Typical kernel size and number of sparse random matrices over Galois fields: a statistical physics approach

Using methods of statistical physics, we study the average number and kernel size of general sparse random matrices over GF(q), with a given connectivity profile, in the thermodynamical limit of large matrices. We introduce a mapping of GF(q) matrices onto spin systems using the representation of the cyclic group of order q as the q-th complex roots of unity. This representation facilitates the derivation of the average kernel size of random matrices using the replica approach, under the replica symmetric ansatz, resulting in saddle point equations for general connectivity distributions. Numerical solutions are then obtained for particular cases by population dynamics. Similar techniques also allow us to obtain an expression for the exact and average number of random matrices for any general connectivity profile. We present numerical results for particular distributions.

Spectra of modular and small-world matrices

We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter- and intra-module degree distributions, subject only to the constraint of finite mean connectivities. We also evaluate spectra of a certain class of small-world matrices generated from random graphs by introducing shortcuts via additional random connectivity components. Both adjacency matrices and the associated graph Laplacians are investigated. For the Laplacians, we find Lifshitz-type singular behaviour of the spectral density in a localized region of small |?| values. In the case of modular networks, we can identify contributions of local densities of state from individual modules. For small-world networks, we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states, exhibiting only localized states in the band gaps. Results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances, and with direct diagonalization results.

Typical performance of low-density parity-check codes over general symmetric channels

Typical performance of low-density parity-check (LDPC) codes over a general binary-input output-symmetric memoryless channel is investigated using methods of statistical mechanics. The binary-input additive-white-Gaussian-noise channel and the binary-input Laplace channel are considered as specific channel noise models.

Magnetization enumerator for real-valued symmetric channels in Gallager error-correcting codes

Using the magnetization enumerator method, we evaluate the practical and theoretical limitations of symmetric channels with real outputs. Results are presented for several regular Gallager code constructions.

Importance of tissue transglutaminase in repair of extracellular matrices and cell death of dermal fibroblasts after exposure to a solarium ultraviolet A source

Investigations were undertaken to study the role of the protein cross-linking enzyme tissue transglutaminase in changes associated with the extracellular matrix and in the cell death of human dermal fibroblasts following exposure to a solarium ultraviolet A source consisting of 98.8% ultraviolet A and 1.2% ultraviolet B. Exposure to nonlethal ultraviolet doses of 60 to 120 kJ per m2 resulted in increased tissue transglutaminase activity when measured either in cell homogenates, "in situ" by incorporation of fluorescein-cadaverine into the extracellular matrix or by changes in the epsilon(gamma-glutamyl) lysine cross-link. This increase in enzyme activity did not require de novo protein synthesis. Incorporation of fluorescein-cadaverine into matrix proteins was accompanied by the cross-linking of fibronectin and tissue transglutaminase into nonreducible high molecular weight polymers. Addition of exogenous tissue transglutaminase to cultured cells mimicking extensive cell leakage of the enzyme resulted in increased extracellular matrix deposition and a decreased rate of matrix turnover. Exposure of cells to 180 kJ per m2 resulted in 40% to 50% cell death with dying cells showing extensive tissue transglutaminase cross-linking of intracellular proteins and increased cross-linking of the surrounding extracellular matrix, the latter probably occurring as a result of cell leakage of tissue transglutaminase. These cells demonstrated negligible caspase activation and DNA fragmentation but maintained their cell morphology. In contrast, exposure of cells to 240 kJ per m2 resulted in increased cell death with caspase activation and some DNA fragmentation. These cells could be partially rescued from death by addition of caspase inhibitors. These data suggest that changes in cross-linking both in the intracellular and extracellular compartments elicited by tissue transglutaminase following exposure to ultraviolet provides a rapid tissue stabilization process following damage, but as such may be a contributory factor to the scarring process that results.

Integration of the Manakov-PMD Equation with Precomputed M(w) Matrices for PMD Simulation

A novel direct integration technique of the Manakov-PMD equation for the simulation of polarisation mode dispersion (PMD) in optical communication systems is demonstrated and shown to be numerically as efficient as the commonly used coarse-step method. The main advantage of using a direct integration of the Manakov-PMD equation over the coarse-step method is a higher accuracy of the PMD model. The new algorithm uses precomputed M(w) matrices to increase the computational speed compared to a full integration without loss of accuracy. The simulation results for the probability distribution function (PDF) of the differential group delay (DGD) and the autocorrelation function (ACF) of the polarisation dispersion vector for varying numbers of precomputed M(w) matrices are compared to analytical models and results from the coarse-step method. It is shown that the coarse-step method achieves a significantly inferior reproduction of the statistical properties of PMD in optical fibres compared to a direct integration of the Manakov-PMD equation.

Symmetric and asymmetric evaluation of the Tokamap in comparison with direct symplectic integration

Comparing the Poincaré plots of the Tokamap and the underlying Hamiltonian system reveals large differences. This stems from the particular choice of evaluation of the singular perturbations present in the system (a series of δ functions). A symmetric evaluation approach is proposed and shown to yield results that almost perfectly match the Hamiltonian system. © 2005 The American Physical Society.

Numerical implementation of the Manakov-PMD equation with precomputed M(Ω) matrices

The Manakov-PMD equation can be integrated with the same numerical efficiency as the coarse-step method by using precomputed M(Ω) matrices, which entirely avoids the somewhat ad-hoc rescaling of coefficients necessary in the coarse-step method.

Properties of sparse random matrices over finite fields

Typical properties of sparse random matrices over finite (Galois) fields are studied, in the limit of large matrices, using techniques from the physics of disordered systems. For the case of a finite field GF(q) with prime order q, we present results for the average kernel dimension, average dimension of the eigenvector spaces and the distribution of the eigenvalues. The number of matrices for a given distribution of entries is also calculated for the general case. The significance of these results to error-correcting codes and random graphs is also discussed.

Modified hydrogel matrices in fibre optic sensors

Common problems encountered in clinical sensing are those of non-biocompatibility, and slow response time of the device. The latter, also applying to chemical sensors, is possibly due to a lack of understanding of polymer support or membrane properties and hence failure to optimise membranes chosen for specific sensor applications. Hydrogels can be described as polymers which swell in water. In addition to this, the presence of water in the polymer matrix offers some control of biocompatibility. They thus provide a medium through which rapid transport of a sensed species to an incorporated reagent could occur. This work considers the feasibility of such a system, leading to the design and construction of an optical sensor test bed. The development of suitable membrane systems and of suitable coating techniques in order to apply them to the fibre optics is described. Initial results obtained from hydrogel coatings implied that the refractive index change in the polymer matrix, due to a change in water content with pH is the major factor contributing to the sensor response. However the presence of the colourimetric reagent was also altering the output signal obtained. An analysis of factors contributing to the overall response, such as colour change and membrane composition were made on both the test bed, via optical response, and on whole membranes via measurement of water content change. The investigation of coatings with low equilibrium water contents, of less than 10% was carried out and in fact a clearer signal response from the test bed was noted. Again these membranes were suprisingly responding via refractive index change, with the reagent playing a primary role in obtaining a sensible or non-random response, although not in a colourimetric fashion. A photographic study of these coatings revealed some clues as to the physical nature of these coatings and hence partially explained this phenomenon. A study of the transport properties of the most successful membrane, on a coated wire electrode and also on the fibre optic test bed, in a series of test environments, indicated that the reagent was possibly acting as an ion exchanger and hence having a major influence on transport and therefore sensor characteristics.

Product cipher negotiation with on-line evaluation for private communication over computer networks

A method is proposed to offer privacy in computer communications, using symmetric product block ciphers. The security protocol involved a cipher negotiation stage, in which two communicating parties select privately a cipher from a public cipher space. The cipher negotiation process includes an on-line cipher evaluation stage, in which the cryptographic strength of the proposed cipher is estimated. The cryptographic strength of the ciphers is measured by confusion and diffusion. A method is proposed to describe quantitatively these two properties. For the calculation of confusion and diffusion a number of parameters are defined, such as the confusion and diffusion matrices and the marginal diffusion. These parameters involve computationally intensive calculations that are performed off-line, before any communication takes place. Once they are calculated, they are used to obtain estimation equations, which are used for on-line, fast evaluation of the confusion and diffusion of the negotiated cipher. A technique proposed in this thesis describes how to calculate the parameters and how to use the results for fast estimation of confusion and diffusion for any cipher instance within the defined cipher space.

Drug/Water interactions in hydrogel matrices

The primary aim of this research has been the investigation of the role of water structuring effects in the widely different extents of irritancy displayed by certain antibiotics. The compounds involved were members of the Lincomycin group of antibiotics. The aqueous solution behaviour of these co~pounds was studied using techniques such as vapour pressure osmometry end differential scanning calorimetry (D.S.C.). The effects of the antibiotics on water structure in hydrogel membrane preparations In which the equilibrium water content (E.W.C.) and constituent amounts of freezing and non-freezing water ware varied were also investigated using D.S.C. The permeability of water swollen hydrogel preparations to aqueous antibiotic solutions as well as other solutes were studied. A series of hydrogel preparations into which the antibiotics had been incorporated during polymerisation were developed and used in studies of the effects of the antibiotics end their water structure modifications on the permeation of a range of solutes.

The controlled release of macromolecules from macroporous hydrophyllic polymer matrices

This work has used novel polymer design and fabrication technology to generate bead form polymer based systems, with variable, yet controlled release properties, specifically for the delivery of macromolecules, essentially peptides of therapeutic interest. The work involved investigation of the potential interaction between matrix ultrastructural morphology, in vitro release kinetics, bioactivity and immunoreactivity of selected macromolecules with limited hydrolytic stability, delivered from controlled release vehicles. The underlying principle involved photo-polymerisation of the monomer, hydroxyethyl methacrylate, around frozen ice crystals, leading to the production of a macroporous hydrophilic matrix. Bead form matrices were fabricated in controllable size ranges in the region of 100µm - 3mm in diameter. The initial stages of the project involved the study of how variables, delivery speed of the monomer and stirring speed of the non solvent, affectedthe formation of macroporous bead form matrices. From this an optimal bench system for bead production was developed. Careful selection of monomer, solvents, crosslinking agent and polymerisation conditions led to a variable but controllable distribution of pore sizes (0.5 - 4µm). Release of surrogate macromolecules, bovine serum albumin and FITC-linked dextrans, enabled factors relating to the size and solubility of the macromolecule on the rate of release to be studied. Incorporation of bioactive macromolecules allowed retained bioactivity to be determined (glucose oxidase and interleukin-2), whilst the release of insulin enabled determination of both bioactivity (using rat epididymal fat pad) and immunoreactivity (RIA). The work carried out has led to the generation of macroporous bead form matrices, fabricated from a tissue biocompatible hydrogel, capable of the sustained, controlled release of biologically active peptides, with potential use in the pharmaceutical and agrochemical industries.

The durability of glass fibre reinforced cements made with new cementitious matrices

Widespread use of glass fibre reinforced cement (GRC) has been impeded by concerns over its durability. Three degradation mechanisms are proposed - fibre corrosion, Ca(OHh precipitation and matrix densification - although their relative importance is debated. Matrices with reduced alkalinities and Ca(OH)2 contents are being developed; the aim of this study was to investigate their hydration and interaction with alkali-resistant fibres to determine the factors controlling their long-term durability, and assess the relevancy of accelerated ageing. The matrices studied were: OPC/calcium-sulphoaluminate cement plus metakaolin (C); OPC plus metakaolin (M); blast-furnace slag cement plus a micro-silica based additive (D); and OPC (O). Accelerated ageing included hot water and cyclic regimes prior to tensile testing. Investigations included pore solution expression, XRD, DTA/TG, SEM and optical petrography. Bond strength was determined from crack spacings using microstructural parameters obtained from a unique image analysis technique. It was found that, for the new matrices - pore solution alkalinities were lower; Ca(OH)2 was absent or quickly consumed; different hydrates were formed at higher immersion temperatures; degradation under 65°C immersion was an order of magnitude slower, and no interfilamental Ca(OH)2 was observed .It was concluded that: fibre weakening caused by flaw growth was the primary degradation mechanism and was successfully modelled on stress corrosion/static fatigue principles. OPC inferiority was attributed partly to its higher alkalinity but chiefly to the growth of Ca(OH)2 aggravating the degradation; and hot water ageing although useful in model formulation and contrasting the matrices, changed the intrinsic nature of the composites rather than simply accelerating the degradation mechanisms.