139,396 research outputs found
Some local--global phenomena in locally finite graphs
In this paper we present some results for a connected infinite graph with
finite degrees where the properties of balls of small radii guarantee the
existence of some Hamiltonian and connectivity properties of . (For a vertex
of a graph the ball of radius centered at is the subgraph of
induced by the set of vertices whose distance from does not
exceed ). In particular, we prove that if every ball of radius 2 in is
2-connected and satisfies the condition for
each path in , where and are non-adjacent vertices, then
has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017).
Furthermore, we prove that if every ball of radius 1 in satisfies Ore's
condition (1960) then all balls of any radius in are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio
An observation on highest weight crystals
As shown by Stembridge, crystal graphs can be characterized by their local
behavior. In this paper, we observe that a certain local property on highest
weight crystals forces a more global property. In type , this statement says
that if a node has a single parent and single grandparent, then there is a
unique walk from the highest weight node to it. In other classical types, there
is a similar (but necessarily more technical) statement. This walk is obtained
from the associated level 1 perfect crystal, . (It is unique unless
the Dynkin diagram contains that of as a subdiagram.)
This crystal observation was motivated by representation-theoretic behavior
of the affine Hecke algebra of type , which is known to be captured by
highest weight crystals of type by results of Grojnowski. As
discussed below, the proofs in either setting are straightforward, and so the
theorem linking the two phenomena is not needed. However, the result is
presented here for crystals as one can say something in all types (Grojnowski's
theorem is only in type ), and because the statement seems more surprising
in the language of crystals than it does for affine Hecke algebra modules
Critical phenomena in exponential random graphs
The exponential family of random graphs is one of the most promising class of
network models. Dependence between the random edges is defined through certain
finite subgraphs, analogous to the use of potential energy to provide
dependence between particle states in a grand canonical ensemble of statistical
physics. By adjusting the specific values of these subgraph densities, one can
analyze the influence of various local features on the global structure of the
network. Loosely put, a phase transition occurs when a singularity arises in
the limiting free energy density, as it is the generating function for the
limiting expectations of all thermodynamic observables. We derive the full
phase diagram for a large family of 3-parameter exponential random graph models
with attraction and show that they all consist of a first order surface phase
transition bordered by a second order critical curve.Comment: 14 pages, 8 figure
A minimal model for congestion phenomena on complex networks
We study a minimal model of traffic flows in complex networks, simple enough
to get analytical results, but with a very rich phenomenology, presenting
continuous, discontinuous as well as hybrid phase transitions between a
free-flow phase and a congested phase, critical points and different scaling
behaviors in the system size. It consists of random walkers on a queueing
network with one-range repulsion, where particles can be destroyed only if they
can move. We focus on the dependence on the topology as well as on the level of
traffic control. We are able to obtain transition curves and phase diagrams at
analytical level for the ensemble of uncorrelated networks and numerically for
single instances. We find that traffic control improves global performance,
enlarging the free-flow region in parameter space only in heterogeneous
networks. Traffic control introduces non-linear effects and, beyond a critical
strength, may trigger the appearance of a congested phase in a discontinuous
manner. The model also reproduces the cross-over in the scaling of traffic
fluctuations empirically observed in the Internet, and moreover, a conserved
version can reproduce qualitatively some stylized facts of traffic in
transportation networks
Quantum calcium-ion interactions with EEG
Previous papers have developed a statistical mechanics of neocortical
interactions (SMNI) fit to short-term memory and EEG data. Adaptive Simulated
Annealing (ASA) has been developed to perform fits to such nonlinear stochastic
systems. An N-dimensional path-integral algorithm for quantum systems,
qPATHINT, has been developed from classical PATHINT. Both fold short-time
propagators (distributions or wave functions) over long times. Previous papers
applied qPATHINT to two systems, in neocortical interactions and financial
options. \textbf{Objective}: In this paper the quantum path-integral for
Calcium ions is used to derive a closed-form analytic solution at arbitrary
time that is used to calculate interactions with classical-physics SMNI
interactions among scales. Using fits of this SMNI model to EEG data, including
these effects, will help determine if this is a reasonable approach.
\textbf{Method}: Methods of mathematical-physics for optimization and for path
integrals in classical and quantum spaces are used for this project. Studies
using supercomputer resources tested various dimensions for their scaling
limits. In this paper the quantum path-integral is used to derive a closed-form
analytic solution at arbitrary time that is used to calculate interactions with
classical-physics SMNI interactions among scales. \textbf{Results}: The
mathematical-physics and computer parts of the study are successful, in that
there is modest improvement of cost/objective functions used to fit EEG data
using these models. \textbf{Conclusion}: This project points to directions for
more detailed calculations using more EEG data and qPATHINT at each time slice
to propagate quantum calcium waves, synchronized with PATHINT propagation of
classical SMNI.Comment: published in Sc
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