Error-correcting codes on a Bethe-like lattice

We analyse Gallager codes by employing a simple mean-field approximation that distorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decoding algorithm as a minimization of a proper free-energy. We find a thermodynamical phase transition that coincides with information theoretical upper-bounds and explain the practical code performance in terms of the free-energy landscape.

Lattice approach to the dynamics of phase-coded soliton trains

We address the collective dynamics of a soliton train propagating in a medium described by the nonlinear Schrödinger equation. Our approach uses the reduction of train dynamics to the discrete complex Toda chain (CTC) model for the evolution of parameters for each train constituent: such a simplification allows one to carry out an approximate analysis of the dynamics of positions and phases of individual interacting pulses. Here, we employ the CTC model to the problem which has relevance to the field of fibre optics communications where each binary digit of transmitted information is encoded via the phase difference between the two adjacent solitons. Our goal is to elucidate different scenarios of the train distortions and the subsequent information garbling caused solely by the intersoliton interactions. First, we examine how the structure of a given phase pattern affects the initial stage of the train dynamics and explain the general mechanisms for the appearance of unstable collective soliton modes. Then we further discuss the nonlinear regime concentrating on the dependence of the Lax scattering matrix on the input phase distribution; this allows one to classify typical features of the train evolution and determine the distance where the soliton escapes from its slot. In both cases, we demonstrate deep mathematical analogies with the classical theory of crystal lattice dynamics.

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.

Computing with noise: phase transitions in Boolean formulas

Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting. Existing bounds on their performance are straightforwardly retrieved, generalized, and identified as the corresponding typical-case phase transitions. Results on error rates, function depth, and sensitivity, and their dependence on the gate-type and noise model used are also obtained.

Collective behaviour in a square lattice of driven Duffing resonators coupled to van der Pol oscillators

The global and local synchronisation of a square lattice composed of alternating Duffing resonators and van der Pol oscillators coupled through displacement is studied. The lattice acts as a sensing device in which the input signal is characterised by an external driving force that is injected into the system through a subset of the Duffing resonators. The parameters of the system are taken from MEMS devices. The effects of the system parameters, the lattice architecture and size are discussed.

A preliminary study into emergent behaviours in a lattice of interacting nonlinear resonators and oscillators

Future sensor arrays will be composed of interacting nonlinear components with complex behaviours with no known analytic solutions. This paper provides a preliminary insight into the expected behaviour through numerical and analytical analysis. Specically, the complex behaviour of a periodically driven nonlinear Duffing resonator coupled elastically to a van der Pol oscillator is investigated as a building block in a 2D lattice of such units with local connectivity. An analytic treatment of the 2-device unit is provided through a two-time-scales approach and the stability of the complex dynamic motion is analysed. The pattern formation characteristics of a 2D lattice composed of these units coupled together through nearest neighbour interactions is analysed numerically for parameters appropriate to a physical realisation through MEMS devices. The emergent patterns of global and cluster synchronisation are investigated with respect to system parameters and lattice size.

Studies of the effects of molecular encounters on N.M.R. spin-lattice relaxation times

This thesis is concerned with investigations of the effects of molecular encounters on nuclear magnetic resonance spin-lattice relaxation times, with particular reference to mesitylene in mixtures with cyclohexane and TMS. The purpose of the work was to establish the best theoretical description of T1 and assess whether a recently identified mechanism (buffeting), that influences n.m.r. chemical shifts, governs Tl also. A set of experimental conditions are presented that allow reliable measurements of Tl and the N. O. E. for 1H and 13C using both C. W. and F.T. n.m.r. spectroscopy. Literature data for benzene, cyclohexane and chlorobenzene diluted by CC14 and CS2 are used to show that the Hill theory affords the best estimation of their correlation times but appears to be mass dependent. Evaluation of the T1 of the mesitylene protons indicates that a combined Hill-Bloembergen-Purcell-Pound model gives an accurate estimation of T1; subsequently this was shown to be due to cancellation of errors in the calculated intra and intemolecular components. Three experimental methods for the separation of the intra and intermolecular relaxation times are described. The relaxation times of the 13C proton satellite of neat bezene, 1,4 dioxane and mesitylene were measured. Theoretical analyses of the data allow the calculation of Tl intra. Studies of intermolecular NOE's were found to afford a general method of separating observed T1's into their intra and intermolecular components. The aryl 1H and corresponding 13C T1 values and the NOE for the ring carbon of mesitylene in CC14 and C6H12-TMS have been used in combination to determine T1intra and T1inter. The Hill and B.P.P. models are shown to predict similarly inaccurate values for T1linter. A buffeting contribution to T1inter is proposed which when applied to the BPP model and to the Gutowsky-Woessner expression for T1inter gives an inaccuracy of 12% and 6% respectively with respect to theexperimentally based T1inter.

Dynamics of Boolean networks: an exact solution

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology. We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.

Lattice approach to the dynamics of phase-coded soliton trains

We address the collective dynamics of a soliton train propagating in a medium described by the nonlinear Schrödinger equation. Our approach uses the reduction of train dynamics to the discrete complex Toda chain (CTC) model for the evolution of parameters for each train constituent: such a simplification allows one to carry out an approximate analysis of the dynamics of positions and phases of individual interacting pulses. Here, we employ the CTC model to the problem which has relevance to the field of fibre optics communications where each binary digit of transmitted information is encoded via the phase difference between the two adjacent solitons. Our goal is to elucidate different scenarios of the train distortions and the subsequent information garbling caused solely by the intersoliton interactions. First, we examine how the structure of a given phase pattern affects the initial stage of the train dynamics and explain the general mechanisms for the appearance of unstable collective soliton modes. Then we further discuss the nonlinear regime concentrating on the dependence of the Lax scattering matrix on the input phase distribution; this allows one to classify typical features of the train evolution and determine the distance where the soliton escapes from its slot. In both cases, we demonstrate deep mathematical analogies with the classical theory of crystal lattice dynamics.

Structure and physical properties of [µ-Tris(1,4-bis(tetrazol-1-yl)butane-N4,N4‘')iron(II)] Bis(hexafluorophosphate), a new Fe(II) spin-crossover compound with a three-dimensional threefold interlocked crystal lattice

[μ-Tris(1,4-bis(tetrazol-1-yl)butane-N4,N4‘)iron(II)] bis(hexafluorophosphate), [Fe(btzb)3](PF6)2, crystallizes in a three-dimensional 3-fold interlocked structure featuring a sharp two-step spin-crossover behavior. The spin conversion takes place between 164 and 182 K showing a discontinuity at about T1/2 = 174 K and a hysteresis of about 4 K between T1/2 and the low-spin state. The spin transition has been independently followed by magnetic susceptibility measurements, 57Fe-Mössbauer spectroscopy, and variable temperature far and midrange FTIR spectroscopy. The title compound crystallizes in the trigonal space group P30¯(No. 147) with a unit cell content of one formula unit plus a small amount of disordered solvent. The lattice parameters were determined by X-ray diffraction at several temperatures between 100 and 300 K. Complete crystal structures were resolved for 9 of these temperatures between 100 (only low spin, LS) and 300 K (only high spin, HS), Z = 1 [Fe(btzb)3](PF  6)2:  300 K (HS), a = 11.258(6) Å, c = 8.948(6) Å, V = 982.2(10) Å3; 100 K (LS), a = 10.989(3) Å, c = 8.702(2) Å, V = 910.1(4) Å3. The molecular structure consists of octahedral coordinated iron(II) centers bridged by six N4,N4‘ coordinating bis(tetrazole) ligands to form three 3-dimensional networks. Each of these three networks is symmetry related and interpenetrates each other within a unit cell to form the interlocked structure. The Fe−N bond lengths change between 1.993(1) Å at 100 K in the LS state and 2.193(2) Å at 300 K in the HS state. The nearest Fe separation is along the c-axis and identical with the lattice parameter c.