On the diameter and radius of randon subgraphs of the cube

The n-dimensional cube Qⁿ is the graph whose vertices are the subsets of {1,…n}, with two vertices adjacent if and only if their symmetric difference is a singleton. Clearly Qⁿ has diameter and radious n. Write M = n2^n-1 = e(Qⁿ) for the size of Qⁿ. Let Q = (Q_t)_o^M be a random Qⁿ-process. Thus Q^t is a spanning subgraph of Qⁿ of size t, and Q_t is obtained from Q_t–1 by the random addition of an edge of Qⁿ not in Q_t–1, Let t^(k) = τ(Q;δ?k) be the hitting time of the property of having minimal degree at least k. We show that the diameter d_t = diam (Q_t) of Q_t in almost every Q? behaves as follows: d_t starts infinite and is first finite at time t⁽¹⁾, it equals n + 1 for t⁽¹⁾ ? t⁽²⁾ and d_t, = n for t ? t⁽²⁾. We also show that the radius of Q_t, is first finite for t = t⁽¹⁾, when it assumes the value n. These results are deduced from detailed theorems concerning the diameter and radius of the almost surely unique largest component of Q_t, for t = Ω(M). © 1994 John Wiley & Sons, Inc.     

Book Reviews

Unsigned book reviews are by the Book Review Editor.     

Energy transfer between the flavin chromophores of electron-transferring flavoprotein fromMegasphaera elsdenii as inferred from time-resolved red-edge and blue-edge fluorescence spectroscopy

Both a mode-locked argon-ion laser and synchrotron radiation were used as excitation sources to obtain time-resolved polarized fluorescence of the two FAD cofactors in electron transferring flavoprotein fromMegasphaera elsdenii. Red-edge excited and blue-edge detected fluorescence anisotropy decay curves did not contain a fast relaxation process which was observed upon mainband excitation and detection. This relaxation was assigned to homo-energy transfer between the two FAD cofactors. Failure of energy transfer as observed with edge spectroscopy on this protein excludes restricted reorientational motion of the flavins as a possible mechanism of depolarization. From the global analysis of the fluorescence anisotropy decay surface obtained at multiple excitation and detection wavelengths, the distance between and the relative orientation of the flavins could be estimated. The methodology described has general applicability in other multichromophoric biopolymers and has the potential to acquire accurate geometrical parameters in these systems.     

Synthesis and thermal crosslinking of oxazolidine‐functionalized polyurethanes

A new oxazolidine derivative was obtained from phenol, 2‐amino‐2‐methylpropane‐1,3‐diol and paraformaldehyde. The reaction of this novel oxazolidine diol with phenylisocyanate lead to a urethane model compound which can be polymerized thermally by oxazolidine ring opening to give a Mannich bridge structure. Linear segmented polyurethanes were prepared by reaction of different ratios of oxazolidine diol and commercial polyethylenglycol (M_w ～ 400) with 4,4′‐methylenbis (cyclohexylisocyanate) (HMDI, 90% isomers mixture). The polyurethanes were thermally characterized and crosslinked by oxazolidine ring opening to obtain materials which showed improved thermal stability. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 4965–4973, 2007     

Synthesis and polymerization of o-hexylaniline: Characterization of the corresponding polyaniline

In the field of research on soluble conducting polymers, the poly(o-alkylanilines) are very interesting because we can expect them to give more soluble polymers and new properties. Like poly(o-propylaniline) (POP), which is more soluble than polyaniline (PANi), poly(o-hexylaniline) (POH) appears to be more soluble in organic solvents than POP because of the longer alkyl groups in the 2-position. The higher solubility confers better processability on this new polymer, and because of this solubility, an NMR study in solution became possible.The nitration of hexylbenzene and the reduction of the resulting product to o-hexylaniline were performed according to the literature. The chemical polymerization was easy and it is possible to produce this polymer in large quantities.The polymerization carried out in anhydrous NH₄F, 2.35 HF medium and in 5 M perchloric acid gave a polymer with almost quantitative yield. The electrochemical behaviour of POH displayed faster electron transfers than PANi in organic solvents, depending on the acido-basicity level of the aqueous solutions. Unlike PANi, fractal growth was not observed.     

Inequalities for a unified family of Voigt functions in several variables

The classical Voigt functions occur frequently in a wide variety of problems in astrophysical spectroscopy, emission, absorption and transfer of radiation in heated atmosphere, and plasma dispersion, and indeed also in the theory of neutron reactions. Here, in the present paper, by applying several known upper bounds for the first-kind Bessel function J _ν (x) given recently by (for example) Landau, Olenko and Krasikov, sharp bounding inequalities are obtained for the unified multivariable Voigt function V _μ,ν (x; y) in terms of the confluent Fox-Wright function ₁ψ₀ and its incomplete variant ₁ψ₀. Connections of the unified multivariable Voigt function V _μ,ν (x; y) with other unifications and generalizations of the classical Voigt function are also briefly pointed out.     

Lagrangian Duality in Set-Valued Optimization

In this paper, we study optimization problems where the objective function and the binding constraints are set-valued maps and the solutions are defined by means of set-relations among all the images sets (Kuroiwa, D. in Takahashi, W., Tanaka, T. (eds.) Nonlinear analysis and convex analysis, pp. 221–228, 1999). We introduce a new dual problem, establish some duality theorems and obtain a Lagrangian multiplier rule of nonlinear type under convexity assumptions. A necessary condition and a sufficient condition for the existence of saddle points are given. The authors thank the two referees for valuable comments and suggestions on early versions of the paper. The research of the first author was partially supported by Ministerio de Educación y Ciencia (Spain) Project MTM2006-02629 and by Junta de Castilla y León (Spain) Project VA027B06.     

Extraction of strontium using an irradiated extraction agent of crown structure

The partition coefficient of strontium upon its extraction from the aqueous solutions of picric acid into chloroform containing DC-18-crown-6, B-15-crown-5 after their exposition to gamma radiation has been studied. A significant decrease of the partition coefficient in the studied range of doses 10–70 kGy was observed. This effect can be attributed to the radiolytic products of chloroform. The radiation destruction as well as dehydrogenation of crown ethers were not observed.