Determination Of G2 Research Articles

Gauss and Eisenstein Sums of Order Twelve

AbstractLet q = pr with p an odd prime, and Fq denote the finite field of q elements. Let Tr : Fq → Fp be the usual trace map and set ζp = exp(2πi/p). For any positive integer e, define the (modified) Gauss sum gr(e) byRecently, Evans gave an elegant determination of g1(12) in terms of g1(3), g1(4) and g1(6) which resolved a sign ambiguity present in a previous evaluation. Here I generalize Evans' result to give a complete determination of the sum gr(12).

New methods of cell cycle analysis based on the selective tagging of cells either at the beginning or end of S phase.

New techniques for cell cycle analysis are presented. Using HeLa cells, methods are described for the selection of a narrow window or cohort of lightly [3H]-labeled cells located either at the very beginning or the very end of S phase. The cohort cells are tagged by a labeling procedure which entails alternating pulses of high and low levels of [3H]thymidine and are identified autoradiographically. Additional methods are described for following the progress of cohort cells through the cell cycle. Theoretically, with the methods described, it should be possible to follow the "early S cohort' cells as they exit from S phase, as they enter and exit M and as they enter the subsequent S phase. This would allow a determination of S, S + G2, S + G2 + M and T. It should theoretically be possible to follow "late S cohort' cells in a similar manner, allowing a determination of G2, G2 + M and G2 + M + G1. To test these predictions, several experiments are presented in which the progress of the two cohorts is monitored. The best data were obtained from the mitotic curves of cohort cells. For each of the cohorts, values were obtained for the time required for peak concentration of cells in mitosis, the coefficients of variation and of skew. The curve of cohort cells passing through mitosis is shown to fit a log-normal curve better than a normal curve. In addition, the mitotic curves are used to estimate the length of M and to estimate the loss of cohort synchrony. Other uses of these methods are discussed.

The Vapor-Pressure Equations of Solutions and the Osmotic Pressure of Rubber

Abstract 1. The laws which are obeyed by ideally dilute and by perfect solutions have been obtained by Guggenheim, and he has extended his analysis to obtain the properties of a class of solutions to which the name regular has been given. It is assumed that the solution consists of molecules of two types, α and β, of approximately the same size, and that each molecule, whether of type α or of type β, is surrounded by the same number z of other molecules, and the potential energy is regarded as the sum of contributions from each pair of molecules in direct contact. That is, the molecules are regarded as occupying sites on a regular lattice, and the potential energy arises from the interactions between molecules which occupy closest neighbor sites. Rushbrooke has examined the properties of such a solution, using the Bethe method to set up the partition function for the assembly. The properties of solutions in which the molecules of one type are of such a size and shape that they have to be regarded as occupying two lattice sites have been investigated by Fowler and Rushbrooke, who determined the limiting form of the vapor-pressure equations for extremely dilute and for extremely concentrated solutions. Chang has determined, for a two-dimensional lattice, the value of the combinatory factor, which can be written as g2,1(Nα,Nβ) for the case in which each molecule of type a occupies two closest neighbor sites and each molecule of type β occupies one site. Chang's determination of g2,1(Nα,Nβ) for a two-dimensional lattice is based on an analysis of the corresponding adsorption problem, in which he superimposes some arbitrary assumptions on the Bethe method. It has, however, been shown that the value which Chang obtained for g2,1(Nα,Nβ) is applicable to two- and to three-dimensional lattices in which no two closest neighbors of a given site are also closest neighbors of one another. Fowler and Guggenheim have shown how, once the combinatory factor has been evaluated, it can be used to determine the configurational entropy of mixing and thence the vapor-pressure equations when the energy of mixing is zero. Using Chang's value for g2,1(Nα,Nβ), they obtained for the vapor-pressure equations, when each solute molecule occupies two closest neighbor sites on the lattice and each solvent molecule occupies only one lattice site, the following equations: