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.

Hydrogels containing linear and cyclic polyethers

Hydrogels are a unique class of polymer which swell, but do not dissolve in, water. A range of 2-hydroxyethyl methacrylate based copolymer hydrogels containing both cyclic and linear polyethers have been synthesised and are described in this thesis. Initially, cyclic polyethers were occluded within the polymer matrix and the transport properties investigated. The results indicated that the presence of an ionophore can be used to modulate ion transport and that ion transport is described by a dual-sorption mechanism. However, these studies were limited due to ionophore loss during hydration. Hence, the synthesis of a range of acrylate based crown ether monomers was considered. A pure sample of 4-acryolylaminobenzo-15-crown-5 was obtained and a terpolymer containing this monomer was prepared. Transport studies illustrated that the presence of a `bound' ionophore modulates ion transport in a similar way to the occluded systems. The transport properties of a series of terpolymers containing linear polyethers were then investigated. The results indicated that the dual-sorption mechanism is observed for these systems with group II metal cations while the transport of group I metal cations, with the exception of sodium, is enhanced. Finally, the equilibrium water contents (EWC) surface and mechanical properties of these terpolymers containing linear polyethers were examined. Although subtle variations in EWC are observed as the structure of the polyether side chain varies, generally EWC is enhanced due to the hydrophilicity of the polyether side chain. The macroscopic surface properties were investigated using a sessile drop technique and FTIR spectroscopy. At a molecular level surface properties were probed using an in vitro ocular spoilation model and preliminary cell adhesion studies. The results indicate that the polyethylene oxide side chains are expressed at the polymer surface thus reducing the adhesion of biological species.

Cyclic and acyclic modifications of 5-aminoimidazole-4-carboxamide

The Introduction gives a brief resume' of the biologically important aspects of 5 -aminoimidazole -4 -carbozamide (1) and explores., in-depth, the synthetic routes to this imidazole. All documented reactions of 5 -aninoimidanole-4 -carboxamide are reviewed in detail, with particular emphasis on the preparation and subsequent coupling reactions of 5 –diazo-imidazole-4 -carboxamide (6). A series of thirteen novel amide 5-amino-2-arylazoimidazole-4-carboxamide derivatives (117-129) were prepared by the coupling of aryldiazonium salts with 5-aminoimidazole-4-carboxamide. Chemical modification of these azo-dyes resulted in the preparation of eight previously unknown acyl derivatives (136-143) Interaction of 5-amino-2-arylazoimidazole-4-carboxides with ethyl formate in sodium ethoxide effected pyrimidine ring closure to the novel 8-arylazohypoxanthines (144 and 145). Several reductive techniques were employed in an effort to obtain the elusive 2,5-diaminoimidazole-4-carboxamide (71),a candidate chemotherapeutic agent, from the arylazoiridazoles. No success can be reported although 5-amino-2-(3-aminoindazol-2-yl) imidazole-4-carboxamide (151) was isolated due to a partial reduction and intramolecular cyclisation of 5-amino72-(2-cyanaphenylazo)imidazole-4-carboxamide (122) .Further possible synthetic approaches to the diaminoimidazole are discussed in Chapter 4. An interesting degradation of a known unstable nitrohydrazone is described in Chapter 5.This resulted in formation of 1, 1-bis(pyrazol--3-ylazo)-1-nitroethane (164) instead of the expected cyclisation to a bicyclic tetrazine N-oxide. An improved preparation of 5-diazoinidazole-4-carboxamide has been achieved, and the diazo-azole formed cycloadducts with isocyanates to yield the hitherto unknown imidazo[5,1-d][1,2,3,5]tetrazin-7(6H)-ones. Eleven derivatives (167-177) of this new ring-system were prepared and characterised. Chemical and spectroscopic investigation showed this ring-system to be unstable under certain conditions, and a comparative study of stability within the group has been made. "Retro-cycloaddition" under protic and photolytic conditions was an unexpected property of 6-substituted imidazo[5,1-d][1,2,3,5]tetrazin--7(0)-ones.Selected examples of the imidazotetrazinone ring-system were tested for antitumour activity. The results of biological evaluation are given in Chapter 7, and have culminated in a Patent application by the collaborating body, May and Baker Ltd. One compound,3-carbamoyl-6-(2-chloro-ethyl)imidazo[5,1-d][1,2,3,5jtetrazin-7(6H)-one (175),shows striking anti-tumour activity in rodent test systems.