The number of conjugacy classes in pattern groups is not a polynomial function

A famous open problem due to Graham Higman asks if the number of conjugacy classes in the group of n x n unipotent upper triangular matrices over the q-element field can be expressed as a polynomial function of q for every fixed n. We consider the generalization of the problem for pattern groups and prove that for some pattern groups of nilpotency class two the number of conjugacy classes is not a polynomial function of q

On the principle of thermal interaction

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Sur l'extension de la thermodynamique aux phénomènes irréversibles liés aux degrés de liberté internes

The thermodynamics of irreversible processes is applied to systems with internal degrees of freedom. By introducing an internal configuration space and assuming continuous motion therein, the second principle of thermodynamics can be stated in the following way: the entropy creation per unit volume of the internal configuration space is essentially positive definite. Explicit evaluation of this entropy creation then at once leads to differential equations of diffusion type for the motion in configuration space. For ideally dilute systems, these equations reduce to generalized FokkerPlanck equations. As a consequence of Onsager reciprocal relations i't is shown that the phenomenological friction tensor in configuration space must be symmetric. The formalism thus obtained can be applied to a great variety of irreversible phenomena related to internal degrees of freedom. Finally some very simple applications are discussed as an illustration. © 1953.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Sur l'hydrodynamique des mélanges liquides d'hélium 3 et d'hélium 4

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Sur deux formulations de l'hydrodynamique et le problème de l'Hélium liquide II

The hydrodynamics of a fluid consisting of several components can be formulated in two different ways by applying the methods of the thermodynamics of irreversible processes. The first formulation is obtained if Gibbs' fundamental formula for the entropy per unit of mass of the system is assumed to be still valid when applied to the movement of the center of gravity of the system; statistically this corresponds to an approximate equilibrium velocity distribution function around the mean mass velocity, that is, to a normal transfer of impulse between the various components. This first formulation contains the classical Stokes-Navier equation for the hydrodynamics of a fluid. A second formulation is obtained if we assume the transfer of impulse between the various components to be a small perturbation or even negligible. This corresponds statistically to approximate equilibrium velocity distribution function around the macroscopical velocity of each component or "fluid" as we say in this case. Assuming therefore Gibbs' formula for the entropy sα per unit of mass of each fluid to be still valid when applied to the macroscopical movement of each fluid, we can derive a new set of hydrothermodynamical relations. When specialising, these equations to the two fluid model of liquid Helium II, Gorter's equations of motion for the normal and the superfluid are obtained. © 1951.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Sur l'hydrodynamique des mélanges liquides de He3 et He4

The hydrothermodynamic treatment of several-fluid systems, based on the assumption of an inhibition of the transfer of impulse, is extended to the case that each fluid consists of more than one component. The theory shows that equations of motion can be set up for every fluid whereas equations of diffusion describe the relative movements of the components belonging to one fluid around the center of gravity of that fluid. The results can be applied to liquid mixture of He3 and He4, assuming these mixtures to consist of a normal fluid containing He3 and the normal fraction of He4, and a superfluid of He4 only. A quantitative discussion is given of several effects arising in this system. © 1951.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Contribution à la thermodynamique de la matière dans un champ électromagnétique -- 28

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Sur deux formulations de l'hydrodynamique et le problème de l'hélium liquide II

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Bilans thermodynamiques locaux dans les systèmes électrochimiques

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Relativistic thermodynamics of irreversible processes I

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