Application of the ellipsoid method in an interactive procedure for multicriteria linear programming

The research reported in this paper is concerned with an application of the ellipsoid algorithm in the interactive multicriteria linear programming step method (STEM) byBenayoun et al. [1971]. Due to this application we eliminate some drawbacks of the original version of STEM and, moreover, we avoid extra computations connected with sensitivity analysis in every iteration. Specifically, we use the ellipsoid algorithm to minimize the Euclidean norm in the criterion space instead of the Chebyshev norm, which ensures that every solution submitted to the decision maker is efficient. As follows from a computational experiment, in comparison with the application of the simplex method, the proposed modification of STEM shows a smaller increase of the computational effort when the number of criteria increases. However, the absolute computation time becomes worse for problems of larger size.

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Zusammenfassung In dieser Arbeit wird über eine Anwendungsmöglichkeit der Khachiyan-Shor-Algorithmus (Ellipsoid-Algorithmus) im Rahmen des STEM-Verfahrens zur interaktiven Lösung linearer Vektoroptimierungsmodelle berichtet. Auf diese Weise können einige spezifische Nachteile des STEM-Verfahrens in seiner Originalversion vermieden werden. Durch die Verwendung der Euklidischen Norm anstelle der beim STEM-Verfahren üblichen Tschebyscheff-Norm wird garantiert, daß dem Entscheidungsträger nur effiziente Lösungen vorgeschlagen werden. Die numerischen Erfahrungen zeigen, daß der Lösungsaufwand der hier vorgeschlagenen Modifikation des STEM-Verfahrens mit steigender Anzahl von Zielfunktionen weniger stark zunimmt als bei der üblichen Version. Dies gilt jedoch nicht hinsichtlich der allgemeinen Problemgröße.

     

Heterocyclen aus α-nitroolefinen. XI . 1,2-oxazine aus α-nitroolefinen und β-dicarbonylverbindungen

Dihydro-1,2-oxazines 8 and 12-oxazines 12 are formed by the reaction of the nitronic acids of the adducts of α-nitroolefins and β-dicarbonyl compounds ( 6 ) by two ways: A ) The nitronic acids 6 are heated in a boiling solution of urea in ethanol or methanol. B ) The nitronic acids 6 are reduced with an aqueous solution of ammoniachloride and sodium sulfide. ¹H-nmr and ¹³C-nmr investigations prove the constitutions. The mechanism is discussed.     

Strukturviskosität von verdünnten Lösungen von Biopolymeren

The importance of the rheological behaviour of solutions of macromolecules is briefly evaluated. The viscosity of the solutions depends on concentration, shear rate and time of shear, this relation being determined by the structure of the dissolved molecules. In dilute solutions shear dependence of viscosity is very frequently caused by the preferential orientation of anisotropic molecules. In such a case the particle dimensions can be calculated from the true limiting viscosity number, an anisotropy factor, the rotational diffusion constant and the effective particle density. These numbers can be derived from the flow curve, which has been extrapolated to zero concentration. It is necessary to measure the flow curve at shear gradients, which are sufficiently low to allow for an extrapolation to vanishing shear rate. By comparing the experimental flow curve with a choice of theoretical ones, the rotational diffusion constant and the anisotropy factor (axial ratio) can be found. From the limiting viscosity number and the axial ratio, the particle density can be calculated.     

Komplexe von 2,6-Dimethylpyridin-1-oxid mit Lanthanidjodiden

Complexes of 2,6-dimethylpyridine 1-oxide with lanthanide iodides of the formulaeLn(2,6-LTNO)₅I₃ whereLn=La, Tb and Yb,Ln(2,6-LTNO)₄I₃ whereLn=Pr and Nd and Er(2,6-LTNO)_4.5I₃ have been prepared and characterised by chemical analysis, infrared and conductance studies. Infrared and conductance data have been interpreted in terms of dimeric (or polymeric) structures involving bridging amine oxide groups.

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Über die Synthese von 5-(o-Trifluormethylphenyl)-1H-thieno[3,4—e]-1,4-diazepin-2(3H)-onen

The synthesis of 5-(o-trifluoromethylphenyl)-1H-thieno-[3,4-e]1,4-diazepin-2(3H)-one (7) and its nitration and chlorination in pos. 8 are described.     

Berechnung von mittleren Schwingungsamplituden einiger tetraedrischer Moleküle und Ionen nach der Methode der charakteristischen Schwingungen

Mean amplitudes of vibration of a series of tetrahedralXY ₄ molecules and ions (hydrides, halides, oxides and oxoanions) have been calculated using the “Method of the Characteristic Vibrations” ofA. Müller. The results indicate that this method leads to very good values for most of the investigated species, and especially in the cases of highM _X/M_Y mass ratio.     

Die Hydrohalogenierung von Hexaphenyldigerman und die Schwingungsspektren einiger Ge2(C6H5) n X6−n -Verbindungen

The reactions of Ge₂(C₆H₅)₆ with HCl and HBr lead in nearly quantitative yields to the 1,1,2,2-tetrahalo derivatives Cl₂(C₆H₅)GeGe(C₆H₅)Cl₂ (I) and Br₂(C₆H₅)GeGe(C₆H₅)Br₂ (II), resp. The Si?Ge bond of (C₆H₅)₃SiGe(C₆H₅)₃ is cleaved under the conditions of hydrohalogenation. The vibrational spectra of Ge₂Br₆, Ge₂(C₆H₅)₆, I, and II are reported. The influence of vibrational coupling on ν GeGe in these compounds is discussed in detail, including vibrational calculations.     

Gleichverteilung und Quadratwurzelschnecke

A new proof of a theorem of W. Neiss is given that the rays of the Quadratwurzelschnecke are uniformly distributed mod 2 . The exact order of the discrepancy of the sequence is determined. Multidimensional generalizations of this sequence are also considered. It is shown that the Quadratwurzelschnecke is something like a roulette.

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Herrn Prof. R. M. Redheffer in Freundschaft zum 60. Geburtstag gewidmet     

Eine Topologie auf der Menge der endlichdimensionalen nichtassoziativen Algebren über einem lokalen Körper

Motivated by an investigation ofKuz'min [2], in this note we introduce a topology on the set of nonassociative algebras of a fixed dimensionr< over a local fieldF. We obtain a space, which is notT ₁. If the trivial algebra is removed, the remaining subspace is quasicompact. The space is homeomoprhic to the group of square classes ofF ^*; forr3 the space is notT ₁. The set of division algebras is open in.