The probabilistic solution of the dirichlet problem for\tfrac{1}{2}\Delta + \left\langle {a,\nabla } \right\rangle + b with singular coefficients

We investigate weak solutions ofLu=?p, whereL= \(\tfrac{1}{2}\Delta + \left\langle {a,\nabla } \right\rangle + b\) and |a|²,b andp belong to the Kato classK ^d (d≥3). We shall characterize the weak solutions by a probabilistic mean value property. This characterization includes a continuity principle. Our second regularity result states that for a weak solutionu, |?u|² belongs locally to K^d. Regarding the inhomogeneous Dirichlet problem forL, we shall prove the corresponding gauge theorem and an existence and uniqueness result.     

Approximation durch Polynomfunktionen auf universellen Algebren

LetA be a topological universal algebra,F(A)=A ^A the full function algebra endowed with the producttopology andP(A) the algebra of polynomial functions onA, i. e. the subalgebra ofF(A) generated by the constant mappings and the identity.A has the local approximation property ifP(A) is dense inF(A)., TopologicalT ₂-groupsG (resp.T ₂-ringsR) of order >2 have the local approximation property if an only if all proper normal subgroups (resp. ideals) are dense inG (resp.R) andG is non-abelian (R is not a zeroring). In particular all subringsR of satisfying R ( resp. denoting the field of rational resp. real numbers) yield examples of rings having this property thus showing that the local approximation property is a far-reaching generalization of the well-known interpolation property.

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Meinem verehten Lehrer, Herrn Professor Dr. E. Hlawka, zum 65. Geburtstag gewidmet     

Zur Synthese von Human-Big-Gastrin I

The synthesis of the tetratriacontapeptide amide corresponding to the revised primary structure of human big gastrin I is described. For this purpose the fragments were designed in view of the maximum use of those utilized in our previous synthesis of human big gastrin I according to the first sequence proposal. Consequently the key tripeptide-Pro-Pro-His- (sequence 7–9) was prepared in suitably protected form to be used as amino or carboxyl component for assembly of the segments 1–9 and 1–14, respectively. Final condensation of the latter nona- and tetradecapeptide derivatives with the C-terminal segments 10–34 and 15–34 via the azide and the dicyclohexylcarbodiimide/N-hydroxysuccinimide procedure, respectively, leads to crude fully protected human big gastrin I. Upon deprotection by exposure to trifluoroacetic acid in presence of ethanedithiol-(1,2) as scavanger, ion exchange chromatography and partition chromatography, the desired tetratriacontapeptide amide was isolated in satisfactory yield with a high degree of purity. The identical immunological behaviour of the synthetic material, if compared with that of natural human big gastrin I, represents ulterior strong evidence for the correctness of the newly proposed structure for this putative prohormonal form of the gastrins.

Kurzmitteilung:Wünsch E., Wendlberger G., Mladenova-Orlinova L., Göhring W., Jaeger E., Scharf R., Gregory R. A., Dockray G. J., Hoppe-Seyler's Z. Physiol. Chem.362, 179 (1981).     

Die photometrische Bestimmung des Wismuts mit Thioharnstoff

Zusammenfassung Es wird gezeigt, daß die Bestimmung des Wismuts mit Thioharnstoff in bromwasserstoffsaurer Lösung auf photometrischem Wege mit dem Gerät ELKO II durchgeführt werden kann. Hierbei benutzt man zweckmäßig das Gerät mit der Glühlampe und Filter S 47 E oder den Quecksilberbrenner in Verbindung mit Filter Hg 436. Damit die Extinktion des gelbgefärbten Komplexes von der Thioharnstoffkonzentration unabhängig wird, muß die Lösung bei Benutzung des Filters S 45 E mindestens 10% Thiohamstoff, bei Benutzung des Filters Hg 436 mindestens 5% Thioharnstoff enthalten. Auch die Bromwasserstoffsäure übt einen Einfluß auf die Ausbildung der Extinktion aus. Wenn man die Messung mit dem Filter S 45 E durchführt und den Thioharnstoffgehalt mit 12% konstant hält, so darf die Bromwasserstoffkonzentration zwischen 1,2–9,2% betragen, um gleichbleibende Extinktionswerte zu erhalten. Benutzt man das Filter Hg 436 bei der gleichen Thioharnstoffkonzentration, so muß der Bromwasserstoffgehalt zwischen 3,0–7,5% liegen. Die Temperatur übt nur einen geringen Einfluß auf die Ausbildung des ThioharnstoffWismutkomplexes aus. Die Extinktion ist zeitunabhängig. Die Extinktionswerte ändern sich innerhalb 3 Std nach Zusammengeben der Reagenslösungen nicht. Das BEERsche Gesetz ist erfüllt. Das Verfahren besitzt erhebliche praktische Bedeutung, da sich sehr viele Metalllegierungen in denen Wismut zu bestimmen ist, mit Hilfe von Bromwasserstoffsäure und Brom sehr schnell lösen lassen.     

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.