Some examples of random walks on free products of discrete groups

Summary We consider the random walk (X_n) associated with a probability p on a free product of discrete groups. Knowledge of the resolvent (or Green's function) of p yields theorems about the asymptotic behaviour of the n-step transition probabilities p^*n(x)=P(X_n= x¦ X₀=e) as n. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour of the type p^*n(x)C_x^– n n^– 3/2. Here we show on the other hand that if G is a free product of m copies ofZ ^r and if (X_n) is the « average » of the classical nearest neighbour random walk on each of the factorsZ ^r, then while it satisfies an « n^–3/2 — law » for r small relative to m, it switches to an n^– r/2 -law for large r. Using the same techniques, we give examples of irreducible probabilities (of infinite support) on the free groupZ ^*m which satisfyn ^– for .     

Characterization of a Nudix hydrolase from <Emphasis Type="Italic">Deinococcus radiodurans</Emphasis> with a marked specificity for (deoxy)ribonucleoside 5'-diphosphates

Background  

Nudix hydrolases form a protein family whose function is to hydrolyse intracellular nucleotides and so regulate their levels and eliminate potentially toxic derivatives. The genome of the radioresistant bacterium Deinococcus radiodurans encodes 25 nudix hydrolases, an unexpectedly large number. These may contribute to radioresistance by removing mutagenic oxidised and otherwise damaged nucleotides. Characterisation of these hydrolases is necessary to understand the reason for their presence. Here, we report the cloning and characterisation of the DR0975 gene product, a nudix hydrolase that appears to be unique to this organism.     

Calculations of the electronic structures of cage molecules using free-electron orbitals as a basis

A non-empirical molecular orbital method, particularly suitable for calculations on cage-like molecules, is described. The method uses as basis functions the set of free-electron functions which are the solutions of Schrödinger's equation for an electron confined between two concentric, spherical potential energy barriers. Application of the theory to the SCF calculation of the energies of the delocalized electrons in benzene and tetrasulphur tetranitride shows that the model is capable of interpreting the properties of such systems. However, it does highlight a difficulty in the calculation of excited state energies with one-centre models which appears to be largely unrecognized.Extension of the method to a consideration of all the valence electrons, using P₄ as an example, reveals problems the origin of which is an inadequate treatment of the core electrons. It is suggested that these problems may best be dealt with by use of a suitable pseudo potential.     

Organophotoredox/palladium dual catalytic decarboxylative Csp3–Csp3 coupling of carboxylic acids and π-electrophiles

A dual catalytic decarboxylative allylation and benzylation method for the construction of new C(sp³)–C(sp³) bonds between readily available carboxylic acids and functionally diverse carbonate electrophiles has been developed. The new process is mild, operationally simple, and has greatly improved upon the efficiency and generality of previous methodology. In addition, new insights into the reaction mechanism have been realized and provide further understanding of the harnessed reactivity.

A dual catalytic decarboxylative allylation and benzylation method for the construction of new C(sp³)–C(sp³) bonds between readily available carboxylic acids and functionally diverse carbonate electrophiles has been developed.     

Imidazopyridine and pyrimidinopyridine systems from perfluorinated pyridine derivatives

Annelation of pentafluoropyridine via an intramolecular nucleophilic aromatic substitution process with benzamidine gave an imidazopyridine system in high yield in a two step process whilst alkyl amidines gave 4-aminotetrafluoropyridine by a competing elimination reaction. 4-Phenylsulfonyl tetrafluoropyridine reacts with amidines to give the corresponding imidazo[4,5-b]pyridine systems. In contrast, 4-cyanotetrafluoropyridine gave a [6,6]-fused pyrimidinopyridine system arising from initial nucleophilic substitution at the C-3 position of the pyridine ring followed by intramolecular cyclization onto the pendant cyano group. The systems prepared by this annelation methodology further demonstrate the utility of perfluorinated heterocyclic substrates for the synthesis of heterocyclic scaffolds that possess multiple functionality and have potential applications in the drug discovery arena.     

The heat of formation of sulphuryl fluoride and the stability of fluorosulphates

A standard enthalphy of -767.2 kJ mol^-1 was determined for sulphuryl fluoride from the heat of alkaline hydrolysis, and a standard free energy of -720.8 kJ mol^-1 derived. The thermodynamically preferred product of ionic fluorosulphate decompositions is always sulphuryl fluoride. Kinetic control must predominate when sulphur trioxide is the main product.     

Narrow band quantitative and multivariate electroencephalogram analysis of peri-adolescent period

ABSTRACT: BACKGROUND: The peri-adolescent period is a crucial developmental moment of transition from childhood to emergent adulthood. The present report analyses the differences in Power Spectrum (PS) of the Electroencephalogram (EEG) between late childhood (24 children between 8 and 13 years old) and young adulthood (24 young adults between 18 and 23 years old). RESULTS: The narrow band analysis of the Electroencephalogram was computed in the frequency range of 0--20 Hz. The analysis of mean and variance suggested that six frequency ranges presented a different rate of maturation at these ages, namely: low delta, delta-theta, low alpha, high alpha, low beta and high beta. For most of these bands the maturation seems to occur later in anterior sites than posterior sites. Correlational analysis showed a lower pattern of correlation between different frequencies in children than in young adults, suggesting a certain asynchrony in the maturation of different rhythms. The topographical analysis revealed similar topographies of the different rhythms in children and young adults. Principal Component Analysis (PCA) demonstrated the same internal structure for the Electroencephalogram of both age groups. Principal Component Analysis allowed to separate four subcomponents in the alpha range. All these subcomponents peaked at a lower frequency in children than in young adults. CONCLUSIONS: The present approaches complement and solve some of the incertitudes when the classical brain broad rhythm analysis is applied. Children have a higher absolute power than young adults for frequency ranges between 0-20 Hz, the correlation of Power Spectrum (PS) with age and the variance age comparison showed that there are six ranges of frequencies that can distinguish the level of EEG maturation in children and adults. The establishment of maturational order of different frequencies and its possible maturational interdependence would require a complete series including all the different ages.     

Photoredox/Cobalt Dual-Catalyzed Decarboxylative Elimination of Carboxylic Acids: Development and Mechanistic Insight

Recently, dual-catalytic strategies towards the decarboxylative elimination of carboxylic acids have gained attention. Our lab previously reported a photoredox/cobaloxime dual catalytic method that allows the synthesis of enamides and enecarbamates directly from N-acyl amino acids and avoids the use of any stoichiometric reagents. Further development, detailed herein, has improved upon this transformation's utility and further experimentation has provided new insights into the reaction mechanism. These new developments and insights are anticipated to aid in the expansion of photoredox/cobalt dual-catalytic systems.