Solvable model for distribution networks on random graphs

We propose a simple model that captures the salient properties of distribution networks, and study the possible occurrence of blackouts, i.e., sudden failings of large portions of such networks. The model is defined on a random graph of finite connectivity. The nodes of the graph represent hubs of the network, while the edges of the graph represent the links of the distribution network. Both, the nodes and the edges carry dynamical two state variables representing the functioning or dysfunctional state of the node or link in question. We describe a dynamical process in which the breakdown of a link or node is triggered when the level of maintenance it receives falls below a given threshold. This form of dynamics can lead to situations of catastrophic breakdown, if levels of maintenance are themselves dependent on the functioning of the net, once maintenance levels locally fall below a critical threshold due to fluctuations. We formulate conditions under which such systems can be analyzed in terms of thermodynamic equilibrium techniques, and under these conditions derive a phase diagram characterizing the collective behavior of the system, given its model parameters. The phase diagram is confirmed qualitatively and quantitatively by simulations on explicit realizations of the graph, thus confirming the validity of our approach. © 2007 The American Physical Society.

Noisy random Boolean formulae:a statistical physics perspective

Properties of computing Boolean circuits composed of noisy logical gates are studied using the statistical physics methodology. A formula-growth model that gives rise to random Boolean functions is mapped onto a spin system, which facilitates the study of their typical behavior in the presence of noise. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding macroscopic phase transitions. The framework is employed for deriving results on error-rates at various function-depths and function sensitivity, and their dependence on the gate-type and noise model used. These are difficult to obtain via the traditional methods used in this field.

On the robustness of random Boolean formulae

Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics. We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models. © 2010 IOP Publishing Ltd.

Phase transitions and memory effects in the dynamics of Boolean networks

The generating functional method is employed to investigate the synchronous dynamics of Boolean networks, providing an exact result for the system dynamics via a set of macroscopic order parameters. The topology of the networks studied and its constituent Boolean functions represent the system's quenched disorder and are sampled from a given distribution. The framework accommodates a variety of topologies and Boolean function distributions and can be used to study both the noisy and noiseless regimes; it enables one to calculate correlation functions at different times that are inaccessible via commonly used approximations. It is also used to determine conditions for the annealed approximation to be valid, explore phases of the system under different levels of noise and obtain results for models with strong memory effects, where existing approximations break down. Links between Boolean networks and general Boolean formulas are identified and results common to both system types are highlighted. © 2012 Copyright Taylor and Francis Group, LLC.

Appearance of bound states in random potentials with applications to soliton theory

We analyze the stochastic creation of a single bound state (BS) in a random potential with a compact support. We study both the Hermitian Schrödinger equation and non-Hermitian Zakharov-Shabat systems. These problems are of special interest in the inverse scattering method for Korteveg–de-Vries and the nonlinear Schrödinger equations since soliton solutions of these two equations correspond to the BSs of the two aforementioned linear eigenvalue problems. Analytical expressions for the average width of the potential required for the creation of the first BS are given in the approximation of delta-correlated Gaussian potential and additionally different scenarios of eigenvalue creation are discussed for the non-Hermitian case.

Phase transitions in wave turbulence

We consider turbulence within the Gross-Pitaevsky model and look into the creation of a coherent condensate via an inverse cascade originating at small scales. The growth of the condensate leads to a spontaneous breakdown of statistical symmetries of overcondensate fluctuations: First, isotropy is broken, then a series of phase transitions marks the changing symmetry from twofold to threefold to fourfold. We describe respective anisotropic flux flows in the k space. At the highest level reached, we observe a short-range positional and long-range orientational order (as in a hexatic phase). In other words, the more one pumps the system, the more ordered the system becomes. The phase transitions happen when the system is pumped by an instability term and does not occur when pumped by a random force. We thus demonstrate nonuniversality of an inverse-cascade turbulence with respect to the nature of small-scale forcing.

Contrast sensitivity for complex and random gratings

This thesis studied the effect of (i) the number of grating components and (ii) parameter randomisation on root-mean-square (r.m.s.) contrast sensitivity and spatial integration. The effectiveness of spatial integration without external spatial noise depended on the number of equally spaced orientation components in the sum of gratings. The critical area marking the saturation of spatial integration was found to decrease when the number of components increased from 1 to 5-6 but increased again at 8-16 components. The critical area behaved similarly as a function of the number of grating components when stimuli consisted of 3, 6 or 16 components with different orientations and/or phases embedded in spatial noise. Spatial integration seemed to depend on the global Fourier structure of the stimulus. Spatial integration was similar for sums of two vertical cosine or sine gratings with various Michelson contrasts in noise. The critical area for a grating sum was found to be a sum of logarithmic critical areas for the component gratings weighted by their relative Michelson contrasts. The human visual system was modelled as a simple image processor where the visual stimuli is first low-pass filtered by the optical modulation transfer function of the human eye and secondly high-pass filtered, up to the spatial cut-off frequency determined by the lowest neural sampling density, by the neural modulation transfer function of the visual pathways. The internal noise is then added before signal interpretation occurs in the brain. The detection is mediated by a local spatially windowed matched filter. The model was extended to include complex stimuli and its applicability to the data was found to be successful. The shape of spatial integration function was similar for non-randomised and randomised simple and complex gratings. However, orientation and/or phase randomised reduced r.m.s contrast sensitivity by a factor of 2. The effect of parameter randomisation on spatial integration was modelled under the assumption that human observers change the observer strategy from cross-correlation (i.e., a matched filter) to auto-correlation detection when uncertainty is introduced to the task. The model described the data accurately.

DPSK signal regeneration with a dual-pump nondegenerate phase-sensitive amplifier

We demonstrate, for the first time to our knowledge, regeneration of a 42.66-Gb/s differential phase-shift keyed signal using a dual-pump nondegenerate four-wave-mixing-based fiber-optic parametric amplifier. The regenerative performance of the subsystem is characterized in terms of bit-error rate against narrowband and wideband introduced noise. While a strong receiver sensitivity improvement, up to 20 dB, is noticed against narrowband noise, against quasi-random (wideband) noise we observe a regeneration of 2.7 dB.

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.

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.

A new three phase method (SDP method) for the multi-objective vehicle routing problem with simultaneous delvery and pickup (VRPSDP)

Transportation service operators are witnessing a growing demand for bi-directional movement of goods. Given this, the following thesis considers an extension to the vehicle routing problem (VRP) known as the delivery and pickup transportation problem (DPP), where delivery and pickup demands may occupy the same route. The problem is formulated here as the vehicle routing problem with simultaneous delivery and pickup (VRPSDP), which requires the concurrent service of the demands at the customer location. This formulation provides the greatest opportunity for cost savings for both the service provider and recipient. The aims of this research are to propose a new theoretical design to solve the multi-objective VRPSDP, provide software support for the suggested design and validate the method through a set of experiments. A new real-life based multi-objective VRPSDP is studied here, which requires the minimisation of the often conflicting objectives: operated vehicle fleet size, total routing distance and the maximum variation between route distances (workload variation). The former two objectives are commonly encountered in the domain and the latter is introduced here because it is essential for real-life routing problems. The VRPSDP is defined as a hard combinatorial optimisation problem, therefore an approximation method, Simultaneous Delivery and Pickup method (SDPmethod) is proposed to solve it. The SDPmethod consists of three phases. The first phase constructs a set of diverse partial solutions, where one is expected to form part of the near-optimal solution. The second phase determines assignment possibilities for each sub-problem. The third phase solves the sub-problems using a parallel genetic algorithm. The suggested genetic algorithm is improved by the introduction of a set of tools: genetic operator switching mechanism via diversity thresholds, accuracy analysis tool and a new fitness evaluation mechanism. This three phase method is proposed to address the shortcoming that exists in the domain, where an initial solution is built only then to be completely dismantled and redesigned in the optimisation phase. In addition, a new routing heuristic, RouteAlg, is proposed to solve the VRPSDP sub-problem, the travelling salesman problem with simultaneous delivery and pickup (TSPSDP). The experimental studies are conducted using the well known benchmark Salhi and Nagy (1999) test problems, where the SDPmethod and RouteAlg solutions are compared with the prominent works in the VRPSDP domain. The SDPmethod has demonstrated to be an effective method for solving the multi-objective VRPSDP and the RouteAlg for the TSPSDP.

Analytical BER performance in differential n-PSK coherent transmission system influenced by equalization enhanced phase noise

Long-haul high speed optical transmission systems are significantly distorted by the interplay between the electronic chromatic dispersion (CD) equalization and the local oscillator (LO) laser phase noise, which leads to an effect of equalization enhanced phase noise (EEPN). The EEPN degrades the performance of optical communication systems severely with the increment of fiber dispersion, LO laser linewidth, symbol rate, and modulation format. In this paper, we present an analytical model for evaluating the performance of bit-error-rate (BER) versus signal-to-noise ratio (SNR) in the n-level phase shift keying (n-PSK) coherent transmission system employing differential carrier phase estimation (CPE), where the influence of EEPN is considered. Theoretical results based on this model have been investigated for the differential quadrature phase shift keying (DQPSK), the differential 8-PSK (D8PSK), and the differential 16-PSK (D16PSK) coherent transmission systems. The influence of EEPN on the BER performance in term of the fiber dispersion, the LO phase noise, the symbol rate, and the modulation format are analyzed in detail. The BER behaviors based on this analytical model achieve a good agreement with previously reported BER floors influenced by EEPN. Further simulations have also been carried out in the differential CPE considering EEPN. The results indicate that this analytical model can give an accurate prediction for the DQPSK system, and a leading-order approximation for the D8PSK and the D16PSK systems.

Sums of a Random Number of Random Variables and their Approximations with ν- Accompanying Infinitely Divisible Laws

* Research supported by NATO GRANT CRG 900 798 and by Humboldt Award for U.S. Scientists.

Foveation time measure in Congenital Nystagmus through second order approximation of the slow phases

Congenital Nystagmus (CN) is an ocular-motor disorder characterised by involuntary, conjugated ocular oscillations, and its pathogenesis is still unknown. The pathology is de fined as "congenital" from the onset time of its arise which could be at birth or in the first months of life. Visual acuity in CN subjects is often diminished due to nystagmus continuous oscillations, mainly on the horizontal plane, which disturb image fixation on the retina. However, during short periods in which eye velocity slows down while the target image is placed onto the fovea (called foveation intervals) the image of a given target can still be stable, allowing a subject to reach a higher visual acuity. In CN subjects, visual acuity is usually assessed both using typical measurement techniques (e.g. Landolt C test) and with eye movement recording in different gaze positions. The offline study of eye movement recordings allows physicians to analyse nystagmus main features such as waveform shape, amplitude and frequency and to compute estimated visual acuity predictors. This analytical functions estimates the best corrected visual acuity using foveation time and foveation position variability, hence a reliable estimation of this two parameters is a fundamental factor in assessing visual acuity. This work aims to enhance the foveation time estimation in CN eye movement recording, computing a second order approximation of the slow phase components of nystag-mus oscillations. About 19 infraredoculographic eye-movement recordings from 10 CN subjects were acquired and the visual acuity assessed with an acuity predictor was compared to the one measured in primary position. Results suggest that visual acuity measurements based on foveation time estimation obtained from interpolated data are closer to value obtained during Landolt C tests. © 2010 IEEE.