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).     

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.     

A high yielding, one-pot synthesis of O,S-dialkyl dithiocarbonates from alcohols using Mitsunobu’s reagent

A novel Mitsunobu-based protocol has been developed for the synthesis of O,S-dialkyl dithiocarbonates from a variety of primary, secondary and tertiary alcohols using carbon disulfide, in good to excellent yields. This protocol is mild and efficient compared to other reported methods.     

Enantiomeric separation and quantification of fluoxetine (Prozac) in human plasma by liquid chromatography/tandem mass spectrometry using liquid-liquid extraction in 96-well plate format

Fluoxetine (Prozac) is currently one of the widely prescribed selective serotonin reuptake inhibitors (SSRIs) for the treatment of depression. A high-throughput sample preparation procedure using liquid-liquid extraction (LLE) in a 96-well plate format in conjunction with liquid chromatography/tandem mass spectrometry (LC/MS/MS) was developed and validated for quantification of fluoxetine enantiomers in human plasma. After addition of internal standard and ammonium hydroxide, samples were extracted with ethyl acetate. The organic extract was evaporated to dryness and reconstituted in methanol. Where possible, sample transfer and LLE steps were automated using a Tomtec Quadra 96 workstation. Adequate separation of fluoxetine enantiomeric pairs (resolution of 1.17) was achieved on a vancomycin column eluted with methanol containing 0.075% (by weight) ammonium trifluoroacetate. A triple quadrupole mass spectrometer, operated in the multiple reaction monitoring mode at m/z 310-->44 for fluoxetine enantiomeric pairs and m/z 287-->241 for oxazepam (internal standard), was used. Analysis was performed in the positive ion mode using atmospheric pressure chemical ionization (APCI). The standard curve range was 2.0-1000 ng/mL for each fluoxetine enantiomer. The intra- and inter-day precision and accuracy of the quality control (QC) samples were <12.5% (CV) and <13.6% (CV), respectively, for each fluoxetine enantiomer; the correlation coefficient was >0.990. Method ruggedness was demonstrated by the reproducible performance of the assay during a three-day validation period.     

Biyclo[2.2.1]heptane as cyclopentane precursor. Part 4 Stereocontrolled syhthesis of a potential intermediate to chromophycane dolastane and clavularane

The regio– and stereoselective synthesis of the keto-esters 20 and 22 are described, the latter being a potential intermediate to several diterpenes. The key steps involve the Diels-Alder cycloaddition between the benzocycloheptenone 5 and cyclopentadiene followed by a regioselective functionalisation of the adduct 6. A remarkable reversal of regioselectivity was observed during oxymercuration of the unsymmetric double bond in 6 and its reduced product 11 leading to 8 and 12 which were subsequently transformed to 20 and 22 respectively.