The many limits of mixed means

The sequences introduced by Carlson (1971) are variants of the Gauss arithmetic geometric sequences (which have been elegantly discussed by D. A. Cox (1984, 1985)). Given (complex)a ₀,b ₀ we define

On the width of an orientation of a tree

There are 2 ^n-1 ways in which a tree on n vertices can be oriented. Each of these can be regarded as the (Hasse) diagram of a partially ordered set. The maximal and minimal widths of these posets are determined. The maximal width depends on the bipartition of the tree as a bipartite graph and it can be determined in time O(n). The minimal width is one of [/2] or [/2]+1, where is the number of leaves of the tree. An algorithm of execution time O(n + ² log ) to construct the minimal width orientation is given.This research was partially funded by the National Science and Engineering Research Council of Canada under Grant Number A4219.     

Normal subgroups and elementary theories of lattice-ordered groups

We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups H _i which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups H _i can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically nice properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S _i ) of infinite linearly ordered sets (S _i , ) such that each group A(S _i ) has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.     

Dual vectors and lower bounds for the nearest lattice point problem

We prove that given a point outside a given latticeL then there is a dual vector which gives a fairly good estimate for how far from the lattice the vector is. To be more precise, there is a set of translated hyperplanesH _i, such thatL _iH_i andd( _iH_i)(6n ²+1)^–1 d( ,L).Supported by an IBM fellowship.     

The gap between monotone and non-monotone circuit complexity is exponential

A. A. Razborov has shown that there exists a polynomial time computable monotone Boolean function whose monotone circuit complexity is at leastn ^{c losn} . We observe that this lower bound can be improved to exp(cn ^1/6–o(1)). The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grötschel—Lovász—Schrijver on the Lovász — capacity, of a graph.     

分子筛修饰电极的研究（Ⅰ）──氢醌／醌电对在分子筛修饰电极上的电化学行为

考察了中性分子电对Ｈ_２Ｑ／Ｑ在１３Ｘ型分子筛修饰电极上的电化学行为。发现Ｈ_２Ｑ／Ｑ电对在修饰电极上电响应迅速，为双电子可逆反应，不受溶液温度及电极与溶液相对运动的影响，而与溶液的ｐＨ值有关。当ｐＨ≤２．０时，修饰电极对Ｈ_２Ｑ／Ｑ电对的氧化／还原有明显的催化作用．     

QSPR Modeling of the Reactivity Parameters of Monomers in Radical Copolymerizations

For a series of monomers, QSPR test analysis is performed by optimizing the correlation weights of the local invariants of molecular graphs representing monomer structures in order to construct models of the reactivity parameters of monomers Q and e. This approach may be used as a tool in reactivity predictions for monomers for which no experimental data on Q and e are available.Original Russian Text Copyright © 2004 by A. A. Toropov, V. O. Kudyshkin, N. L. Voropaeva, I. N. Ruban, and S. Sh. Rashidova__________Translated from Zhurnal Strukturnoi Khimii, Vol. 45, No. 6, pp. 994–998, November–December, 2004.     

Fluorescence quenching of anthropogenic fulvic acids by Cu(II), Fe(III) and UO(2)(2+)

The quenching of the fluorescence of three anthropogenic fulvic acids (FA) provoked by Cu(II) (pH 6.0), Fe(III) (pH 4.0) and UO₂²⁺ (pH 3.5), was analyzed by a non-linear method and by Stern-Volmer plots. The FA samples were extracted from composted sewage sludges (csFA), composted municipal wastes (mwFA) and composted livestock wastes (lsFA). Synchronous-scan fluorescence (SyF) spectra were collected as a function of metal ion concentration. Spectral data were treated by a self-modeling mixture analysis method (SIMPLISMA) to detect the SyF spectral band with the strongest quenching and to calculate the corresponding quenching profile. The analysis of these profiles by a non-linear method allowed the estimation of conditional stability constants (K) and of the percentage of non-complexing fluorophores. The same quantitative information was obtained by the modified Stern-Volmer equation taking into account the existence of fluorophores that do not participate in the complexation. Good agreement was found between the results of the two procedures. The log K calculated by the non-linear method were (standard deviation in parenthesis): csFA, Cu(II), 4.22 (5); Fe(III), 5.0 (1); UO₂²⁺, 5.2 (2); mwFA, Cu(II), 4.21 (3); Fe(III), 5.6 (2); UO₂²⁺, 4.7 (3); lsFA, Cu(II), 4.51 (8); Fe(III), 5.5 (2); UO₂²⁺, 3.6 (2).     

Phase separation of poly(γ-benzyl L-glutamate) to liquid crystal and isotropic solution in various helicogenic solvents

Solvent effects on the phase separation of poly(-benzyl L-glutamate) to liquid crystal and isotropic solution have been observed in various helicogenic solvents. The temperature-composition phase diagrams have been determined for each solution. The critical concentrations, ₂ ^* , at which the phase separation occours have been compared in various solvents. In dimethylformamide in which the polymer is molecularly dispersed, the observed ₂ ^* value has agreed with that calculated by Flory's theory. In some solvents in which the polymer aggregates in a head-to-tail mode such as chloroform, the observed ₂ ^* values have been considerably small. It is assumed that the polymer aggregates behave as longer particles than the original particles. In dioxane in which the polymer aggregates highly both in a head-to-tail and a side-by-side modes, the ₂ ^* value has been a little larger than that in chloroform. In this case the relationship between the aggregation and the liquid crystal formation is so complicated that further investigation is necessary. In aromatic solvents such asm-cresol that dissolves the polymer almost molecularly, the ₂ ^* is smaller than that in dimethylformamide. Therefore, the intermolecular interactions between the phenyl groups in the side groups of the polymer and those in solvent molecules must be considered.The author is grateful to Mr. K. Sano and Mr. M. Watanabe for their observation of the liquid crystal formation.     

Simultaneous hydrogen and peroxide production by photocatalytic water splitting

Photocatalytic oxidation of water is a promising method to realize large-scale H₂O₂ production without a hazardous and energy-intensive process. In this study, we introduce a Pt/TiO₂(anatase) photocatalyst to construct a simple and environmentally friendly system to achieve simultaneous H₂ and H₂O₂ production. Both H₂ and H₂O₂ are high-value chemicals, and their separation is automatic. Even without the assistance of a sacrificial agent, the system can reach an efficiency of 7410 and 5096 μmol g^–1 h^–1 (first 1 h) for H₂ and H₂O₂, respectively, which is much higher than that of a commercial Pt/TiO₂(anatase) system that has a similar morphology. This exceptional activity is attributed to the more favorable two-electron oxidation of water to H₂O₂, compared with the four-electron oxidation of water to O₂.