On equilibrium distributions and detailed balance relations in non-isothermal systems

The significance of the detailed balance principle and equilibrium solutions of the master equation is discussed from a thermodynamic point of view for isolated and isothermal systems. Starting from a master equation for all the time dependent degrees of freedom it is shown that the uniqueness of the equilibrium distribution as a stationary solution is ensured if the detailed rate constants are balanced with the aid of the distribution which maximizes the entropy subject to the thermodynamic constraints. This procedure should precede physical assumptions which simplify the original master equation, e.g. the assumption that rapidly relaxing modes can be described by canonical distribution functions.     

Thermocapillary Alignment of Gas Bubbles Induced by Convective Transport

The effect of a weak convective heat transfer on the thermocapillary interaction of two bubbles with an arbitrary orientation relative to an externally imposed temperature gradient is examined. Asymptotic analysis of the case of large separation distances, Z, suggests that the corrections to the bubbles' velocities are of (Pe/Z²), rather than (Pe²) previously found for an isolated bubble. Equal-sized bubbles are known to move with the same velocities, as if they were isolated, when heat conduction is the only transport mechanism. However, the convective transport results in a relative motion of the bubbles. The tendency of equal bubbles to line up in a plane perpendicular to the applied thermal gradient is shown analytically in the weakly nonlinear limit of small Pe numbers, and an interesting interaction behavior in the case of unequal bubbles is discussed.     

EVENTS SURROUNDING THE EARLY DEVELOPMENT OF EUGLENA CHLOROPLASTS. 12. SPECTROSCOPIC EXAMINATION OF THE PROTOCHLOROPHYLL(IDE) PHOTOTRANSFORMATION IN INTACT CELLS*

Abstract— Absorption measurements in the 600–720 nm region of dark-grown cells of Euglena gracilis Klebs var. bacillaris Cori were made in vivo at room temperature using computer-assisted spectrophotometry. Dark-grown wild-type cells have a prominent absorption maximum at 634 nm due to protochlorophyll(ide) absorption. Upon illumination, the absorption at 634nm decreases and a peak appears at 674nm, representing the phototransformation of protochlorophyll(ide) to chlorophyll(ide). Using difference spectroscopy, the resynthesis in the dark of protochlorophyll(ide) by previously-illuminated wild-type dark-grown cells was found to begin at about 10min after illumination and reached completion by about 25 min, the amount of protochlorophyll(ide) resynthesized being equivalent to that of dark-grown cells. Resynthesis of protochlorophyll(ide) following a second illumination follows the same kinetics, indicating that protochlorophyll(ide) resynthesis is under tight regulation, possibly via feedback control. Cells of dark-grown wild-type and W₃BUL, a mutant lacking protochlorophyll(ide) contain a component absorbing at 658 nm which does not undergo phototransformation when examined by difference spectroscopy at room and liquid N₂ temperatures. Following the phototransformation of protochlorophyll(ide) 634 to chlorophyll(ide) 674, the chlorophyll(ide) 674 shifts to shorter wavelengths, ultimately to 671 nm. Possible relationships among the various spectroscopic forms of protochlorophyll(ide) and chlorophyll(ide) at room temperature and liquid N₂ temperature in Euglena and higher plants are presented. It is concluded that Euglena, unlike older, etiolated higher plants, contains only protochlorophyll(ide) 634, making it an excellent system in which to examine the phototransformation of this pigment species in the absence of other forms.     

The nilpotency class of the unit group of a modular group algebra II

LetG be a finitep-group, and letU(G) be the group of units of the group algebraFG, whereF is a field of characteristicp. It is shown that, if the commutative subgroup ofG has order at leastp ², then the nilpotency class ofU(G) is at least 2p−1. The authors are grateful to the Dipartimento di Matematica of the Universita di Trento, and to the Mathematical Institute of the University of Oxford, for their hospitality while this paper was being written. Then are also grateful to Robert Sandling, for communication of results, and problems, prior to publication.     

The co-degrees of irreducible characters

LetG be a finite group. The co-degree of an irreducible character χ ofG is defined to be the number |G|/χ(1). The set of all prime divisors of all the co-degrees of the nonlinear irreducible characters ofG is denoted by Σ(G). First we show that Σ(G)=π(G) (the set of all prime divisors of |G|) unlessG is nilpotent-by-abelian. Then we make Σ(G) a graph by adjoining two elements of Σ(G) if and only if their product divides a co-degree of some nonlinear character ofG. We show that the graph Σ(G) is connected and has diameter at most 2. Additional information on the graph is given. These results are analogs to theorems obtained for the graph corresponding to the character degrees (by Manz, Staszewski, Willems and Wolf) and for the graph corresponding to the class sizes (by Bertram, Herzog and Mann). Finally, we investigate groups with some restriction on the co-degrees. Among other results we show that ifG has a co-degree which is ap-power for some primep, then the corresponding character is monomial andO _p (G)≠1. Also we describe groups in which each co-degree of a nonlinear character is divisible by at most two primes. These results generalize results of Chillag and Herzog. Other results are proved as well. The paper was written during this author’s visit at the Technion and the University of Tel Aviv. He would like to thank the departments of mathematics at the Technion and the University of Tel Aviv for their hospitality and support.     

Subgroups of powerful groups

We consider subgroups of powerfulp-groups. In particular, we give a new proof that allp-groups are sections of powerfulp-groups, give necessary and sufficient conditions for a 2-generator group to be a normal subgroup of a powerfulp-groups, and show thatp-groups of class 2, orp-groups with a cyclic commutator subgroup, are such normal subgroups.     

Injectors and normal subgroups of finite groups

A class of subgroups,N-injectors, previously defined only for soluble groups, is here defined for some non-soluble groups. It is proved that injectors have some properties which resemble those of Sylow subgroups.