Regular Subgraphs in Graphs and Rooted Graphs and Definability in Monadic Second - Order Logic

We investigate the definability in monadic ∑₁¹ and monadic Π₁¹ of the problems REG_k, of whether there is a regular subgraph of degree k in some given graph, and XREG_k, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REG_k^k and XREG_k^k, respectively, for k ≥ 2 (all our graphs are undirected). Our motivation partly stems from the fact (which we prove here) that REG_k^k and XREG_k^k are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑₁¹ games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑₁¹ and monadic Π₁¹, and we compare the definability of these problems (in monadic ∑₁¹ and monadic Π₁¹ with their computational complexity (which varies from solvable using logspace to NP - complete).     

The structures of higher boron halides B8X12(X = F, Cl, Br and I) by gas-phase electron diffraction and ab initio calculations

The structure of B8F12 has been shown by gas electron diffraction and computational methods (up to MP2/6-31+G*) to have the same highly asymmetric form observed in crystalline phases. The structure can be regarded as derived from a central B2 group, bridged by two BF2 groups to give a central B4 core that is folded, not planar, and with a very short bond [164.3 pm calculated, 164.2(19) pm experimental] along the fold line. There are also four terminal BF2 groups. One of the other four bonds in the core is consistently 20-30 pm longer than the others. This asymmetry has been attributed to many intra-molecular B...F interactions, particularly those between core boron atoms and fluorines of the terminal BF2 groups. Calculations for the chloro analogue lead to a structure similar to that for B8F12, but with the long core bond extended so that one of the bridging BCl2 groups may now be regarded as terminal. With bromine as the halogen the structure changes again, with one bromine atom taking up a bridging position. With iodine, this process continues further, and there are three bridging iodine atoms. However, in this case this is not the lowest energy structure, and instead a loosely associated dimer of B4I6 is preferred. In all these cases, and particularly with the heavier halogens, there are huge differences between the results obtained with different computational methods.     

Precision model independent determination of |Vub| from B --> pilnu

A precision method for determining |V(ub)| using the full range in q(2) of B --> pilnu data is presented. At large q(2) the form factor is taken from unquenched lattice QCD, at q(2) = 0 we impose a model independent constraint obtained from B --> pipi using the soft-collinear effective theory, and the shape is constrained using QCD dispersion relations. We find |V(ub)| = (3.54 +/- 0.170 +/- 0.44) x 10(-3). With 5% experimental error and 12% theory error, this is competitive with inclusive methods. Theory error is dominated by the input points, with negligible uncertainty from the dispersion relations.     

The structural and spectroscopic characterisation of three actinyl complexes with coordinated and uncoordinated perrhenate: [UO2(ReO4)2(TPPO)3], [[(UO2)(TPPO)3]2(mu2-O2)][ReO4]2 and [NpO2(TPPO)4][ReO4

The first structural characterization of an actinide complex with coordinated perrhenate is reported, [UO2(ReO4)2(TPPO)3] (1). In this [UO2]2+ complex two [ReO4]- anions and three TPPO (triphenylphosphine oxide) P=O donor ligands are coordinated in the equatorial plane in a cisoid arrangement. This bonding arrangement, and apparent strain observed in the equatorially bonded ligands, is attributed to the solid state packing in adjacent molecules in which hydrophobic TPPO ligands form an effective "shell" around a hydrophilic core of two UO2(ReO4)2 moieties. Solid state vibrational spectroscopy (infrared and Raman), 31P CP MAS NMR and elemental analysis are also consistent with the formula of 1. Solution state vibrational spectroscopy and 31P NMR measurements in EtOH indicate the lability of the TPPO and [ReO4]- groups. The photolytic generation of peroxide in EtOH solutions of 1 leads to the formation of trace quantities of [[(UO2)(TPPO)3]2(mu2-O2)][ReO4]2, 2, in which the coordinated [ReO4]- groups of 1 have been displaced by bridging O2(2-), derived from atmospheric O2. Finally, attempts to synthesise a [NpO2]+ analogue of have resulted only in the formation of [NpO2(TPPO)4][ReO4], 3, in which [ReO4]- acts solely as a counter anion. From these results it can be concluded that [ReO4]- will bond to [UO2]2+, but will be readily displaced by a more strongly coordinating ligand (e.g. peroxide) and will not coordinate to an actinyl cation with a lower charge, [NpO2]+, under the same reaction conditions.     

Fast,real-time spectrometer based on a pulsed quantum-cascade laser

We describe a mid-infrared spectrometer that is based on the combination of a multiple-pass absorption cell and a submicrosecond pulsed quantum-cascade laser. The spectrometer is capable of both making sensitive measurements and providing a real-time display of the spectral fingerprint of molecular vapors. For a cell with a path length of 9.6 m, dilution measurements made of the nu9 band transitions of 1,1-difluoroethylene indicate a sensitivity of 500 parts in 10(9), corresponding to a fractional absorbance of 4 x 10(-4).     

Dyad beads and the combinatorial discovery of catalysts

Dyad beads, bearing both a substrate and a catalyst, were prepared to enable direct split and mix bead based screening for catalysis.     

The preparation of hexasilsesquioxane (T6) cages by "non aqueous" hydrolysis of trichlorosilanes

Hexasilsesquioxane cages (T6) have been prepared from a range of alkyl and aryl trichlorosilanes using a "non aqueous" hydrolysis with dimethyl sulfoxide.     

An equivariant Brauer semigroup and the symmetric imprimitivity theorem

Suppose that is a second countable locally compact transformation group. We let denote the set of Morita equivalence classes of separable dynamical systems where is a -algebra and is compatible with the given -action on . We prove that is a commutative semigroup with identity with respect to the binary operation for an appropriately defined balanced tensor product on -algebras. If and act freely and properly on the left and right of a space , then we prove that and are isomorphic as semigroups. If the isomorphism maps the class of to the class of , then is Morita equivalent to .