116 research outputs found
Some of my Favourite Problems in Number Theory, Combinatorics, and Geometry
To the memor!l of m!l old friend Professor George Sved.I heard of his untimel!l death while writing this paper
STAND: A Spatio-Temporal Algorithm for Network Diffusion Simulation
Information, ideas, and diseases, or more generally, contagions, spread over
space and time through individual transmissions via social networks, as well as
through external sources. A detailed picture of any diffusion process can be
achieved only when both a good network structure and individual diffusion
pathways are obtained. The advent of rich social, media and locational data
allows us to study and model this diffusion process in more detail than
previously possible. Nevertheless, how information, ideas or diseases are
propagated through the network as an overall process is difficult to trace.
This propagation is continuous over space and time, where individual
transmissions occur at different rates via complex, latent connections.
To tackle this challenge, a probabilistic spatiotemporal algorithm for
network diffusion (STAND) is developed based on the survival model in this
research. Both time and spatial distance are used as explanatory variables to
simulate the diffusion process over two different network structures. The aim
is to provide a more detailed measure of how different contagions are
transmitted through various networks where nodes are geographic places at a
large scale
Universality for a class of random band matrices
We prove the universality for the eigenvalue gap statistics in the bulk of
the spectrum for band matrices, in the regime where the band width is
comparable with the dimension of the matrix, . All previous results
concerning universality of non-Gaussian random matrices are for mean-field
models. By relying on a new mean-field reduction technique, we deduce
universality from quantum unique ergodicity for band matrices
Bulk Universality of General -Ensembles with Non-convex Potential
We prove the bulk universality of the -ensembles with non-convex
regular analytic potentials for any . This removes the convexity
assumption appeared in our earlier work. The convexity condition enabled us to
use the logarithmic Sobolev inequality to estimate events with small
probability. The new idea is to introduce a "convexified measure" so that the
local statistics are preserved under this convexification.Comment: arXiv admin note: text overlap with arXiv:1104.227
Universality of General -Ensembles
We prove the universality of the -ensembles with convex analytic
potentials and for any , i.e. we show that the spacing distributions
of log-gases at any inverse temperature coincide with those of the
Gaussian -ensembles.Comment: Sep 21, 2011: Lemma 3.9 was corrected. Feb 5: Some typos in Lemma 5.9
were correcte
Recommended from our members
Universality of general β-ensembles
We prove the universality of the β-ensembles with convex analytic potentials and for any β>0; that is, we show that the spacing distributions of log-gases at any inverse temperature β coincide with those of the Gaussian β-ensembles.Mathematic
The Landauer Resistance and Band Spectra for the Counting Quantum Turing Machine
The generalized counting quantum Turing machine (GCQTM) is a machine which,
for any N, enumerates the first integers in succession as binary
strings. The generalization consists of associating a potential with read-1
steps only. The Landauer Resistance (LR) and band spectra were determined for
the tight binding Hamiltonians associated with the GCQTM for energies both
above and below the potential height. For parameters and potentials in the
electron region, the LR fluctuates rapidly between very high and very low
values as a function of momentum. The rapidity and extent of the fluctuations
increases rapidly with increasing N. For N=18, the largest value considered,
the LR shows good transmission probability as a function of momentum with
numerous holes of very high LR values present. This is true for energies above
and below the potential height. It is suggested that the main features of the
LR can be explained by coherent superposition of the component waves reflected
from or transmitted through the potentials in the distribution. If
this explanation is correct, it provides a dramatic illustration of the effects
of quantum nonlocality.Comment: 19 pages Latex, elsart.sty file included, 12 postscript figures,
Submitted to PhysComp96 for publication in Physica-
Transmission and Spectral Aspects of Tight Binding Hamiltonians for the Counting Quantum Turing Machine
It was recently shown that a generalization of quantum Turing machines
(QTMs), in which potentials are associated with elementary steps or transitions
of the computation, generates potential distributions along computation paths
of states in some basis B. The distributions are computable and are thus
periodic or have deterministic disorder. These generalized machines (GQTMs) can
be used to investigate the effect of potentials in causing reflections and
reducing the completion probability of computations. This work is extended here
by determination of the spectral and transmission properties of an example GQTM
which enumerates the integers as binary strings. A potential is associated with
just one type of step. For many computation paths the potential distributions
are initial segments of a quasiperiodic distribution that corresponds to a
substitution sequence. The energy band spectra and Landauer Resistance (LR) are
calculated for energies below the barrier height by use of transfer matrices.
The LR fluctuates rapidly with momentum with minima close to or at band-gap
edges. For several values of the parameters, there is good transmission over
some momentum regions.Comment: 22 pages Latex, 13 postscript figures, Submitted to Phys. Rev.
Quantum Ballistic Evolution in Quantum Mechanics: Application to Quantum Computers
Quantum computers are important examples of processes whose evolution can be
described in terms of iterations of single step operators or their adjoints.
Based on this, Hamiltonian evolution of processes with associated step
operators is investigated here. The main limitation of this paper is to
processes which evolve quantum ballistically, i.e. motion restricted to a
collection of nonintersecting or distinct paths on an arbitrary basis. The main
goal of this paper is proof of a theorem which gives necessary and sufficient
conditions that T must satisfy so that there exists a Hamiltonian description
of quantum ballistic evolution for the process, namely, that T is a partial
isometry and is orthogonality preserving and stable on some basis. Simple
examples of quantum ballistic evolution for quantum Turing machines with one
and with more than one type of elementary step are discussed. It is seen that
for nondeterministic machines the basis set can be quite complex with much
entanglement present. It is also proved that, given a step operator T for an
arbitrary deterministic quantum Turing machine, it is decidable if T is stable
and orthogonality preserving, and if quantum ballistic evolution is possible.
The proof fails if T is a step operator for a nondeterministic machine. It is
an open question if such a decision procedure exists for nondeterministic
machines. This problem does not occur in classical mechanics.Comment: 37 pages Latexwith 2 postscript figures tar+gzip+uuencoded, to be
published in Phys. Rev.
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