955 research outputs found
On the Cauchy problem for Gross-Pitaevskii hierarchies
The purpose of this paper is to investigate the Cauchy problem for the
Gross-Pitaevskii infinite linear hierarchy of equations on We prove local existence and uniqueness of solutions in certain
Sobolev type spaces of sequences of marginal
density operators with In particular, we give a clear
discussion of all cases which covers the local well-posedness
problem for Gross-Pitaevskii hierarchy in this situation.Comment: 17 pages. The referee's comments and suggestions have been
incorporated into this version of the pape
On graphs with a large chromatic number containing no small odd cycles
In this paper, we present the lower bounds for the number of vertices in a
graph with a large chromatic number containing no small odd cycles
Statistical Analysis of Airport Network of China
Through the study of airport network of China (ANC), composed of 128 airports
(nodes) and 1165 flights (edges), we show the topological structure of ANC
conveys two characteristics of small worlds, a short average path length
(2.067) and a high degree of clustering (0.733). The cumulative degree
distributions of both directed and undirected ANC obey two-regime power laws
with different exponents, i.e., the so-called Double Pareto Law. In-degrees and
out-degrees of each airport have positive correlations, whereas the undirected
degrees of adjacent airports have significant linear anticorrelations. It is
demonstrated both weekly and daily cumulative distributions of flight weights
(frequencies) of ANC have power-law tails. Besides, the weight of any given
flight is proportional to the degrees of both airports at the two ends of that
flight. It is also shown the diameter of each sub-cluster (consisting of an
airport and all those airports to which it is linked) is inversely proportional
to its density of connectivity. Efficiency of ANC and of its sub-clusters are
measured through a simple definition. In terms of that, the efficiency of ANC's
sub-clusters increases as the density of connectivity does. ANC is found to
have an efficiency of 0.484.Comment: 6 Pages, 5 figure
On the Consensus Threshold for the Opinion Dynamics of Krause-Hegselmann
In the consensus model of Krause-Hegselmann, opinions are real numbers
between 0 and 1 and two agents are compatible if the difference of their
opinions is smaller than the confidence bound parameter \epsilon. A randomly
chosen agent takes the average of the opinions of all neighbouring agents which
are compatible with it. We propose a conjecture, based on numerical evidence,
on the value of the consensus threshold \epsilon_c of this model. We claim that
\epsilon_c can take only two possible values, depending on the behaviour of the
average degree d of the graph representing the social relationships, when the
population N goes to infinity: if d diverges when N goes to infinity,
\epsilon_c equals the consensus threshold \epsilon_i ~ 0.2 on the complete
graph; if instead d stays finite when N goes to infinity, \epsilon_c=1/2 as for
the model of Deffuant et al.Comment: 15 pages, 7 figures, to appear in International Journal of Modern
Physics C 16, issue 2 (2005
Quantifying the connectivity of a network: The network correlation function method
Networks are useful for describing systems of interacting objects, where the
nodes represent the objects and the edges represent the interactions between
them. The applications include chemical and metabolic systems, food webs as
well as social networks. Lately, it was found that many of these networks
display some common topological features, such as high clustering, small
average path length (small world networks) and a power-law degree distribution
(scale free networks). The topological features of a network are commonly
related to the network's functionality. However, the topology alone does not
account for the nature of the interactions in the network and their strength.
Here we introduce a method for evaluating the correlations between pairs of
nodes in the network. These correlations depend both on the topology and on the
functionality of the network. A network with high connectivity displays strong
correlations between its interacting nodes and thus features small-world
functionality. We quantify the correlations between all pairs of nodes in the
network, and express them as matrix elements in the correlation matrix. From
this information one can plot the correlation function for the network and to
extract the correlation length. The connectivity of a network is then defined
as the ratio between this correlation length and the average path length of the
network. Using this method we distinguish between a topological small world and
a functional small world, where the latter is characterized by long range
correlations and high connectivity. Clearly, networks which share the same
topology, may have different connectivities, based on the nature and strength
of their interactions. The method is demonstrated on metabolic networks, but
can be readily generalized to other types of networks.Comment: 10 figure
Characterizing the network topology of the energy landscapes of atomic clusters
By dividing potential energy landscapes into basins of attractions
surrounding minima and linking those basins that are connected by transition
state valleys, a network description of energy landscapes naturally arises.
These networks are characterized in detail for a series of small Lennard-Jones
clusters and show behaviour characteristic of small-world and scale-free
networks. However, unlike many such networks, this topology cannot reflect the
rules governing the dynamics of network growth, because they are static spatial
networks. Instead, the heterogeneity in the networks stems from differences in
the potential energy of the minima, and hence the hyperareas of their
associated basins of attraction. The low-energy minima with large basins of
attraction act as hubs in the network.Comparisons to randomized networks with
the same degree distribution reveals structuring in the networks that reflects
their spatial embedding.Comment: 14 pages, 11 figure
Maximum flow and topological structure of complex networks
The problem of sending the maximum amount of flow between two arbitrary
nodes and of complex networks along links with unit capacity is
studied, which is equivalent to determining the number of link-disjoint paths
between and . The average of over all node pairs with smaller degree
is for large with a constant implying that the statistics of is related to the
degree distribution of the network. The disjoint paths between hub nodes are
found to be distributed among the links belonging to the same edge-biconnected
component, and can be estimated by the number of pairs of edge-biconnected
links incident to the start and terminal node. The relative size of the giant
edge-biconnected component of a network approximates to the coefficient .
The applicability of our results to real world networks is tested for the
Internet at the autonomous system level.Comment: 7 pages, 4 figure
Giant Clusters in Random Ad Hoc Networks
The present paper introduces ad hoc communication networks as examples of
large scale real networks that can be prospected by statistical means. A
description of giant cluster formation based on the single parameter of node
neighbor numbers is given along with the discussion of some asymptotic aspects
of the giant cluster sizes.Comment: 6 pages, 5 figures; typos and correction
Vibrational modes and spectrum of oscillators on a scale-free network
We study vibrational modes and spectrum of a model system of atoms and
springs on a scale-free network in order to understand the nature of
excitations with many degrees of freedom on the scale-free network. We assume
that the atoms and springs are distributed as nodes and links of a scale-free
network, assigning the mass and the specific oscillation frequency
of the -th atom and the spring constant between the
-th and -th atoms.Comment: 8pages, 2 figure
On the Rigorous Derivation of the 3D Cubic Nonlinear Schr\"odinger Equation with A Quadratic Trap
We consider the dynamics of the 3D N-body Schr\"{o}dinger equation in the
presence of a quadratic trap. We assume the pair interaction potential is
N^{3{\beta}-1}V(N^{{\beta}}x). We justify the mean-field approximation and
offer a rigorous derivation of the 3D cubic NLS with a quadratic trap. We
establish the space-time bound conjectured by Klainerman and Machedon [30] for
{\beta} in (0,2/7] by adapting and simplifying an argument in Chen and
Pavlovi\'c [7] which solves the problem for {\beta} in (0,1/4) in the absence
of a trap.Comment: Revised according to the referee report. Accepted to appear in
Archive for Rational Mechanics and Analysi
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