The Arrow-Barankin-Blackwell theorem in a dual space setting

In 1953 Arrow, Barankin, and Blackwell proved that, ifR ⁿ is equipped with its natural ordering and ifF is a closed convex subset ofR ⁿ , then the set of points inF that can be supported by strictly positive linear functionals is dense in the set of all efficient (maximal) points ofF. Many generalizations of this density result to infinite-dimensional settings have been given. In this note, we consider the particular setting where the setF is contained in the topological dualY ^* of a partially ordered, nonreflexive normed spaceY, and the support functionals are restricted to be either nonnegative or strictly positive elements in the canonical embedding ofY inY ^*. Three alternative density results are obtained, two of which generalize a space-specific result due to Majumdar for the dual system (Y,Y ^*)=(L ¹,L ).This research was supported in part by funds provided by the Provident Chair of Excellence in Applied Mathematics at the University of Tennessee, Chattanooga, Tennessee.     

Developments in Arylnitrene Chemistry: Syntheses and Mechanisms [New synthetic methods (31)]

Aryl- and heteroaryl-nitrenes can take part in intra-and intermolecular reactions in both of their possible electronic states (triplet and singlet). In this review we have endeavored to high-light recent synthetic uses of these reactive intermediates as well as draw attention to avenues open to further exploration in this field. Singlet arylnitrenes will interact with suitable ortho-positioned substituents to give a variety of fused azoles, some in excellent yield. In suitable solvents and in presence of amines and alcohols, phenylnitrenes undergo ring expansion to azepines which can also occur in nitrenes of certain fused bicyclic aromatics (naphthalene, quinoline, isoquinoline, benzo[b]thiophene). The latter nitrenes may also give rise to o-diamines which are useful starters for further heterocyclic synthesis. Triplet arylnitrenes (usually regarded as having only a nuisance effect in synthetic work) may also be utilized in practicable heterocyclic syntheses within a suitable molecular framework. Decomposition of aryl azides in a mixture of an organic and polyphosphoric acid leads to fused oxazoles. The mechanism is discussed for all the reactions considered.     

Advantages of high-resolution and high-mass range magnetic-sector mass spectrometry for electrospray ionization

Electrospray ionization (ESI) mass spectra have been measured on a magnetic-sector double-focusing mass spectrometer for a number of proteins and peptides. It is pointed out how in theory raising the mass resolution of a mass spectrometer from 800–1000 to 2400–3000 significantly increases the precision with which the envelope of isotopic peaks of a protein ion (or other organic ion) can be defined, particularly at higher masses. Better definition of the isotopic envelope ought to lead to higher precision in the experimental determination of molecular mass, which has been demonstrated. It is shown how ESI mass spectra of high-mass molecules are significantly less congested at higher m/z values, so that for these molecules (RMM > 40 000) there is an advantage in being able to record peaks at higher m/z values (m/z > 2000) representing ions with fewer charges. Fragmentation of a small peptide in the ESI source has been found to provide sequence information.