On the number of conjugacy classes of a finite solvable group II

We prove that a finite solvable group G has at least (49p+1)/60 conjugacy classes whenever p is a prime such that p² divides the order of G. We also construct an infinite family of finite solvable groups, where this bound is attained.     

Natural convections in conjugated single and double enclosures

The natural convection in single and double conjugated enclosures are numerically investigated. The single and double enclosures are formed by low conductance walls with finite thickness. The outside vertical surfaces of the conducting walls are of the third kind of boundary condition while the top and bottom outside surfaces are adiabatic. The problem studied is characterized by a dominant horizontal temperature gradient and the thermal boundary conditions at the cavity surfaces can not be specified in priori. Numerical results reveal the characteristics in such kind of enclosures and show the importance of the thermal boundary conditions on the natural convection in enclosures. It is also found that the natural convections in the conjugated double enclosures are basically the same, with a major difference in their fluid temperature levels.     

The Shapley value for cooperative games under precedence constraints

Cooperative games are considered where only those coalitions of players are feasible that respect a given precedence structure on the set of players. Strengthening the classical symmetry axiom, we obtain three axioms that give rise to a unique Shapley value in this model. The Shapley value is seen to reflect the expected marginal contribution of a player to a feasible random coalition, which allows us to evaluate the Shapley value nondeterministically. We show that every exact algorithm for the Shapley value requires an exponential number of operations already in the classical case and that even restriction to simple games is #P-hard in general. Furthermore, we outline how the multi-choice cooperative games of Hsiao and Raghavan can be treated in our context, which leads to a Shapley value that does not depend on pre-assigned weights. Finally, the relationship between the Shapley value and the permission value of Gilles, Owen and van den Brink is discussed. Both refer to formally similar models of cooperative games but reflect complementary interpretations of the precedence constraints and thus give rise to fundamentally different solution concepts.     

Sensitivity of trust-region algorithms to their parameters

In this paper, we examine the sensitivity of trust-region algorithms on the parameters related to the step acceptance and update of the trust region. We show, in the context of unconstrained programming, that the numerical efficiency of these algorithms can easily be improved by choosing appropriate parameters. Recommended ranges of values for these parameters are exhibited on the basis of extensive numerical tests. MSC classification: 65K05, 90C26, 90C30     

A set of axioms for a value for partition function games

Myerson (1977) derived an efficient value for games in partition function form. In this paper, we present a set of axioms which characterize a different efficient value for such games. This latter value assigns value 0 to dummies and assigns nonnegative values to players in monotone simple games.     

Studies on the preparation and characterization of monodisperse polystyrene latices V.: The preparation of cationic latices

Summary Methods are described for the preparation of monodisperse polystyrene latices using as initiators, 2-azo-bis-(2-methylpropamidinium) dichloride and 2-azo-bis-(2-isopropyliminazolium) dichloride. The preparations were carried out in the absence of any added surface active agents. The effect on particle size of varying initiator concentration, monomer concentration, temperature and ionic strength was examined and the conditions defined for preparing monodisperse latices, with particles in the size range 0.2 to 1m. Electrophoretic measurements confirmed that the latex particles were cationic. The evidence obtained suggested that the charged sites on the particle surface were amidinium groupings. The latices were not stable at pH values greater than 11 and prolonged heating at 90° also caused coagulation.

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Zusammenfassung Es wurden Methoden zur Darstellung monodisperser Polystyrol-Latices unter Verwendung von 2-azo-bis-(2-methylpropamidinium)-dichlorid und 2-azo-bis-(2-isopropyliminazolium)-dichlorid als Initiator beschrieben. Die Darstellung erfolgte in Abwesenheit von grenzflächenaktiven Verbindungen. Der Einfluß unterschiedlicher Initiator- und Monomerkonzentrationen, der Temperatur und der lonenstärke auf die Teilchengröße wurde untersucht und die Bedingungen festgelegt, unter denen monodisperse Latices mit Teilchen im Größenbereich von 0,2-1m erhalten werden. Elektrophoretische Messungen bestätigten, daß die Latexteilchen kationisch sind. Die Ladungen auf der Teilchenoberfläche sind Amidiniumgruppen. Die Latices waren bei pH-Werten über 11 nicht stabil; längeres Erhitzen auf 90° führte ebenfalls zur Koagulation.

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With 6 figures and 9 tables