A design and a code invariant under the simple group Co3

Mathematics;mathematics

Permutation Groups and Binary Self-Orthogonal Codes

Let G be a permutation group on an n-element set Ω. We study the binary code C(G,Ω) defined as the dual code of the code spanned by the sets of fixed points of involutions of G. We show that any G-invariant self-orthogonal code of length n is contained in C(G,Ω). Many self-orthogonal codes related to sporadic simple groups, including the extended Golay code, are obtained as C(G,Ω). Some new self-dual codes invariant under sporadic almost simple groups are constructed

Some strongly regular graphs and self-orthogonal codes from the unitary group U4(3)

We construct self-orthogonal codes from the row span over F2 or F3 of the adjacency matrices of some strongly regular graphs defined by the rank-3 action of the simple unitary group U4(3) on the conjugacy classes of some of its maximal subgroups. We establish some properties of these codes and the nature of some classes of codewords

The Electronic Structure of Some Linear Molecules

This thesis contains an account of molecular electronic structure calculations performed on the linear nitrile molecules CN, HCN and FCN. The effect on the CN residue of varying X in the XCN system was observed by analysing the migration of electronic charge with the aid of population analyses and pictorial representations of the electron density distribution. The calculations were performed as accurately as was feasible using the linear combination of atomic orbitals, molecular orbital, self-consistent field procedure# The only approximation made to this scheme was that the three-centre integrals were evaluated according to approximate formulae. All other integrals were evaluated accurately either by numerical integration or by analytical methods. A minimal basis set was employed, each atomic orbital being a linear combination of Slater type orbitals of the 'double zeta' type, the orbital exponents of which had been optimally chosen in atomic calculations. These natural atomic orbitals were transformed to an orthogonalised basis set to simplify the mathematical analysis of the iterative procedure by which the energy minimisation was effected. In this work the iteration scheme, to provide a self-consistent wave function, was elaborated in terms of density matrix theory. In chapter one, a brief account of the various quantum chemical techniques which have been used to study chemical systems, is given, together with an appraisal of the problems involved in such calculations. The standard and aims of the present work are also presented here. Chapter two contains a fuller account of the theory upon which the present calculations are based. Particular attention is paid to the approximate methods by which the multicentre integrals are calculated. Two of these methods, based on an asymmetric partitioning of the overlap charge densities, have not been evaluated to any extent before# Experimental information such as the internuclear separations, the double zeta functions and the various units and conversion factors employed in this work, is detailed in chapter three# The results of the molecular electronic structure calculations performed are presented in chapter four. Population analyses, pictorial representations of the electron density distribution, together with a few molecular properties are presented for each of five internuclear separations of C-N in CN-, of H-C in HCN and of F-C in FCN. Where possible comparisons are made with calculations of other workers. The predicted minimum energies for CN-, HCN and FCN are respectively, -92.18149 a.u. at an internuclear separation of 2.348 a.u.; -92.74494 a.u. at a C-H separation of 2.086 a.u.; and -191.6180 a.u. at a C-F separation of 2.48 a.u. The calculated dipole moments, in Debye's, for HCN and FCN are respectively 2.59 and 1.54 and are to be compared with the experimental values of 3.00 and 2.17. An evaluation is made of the Mulliken and two variants of the Lowdin multicentre integral approximations with special reference to the validity of ab initio calculations performed with their aid in integral evaluation. It is shown that both Lowdin approximations give markedly improved results as compared with the Mulliken approximation, and even in the case of the twenty-two electron system of FCN the results are encouragingly close to those of the most accurate calculations. A comparison of some accurate three-centre one-electron integrals for HCN and FCN with the values resulting from the multicentre integral approximations, shows that the full Lowdin method is the best approximation in most cases and often yields values accurate to about three decimal places. Contour diagrams of the probability charge density and its profile along the internuclear axis are presented for each of the five calculations on CN, HCN and FCN. The trends shown are compared with those arising from the population analyses of Mulliken, Lowdin and Doggett. The gross atom charge densities resulting from two of the population analyses, reproduce the molecular dipole moment when account is taken of any atomic dipoles present. The results of these two different methods of analysing the electronic charge distribution are often at variance, but in general the overall trends are similar. Various contour diagrams of the charge density difference, for example, between the molecular and atomic densities or between two molecular species, are presented, and they clearly show the resultant migration of electronic charge. The system of computer programs which was written to perform the various stages of the SCF calculations, is described in chapter five. Flow diagrams are presented for each program. Finally, proposals as to future work using the information and experience obtained in the present calculations, are put forward in chapter six. There are two appendices. The first describes the method of Gaussian quadrature which was used as the basis of the numerical integration technique by which most of the molecular integrals were evaluated. Appendix two contains the values of all the integrals required to perform the calculation on HCN at its equilibrium configuration, the multicentre integrals being evaluated by the full Lowdin approximation

Fragment-based screening identifies molecules targeting the substrate-binding ankyrin repeat domains of tankyrase.

The PARP enzyme and scaffolding protein tankyrase (TNKS, TNKS2) uses its ankyrin repeat clusters (ARCs) to bind a wide range of proteins and thereby controls diverse cellular functions. A number of these are implicated in cancer-relevant processes, including Wnt/β-catenin signalling, Hippo signalling and telomere maintenance. The ARCs recognise a conserved tankyrase-binding peptide motif (TBM). All currently available tankyrase inhibitors target the catalytic domain and inhibit tankyrase's poly(ADP-ribosyl)ation function. However, there is emerging evidence that catalysis-independent "scaffolding" mechanisms contribute to tankyrase function. Here we report a fragment-based screening programme against tankyrase ARC domains, using a combination of biophysical assays, including differential scanning fluorimetry (DSF) and nuclear magnetic resonance (NMR) spectroscopy. We identify fragment molecules that will serve as starting points for the development of tankyrase substrate binding antagonists. Such compounds will enable probing the scaffolding functions of tankyrase, and may, in the future, provide potential alternative therapeutic approaches to inhibiting tankyrase activity in cancer and other conditions

Mini-Workshop: Amalgams for Graphs and Geometries

[no abstract available