A note on some piecewise-linear difference equations with Mersenne-type periodic solutions

This paper studies solutions of some piecewise-linear difference equations. In two particular cases, a descent argument is used to show that all solutions are periodic with either prime period 3(2 ^k  ? 1) or 6(2 ^k  ? 1) for some k ≥ 1. The existence of solutions with such periods is also considered.     

Bounded matrix groups

The principal result is that if G is any irreducible subgroup of GL(n, C) (C the complex field) such that |tr(7^)| is uniformly bounded for all A in G and some fixed n × n nonzero matrix T, then G is equivalent to a unitary group. Similar results are proved for certain associated representations of G.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

Further Thoughts on Turbulent Flow in a Pipe

Nikuradse’s 1932 paper on turbulent flow in a smooth pipe contains a wealth of information on flow resistance (friction factor) and profiles of velocity and eddy viscosity. The goal here is to study this information in detail with the objective of applying it to other turbulent-flow situations. In particular, reverse engineering supports a value of n = 2 for the exponent on the volumetric dissipation in the decay term of the equation of the dissipation theorem. Of equal importance, integration of Nikuradse’s profiles of eddy viscosity does not lead to his formula for the universal resistance law; instead the presence of the viscous sublayer has an overt effect on the result even though we had thought that such a region influenced only the form of mass transfer at high values of the Schmidt number. A formula is proposed for the decay of dissipation for turbulent flow in smooth pipes.

     

Biopolymer Skeleton Produced by Rhizobium radiobacter: Stoichiometric Alternation of Glycosidic and Amidic Bonds in the Lipopolysaccharide O‐Antigen

The lipopolysaccharide (LPS) O‐antigen structure of the plant pathogen Rhizobium radiobacter strain TT9 and its possible role in a plant‐microbe interaction was investigated. The analyses disclosed the presence of two O‐antigens, named Poly1 and Poly2. The repetitive unit of Poly2 constitutes a 4‐α‐l ‐rhamnose linked to a 3‐α‐d ‐fucose residue. Surprisingly, Poly1 turned out to be a novel type of biopolymer in which the repeating unit is formed by a monosaccharide and an amino‐acid derivative, so that the polymer has alternating glycosidic and amidic bonds joining the two units: 4‐amino‐4‐deoxy‐3‐O‐methyl‐d ‐fucose and (2′R,3′R,4′S)‐N‐methyl‐3′,4′‐dihydroxy‐3′‐methyl‐5′‐oxoproline). Differently from the O‐antigens of LPSs from other pathogenic Gram‐negative bacteria, these two O‐antigens do not activate the oxidative burst, an early innate immune response in the model plant Arabidopsis thaliana, explaining at least in part the ability of this R. radiobacter strain to avoid host defenses during a plant infection process.     

Microchip micellar electrokinetic chromatography separation of alkaloids with UV-absorbance spectral detection

A microchip device is demonstrated for the electrophoretic separation and UV-absorbance spectral detection of four toxic alkaloids: colchicine, aconitine, strychnine, and nicotine. A fused-silica (quartz) microchip containing a simple cross geometry is utilized to perform the separations, and a miniature, fiber-optic CCD spectrometer is coupled to the microchip for detection. Sensitive UV-absorbance detection is achieved via the application of online preconcentration techniques in combination with the quartz microchip substrate which contains an etched bubble-cell for increased pathlength. The miniature CCD spectrometer is configured to detect light between 190 and 645 nm and LabView programming written in-house enables absorbance spectra as well as separations to be monitored from 210 to 400 nm. Consequently, the configuration of this microchip device facilitates qualitative and quantitative separations via simultaneous spatial and spectral resolution of solutes. UV-absorbance limits of quantification for colchicine, 20 microM (8 mg/L); strychnine, 50 microM (17 mg/L); aconitine, 50 microM (32 mg/L); and nicotine, 100 microM (16 mg/L) are demonstrated on the microchip. With the exception of aconitine, these concentrations are > or =20-times more sensitive than lethal dose monitoring requirements. Finally, this device is demonstrated to successfully detect each toxin in water, skim milk, and apple juice samples spiked at sublethal dose concentrations after a simple, SPE procedure.     

Cyclometallated platinum(II) complexes: oxidation to, and C-H activation by, platinum(IV)

A series of cyclometallated phenylpyridine platinum(II) complexes have been synthesised with a systematic variation in both the phenylpyridine and the ancillary ligand. Oxidation of one of the cyclometallated species leads to a number of isomeric platinum(IV) complexes, all of which eventually isomerize to a single compound. The route to these new compounds has been demonstrated to involve an initial slow oxidation followed by a rapid C-H activation to give doubly cyclometallated complexes. The solid state structures of a number of both the platinum(II) and the platinum(IV) species have been solved; many of the structures exhibited extended interactions that result in complex three dimensional packing.     

Cd13−xInySb10 (x≈2.7, y≈1.5): An Interstitial‐Free Variant of Thermoelectric β‐Zn4Sb3

Although free from structural disorder , the new intermetallic compound Cd_13?xIn_ySb₁₀ (see figure) displays similarly low thermal conductivity values as disordered thermoelectric β‐Zn₄Sb₃ with an isostructural framework.

     

Calculation of the streaming potential near a rotating disk

A corrected theory of the streaming potential in the vicinity of a disk-shaped sample rotating in an electrolytic solution is presented. When streaming-potential measurements on a variety of materials were reduced to a zeta potential according to a prior theory, the results exceeded expected values by a factor of approximately 2, even though other aspects of the same experiments seemed to confirm the theory. Investigation of the source of the discrepancy revealed a flaw in the prior theory. The crucial understanding is that the surface current produced by the rotation of the disk emerges from the diffuse layer and enters the bulk solution at the periphery of the disk. The new treatment accounts entirely for the discrepancy between literature data and results based on the prior theory.