1,573 research outputs found
The braiding for representations of q-deformed affine
We compute the braiding for the `principal gradation' of for from first principles, starting from the idea of a rigid
braided tensor category. It is not necessary to assume either the crossing or
the unitarity condition from S-matrix theory. We demonstrate the uniqueness of
the normalisation of the braiding under certain analyticity assumptions, and
show that its convergence is critically dependent on the number-theoretic
properties of the number in the deformation parameter . We also examine the convergence using probability, assuming a uniform
distribution for on the unit circle.Comment: LaTeX, 10 pages with 2 figs, uses epsfi
Predicting criticality and dynamic range in complex networks: effects of topology
The collective dynamics of a network of coupled excitable systems in response
to an external stimulus depends on the topology of the connections in the
network. Here we develop a general theoretical approach to study the effects of
network topology on dynamic range, which quantifies the range of stimulus
intensities resulting in distinguishable network responses. We find that the
largest eigenvalue of the weighted network adjacency matrix governs the network
dynamic range. Specifically, a largest eigenvalue equal to one corresponds to a
critical regime with maximum dynamic range. We gain deeper insight on the
effects of network topology using a nonlinear analysis in terms of additional
spectral properties of the adjacency matrix. We find that homogeneous networks
can reach a higher dynamic range than those with heterogeneous topology. Our
analysis, confirmed by numerical simulations, generalizes previous studies in
terms of the largest eigenvalue of the adjacency matrix.Comment: 4 pages, 3 figure
Effects of network topology, transmission delays, and refractoriness on the response of coupled excitable systems to a stochastic stimulus
We study the effects of network topology on the response of networks of
coupled discrete excitable systems to an external stochastic stimulus. We
extend recent results that characterize the response in terms of spectral
properties of the adjacency matrix by allowing distributions in the
transmission delays and in the number of refractory states, and by developing a
nonperturbative approximation to the steady state network response. We confirm
our theoretical results with numerical simulations. We find that the steady
state response amplitude is inversely proportional to the duration of
refractoriness, which reduces the maximum attainable dynamic range. We also
find that transmission delays alter the time required to reach steady state.
Importantly, neither delays nor refractoriness impact the general prediction
that criticality and maximum dynamic range occur when the largest eigenvalue of
the adjacency matrix is unity
Bandit Online Optimization Over the Permutahedron
The permutahedron is the convex polytope with vertex set consisting of the
vectors for all permutations (bijections) over
. We study a bandit game in which, at each step , an
adversary chooses a hidden weight weight vector , a player chooses a
vertex of the permutahedron and suffers an observed loss of
.
A previous algorithm CombBand of Cesa-Bianchi et al (2009) guarantees a
regret of for a time horizon of . Unfortunately,
CombBand requires at each step an -by- matrix permanent approximation to
within improved accuracy as grows, resulting in a total running time that
is super linear in , making it impractical for large time horizons.
We provide an algorithm of regret with total time
complexity . The ideas are a combination of CombBand and a recent
algorithm by Ailon (2013) for online optimization over the permutahedron in the
full information setting. The technical core is a bound on the variance of the
Plackett-Luce noisy sorting process's "pseudo loss". The bound is obtained by
establishing positive semi-definiteness of a family of 3-by-3 matrices
generated from rational functions of exponentials of 3 parameters
Differential and holomorphic differential operators on noncommutative algebras
Abstract This paper deals with sheaves of differential operators on noncommutative algebras, in a manner related to the classical theory of D-modules. The sheaves are defined by quotienting the tensor algebra of vector fields (suitably deformed by a covariant derivative). As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of a category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considered
Periodic Neural Activity Induced by Network Complexity
We study a model for neural activity on the small-world topology of Watts and
Strogatz and on the scale-free topology of Barab\'asi and Albert. We find that
the topology of the network connections may spontaneously induce periodic
neural activity, contrasting with chaotic neural activities exhibited by
regular topologies. Periodic activity exists only for relatively small networks
and occurs with higher probability when the rewiring probability is larger. The
average length of the periods increases with the square root of the network
size.Comment: 4 pages, 5 figure
Adaptive self-organization in a realistic neural network model
Information processing in complex systems is often found to be maximally
efficient close to critical states associated with phase transitions. It is
therefore conceivable that also neural information processing operates close to
criticality. This is further supported by the observation of power-law
distributions, which are a hallmark of phase transitions. An important open
question is how neural networks could remain close to a critical point while
undergoing a continual change in the course of development, adaptation,
learning, and more. An influential contribution was made by Bornholdt and
Rohlf, introducing a generic mechanism of robust self-organized criticality in
adaptive networks. Here, we address the question whether this mechanism is
relevant for real neural networks. We show in a realistic model that
spike-time-dependent synaptic plasticity can self-organize neural networks
robustly toward criticality. Our model reproduces several empirical
observations and makes testable predictions on the distribution of synaptic
strength, relating them to the critical state of the network. These results
suggest that the interplay between dynamics and topology may be essential for
neural information processing.Comment: 6 pages, 4 figure
Self-Organized Criticality model for Brain Plasticity
Networks of living neurons exhibit an avalanche mode of activity,
experimentally found in organotypic cultures. Here we present a model based on
self-organized criticality and taking into account brain plasticity, which is
able to reproduce the spectrum of electroencephalograms (EEG). The model
consists in an electrical network with threshold firing and activity-dependent
synapse strenghts. The system exhibits an avalanche activity power law
distributed. The analysis of the power spectra of the electrical signal
reproduces very robustly the power law behaviour with the exponent 0.8,
experimentally measured in EEG spectra. The same value of the exponent is found
on small-world lattices and for leaky neurons, indicating that universality
holds for a wide class of brain models.Comment: 4 pages, 3 figure
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