Different thermal reactivity of a 1,2-thiaphospholo[a]phosphirane in free and metal carbonyl complexed form

The W(CO)5 and Fe(CO)4 complexes of the bicyclic phosphirane 3,5,6,6-tetraphenyl-1-phospha-2-thiabicyclo[3.1.0]hex-3-ene undergo a thermal 2-phenylphosphirane --> dihydrophosphaisoindole ring expansion, while the free phosphirane suffers both a [2 + 1] cycloreversion and a fragmentation yielding a butadienyl sulfide.     

A general formalism of deriving the pair potential of polymer chains for arbitrary interaction potentials between isolated chain segments at and close to the theta-point

A formalism has been worked out which allows to transform any non-punctiform segment-segment potential of isolated polymer segments ε of fairly short-ranged character into the pair-potentialU operating between linear polymer chains with a certain reference to the arguments as they have been originally put forward byFlory andKrigbaum. Although no restrictions are made in the derivation as to the repulsive or attractive contribution of the segment-segment potential ε because of some known general deficiencies of theFlory-Krigbaum treatment for exclusively repulsive interaction, the resulting equations are primarily intended to describe the thermodynamic situation at and close to the θ-point where repulsion and attraction—though working at different ranges of segment separation—cancel. As the equation derived is somewhat complicated two different approximate forms have been developed: The first one is based on aTaylor series expansion retaining terms up to the fourth order which allows to characterizeU by the second and the fourth moment of the pair segment-segment distribution function, β and γ (β being the so-called binary cluster integral of segment-segment interaction, which is considered to be zero for θ-conditions). In this caseU may be represented by an expression of the general form $$U/kT = A(1 - BR^2 )\exp \{ - bR^2 \} .$$ The second method is based on a separate integration over the repulsive and attractive ranges of ε giving the repulsive (U ₊) and the attractive (U _?) part ofU finally after some approximations leading to an equation of the general form $$U/kT = (U_ + + U_ - )/kT = A_1 \exp \{ - b_1 R^2 \} - A_2 \exp \{ - b_2 R^2 \} .$$ In both cases the knowledge of the exact form of ε is dispensable, only β and γ—or for the second case their repulsive (β₊ and γ₊) and attractive (β_? and γ_?) parts have to be known. It is shown that the approximations are in excellent accordance with the exact form so that they may be conveniently used to describe pair potentials of polymer chains and to analyze pair potentials of segment-segment interactions under the limitations and conditions indicated.     

Ansamycine, 5. Mitt., Chemie der Metacycloprodigiosine

In order to obtain a better partition of a prodigiosin derivative in biological media, two hydroxy groups were introduced into theansa-part of the molecule. The synthesis of the title compound is described in detail.     

Efficient kinetic resolution of a racemic 7-membered biaryl lactone: enantioselective synthesis of 2,2′-dihydroxymethyl-1,1′-binaphthyl

A novel pathway to enantiomerically pure 2,2′-dihydroxymethyl-1,1′-binaphthyl is described, through kinetic resolution of its racemic 7-membered biaryl lactone precursor. Oxazaborolidine-assisted borane reduction of this lactone proceeds with very high enantioselectivity (k_rel=50). For optimum results on a preparative scale, a ‘three fractions strategy’ is suggested, which combines kinetic resolution and fractional crystallization and leads to both atropisomers in high enantiomeric excess.     

Electrophoretic clean-up of organic acids from coffee for the GC/MS analysis

The clean-up presented here includes a free flow field step electrophoresis followed by ultrafiltration. Thus, organic acids can be separated from non-acidic and high-molecular compounds in roasted and instant coffee. The acids are identified by gas chromatography/mass spectrometry after freeze-drying and trimethylsilylation. With the method presented, 31 acids could be identified in commercial roasted coffee blends and in instant coffee, among them for the first time in coffee: 3-hydroxypropionic, 2-oxobutyric, glyceric, 2,4-dihydroxybutyric, 5-hydroxymethylfuran-2-carboxylic and 2-hydroxyglutaric acid.     

Die Klassengruppen quadratischer und kubischer Zahlkörper und die Selmergruppen gewisser elliptischer Kurven

For k∈?\0 we define the elliptic curve A_k, by the equation: Y²=X³+k. Let C(k) be the class group of \(K_o = Q(\sqrt[6]{k})\) . If k is a square we find a close connection between the elements of order 2 of the Selmer group of A_k and C(k); there is a corresponding connection between the elements of order 3 if k is a cube and satisfies some additional conditions. The main tool to prove the statements is the global and local Galois cohomology of elliptic curves; it seems remarkable that nearly no “explicit” number theory has to be used.     

A very special nonexistence theorem for abelian difference sets

It is wellknown that the technique of character sums together with the tools of algebraic number theory is the adequate method for the study of difference sets in abelian groups, compare for instance Ott [5] or Turyn [6]. In this paper we use this method to prove a new non-existence theorem for certain difference sets in abelian groups of order rp^a rp^a , where r 1 2 r \neq 2 and p are distinct primes.     

Existence of stationary,collisionless plasmas in bounded domains

We consider a collisionless plasma, which consists of electrons and positively charged ions and is confined to a bounded domain in ?³. The distribution functions of the particles are assumed to satisfy specular reflections on the boundary of the domain and the boundary is assumed to be perfectly conducting. We establish the existence of stationary plasmas in the non-relativistic, electrostatic case described by the Vlasov–Poisson system as well as in the relativistic, electrodynamic case described by the relativistic Vlasov–Maxwell system.     

Finding minimum areak-gons

Given a setP ofn points in the plane and a numberk, we want to find a polygon with vertices inP of minimum area that satisfies one of the following properties: (1) is a convexk-gon, (2) is an empty convexk-gon, or (3) is the convex hull of exactlyk points ofP. We give algorithms for solving each of these three problems in timeO(kn ³). The space complexity isO(n) fork=4 andO(kn ²) fork5. The algorithms are based on a dynamic programming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.This paper includes work done while David Eppstein was at Columbia University, Department of Computer Science, and while Günter Rote and Gerhard Woeginger were at the Freie Universität Berlin, Fachbereich Mathematik, Institut für Informatik. Research was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM).