Dinitration studies of 2,4,5-tribromo-3,6-dimethylphenol: the formation of an acyloin rearrangement product

Dinitration of 2,4,5-tribromo-3,6-dimethylphenol(1) gives either the tribromodinitro-compound(2) or its acyloin rearrangement product(3), depending on the reaction conditions: X-ray crystal structure analyses are reported for compounds (2) and (3).     

The numbers of chiral and achiral alkanes and monosubstituted alkanes

Whereas the theory for the enumeration of the optical isomers of the lakyl radicals and the alkanes has long been understood, this is not the case for the corresponding archiral isomers. We present for the first time recurrence formulae for counting the number of archiral isomers of the alkyl radicals and the alkanes. For chiral and archiral alkanes and monosubstituted alkanes, numerical results up to C₁₄ are tabulated.After presenting the history of the problem and the necessary definitions, we proceed to derive functional equations on the various generating functions, which readily yield the more explicit recurrence formulae usefule for numerical calculations. In the process, we first re-derive Pólya's expression for planted steric trees using his classical enumeration theorem. This result is then extended to the enumeration of free steric trees using the now standard tree-counting method due to Otter and known as a dissimilarity characteristic equation.By definition, a steric tree is a quartic tree (all points having degree 1 or 4) in which the four neighbors of every carbon point are given a tetrahedral configuration. Building on the methods of the first two authors for counting chiral and archiral trees in the plane, we obtain the formula for counting achiral steric trees, thus setting a problem first enunciated by van't Hoff and Le Bel in 1874.     

Micelle studies using a metal ion/murexide indicator system

Summary Complex formation between divalent metal ions and the hydrophilic murexide anion in the presence of anionic micelles has been employed to study aspects of micelle formation, the binding of divalent metal ions to micelles, and the kinetics of metal-complex formation in the presence of micelles.

---

Zusammenfassung Die Komplexbildung zweiwertiger Metallionen mit dem hydrophilen Indikator Murexid wurde in Gegenwart anionischer Mizellen untersucht. Die Indikatorreaktion wurde benutzt, um Aussagen über Mizellbildung, Bindung zweiwertiger Metallionen an Mizellen und die Kinetik der Metallkomplexbildung bei Anwesenheit von Mizellen zu erhalten.

---

---

With 8 figures and 1 table

---

Dedicated to Prof.G. Rehage on the occasion of his 60th birthday.     

Rates of convergence and optimal spectral bandwidth for long range dependence

Summary For a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like ^1–2H as 0+ for 1/2<H<1 (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies 2j/n,j=1,..., m, wherem andm/n0 asn. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice ofm.