On the maximum of the minimum of a sum

LetB denote the closure of a bounded open set of points inE ⁿ with Jordan content |B|>0 and letc>0 be constant. Typical of the expressions considered is $$M(N,c) = \max _{\left\{ {x_j } \right\}} \min _{x \in B} \sum\limits_{j = 1}^N {\left| {x - x_j } \right|^{ - c} } ,x_j \in E^n$$ Together with its analogs and extensions, the problem forc has a long history, associated with the names of Fekete, Leja, Pólya, Szegö, Frostman and Carleson, to mention just a few. It involves the notions of generalized capacity, transfinite diameter, and equilibrium potential. Here we consider the casec≧n and its extensions, for which the prior history seems less comprehensive. Illustrative of the results obtained are the three equations $$\mathop {\lim }\limits_{N \to \infty } \frac{{M(N,n)}}{{N\log N}} = \frac{{\omega (n)}}{{\left| B \right|}},\mathop {\lim }\limits_{N \to \infty } \frac{{M(N,c)}}{{N^{{c \mathord{\left/ {\vphantom {c n}} \right. \kern-\nulldelimiterspace} n}} }} = \frac{{L(n,c)}}{{\left| B \right|^{{c \mathord{\left/ {\vphantom {c n}} \right. \kern-\nulldelimiterspace} n}} }},\mathop {\lim }\limits_{c \to \infty } L(n,c)^{{1 \mathord{\left/ {\vphantom {1 c}} \right. \kern-\nulldelimiterspace} c}} = \mathop {\lim }\limits_{N \to \infty } \frac{{\left( {\left| B \right|/N} \right)^{{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}} }}{{\varrho (N)}}$$ In the firstc=n and ω (n) is the volume of the unit ball. In the secondc>n and existence of the limit is asserted, 0<L(n,c)<∞. In the third, ? (N) is the smallest value such thatN spheres of radius ? (N) can coverB. The results would be unchanged if we requiredx _j ∈B instead ofx _j ∈E ⁿ in the definition ofM(N, c).     

N-labelling studies of the reaction of 2,6-di-t-butyl-4-methyl-4-nitrocyclohexa-2,5-dienone with nitrogen dioxide; A revised mechanism

Reaction of the ¹⁵N-4-nitrodienone (1) with excess nitrogen dioxide in benzene for 16 h gives the 4,5,6-trinitro ketone (2), labelled (¹⁵NO₂) at C6 but not at C4, and recovered 4-nitrodienone (1) in which extensive loss of ¹⁵NO₂ has occurred. The mechanistic implications of these and related results are discussed.     

Where is the sodium in self-assembled monolayers of single-stranded DNA?

Monolayers of single-stranded DNA (ssDNA) immobilized on surfaces form the basis of a number of important biotechnology applications, including DNA microarrays and biosensors. The organization of ssDNA as layer on a solid substrate allows one to investigate various properties of the DNA in a controlled manner and to use DNA for analytical applications as well as for exploring futuristic schemes for molecular electronics. It is commonly assumed that the adsorbed DNA layer contains some structural water and the cations. Here we show, based on XPS studies, that when monolayers of ssDNA are formed from sodium phosphate buffer and washed thoroughly, no Na+ signal is detected. A finite concentration of ions is observed when the DNA is made from a solution of Mg2+ ions, but it is still only a fifth of what it would be if all the phosphate ions were fully neutralized by the metal cations.     

Silicon clusters: chemistry and structure

The chemical reactions of size selected silicon cluster ions (containing up to 70 atoms) have been studied with a number of different reagents using injected ion drift tube techniques. Both kinetic and equilibrium measurements have been performed as a function of temperature, and the influence of cluster annealing on chemical reactivity explored. Unlike metal clusters, where bulk behavior appears to be approached with around 30 atoms, large silicon clusters (n up to 70) are much less reactive than bulk silicon surfaces. These results suggest that the clusters in the size range examined here are not small crystals of bulk silicon, but have compact, high coordination number structures with few dangling bonds.     

Intermolecular multiple quantum coherences at high magnetic field: the nonlinear regime

Experiments have been carried at magnetic-field strengths of 9.4, 14.1, and 17.6 T to explore the evolution of intermolecular multiple quantum coherences in the nonlinear regime where the system evolves for times that are much greater than the characteristic time of action of the long-range dipolar field, tau(d). The results show the expected Bessel function form of the recorded signal as a function of time of evolution, with evident zeros and sign changes. As expected, the rate of signal evolution increases at higher-field strengths as a result of the increased equilibrium magnetization. A numerical method for calculating the evolution of magnetization under the action of the distant dipolar field, relaxation, and diffusion that is based on Fourier analysis of the magnetization distribution has been applied to the correlated two-dimensional spectroscopy revamped by asymmetric z-gradient echo detection sequence in the nonlinear regime and shown to produce results that are in good agreement with experimental data acquired at different magnetic fields and rates of spatial modulation. Experiments and simulations have also been used to explore the evolution of magnetization in a mixture of two interacting spin species in the nonlinear regime.