Typical case behaviour of spin systems in random graph and composite ensembles

This thesis includes analysis of disordered spin ensembles corresponding to Exact Cover, a multi-access channel problem, and composite models combining sparse and dense interactions. The satisfiability problem in Exact Cover is addressed using a statistical analysis of a simple branch and bound algorithm. The algorithm can be formulated in the large system limit as a branching process, for which critical properties can be analysed. Far from the critical point a set of differential equations may be used to model the process, and these are solved by numerical integration and exact bounding methods. The multi-access channel problem is formulated as an equilibrium statistical physics problem for the case of bit transmission on a channel with power control and synchronisation. A sparse code division multiple access method is considered and the optimal detection properties are examined in typical case by use of the replica method, and compared to detection performance achieved by interactive decoding methods. These codes are found to have phenomena closely resembling the well-understood dense codes. The composite model is introduced as an abstraction of canonical sparse and dense disordered spin models. The model includes couplings due to both dense and sparse topologies simultaneously. The new type of codes are shown to outperform sparse and dense codes in some regimes both in optimal performance, and in performance achieved by iterative detection methods in finite systems.

Design of a flat-top fiber Bragg filter via quasi-random modulation of the refractive index

The statistics of the reflection spectrum of a short-correlated disordered fiber Bragg grating are studied. The averaged spectrum appears to be flat inside the bandgap and has significantly suppressed sidelobes compared to the uniform grating of the same bandwidth. This is due to the Anderson localization of the modes of a disordered grating. This observation prompts a new algorithm for designing passband reflection gratings. Using the stochastic invariant imbedding approach it is possible to obtain the probability distribution function for the random reflection coefficient inside the bandgap and obtain both the variance of the averaged reflectivity as well as the distribution of the time delay of the grating.

Random input problem for the nonlinear Schrödinger equation

We consider the random input problem for a nonlinear system modeled by the integrable one-dimensional self-focusing nonlinear Schrödinger equation (NLSE). We concentrate on the properties obtained from the direct scattering problem associated with the NLSE. We discuss some general issues regarding soliton creation from random input. We also study the averaged spectral density of random quasilinear waves generated in the NLSE channel for two models of the disordered input field profile. The first model is symmetric complex Gaussian white noise and the second one is a real dichotomous (telegraph) process. For the former model, the closed-form expression for the averaged spectral density is obtained, while for the dichotomous real input we present the small noise perturbative expansion for the same quantity. In the case of the dichotomous input, we also obtain the distribution of minimal pulse width required for a soliton generation. The obtained results can be applied to a multitude of problems including random nonlinear Fraunhoffer diffraction, transmission properties of randomly apodized long period Fiber Bragg gratings, and the propagation of incoherent pulses in optical fibers.

Persistence in a random bond ising model of socio-econo dynamics

We study the persistence phenomenon in a socio-econo dynamics model using computer simulations at a nite temperature on hypercubic lattices in dimensions up to ve. The model includes a \social" local eld which contains the magnetization at time t. The nearest neighbour quenched interactions are drawn from a binary distribution which is a function of the bond concentration, p. The decay of the persistence probability in the model depends on both the spatial dimension and p. We nd no evidence of \blocking" in this model. We also discuss the implications of our results for possible applications in the social and economic elds. It is suggested that the absence, or otherwise, of blocking could be used as a criterion to decide on the validity of a given model in dierent scenarios.

The throughput order of multicast traffics with physical-layer network coding in random wireless ad hoc networks

This paper attempts to address the effectiveness of physical-layer network coding (PNC) on the throughput improvement for multi-hop multicast in random wireless ad hoc networks (WAHNs). We prove that the per session throughput order with PNC is tightly bounded as T((nvmR (n))-1) if m = O(R-2 (n)), where n is the total number of nodes, R(n) is the communication range, and m is the number of destinations for each multicast session. We also show that per-session throughput order with PNC is tight bounded as T(n-1), when m = O(R-2(n)). The results of this paper imply that PNC cannot improve the throughput order of multicast in random WAHNs, which is different from the intuition that PNC may improve the throughput order as it allows simultaneous signal access and combination.

From rogue waves to random walks:nonlinear instabilities in supercontinuum generation

The noise properties of supercontinuum generation continue to be a subject of wide interest within both pure and applied physics. Aside from immediate applications in supercontinuum source development, detailed studies of supercontinuum noise mechanisms have attracted interdisciplinary attention because of links with extreme instabilities in other physical systems, especially the infamous and destructive oceanic rogue waves. But the instabilities inherent in supercontinuum generation can also be interpreted in terms of natural links with the general field of random processes, and this raises new possibilities for applications in areas such as random number generation. In this contribution we will describe recent work where we interpret supercontinuum intensity and phase fluctuations in this way.

Tunable random fiber laser

An optical fiber is treated as a natural one-dimensional random system where lasing is possible due to a combination of Rayleigh scattering by refractive index inhomogeneities and distributed amplification through the Raman effect. We present such a random fiber laser that is tunable over a broad wavelength range with uniquely flat output power and high efficiency, which outperforms traditional lasers of the same category. Outstanding characteristics defined by deep underlying physics and the simplicity of the scheme make the demonstrated laser a very attractive light source both for fundamental science and practical applications.

Graphene based widely-tunable and singly-polarized pulse generation with random fiber lasers

Pulse generation often requires a stabilized cavity and its corresponding mode structure for initial phase-locking. Contrastingly, modeless cavity-free random lasers provide new possibilities for high quantum efficiency lasing that could potentially be widely tunable spectrally and temporally. Pulse generation in random lasers, however, has remained elusive since the discovery of modeless gain lasing. Here we report coherent pulse generation with modeless random lasers based on the unique polarization selectivity and broadband saturable absorption of monolayer graphene. Simultaneous temporal compression of cavity-free pulses are observed with such a polarization modulation, along with a broadly-tunable pulsewidth across two orders of magnitude down to 900 ps, a broadly-tunable repetition rate across three orders of magnitude up to 3 MHz, and a singly-polarized pulse train at 41 dB extinction ratio, about an order of magnitude larger than conventional pulsed fiber lasers. Moreover, our graphene-based pulse formation also demonstrates robust pulse-to-pulse stability and widewavelength operation due to the cavity-less feature. Such a graphene-based architecture not only provides a tunable pulsed random laser for fiber-optic sensing, speckle-free imaging, and laser-material processing, but also a new way for the non-random CW fiber lasers to generate widely tunable and singly-polarized pulses.

Random walks and random numbers from supercontinuum generation

We report a numerical study showing how the random intensity and phase fluctuations across the bandwidth of a broadband optical supercontinuum can be interpreted in terms of the random processes of random walks and Lévy flights. We also describe how the intensity fluctuations can be applied to physical random number generation. We conclude that the optical supercontinuum provides a highly versatile means of studying and generating a wide class of random processes at optical wavelengths. © 2012 Optical Society of America.

The dawn of the RNA World:toward functional complexity through ligation of random RNA oligomers

A main unsolved problem in the RNA World scenario for the origin of life is how a template-dependent RNA polymerase ribozyme emerged from short RNA oligomers obtained by random polymerization on mineral surfaces. A number of computational studies have shown that the structural repertoire yielded by that process is dominated by topologically simple structures, notably hairpin-like ones. A fraction of these could display RNA ligase activity and catalyze the assembly of larger, eventually functional RNA molecules retaining their previous modular structure: molecular complexity increases but template replication is absent. This allows us to build up a stepwise model of ligation- based, modular evolution that could pave the way to the emergence of a ribozyme with RNA replicase activity, step at which information-driven Darwinian evolution would be triggered. Copyright © 2009 RNA Society.

On the structural repertoire of pools of short, random RNA sequences

A detailed knowledge of the mapping between sequence and structure spaces in populations of RNA molecules is essential to better understand their present-day functional properties, to envisage a plausible early evolution of RNA in a prebiotic chemical environment and to improve the design of in vitro evolution experiments, among others. Analysis of natural RNAs, as well as in vitro and computational studies, show that certain RNA structural motifs are much more abundant than others, pointing out a complex relation between sequence and structure. Within this framework, we have investigated computationally the structural properties of a large pool (10 molecules) of single-stranded, 35 nt-long, random RNA sequences. The secondary structures obtained are ranked and classified into structure families. The number of structures in main families is analytically calculated and compared with the numerical results. This permits a quantification of the fraction of structure space covered by a large pool of sequences. We further show that the number of structural motifs and their frequency is highly unbalanced with respect to the nucleotide composition: simple structures such as stem-loops and hairpins arise from sequences depleted in G, while more complex structures require an enrichment of G. In general, we observe a strong correlation between subfamilies-characterized by a fixed number of paired nucleotides-and nucleotide composition. Our results are compared to the structural repertoire obtained in a second pool where isolated base pairs are prohibited. © 2008 Elsevier Ltd. All rights reserved.

Algorithms for solving large random graphs

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Investigations into responses of cylindrical shells to random acoustic excitation

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Changes in neural complexity during the perception of 3D images using random dot stereograms

One developing theme in consciousness research is that consciousness is not the product of any specific component of the brain, rather it is an emergent property of the changing patterns of connectivity between different specialised functional components. For example, the dynamic core hypothesis proposes that conscious experience requires high levels of neural complexity, where complexity is defined in terms of functional connectivity. To test this hypothesis, electroencephalography was recorded while participants were shown random dot-stereograms. Consistent with the dynamic core hypothesis, neural complexity increased as the participants changed from simply viewing the stereogram to consciously perceiving the hidden 3D image.

Dynamics of linear polymers in random media

We study phenomenological scaling theories of the polymer dynamics in random media, employing the existing scaling theories of polymer chains and the percolation statistics. We investigate both the Rouse and the Zimm model for Brownian dynamics and estimate the diffusion constant of the center-of-mass of the chain in such disordered media. For internal dynamics of the chain, we estimate the dynamic exponents. We propose similar scaling theory for the reptation dynamics of the chain in the framework of Flory theory for the disordered medium. The modifications in the case of correlated disorders are also discussed. .