The time-symmetric initial value problem for a homogeneous,anisotropic, empty,closed universe,using Regge calculus

Models for a homogeneous, anisotropic, empty, closed universe at a particular moment of time are constructed from sets of empty tetrahedra. It is shown using Regge calculus that only two such models have a moment of time symmetry.On leave from Girton College, Cambridge CB3 OJG, and Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW, England.     

Bulky 4-phosphacyclohexanones: diastereoselective complexations, orthometallations and unprecedented [3.1.1]metallabicycles

The 4-phosphacyclohexanones, 2,2,6,6-tetramethyl-1-phenyl-4-phosphorinanone (La), 1,2,6-triphenyl-4-phosphorinanone ((Ph)Lb), 1-cyclohexyl-2,6-diphenyl-4-phosphorinanone ((Cy)Lb) and 1-tert-butyl-2,6-diphenyl-4-phosphorinanone ((Bu)Lb) have been made by modifications of literature methods. Phosphines (R)Lb are each formed as mixtures of meso- and rac-diastereoisomers. Isomerically pure rac-(Ph)Lb, rac-(Cy)Lb and meso-(Bu)Lb can be isolated by recrystallisation from MeCN. Heating mixtures of isomers of (R)Lb with TsOH leads to isomerisations to give predominantly the meso-(R)Lb. The complex trans-[PdCl2(La)2] (1) is readily made from [PdCl2(NCPh)2] but the analogous platinum complex 2 has not been detected and instead, cyclometallation at the 3-position (alpha to the ketone) in the phosphacycle occurs to give trans-[PtCl(La)(La-3H)] (3) (where La-3H = La deprotonated at the 3-position) featuring a [3.1.1]metallabicycle as confirmed by X-ray crystallography. The analogous palladabicycle 4 has been detected upon treatment of 1 with Et3N in refluxing toluene. The type of complex formed by (R)Lb depends on which diastereoisomer (meso or rac) is involved. rac-(Ph)Lb (a mixture of R,R- and S,S-enantiomers, labelled alpha and beta) forms trans-[MCl2(rac-(Ph)Lb)2], M = Pd (5) or Pt (6), as mixtures of diastereoisomers (alphaalpha/betabeta and alphabeta forms). The structure of alphaalpha-6 has been determined by X-ray crystallography. Ligand competition experiments monitored by 31P NMR showed that Pd(II) and Pt(II) have a significant preference to bind rac-(Ph)Lb over meso-(Ph)Lb. meso-(Bu)Lb reacts with [PtCl2(NCBu(t))2] under ambient conditions to give the binuclear complex [Pt2Cl2(meso-(Bu)Lb-2'H)2] (7) where orthometallation has occurred on one of the exocyclic phenyl substituents as confirmed by X-ray crystallography. rac-(Bu)Lb reacts with [PtCl2(NCBu(t))2] to give a mononuclear cyclometallated species assigned the structure trans-[PtCl(rac-(Bu)Lb-2'H)((Bu)Lb)] (8) on the basis of its 31P NMR spectrum. rac-(Cy)Lb reacts with [PtCl2(NCBu(t))2] in refluxing toluene to give trans-[PtCl2(rac-(Cy)Lb)2] (9) and the crystal structure of alphabeta-9 has been determined.     

Modification of monomolecular self-assembled films by nitrogen-oxygen plasma

The modification of octadecanethiolate self-assembled monolayers on Au and Ag by nitrogen-oxygen downstream microwave plasma with variable oxygen content (up to 1%) has been studied by synchrotron-based high-resolution X-ray photoelectron spectroscopy. The primary processes were dehydrogenation, desorption of hydrocarbon and sulfur-containing species, and the oxidation of the alkyl matrix and headgroup-substrate interface. The exact character and the rates of the plasma-induced changes were found to be dependent on the substrate and plasma composition, with the processes in the aliphatic matrix and headgroup-substrate interface being mostly decoupled. In particular, the rates of all major plasma-induced processes were found to be directly proportional to the oxygen content in the plasma, which can be, thus, considered as a measure of the plasma reactivity. Along with the character of the observed changes, exhibiting a clear dominance of the oxidative processes, this suggests that the major effect of the oxygen-nitrogen downstream microwave plasma is provided by reactive oxygen-derived species in the downstream region, viz. long-living oxygen radicals and metastable species.     

Large molecular aggregates: From atmospheric aerosols to drug nanoparticles

Large molecular aggregates with sizes ranging from subnanometers to microns are ubiquitous. As atmospheric aerosols they influence our climate, in interstellar space they are discussed as reactive sites, and in medicine small particles are considered as promising candidates to achieve a targeted drug delivery. The present contribution is focused on the characterization of the physical-chemical properties of these particles and on their targeted generation. One of the greatest challenges is to understand the properties of these aggregates on a molecular level. The latter point is discussed in detail focussing on the vibrational dynamics of these particles.     

Backbone conformational preferences and pseudorotational ring puckering of 1-aminocyclopentane-1-carboxylic acid

We have used quantum mechanical calculations at the B3LYP/6-311G(d,p) level to determine the conformational preferences of the N-acetyl-N'-methylamide derivative of 1-aminocyclopentane-1-carboxylic acid in the gas phase, chloroform solution, and water solution. The backbone conformation of this dipeptide has been described through the dihedral angles varphi and psi, while the pseudorotational phase angle was used to define the conformation of the cyclopentane ring. Results indicate that the backbone flexibility of this amino acid is restricted by the cyclic nature of the side chain, the relative stability of the different conformations depending on the polarity of the environment. The potential energy of the pseudorotation was also studied as a function of the backbone conformation. Interestingly, the conformation of the cyclic side chain depends on the backbone arrangement. Furthermore, the number of pseudorotational states accessible at room temperature is high in all the investigated environments, especially in aqueous solution. Finally, a set of force-field parameters for classical molecular mechanics calculations was developed for the investigated amino acid. Molecular dynamics simulations in both chloroform and aqueous solutions were performed to demonstrate the reliability of such parameters.     

Carotenoid radical anions and their protonated derivatives

In this study, we report the protonation reactions for astaxanthin and canthaxanthin radical anions in methanol, alkaline methanol, and aqueous 2% Triton X-100 at different pH values. The pKa values for the corresponding alpha-hydroxy radical derivatives of astaxanthin, canthaxanthin, and beta-apo-8'-carotenal were estimated in 2% Triton X-100. Also, the effects of the microenvironment and the structure of the carotenoids on the protonation rate constant are discussed.     

Detection of chemical warfare simulants by phosphorylation of a coumarin oximate

The detection of chemical warfare simulants is attained by the PET mechanism that gives an "off-on" fluorescent response with a half-life of approximately 50 ms upon phosphorylation of a reactive oximate functionality; the X-ray crystal structure of the oximate was also obtained and is discussed.     

A forest building process on simple graphs

Consider the following process on a simple graph without isolated vertices: order the edges randomly and keep an edge if and only if it contains a vertex which is not contained in some preceding edge. The resulting set of edges forms a spanning forest of the graph.The probability of obtaining $k$ components in this process for complete bipartite graphs is determined as well as a formula for the expected number of components in any graph. A generic recurrence and some additional basic properties are discussed.     

Reconfiguration graphs of shortest paths

For a graph $G$ and$a,b\in V\left(G\right)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and$b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and$b$ in $G$. Two vertices in $V\left(S(G,a,b)\right)$ are adjacent, if their corresponding paths in $G$ differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth 5 or greater. We include an infinite family of well structured examples, showing that the shortest path graph of a grid graph is an induced subgraph of a lattice.     

Stability in the Erdős–Gallai Theorem on cycles and paths,II

The Erd?s–Gallai Theorem states that for $k\ge 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k?1)(n?1)$ edges. A stronger version of the Erd?s–Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e\left(G\right)\le max\{h(n,k,2),h(n,k,?\frac{k?1}{2}?)\}$, where $h(n,k,a)?\left(\genfrac{}{}{0ex}{}{k?a}{2}\right)+a(n?k+a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,?\frac{k?1}{2}?)$ edges.In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for $k\ge 3$ odd and all $n\ge k$, every $n$-vertex 2-connected graph $G$ with no cycle of length at least $k$ is a subgraph of one of the two extremal graphs or $e\left(G\right)\le max\{h(n,k,3),h(n,k,\frac{k?3}{2})\}$. The upper bound for $e\left(G\right)$ here is tight.