Comment on “Zhang–Zhang polynomials of cyclo-polyphenacenes” by Q. Guo, H. Deng, and D. Chen

Explicit formulas for computing the Zhang–Zhang polynomial of cyclo-polyphenacenes, presented by Guo et?al. (J. Math. Chem. 46:347, 2009) are found to be erroneous. In the present comment, a corrected version of Theorem 4.2 is given. The new formulation has been extensively tested by comparison with the Zhang–Zhang polynomials computed by brute force using a new, completely automatized computer code.     

H/D isotopic recognition in hydrogen bonded systems: H/D isotopic self-organization effects in the IR spectra of the hydrogen bond in 2-methylimidazole crystals

Polarized IR spectra of H12(3)45 2-methylimidazole and of its H1D2(3)45, D1H2(3)45 and D12(3)45 deuterium derivative crystals are reported and interpreted within the limits of the "strong-coupling" theory. The spectra interpretation facilitated the recognition of the H/D isotopic "self-organization" phenomenon, which depends on a non-random distribution of protons and deuterons in the lattices of isotopically diluted crystal samples. The H/D isotopic "self-organization" mechanism engaged all four hydrogen bonds from each unit cell. These effects basically resulted from the dynamical co-operative interactions involving adjacent hydrogen bonds in each hydrogen bond chain. A weaker exciton coupling involved the closely spaced hydrogen bonds; each belonging to a different chain of associated 2-methylimidazole molecules. The high intensity of the narrow band at ca. 1880cm(-1) was interpreted as the result of coupling between the γ(N-H?N) proton bending "out of plane" vibration overtone and the ν(N-H) proton stretching vibration.     

Modification of the Two-Point Scaling Theory for the Description of the Phase Transition in Solution. Analysis of Sodium Octanoate Aqueous Solutions

On the basis of conventional scaling theory, the two-point scaling theory was modified in order to describe the influence of composition on the partial molar heat capacity and volume during the micellization process. To verify the theory, isobaric heat capacities and densities of aqueous sodium octanoate solutions were measured over wide composition and temperature ranges and the modified approach was used to analyze the calculated partial molar heat capacities and volumes of the surfactant in water. The results obtained indicate that the micellization process is subject to the scaling laws. The results were compared with those for other systems. Peculiar behavior of the critical indices was observed and correlated with the structure of the micelles.     

Gas-Phase Hydration Thermochemistry of Sodiated and Potassiated Nucleic Acid Bases

Hydration reactions of sodiated and potassiated nucleic acid bases (uracil, thymine, cytosine, and adenine) produced by electrospray have been studied in a gas phase using the pulsed ion-beam high-pressure mass spectrometer. The thermochemical properties, ΔH ( o ) ( n ), ΔS ( o ) ( n ), and ΔG ( o ) ( n ), for the hydrated systems were obtained from hydration equilibrium measurement. The structural aspects of the hydrated complexes are discussed in conjunction with available literature data. The correlation between water binding energies in the hydrated complexes and the corresponding metal ion affinities of nucleobases suggests that a significant (if not dominant) amount of the canonical structure of cytosine undergoes tautomerization during electrospray ionization, and the thermochemical values for cationized cytosine probably correspond to a mixture of tautomeric complexes.     

Tractability of Multivariate Integration for Weighted Korobov Classes

We study the worst-case error of multivariate integration in weighted Korobov classes of periodic functions of d coordinates. This class is defined in terms of weights γ_j which moderate the behavior of functions with respect to successive coordinates. We study two classes of quadrature rules. They are quasi-Monte Carlo rules which use n function values and in which all quadrature weights are 1/n and rules for which all quadrature weights are non-negative. Tractability for these two classes of quadrature rules means that the minimal number of function values needed to guarantee error in the worst-case setting is bounded by a polynomial in d and ⁻¹. Strong tractability means that the bound does not depend on d and depends polynomially on ⁻¹. We prove that strong tractability holds iff ∑^∞_j=1 γ_j<∞, and tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/log d<∞. Furthermore, strong tractability or tractability results are achieved by the relatively small class of lattice rules. We also prove that if ∑^∞_j=1 γ^1/α_j<∞, where α measures the decay of Fourier coefficients in the weighted Korobov class, then for d1, n prime and δ>0 there exist lattice rules that satisfy an error bound independent of d and of order n^−α/2+δ. This is almost the best possible result, since the order n^−α/2 cannot be improved upon even for d=1. A corresponding result is deduced for weighted non-periodic Sobolev spaces: if ∑^∞_j=1 γ^1/2_j<∞, then for d1, n prime and δ>0 there exist shifted lattice rules that satisfy an error bound independent of d and of order n^−1+δ. We also check how the randomized error of the (classical) Monte Carlo algorithm depends on d for weighted Korobov classes. It turns out that Monte Carlo is strongly tractable iff ∑^∞_j=1 log γ_j<∞ and tractable iff lim sup_d→∞ ∑^d_j=1 log γ_j/log d<∞. Hence, in particular, for γ_j=1 we have the usual Korobov space in which integration is intractable for the two classes of quadrature rules in the worst-case setting, whereas Monte Carlo is strongly tractable in the randomized setting.     

Complexity of Weighted Approximation over

We study approximation of univariate functions defined over the reals. We assume that the rth derivative of a function is bounded in a weighted L_p norm with a weight ψ. Approximation algorithms use the values of a function and its derivatives up to order r−1. The worst case error of an algorithm is defined in a weighted L_q norm with a weight ρ. We study the worst case (information) complexity of the weighted approximation problem, which is equal to the minimal number of function and derivative evaluations needed to obtain error . We provide necessary and sufficient conditions in terms of the weights ψ and ρ, as well as the parameters r, p, and q for the weighted approximation problem to have finite complexity. We also provide conditions which guarantee that the complexity of weighted approximation is of the same order as the complexity of the classical approximation problem over a finite interval. Such necessary and sufficient conditions are also provided for a weighted integration problem since its complexity is equivalent to the complexity of the weighted approximation problem for q=1.     

Hyperbranched Aromatic Polyesters

Summary: Having developed a theoretical model of hyperbranched polymerization involving AB₂ and B₂ monomers we present preliminary results of a test of its validity for the system consisting of 4,4-bis(4-hydroxyphenyl)pentanoic acid (AB₂ monomer) and bisphenol A (B₂ monomer). The monomers reacted at room temperature to yield the title hyperbranched polyesters. Diisopropylcarbodiimide (condensation water remover) along with an effective catalyst were used for that purpose. The total of AB₂ monomer was introduced at once to a reactor containing B₂ core monomer or it was divided into 3 or 5 equal portions and each of them introduced after the monomers previously present have reacted to a predetermined conversion. GPC measurements of the resulting molecular mass distribution gave inconclusive results because of poor reproducibility of polymer isolation/purification procedure and/or insufficient accuracy of GPC technique.     

Recurrence formulas for permanents of (0, 1)-circulants

A general method is developed for constructing linear recurrence formulas expressing the permanents of (0, 1)-circulants in terms of permanents of (0, 1)-circulants of the same type of lower orders. The method extends to all (0, 1)-circulants a result of N. Metropolis, M. L. Stein, and P. R. Stein on permanents of (0, 1)-circulants of a special type.     

Incremental energy minimization in dissipative solidsMinimisation d'énergie excédentaire dans solides dissipatifs

The incremental energy minimization is examined as a method of determining solution paths for time-independent dissipative solids. Isothermal quasi-static deformations are considered, and the deformation work is locally decomposed into the increments in free energy and intrinsic dissipation. General conditions necessary for the applicability of the minimization procedure are derived and discussed. To cite this article: H. Petryk, C. R. Mecanique 331 (2003).