Disturbing the Dyson conjecture, in a generally GOOD way

Dyson's celebrated constant term conjecture [F.J. Dyson, Statistical theory of the energy levels of complex systems I, J. Math. Phys. 3 (1962) 140-156] states that the constant term in the expansion of _∏1≦i≠j≦naj(1−xi/xj) is the multinomial coefficient (a₁+a₂+?+an)!/(a₁!a₂!?an!). The definitive proof was given by I.J. Good [I.J. Good, Short proof of a conjecture of Dyson, J. Math. Phys. 11 (1970) 1884]. Later, Andrews extended Dyson's conjecture to a q-analog [G.E. Andrews, Problems and prospects for basic hypergeometric functions, in: R. Askey (Ed.), The Theory and Application of Special Functions, Academic Press, New York, 1975, pp. 191-224]. In this paper, closed form expressions are given for the coefficients of several other terms in the Dyson product, and are proved using an extension of Good's idea. Also, conjectures for the corresponding q-analogs are supplied. Finally, perturbed versions of the q-Dixon summation formula are presented.     

Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions

Functional central limit theorems for triangular arrays of rowwise independent stochastic processes are established by a method replacing tail probabilities by expectations throughout. The main tool is a maximal inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by an additional inequality due to M. Ledoux and M. Talagrand. The entropy condition emerging in our theorems was introduced by K. S. Alexander, whose functional central limit theorem for so-calledmeasure-like processeswill be also regained. Applications concern, in particular, so-calledrandom measure processeswhich include function-indexed empirical processes and partial-sum processes (with random or fixed locations). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examples include nonparametric regression and intensity estimation for spatial Poisson processes.     

Diagrams,orientations, and varieties

One of the central problems in the theory of ordered sets is to describe the orientations of the covering graph of an ordered set. We show that the particular operation called inversion, together with the classical constructions of retraction and product, provide a context for the classification of all such orientations. AMS subject classifications (1980). 06A10, 05C75, 05C20.     

Product or sum: comparative tests of Voigt,and product or sum of Gaussian and Lorentzian functions in the fitting of synthetic Voigt‐based X‐ray photoelectron spectra

A comparative study for the fitting of X‐ray photoelectron spectra (XPS) using different model functions is presented. Synthetically generated test spectra using Gaussian/Lorentzian convolution and a real measured spectrum are fitted with the three commonly used models: product, sum and Gaussian/Lorentzian convolution functions. In these limited tests, it was found that the sum function is superior to the product function, particularly for low‐noise spectra. Copyright © 2007 John Wiley & Sons, Ltd.     

Alternating copolymers consisting of arylalkyl methacrylates. I. Fluorescence properties of poly(arylalkyl methacrylate-alt-styrene) in organic solution

Alternating and random copolymers of 9-phenanthrylmethyl methacrylate or 2-(9-carbazolyl)ethyl methacrylate with styrene were synthesized and their fluorescence properties were examined. There was no noticeable difference in the spectral features of the alternating and random copolymers in tetrahydrofuran (THF), demonstrating that this type of polymers have no quenching sites in the polymer chains. The fluorescence quenching studies indicated that the alternating copolymers permitted singlet-state energy migration as efficiently as the corresponding random copolymers but less efficiently than the random copolymers with higher chromophore contents. These results strongly suggest that to be chromophores close to each other is most important for facilitation of an intramolecular energy migration. © 1995 John Wiley & Sons, Inc.     

Asymmetric polymerization using C1 resources

Two examples of asymmetric alternating copolymerization, (1) the alternating copolymerization of α‐olefins (monosubstituted ethenes) with carbon monoxide and (2) the alternating copolymerization of meso‐epoxide with carbon dioxide, are described, and the meaning of chirality in polymer synthesis is emphasized. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 215–221, 2004     

Determination of complex formation constants by phase sensitive alternating current polarography: Cadmium-polymethacrylic acid and cadmium-polygalacturonic acid

The use of phase sensitive alternating current polarography (ACP) for the evaluation of complex formation constants of systems where electrodic adsorption is present has been proposed. The applicability of the technique implies the previous selection of the phase angle where contribution of capacitive current is minimized. This is made using Multivariate Curve Resolution by Alternating Least Squares (MCR-ALS) in the analysis of ACP measurements at different phase angles. The method is checked by the study of the complexation of Cd by polymethacrylic (PMA) and polygalacturonic (PGA) acids, and the optimal phase angles have been ca. −10° for Cd-PMA and ca. −15° for Cd-PGA systems. The goodness of phase sensitive ACP has been demonstrated comparing the determined complex formation constants with those obtained by reverse pulse polarography, a technique that minimizes the electrode adsorption effects on the measured currents.     

Hierarchical Structure‐Within‐Structure Morphologies in A2‐star‐(B‐alt‐C) Molecules

We employ dissipative particle dynamics (DPD) to examine the self‐assembly behavior of A₂‐star‐(B‐alt‐C) molecules. We successfully observe various types of hierarchical structure‐within‐structures, such as A‐formed spheres in the matrix formed by B and C alternating layers, hexagonally packed A‐formed cylinders in the matrix with B and C segregated layers, B and C alternating layers‐within‐lamellae, coaxial B and C alternating domains within hexagonally packed BC‐formed cylinders in the A‐matrix, and co‐centric BC‐alternating domains within BC‐formed spheres in the A‐matrix, by increasing the A composition. Generally speaking, the small length‐scale B and C segregated domains are in parallel to the large length‐scale structures. This hierarchical periodicity along the same axis as well as the various characteristic structures, that the A₂‐star‐(B‐alt‐C) copolymers display, are quite different from those in A‐block‐(B‐graft‐C) coil‐comb copolymers. Moreover, it is interesting to find that when the copolymer chain length increases, though the hierarchical structure type is maintained, the number of small length‐scale lamellae that can form within the large length‐scale structure increases. These hierarchical structures under various compositions are reported theoretically for the first time in the copolymer systems consisting of the alternating blocks, and are in good agreement with the most recent experimental work by Matsushita and co‐workers (Macromolecules 2007 , 40, 4023).     

马来酸酐与乙酸乙烯酯交替共聚合反应的研究

本文研究了马来酸酐（MAn）在过氧化苯甲酰（BP）引发作用下与醋酸乙烯酯（VAc）的交替共聚反应。红外光谱证明了交替共聚物的结构，分析结果表明共聚物是由反应单体技1：1摩尔比例组成。当c（BPO）＝6．8×10~（－3）moi／L，c（VAc）＝3．4×10~（－3）mol／L，p（MAn）＝32．7g／L，63～65℃反应18小时，转化率可达92％以上。