Congruence properties of Apéry numbers

Apéry introduced a recurrence relation for a proof of the irrationality of ζ(3). Let a_n (n ≥ 0) satisfy the relation n³a_n ? (34n³ ? 51n² + 27n ? 5)a_{n ? 1} + (n ? 1)³a_{n ? 2} = 0. Which values of a₀ and a₁ cause each a_n to be an integer? This question is answered and some congruence properties of the a_n are given.     

On two expressions of the MLE for a special case of the extended growth curve models

An algebraic proof is given on the equivalence between two solutions of MLEs for a special case of the extended growth curve models called the Banken model. One solution given by Verbyla and Venables is an iterative solution in the general case but reduces to a non-iterative one in the case of the Banken model. The other solution given by von Rosen is a closed-form solution specifically targeted at the Banken model. The proof has turned out to be quite challenging yet intriguing as it touches on many aspects of intricate matrix theory involving projection matrices.     

On the necessary and sufficient condition for the extended Wedderburn–Guttman theorem

Let A be a u by v matrix, and let M and N be u by p and v by q matrices, where p may not be equal to q or rank(M^′AN)<min(p,q). Recently, Galantai [A. Galantai, A note on the generalized rank reduction, Acta Math. Hungarica 116 (2007) 239–246] presented what he claimed to be the necessary and sufficient condition for rank(A-AN(M^′AN)^-M^′A)=rank(A)-rank(AN(M^′AN)^-M^′A) to hold. This rank subtractivity formula along with the condition under which it holds is called the extended Wedderburn–Guttman theorem. In this paper, we show that some of Galantai’s assertions are incorrect.     

High oxygen-reduction activity of silk-derived activated carbon

Activated carbon prepared from silk fibroin, which is free of metal elements, showed a high catalytic activity for the oxygen-reduction reaction (ORR). The activated carbon had a very high onset potential of E_onset = 0.83 V (vs. RHE) in oxygen-saturated 0.5 M H₂SO₄ at 60 °C. The ORR on the activated carbon proceeded by a four-electron process in the high-electrode-potential region; this gradually decreased to a 3.5-electron reaction below about 0.6 V (vs. RHE). Only about 1% of nitrogen atoms (mostly quaternary) remained in the activated carbon by heat-treatment at up to 1200 °C are responsible for the high catalytic activity. The open circuit voltage of a polymer electrolyte fuel cell using the activated carbon as the cathode and a platinum/carbon black anode under pure oxygen and hydrogen gases, respectively, both at one atmosphere, was 0.96 V at 27 °C.     

Binding enhancements of nicotinic acid to water-soluble zinc porphyrins based on triple attractions of coordination, Coulomb, and CH-pi interactions

The binding of amines to artificial zinc-porphyrin receptors 1-4 was examined in basic aqueous solutions. For nicotinic acid and 3,5-dicarboxypyridine, substantial binding enhancements were observed compared to other amines with no pi system or carboxyl group. This observation suggested that interligand attractions of Coulomb and CH-pi interactions in addition to N-atom coordination can act effectively as recognition factors. The differences in the Coulomb interaction between carboxylate and sulfonate anions were also discussed.     

Double helical oligoresorcinols specifically recognize oligosaccharides via heteroduplex formation through noncovalent interactions in water

We report on the oligosaccharide recognition through noncovalent interactions in water based on a unique supramolecular homoduplex-to-heteroduplex transformation of the oligoresorcinol nonamer as a fully artificial receptor. The oligoresorcinol forms a double helix in water, which unravels and entwines upon complexation with specific oligosaccharides with a particular chain length and glucosidic linkage pattern, thus generating the heteroduplex with an excess one-handed helical conformation that can be readily monitored and further quantified by absorption, circular dichroism, and NMR spectroscopies.     

Comparative investigation of the cleavage step in the synthesis of model peptide resins: implications for Nalpha-9-fluorenylmethyloxycarbonyl-solid phase peptide synthesis

Based on our studies of the stability of model peptide-resin linkage in acid media, we previously proposed a rule for resin selection and a final cleavage protocol applicable to the Nalpha-tert-butyloxycarbonyl (Boc)-peptide synthesis strategy. We found that incorrect choices resulted in decreases in the final synthesis yield, which is highly dependent on the peptide sequence, of as high as 30%. The present paper continues along this line of research but examines the Nalpha-9-fluorenylmethyloxycarbonyl (Fmoc)-synthesis strategy. The vasoactive peptide angiotensin II (AII, DRVYIHPF) and its [Gly8]-AII analogue were selected as model peptide resins. Variations in parameters such as the type of spacer group (linker) between the peptide backbone and the resin, as well as in the final acid cleavage protocol, were evaluated. The same methodology employed for the Boc strategy was used in order to establish rules for selection of the most appropriate linker-resin conjugate or of the peptide cleavage method, depending on the sequence to be assembled. The results obtained after treatment with four cleavage solutions and with four types of linker groups indicate that, irrespective of the circumstance, it is not possible to achieve complete removal of the peptide chains from the resin. Moreover, the Phe-attaching peptide at the C-terminal yielded far less cleavage (50-60%) than that observed with the Gly-bearing sequences at the same position (70-90%). Lastly, the fastest cleavage occurred with reagent K acid treatment and when the peptide was attached to the Wang resin.     

Test of athermal terms of activity coefficient models by Monte Carlo simulation with hard-core models

The infinite dilution activity coefficients of exactly athermal fluids were calculated by Monte Carlo simulation with hard-core models. The hard-core models used in this work were hard-sphere and hard-spherocylinder models. The Widom test particle method was adopted to calculate the residual chemical potentials of solutes in pure solvent and in pure solute solutions. The infinite dilution activity coefficients of solutes were obtained from the residual chemical potentials of solutes. The infinite dilution activity coefficients calculated by Monte Carlo simulation were compared with those of athermal terms in activity coefficient equations. Staverman–Guggenheim equation overestimates the activity coefficients. The deviations of activity coefficients increase with increasing the hard-core volume of solute. Flory–Huggins equation based on molar volume gives good results for the hard-spherocylinder systems. Elbro-FV equation gives good results for both the hard-sphere and hard-spherocylinder systems.     

Single-molecule force spectroscopy for studying kinetics of enzymatic dextran elongations

Catalytic elongation of dextran by a single molecule of dextransucrase (DSase) was directly monitored by observing the movements of the positions of a rupture peak, which represented the adhesive force between an isomaltoheptaose (dextran 7-mer)-immobilized probe and a DSase-immobilized mica surface. This was initiated with the addition of sucrose monomers. From the histograms of the rupture peaks after elongation reactions on each individual enzyme and the continuous peak shift of certain single enzymes, the catalytic elongation rate constant (k(cat)) was ascertained to be 1.2-2.7 s(-1).