24,980 research outputs found
Matrices coupled in a chain. I. Eigenvalue correlations
The general correlation function for the eigenvalues of complex hermitian
matrices coupled in a chain is given as a single determinant. For this we use a
slight generalization of a theorem of Dyson.Comment: ftex eynmeh.tex, 2 files, 8 pages Submitted to: J. Phys.
Minimizing Higgs Potentials via Numerical Polynomial Homotopy Continuation
The study of models with extended Higgs sectors requires to minimize the
corresponding Higgs potentials, which is in general very difficult. Here, we
apply a recently developed method, called numerical polynomial homotopy
continuation (NPHC), which guarantees to find all the stationary points of the
Higgs potentials with polynomial-like nonlinearity. The detection of all
stationary points reveals the structure of the potential with maxima,
metastable minima, saddle points besides the global minimum. We apply the NPHC
method to the most general Higgs potential having two complex Higgs-boson
doublets and up to five real Higgs-boson singlets. Moreover the method is
applicable to even more involved potentials. Hence the NPHC method allows to go
far beyond the limits of the Gr\"obner basis approach.Comment: 9 pages, 4 figure
Universality in Complex Networks: Random Matrix Analysis
We apply random matrix theory to complex networks. We show that nearest
neighbor spacing distribution of the eigenvalues of the adjacency matrices of
various model networks, namely scale-free, small-world and random networks
follow universal Gaussian orthogonal ensemble statistics of random matrix
theory. Secondly we show an analogy between the onset of small-world behavior,
quantified by the structural properties of networks, and the transition from
Poisson to Gaussian orthogonal ensemble statistics, quantified by Brody
parameter characterizing a spectral property. We also present our analysis for
a protein-protein interaction network in budding yeast.Comment: 4+ pages, 4 figures, to appear in PRE, major change in the paper
including titl
Chiron: A Robust Recommendation System with Graph Regularizer
Recommendation systems have been widely used by commercial service providers
for giving suggestions to users. Collaborative filtering (CF) systems, one of
the most popular recommendation systems, utilize the history of behaviors of
the aggregate user-base to provide individual recommendations and are effective
when almost all users faithfully express their opinions. However, they are
vulnerable to malicious users biasing their inputs in order to change the
overall ratings of a specific group of items. CF systems largely fall into two
categories - neighborhood-based and (matrix) factorization-based - and the
presence of adversarial input can influence recommendations in both categories,
leading to instabilities in estimation and prediction. Although the robustness
of different collaborative filtering algorithms has been extensively studied,
designing an efficient system that is immune to manipulation remains a
significant challenge. In this work we propose a novel "hybrid" recommendation
system with an adaptive graph-based user/item similarity-regularization -
"Chiron". Chiron ties the performance benefits of dimensionality reduction
(through factorization) with the advantage of neighborhood clustering (through
regularization). We demonstrate, using extensive comparative experiments, that
Chiron is resistant to manipulation by large and lethal attacks
Lax Operator for the Quantised Orthosymplectic Superalgebra U_q[osp(2|n)]
Each quantum superalgebra is a quasi-triangular Hopf superalgebra, so
contains a \textit{universal -matrix} in the tensor product algebra which
satisfies the Yang-Baxter equation. Applying the vector representation ,
which acts on the vector module , to one side of a universal -matrix
gives a Lax operator. In this paper a Lax operator is constructed for the
-type quantum superalgebras . This can in turn be used to
find a solution to the Yang-Baxter equation acting on
where is an arbitrary module. The case is included
here as an example.Comment: 15 page
Generalized Mixability via Entropic Duality
Mixability is a property of a loss which characterizes when fast convergence
is possible in the game of prediction with expert advice. We show that a key
property of mixability generalizes, and the exp and log operations present in
the usual theory are not as special as one might have thought. In doing this we
introduce a more general notion of -mixability where is a general
entropy (\ie, any convex function on probabilities). We show how a property
shared by the convex dual of any such entropy yields a natural algorithm (the
minimizer of a regret bound) which, analogous to the classical aggregating
algorithm, is guaranteed a constant regret when used with -mixable
losses. We characterize precisely which have -mixable losses and
put forward a number of conjectures about the optimality and relationships
between different choices of entropy.Comment: 20 pages, 1 figure. Supersedes the work in arXiv:1403.2433 [cs.LG
Competition and cooperation:aspects of dynamics in sandpiles
In this article, we review some of our approaches to granular dynamics, now
well known to consist of both fast and slow relaxational processes. In the
first case, grains typically compete with each other, while in the second, they
cooperate. A typical result of {\it cooperation} is the formation of stable
bridges, signatures of spatiotemporal inhomogeneities; we review their
geometrical characteristics and compare theoretical results with those of
independent simulations. {\it Cooperative} excitations due to local density
fluctuations are also responsible for relaxation at the angle of repose; the
{\it competition} between these fluctuations and external driving forces, can,
on the other hand, result in a (rare) collapse of the sandpile to the
horizontal. Both these features are present in a theory reviewed here. An arena
where the effects of cooperation versus competition are felt most keenly is
granular compaction; we review here a random graph model, where three-spin
interactions are used to model compaction under tapping. The compaction curve
shows distinct regions where 'fast' and 'slow' dynamics apply, separated by
what we have called the {\it single-particle relaxation threshold}. In the
final section of this paper, we explore the effect of shape -- jagged vs.
regular -- on the compaction of packings near their jamming limit. One of our
major results is an entropic landscape that, while microscopically rough,
manifests {\it Edwards' flatness} at a macroscopic level. Another major result
is that of surface intermittency under low-intensity shaking.Comment: 36 pages, 23 figures, minor correction
- …