Solving Sequences of Refined Multistage Stochastic Linear Programs

Multistage stochastic programs with continuous underlying distributions involve the obstacle of high-dimensional integrals where the integrands' values again are given by solutions of stochastic programs. A common solution technique consists of discretizing the support of the original distributions leading to scenario trees and corresponding LPs which are – up to a certain size – easy to solve. In order to improve the accuracy of approximation, successive refinements of the support result in rapidly expanding scenario trees and associated LPs. Hence, the solvability of the multistage stochastic program is limited by the numerical solvability of sequences of such expanding LPs. This work describes an algorithmic technique for solving the large-scale LP of refinement ν based on the solutions at the previous ν?1 refinements. Numerical results are presented for practical problem statements within financial applications demonstrating significant speedup (depending on the size of the LP instances).     

G-invariante Morse-funktionen

If f is a Morse function on a smooth manifold M there exists a homotopy equivalence from M to a CW complex X such that the critical points of f with index are in a one-one correspondence to the -cells of X. In the equivariant case, a similar result holds for a special type of invariant Morse functions. In this paper we prove the existence of such special invariant Morse functions on compact smooth G-manifolds. As a consequence, any compact smooth G-manifold is homotopy equivalent to a G-CW complex. Other applications deal with the Euler number of the fixed point set and Morse inequalities in equivariant homology theory.     

On the stereospecific conversion of proximally-oxygenated trisubstituted vinyltriphenylstannanes into stereodefined trisubstituted alkenes

[reaction: see text] Allylically oxygenated vinyl alpha-triphenylstannanes such as 22 can be readily converted into vinyl iodides and thereafter stereodefined trisubstituted alkenes with retention of configuration.     

4-Amino-imidazole durchThorpe-Cyclisierung

The alkylation of arylaminomethylenecyanamides1 or cyano-imidothiocarbamates2 with -halogen carbonyl compounds followed by base catalysed cyclization yields substituted 4-amino-imidazoles4. Imidazo[4,5-d]pyrimidones5, 6 and imidazo[4,5-b]pyridines7 can be obtained from4.     

Formulas for \bar \beta parameters and their limitations

The operator relationship p=[r, h] between linear momentum, position vector and Hamiltonian is the basis for an investigation of formulas for ¦ atomic orbitals is discussed. Formulas for parameters useful in extended Hückel methods are presented.

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Zusammenfassung Die Operatorengleichung p=[r, h] zwischen Impuls, Ortsvektor und Hamiltonoperator ist die Grundlage für eine Untersuchung von Formeln für -Parameter in der erweiterten Hückel[1]- und CNDO [2]-Methode. Das Eichproblem dieser Gleichung in Verbindung mit abgebrochenen Entwicklungen für Matrixelemente ¦P¦ über Atomfunktionen wird diskutiert. Nützliche Formeln für die erweiterte Hückel-Methode werden abgeleitet.

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Résumé La relation entre opérateurs moment linéaire, position et hamiltonien: p=[r, h] sert de base à une étude de formules pour les paramètres dans les méthodes de type Hückel étendu [1] et CNDO [2]. Le problème de jauge de cette équation est discuté par rapport au développement limité des éléments de matrice ¦P¦ dans une base d'orbitales atomiques. Présentation de formules pour les paramètres utiles dans les méthodes Hückel étendu.