The spectral density of various ensembles sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: whose associated graphs are locally treelike, and covariance matrices. derive a closed set equations from which eigenvalues can be efficiently calculated. Within this approach, Wigner semicircle law for Gaussian Marcenko-Pastur recovered easily. Our results compared with numerical diagonalization, showing excellent agreement.

I INTRODUCTION TO NEURAL NETWORKS 1. General introduction 2. Layered networks 3. Recurrent with binary neurons II ADVANCED 4. Competitive unsupervised learning processes 5. Bayesian techniques in supervised 6. Gaussian 7. Support vector machines for classification III INFORMATION THEORY AND 8. Measuring information 9. Identification of entropy as an measure 10. Building blocks Shannon's theory 11. Information and statistical inference 12. Applications to neural IV MACROSCOPIC ANALYSIS OF...

We compute the spectral density for ensembles of sparse symmetric random matrices using replica. Our formulation replica-symmetric ansatz shares symmetries that suggested in a seminal paper by Rodgers and Bray (symmetry with respect to permutation replica rotation symmetry space replica), but uses different representation terms superpositions Gaussians. It gives rise pair integral equations which can be solved stochastic population-dynamics algorithm. Remarkably our allows us identify...

Exact deterministic equations for the evolution of temporal sequences with a delay stabilization and heat-bath dynamics are derived. The temperature may be arbitrary both parallel sequential updating allowed. display limit-cycle oscillations broad diversity waveforms. For exponential it is shown how an increase transition amplitude leads to local chaos.

Abstract Water transport in an operating PEM fuel cell was investigated with synchrotron X‐ray radiography a spatial resolution of 3 μm and temporal 5 s. This method allows for the detection water accumulations less than 10 diameter. We demonstrate that imaging can dramatically expand possibilities high time resolution, especially as complement to neutron radiography. processes from first appearance small gas diffusion layer their into channel system were analysed situ . Correlations between...

The photodissociation spectrum of Ar+3 between 520 and 620 nm is reported. A broadband peaking near observed with a cross section ∼1.8×10−16 cm2 at the peak, in agreement findings Levinger et al. [J. Chem. Phys. 89, 71 (1988)]. However, present work, shoulder whose prominence highly temperature dependent 545 555 nm. new results are discussed terms various electronic transitions pathways to dissociation. mechanisms studied by translational energy analysis photofragmentation. Three that...

The Hebb rule (Hebb, 1949) indicates how information presented to a neural network during learning session is stored in the synapses, local elements which act as mediators between neurons. In this paper we demonstrate that can be used handle both stationary and dynamic objects such single patterns cycles. two main ideas are: a) broad distribution of delays they occur natural dynamics b) incorporation very same session. Our work shows resulting procedure robust faithful.

We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erdos-R\'enyi networks, based on recursion equations shells around a reference node and paths originating from it. The results are agreement with numerical simulations broad range network sizes connectivities. average standard deviation also obtained. In case that mean degree scales as $N^{\alpha}$ size, becomes extremely narrow asymptotic limit, namely almost all pairs nodes...

We review the problem of how to compute spectral density sparse symmetric random matrices, i.e. weighted adjacency matrices undirected graphs. Starting from Edwards-Jones formula, we illustrate milestones this line research, including pioneering work Bray and Rodgers using replicas. focus first on cavity method, showing that it quickly provides correct recursion equations both for single instances at ensemble level. also describe an alternative replica solution proves be equivalent method....

Abstract We present analytical results for the joint probability distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>FR</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mi>s</mml:mi> stretchy="false">)</mml:mo> </mml:math> of first return (FR) times t and number distinct sites s...

The photodissociation spectrum of the mass selected cluster ion Ar+3, produced by electron impact ionization a supersonic beam, has been measured for first time. Measurements over wavelength range 539 to 620 nm show cross section rise and peak near 545 then decrease steadily throughout longer range; magnitude is found be smaller than corresponding 2Σ+u→2Σ+g transition Ar+2 in agreement with some theoretical predictions. Interestingly, Ar+ only detected photoproduct.

The critical behaviour of the randomly spin-diluted Ising model in two space dimensions is investigated by a new method which combines grand ensemble approach to disordered systems proposed Morita with phenomenological renormalization group scheme Nightingale. Accurate approximations for phase diagram and connectivity length exponent percolation transition are obtained. Our results suggest that thermal system might be different from pure system: we observe continuous variation exponents...

We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter- intra-module degree distributions, subject only to the constraint finite mean connectivities. also evaluate a certain class small-world generated from by introducing shortcuts via additional connectivity components. Both adjacency associated graph Laplacians are investigated. For Laplacians, we find Lifshitz-type singular behaviour spectral density in localized region small...

We present analytical results for the distribution of shortest path lengths between random pairs nodes in configuration model networks. The results, which are based on recursion equations, shown to be good agreement with numerical simulations networks degenerate, binomial, and power-law degree distributions. mean, mode, variance also evaluated. These provide expressions central measures dispersion terms moments distribution, illuminating connection two

Conjugated polymers are employed in a variety of application areas due to their bright fluorescence and strong biocompatibility. However, understanding the structure amorphous conjugated on nanoscale is extremely challenging compared related crystalline phases. Using bespoke classical force field, we study poly(9,9-di-

The decay of (Ar+3)*→Ar+2+Ar is found to be metastable on a time scale longer than 40 μs. Theoretical considerations are presented for these slow dissociation rates reported in this and other studies. From model calculations it inferred that tunneling lifetimes from 10−10 10 s may observed.

Click to increase image sizeClick decrease size Acknowledgements We thank Peter Neu for useful discussions and helpful remarks. This paper has been prepared by the authors as a commentary on topic at September 2006 is not definitive analysis of subject matter covered. It does constitute advice or guidance in this area should be relied upon such. The J.H. his capacity an employee Hymans Robertson LLP views expressed do represent those LLP. Neither nor accept liability any errors omissions.