Random walks with different directions

As an extension of Polya’s classical result on random walks on the square grids (\({\mathbf {Z}}^d\)), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after n steps is at most \(n^{-d/2 - d/(d-2) +o(1) }\), which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is \(n^{-\omega (1) }\), which is much worse than in higher dimensions. In dimension 3, we prove an upper bound of order \(n^{-4 +o(1) }\). We find a new conjecture concerning incidences between spheres and points in \({\mathbf {R}}^3\), which, if holds, would improve the bound to \(n^{-9/2 +o(1) }\), which is consistent to the \(d \ge 4\) case. This conjecture resembles Szemerédi-Trotter type results and is of independent interest.     

An unexpected Pummerer rearrangement in the synthetic route to ethyl (2′-hydroxy-4′,5′-methylenedioxyphenyl)acetate: An alternative approach to 2,3-dimethylthio benzofurans

The synthesis of ethyl (2′-hydroxy-4′,5′-methylendioxophenyl)acetate, a fragment of the antihyperglycemic natural coumarin subcoriacin, is reported. We found an expeditious route to the title compound in five steps. Final metal catalyzed acid ethanolysis of the vinylic 1,1-methylthio methylsulfoxide derivative afforded the required aryl acetic ester, but in the absence of metal catalyst, an unexpected Pummerer rearrangement produced the 2,3-dimethylthiofuran derivative as the major product. This last result provides an alternative entry to 2,3-dimethlythiobenzofurans.     

Bromo­di­methyl(N‐methyl­pyrrolidin‐2‐one‐O)­tin(IV)‐di‐μ‐bromo‐bromo­di­methyl­tin(IV)

The preparation and X‐ray analysis of the title compound, [Sn₂Br₄(CH₃)₄(C₅H₉NO)], are described. The compound contains two Sn atoms in the asymmetric unit, that complexed by N‐methyl­pyrrolidin‐2‐one being hexacoordinated (a), the other exhibiting pentacoordination (b). The most important features are three different Sn—Br bond lengths at both Sn atoms with the following values: (a) 2.5060 (9), 2.7152 (10) and 3.7118 (10) Å; (b) 2.5084 (10), 2.5279 (9) and 3.5841 (10) Å.     

Development of an automated in situ fracture stage for a ToF‐SIMS system

The design philosophy and implementation of an ultra high vacuum (UHV), PC controlled, automated in situ fracture stage for a surface analysis system is described. ToF‐SIMS spectra are shown to illustrate the improvement in spectral quality obtained from micro‐compact tension (CT) tests of polymer matrix fracture surfaces produced using the fracture stage in UHV compared to those obtained from a sample tested at air. This system is flexible in that by changing the capacity of the load cell it is possible to reduce or increase maximum loads as the specimen type and material demands. The stage has been designed with instrumental flexibility in mind, utilising commercial SEM‐stub type sample mounts, and can thus be used for AES/SAM and XPS investigations, as well as ToF‐SIMS analysis, in the authors' laboratory. Copyright © 2008 John Wiley & Sons, Ltd.     

Computational studies of enzyme mechanism: linking theory with experiment in the analysis of enzymic H-tunnelling

Hydrogen transfer--an essential component of most biological reactions--is a quantum problem. A crucial question of great current interest is how enzymes modulate the quantum dynamics of hydrogen transfer to achieve their outstanding catalytic properties. That tunnelling occurs is now widely accepted, with the conceptual frameworks incorporating protein motion into the enzymic H-tunnelling process. Computational simulation can be used to help elucidate how enzymes work and facilitate H-tunnelling at the atomic level. We review the strength of a multidisciplinary approach--combining computational simulations with enzyme kinetics and structural biology--in revealing tunnelling mechanisms in enzymes. We focus on two paradigm systems--aromatic amine dehydrogenase, in which H-tunnelling is facilitated by fast (sub-picosecond) short range motions, and dihydrofolate reductase, in which a network of long-range coupled motions drives the tunnelling event.     

Muon spin relaxation study of Zr(H2PO4)(PO4).2H2O

Muon spin relaxation has been used to study the muon dynamics in the layered zirconium phosphate Zr(H(2)PO(4))(PO(4)).2H(2)O as a function of temperature. Radiofrequency decoupling was used to establish the origin of the local dipolar field as coupling with (1)H spins. Muons were trapped at two sites, one identified as HMuO and the other consistent with PO-Mu on the basis of their zero-field second moments. Although a small decrease in the local nuclear dipolar field was seen with temperature, the muons remained essentially static over the temperature range 20-300 K.     

A molecular orbital rationalization of ligand effects in N2 activation

Molecular orbital theory has been used to study a series of [(micro-N2){ML3}2] complexes as models for dinitrogen activation, with M=Mo, Ta, W, Re and L=NH2, PH2, AsH2, SbH2 and N(BH2)2. The main aims of this study have been to provide a thorough electronic analysis of the complexes and to extend previous work involving molecular orbital analyses. Molecular orbital diagrams have been used to rationalize why for L=NH2 ligand rotation is important for the singlet state but not the triplet, to confirm the effect of ligand pi donation, and to rationalize the importance of the metal d-electron configuration. The outcomes of this study will assist with a more in-depth understanding of the electronic basis for N2 activation and allow clearer predictions to be made about the structure and multiplicity of systems involved in transition-metal catalysis.     

Porphyrins bearing stable meso-alkylidenyl double bonds. A new family of nonplanar porphyrinoids

[Structure: see text] Unique core-modified porphyrinoids, such as oxabenziporphyrins, oxapyriporphyrins, and thiapyriporphyrins, bearing exocyclic C-C double bonds at meso-positions, have been synthesized and characterized. The synthesis was accomplished by utilizing typical "3+1"-type condensation. Two different stable tautomeric forms were isolated, and the two tautomeric forms can be interconvertible upon treatment with base. In contrast, only the structure bearing an exocyclic double bond was isolated in the case of oxapyriporphyrin and oxabenziporphyrin.     

Determination of interferon-gamma in Chinese hamster ovary cell culture supernatant by coupled-column liquid chromatography

A robust tandem HPLC method coupling size-exclusion (Shodex Asahipak GS-320HQ) and reversed phase (Vydac 218TP54) columns with ultraviolet detection was developed for quantitative determination of interferon-gamma (IFN-gamma) in Chinese hamster ovary cell culture supernatant. The 2D-HPLC system was linked up by a 6-port 2-position low hold-up volume switch valve. Compared to a commercial ELISA kit for IFN-gamma, the coupled column LC approach was able to detect and quantify soluble IFN-gamma, regardless of the glycoprotein's molecular/conformational variability and sample background. Each LC-LC analysis took 90 minutes inclusive of column regeneration. The relative standard deviation of measurements (n = 5) was less than 3%. The limit of detection (LOD) was determined to be 0.35 microg IFN-gamma.     

Adsorption equilibrium of binary methane/ethane mixtures in BPL activated carbon: isotherms and calorimetric heats of adsorption

The adsorption of pure methane and ethane in BPL activated carbon has been measured at temperatures between 264 and 373 K and at pressures up to 3.3 MPa with a bench-scale high-pressure open-flow apparatus. The same apparatus was used to measure the adsorption of binary methane/ethane mixtures in BPL at 301.4 K and at pressures up to 2.6 MPa. Thermodynamic consistency tests demonstrate that the data are thermodynamically consistent. In contrast to two sets of data previously published, we found that the adsorption of binary methane/ethane in BPL behaves ideally (in the sense of obeying ideal adsorbed solution theory, IAST) throughout the pressure and gas-phase composition range studied. A Tian-Calvet type microcalorimeter was used to measure low-pressure isotherms, the isosteric heats of adsorption of pure methane and ethane in BPL activated carbon, and the individual heats of adsorption in binary mixtures, at 297 K and at pressures up to 100 kPa. The mixture heats of adsorption were consistent with IAST.