Weighted majority games and the matrix of homogeneity

If a weighted majority game (not necessarily constant sum or super additive) is described by the weights (voting strength) of the players involved and a majority level, then it is desirable to know whether the game is in addition homogeneous. The paper provides a recursive procedure defining a test for homogeneity. This procedure involves the computation of a number theoretical function, the matrix of homogeneity. If this matrix is knownall majority levels with respect to which the given set of weights represents a homogeneous simple weighted majority game are known at once.

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Zusammenfassung Ein gewichtetes Majoritätsspiel (nicht notwendig superadditiv oder mit Konstantsummeneigenschaft) kann beschrieben werden durch die Gewichte (die Abstimmungsstärke) der beteiligten Spieler und durch eine Mehrheitsschranke. Wenn eine solche Darstellung gegeben ist, so ist es wünschenswert, das Spiel zusätzlich auf die Eigenschaft der Homogenität hin zu untersuchen. Die vorliegende Arbeit stellt eine rekursive Prozedur zur Verfügung, die einen Test auf Homogenität einer vorgelegten Darstellung eines Spieles impliziert. Die Prozedur ermöglicht die Berechnung einer zahlentechnischen Funktion, der Homogenitätsmatrix. Wenn diese Matrix bekannt ist, kann man bei einem vorgegebenen Gewichtssatz alle Majoritätsschranken angeben, bezüglich welcher die vorgelegten Gewichte ein homogenes einfaches gewichtetes Majoritätsspiel darstellen.

     

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.

     

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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Mit 2 Abbildungen     

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.     

Observations on the swelling characteristics of the zwitterionic hydrogel of poly(1-3-sulfopropyl)-2-vinyl-pyridinium-betaine hydrogel

Samples of poly[1-(3-sulfopropyl)-2-vinyl-pyridinium-betaine] (PSPV) have been synthesized to high conversion by free radical polymerization in aqueous solution of the zwitterionic monomer SPV with several concentrations of the crosslinker N,N′-methylene-bis-acrylamide (MBA). The densities of the resultant xerogels increased regularly with the content of MBA. Hydrogels obtained by swelling them in water and aqueous KSCN solution were examined by gravimetric and dimensional analysis. The water contents increased with decreasing content of MBA, the value of 92.7 wt% at the lowest MBA content being higher than that for other zwitterionic hydrogels. Enhanced swelling occurred in 1 M aq. KSCN at each MBA content, the total swelling being 98.1 wt% at the lowest crosslinker content. Swelling increased with increasing temperature. An approximate procedure to formulating swelling equilibrium in term of the volume fraction of water in hydrogel, in conjunction with the van’t Hoff equation, yields a small positive value for the enthalpy of swelling. This is compared with values derived similarly for other hydrogels.     

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.     

Sorption of 241Am by Aspergillus nigerspore and hyphae

Biosorption of ²⁴¹Am by a fungus A. niger, including the spore and hyphae, was investigated. The preliminary results showed that the adsorption of ²⁴¹Am by the microorganism was efficient. More than 96% of the total ²⁴¹Am could be removed from ²⁴¹Am solutions of 5.6-111 MBq/l (C _o) by spore and hyphaeof A. niger, with adsorbed ²⁴¹Am metal (Q) of 7.2-142.4 MBq/g biomass, and 5.2-106.5 MBq/g, respectively. The biosorption equilibrium was achieved within 1 hour and the optimum pH range was pH 1-3. No obvious effects on ²⁴¹Am adsorption by the fungus were observed at 10-45 °C, or in solutions containing Au³⁺ or Ag⁺, even 2000 times above the ²⁴¹Am concentration. The ²⁴¹Am biosorption by the fungus obeys the Freundlich adsorption equation. There was no significant difference between the adsorption behavior of A. nigerspore and hyphae. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.