to
transform one graphical realization into another one of the same degree
sequence. This latter problem got particular emphases in connection of fast
mixing Markov chain approaches to sample uniformly all possible realizations of
a given degree sequence. (This becomes a matter of interest in connection of --
among others -- the study of large social networks.) Earlier there were only
crude upper bounds on the shortest possible length of such swap sequences
between two realizations. In this paper we develop formulae (Gallai-type
identities) for these {\em swap-distance}s of any two realizations of simple
undirected or directed degree sequences. These identities improves considerably
the known upper bounds on the swap-distances.Comment: to be publishe
Asymptotic Structure of Graphs with the Minimum Number of Triangles
We consider the problem of minimizing the number of triangles in a graph of
given order and size and describe the asymptotic structure of extremal graphs.
This is achieved by characterizing the set of flag algebra homomorphisms that
minimize the triangle density.Comment: 22 pages; 2 figure
Local Eigenvalue Density for General MANOVA Matrices
We consider random n\times n matrices of the form
(XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries
with zero mean and variance one. These matrices are the natural generalization
of the Gaussian case, which are known as MANOVA matrices and which have joint
eigenvalue density given by the third classical ensemble, the Jacobi ensemble.
We show that, away from the spectral edge, the eigenvalue density converges to
the limiting density of the Jacobi ensemble even on the shortest possible
scales of order 1/n (up to \log n factors). This result is the analogue of the
local Wigner semicircle law and the local Marchenko-Pastur law for general
MANOVA matrices.Comment: Several small changes made to the tex
The Linear Boltzmann Equation as the Low Density Limit of a Random Schrodinger Equation
We study the evolution of a quantum particle interacting with a random
potential in the low density limit (Boltzmann-Grad). The phase space density of
the quantum evolution defined through the Husimi function converges weakly to a
linear Boltzmann equation with collision kernel given by the full quantum
scattering cross section.Comment: 74 pages, 4 figures, (Final version -- typos corrected
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