On the Brocard–Ramanujan Diophantine Equation n! + 1 = m2

In 1876, H. Brocard posed the problem of finding all integral solutions to n! + 1 = m². In 1913, unaware of Brocard's query, S. Ramanujan gave the problem in the form, The number 1 + n! is a perfect square for the values 4, 5, 7 of n. Find other values. We report on calculations up to n = 10⁹ and briefly discuss a related problem.     

Macroscopic Determinism in Noninteracting Systems Using Large Deviation Theory

We consider a general system of n noninteracting identical particles which evolve under a given dynamical law and whose initial microstates are a priori independent. The time evolution of the n-particle average of a bounded function on the particle microstates is then examined in the large-n limit. Using the theory of large deviations, we show that if the initial macroscopic average is constrained to be near a given value, y, then the macroscopic average at time t converges in probability as n to a value _t(y) given explicitly in terms of a canonical expectation. Some general features of the graph of _t(y) versus t are examined, particularly in regard to continuity, symmetry, and convergence.     

Energetic and structural characterization of 2‐R‐3‐methylquinoxaline‐1,4‐dioxides (R = benzoyl or tert‐butoxycarbonyl): experimental and computational studies

The gaseous standard molar enthalpies of formation of two 2‐R‐3‐methylquinoxaline‐1,4‐dioxides (R = benzoyl or tert‐butoxycarbonyl), at T = 298.15 K, were derived using the values for the enthalpies of formation of the compounds in the condensed phase, measured by static bomb combustion calorimetry, and for the enthalpies of sublimation, measured by Knudsen effusion, using a quartz crystal oscillator. The three dimensional structure of 2‐tert‐butoxycarbonyl‐3‐methylquinoxaline‐1,4‐dioxide has been obtained by X‐ray crystallography showing that the two N? O bond lengths in this compound are identical. The experimentally determined geometry in the crystal is similar to that obtained in the gas‐phase after computations performed at the B3LYP/6‐311 + G(2d,2p) level of theory. The experimental and computational results reported allow to extend the discussion about the influence of the molecular structure on the dissociation enthalpy of the N? O bonds for quinoxaline 1,4‐dioxide derivatives. As found previously, similar N? O bond lengths in quinoxaline‐1,4‐dioxide compounds are not linked with N? O bonds having the same strength. Copyright © 2007 John Wiley & Sons, Ltd.     

Radiation damage to protein specimens from electron beam imaging and diffraction: a mini‐review of anti‐damage approaches,with special reference to synchrotron X‐ray crystallography

Recent research progress using X‐ray cryo‐crystallography with the photon beams from third‐generation synchrotron sources has resulted in recognition that this intense radiation commonly damages protein samples even when they are held at 100 K. Other structural biologists examining thin protein crystals or single particle specimens encounter similar radiation damage problems during electron diffraction and imaging, but have developed some effective countermeasures. The aim of this concise review is to examine whether analogous approaches can be utilized to alleviate the X‐ray radiation damage problem in synchrotron macromolecular crystallography. The critical discussion of this question is preceded by presentation of background material on modern technical procedures with electron beam instruments using 300–400 kV accelerating voltage, low‐dose exposures for data recording, and protection of protein specimens by cryogenic cooling; these practical approaches to dealing with electron radiation damage currently permit best resolution levels of 6 Å (0.6 nm) for single particle specimens, and of 1.9 Å for two‐dimensional membrane protein crystals. Final determination of the potential effectiveness and practical value of using such new or unconventional ideas will necessitate showing, by experimental testing, that these produce significantly improved protection of three‐dimensional protein crystals during synchrotron X‐ray diffraction.     

Thiolated poly(ε-caprolactone) macroligand with vacant coordination sites on gold substrate: Synthesis and surface characterization

Surface-confined telechelic poly(ε-caprolactone) macroligand with two distinct functional groups per polymeric chain has been synthesized and characterized. The molecular microstructure of the macroligand with regard to the properties of the end-capped functionalities and with those on surface substrate has been studied by solution and surface analytical methods (i.e., X-ray photoelectron spectroscopy (XPS), grazing angle reflectance-Fourier transform IR spectroscopy (GA-FTIR), water contact angle measurements, and atomic force microscopy (AFM)) to elucidate the structure and properties of such multifunctional polymer on gold (1 1 1) substrate.     

Solution Properties of the Exopolysaccharide Produced by Pseudomonas Marginalis Strain HT041B 1

Abstract

The solution properties of the exopolysaccharide marginalan produced by Pseudomonas marginalis HT041B were investigated by means of low-angle laser light-scattering, capillary viscometry, and rheology. Potentiometric and viscosimetric data indicated the absence of a cooperative transition of the disorder-to-order type. The experimental findings obtained in dilute solution (Mark-Houwink coefficients, rigidity coefficient, characteristic ratio) suggested that the polymer behaves like a semiflexible chain which adopts a disordered conformation. The rheological behaviour of more concentrated marginalan solutions, as determined by means of both steady shear and oscillatory measurements, further confirmed the disordered conformational state of the polymer in solution.

     

Synthesis of a Trisaccharide Repeating Unit Related to Arabinogalactan-Protein (AGP) Polysaccharides 1

Abstract

Starting from L-arabinose and methyl β-D-galactopyranoside, methyl 2,3,4-tri-O-benzyl-6-O-[2,4,6-tri-O-benzoyl-3-O-(23,5-tri-O-benzoyl-α-L-arabinofuranosyl)-β-D-galactopyranosyl]-β-D-galactopyranoside 10 has been synthesized. Removal of protecting groups gave the methyl glycoside 12 of a trisaccharide representative of a repeating unit of arabinogalactan (AGP) polysaccharides.

     

Poly(arylene ether)s with pendant diphenyl phosphoryl groups: Synthesis,characterization, and thermal properties

A series of poly(arylene ether)s, (PAEs), carrying a pendant diphenyl phosphoryl group were prepared via the nucleophilic aromatic substitution (NAS) reactions of 3,5‐difluorotriphenylphosphine oxide, 6 . The difluoro monomer 6 was synthesized via two‐step reaction sequence and subsequently characterized by ¹H, ¹³C, ¹⁹F, and ³¹P NMR spectroscopy, GC/MS, and elemental analysis. The reactivity of the electrophilic sites in 6 , activated by only a diphenylphosphoryl group located in the meta‐position, in 6 was probed via NMR spectroscopy and model reactions and was determined to be sufficient to undergo typical NAS reactions. High molecular weight, amorphous, organic soluble poly(arylene ether)s, bearing a pendant diphenylphosphoryl group, were prepared via the reaction of 6 with a variety of bis‐phenols under typical NAS conditions. The poly(arylene ether)s were characterized for structure via the use of ¹H, ¹³C, and ³¹P NMR spectroscopy while their thermal properties were evaluated using DSC and TGA analysis. The glass transition temperatures (T_g) of the synthesized PAEs ranged from 143 to 175 °C, while their 5% weight loss temperatures ranged from 467 to 510 °C under nitrogen and from 470 to 526 °C in air. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011