5,213 research outputs found
Collapsing Estimates and the Rigorous Derivation of the 2d Cubic Nonlinear Schr\"odinger Equation with Anisotropic Switchable Quadratic Traps
We consider the 2d and 3d many body Schr\"odinger equations in the presence
of anisotropic switchable quadratic traps. We extend and improve the collapsing
estimates in Klainerman-Machedon [24] and Kirkpatrick-Schlein-Staffilani [23].
Together with an anisotropic version of the generalized lens transform in
Carles [3], we derive rigorously the cubic NLS with anisotropic switchable
quadratic traps in 2d through a modified Elgart-Erd\"os-Schlein-Yau procedure.
For the 3d case, we establish the uniqueness of the corresponding
Gross-Pitaevskii hierarchy without the assumption of factorized initial data.Comment: v6, 32 pages. Added an algebraic explanation of the generalized lens
transform using the metaplectic representation. Accepted to appear in Journal
de Math\'ematiques Pures et Appliqu\'ees. Comments are welcome
Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction
Using a new method it is possible to derive mean field equations from the
microscopic body Schr\"odinger evolution of interacting particles without
using BBGKY hierarchies.
In this paper we wish to analyze scalings which lead to the Gross-Pitaevskii
equation which is usually derived assuming positivity of the interaction. The
new method for dealing with mean field limits presented in [6] allows us to
relax this condition. The price we have to pay for this relaxation is however
that we have to restrict the scaling behavior to and that we have
to assume fast convergence of the reduced one particle marginal density matrix
of the initial wave function to a pure state
Critical Points for Random Boolean Networks
A model of cellular metabolism due to S. Kauffman is analyzed. It consists of
a network of Boolean gates randomly assembled according to a probability
distribution. It is shown that the behavior of the network depends very
critically on certain simple algebraic parameters of the distribution. In some
cases, the analytic results support conclusions based on simulations of random
Boolean networks, but in other cases, they do not.Comment: 19 page
Uniform synchronous criticality of diversely random complex networks
We investigate collective synchronous behaviors in random complex networks of
limit-cycle oscillators with the non-identical asymmetric coupling scheme, and
find a uniform coupling criticality of collective synchronization which is
independent of complexity of network topologies. Numerically simulations on
categories of random complex networks have verified this conclusion.Comment: 8 pages, 4 figure
Clustering in complex networks. II. Percolation properties
The percolation properties of clustered networks are analyzed in detail. In
the case of weak clustering, we present an analytical approach that allows to
find the critical threshold and the size of the giant component. Numerical
simulations confirm the accuracy of our results. In more general terms, we show
that weak clustering hinders the onset of the giant component whereas strong
clustering favors its appearance. This is a direct consequence of the
differences in the -core structure of the networks, which are found to be
totally different depending on the level of clustering. An empirical analysis
of a real social network confirms our predictions.Comment: Updated reference lis
Maximum size of reverse-free sets of permutations
Two words have a reverse if they have the same pair of distinct letters on
the same pair of positions, but in reversed order. A set of words no two of
which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size
of a reverse-free set of words from [n]^k where no letter repeats within a
word. We show the following lower and upper bounds in the case n >= k: F(n,k)
\in n^k k^{-k/2 + O(k/log k)}. As a consequence of the lower bound, a set of
n-permutations each two having a reverse has size at most n^{n/2 + O(n/log n)}.Comment: 10 page
Two-dimensional gauge theories of the symmetric group S(n) and branched n-coverings of Riemann surfaces in the large-n limit
Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge
theory of the symmetric group S(n) defined on a cell discretization of the
surface. We study the theory in the large-n limit, and we find a rich phase
diagram with first and second order transition lines. The various phases are
characterized by different connectivity properties of the covering surface. We
point out some interesting connections with the theory of random walks on group
manifolds and with random graph theory.Comment: Talk presented at the "Light-cone physics: particles and strings",
Trento, Italy, September 200
The asymptotic limits of zero modes of massless Dirac operators
Asymptotic behaviors of zero modes of the massless Dirac operator
are discussed, where
is the triple of Dirac
matrices, , and is a
Hermitian matrix-valued function with
, .
We shall show that for every zero mode , the asymptotic limit of
as exists. The limit is expressed in terms of an
integral of .Comment: 9 page
Evolutionary trees: an integer multicommodity max-flow-min-cut theorem
In biomathematics, the extensions of a leaf-colouration of a binary tree to the whole vertex set with minimum number of colour-changing edges are extensively studied. Our paper generalizes the problem for trees; algorithms and a Menger-type theorem are presented. The LP dual of the problem is a multicommodity flow problem, for which a max-flow-min-cut theorem holds. The problem that we solve is an instance of the NP-hard multiway cut problem
Comparison of Ising magnet on directed versus undirected Erdos-Renyi and scale-free network
Scale-free networks are a recently developed approach to model the
interactions found in complex natural and man-made systems. Such networks
exhibit a power-law distribution of node link (degree) frequencies n(k) in
which a small number of highly connected nodes predominate over a much greater
number of sparsely connected ones. In contrast, in an Erdos-Renyi network each
of N sites is connected to every site with a low probability p (of the orde r
of 1/N). Then the number k of neighbors will fluctuate according to a Poisson
distribution. One can instead assume that each site selects exactly k neighbors
among the other sites. Here we compare in both cases the usual network with the
directed network, when site A selects site B as a neighbor, and then B
influences A but A does not influence B. As we change from undirected to
directed scale-free networks, the spontaneous magnetization vanishes after an
equilibration time following an Arrhenius law, while the directed ER networks
have a positive Curie temperature.Comment: 10 pages including all figures, for Int. J, Mod. Phys. C 1
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