Theoretical studies on microscopic and macroscopic nonuniformities in electric-discharge-excited XeCl lasers

The satbilit r of the pumping discharge for XeCI (Ne/Xe/HCI) lasers is investigated by using different one-dimensional models, considering both bulk discharge instabilities and filamentary ones. It is found that the bulk stability in a XeCl laser discharge can he improved, under certain conditions, due to the vihrational excitation of HCl molecules: this effect is studied by using different models for the HCl vihrational kinetics. It is established that the very short instability time found in experiments cannot he explained as a result of hulk instability of the discharge. On the other hand, a satisfactory comparison with experiments can he obtained by adding to the model a simplified microinstability which develops in the discharge.     

Copper(II) ion hydrolysis in aqueous solution

A copper(II) ion-selective-electrode potentiometric method was used to determine the first and second hydrolysis constants of Cu²⁺. Special techniques prevented copper(II) hydroxide precipitation, and copper(II) carbonate and cipper(II) organic complexation during the titration of the experimental solution over the pH range 6.8–8.4. The large change in the total copper concentration during the titration due to adsorption of copper onto the vessel walls was accounted for by measuring the total copper concentration at each pH by atomic absorption spectrophotometry. The two hydrolysis constants were determined at 25°C in 0.7 and 0.05m NaClO₄ media. The measured stability constants are independent of the copper concentration and yield similar zero ionic strength values. Also, the stepwise equilibrium constants decrease as the ligand number increases.     

Inclusion complexing by water-soluble β-cyclodextrin polymers

Cross-linked -cyclodextrin with a molecular weight of less than 10000 has good solubility in water, and it is a better inclusion complexing agent than the parent -cyclodextrin. By including lipophilic guest molecules into the apolar cyclodextrin cavity, their apparent lipophilicity is reduced because the outer surface of the molecular wrapping (the crosslinked -CD) is highly hydrophilic. The relative stability of the inclusion complexes can be rapidly determined by reversed-phase thin-layer chromatography. The reversed-phase TLC behaviour of 25 triphenylmethane derivatives and analogues were studied in the presence of -cyclodextrin polymers containing neutral and carboxyl groups. Increasing the molecular weight results in an increased complex-forming capacity. The carboxyl group modifies the accessibility of the CD cavity which in turn results in increased or decreased complex stability, depending on the guest molecule. The presence of organic solvents diminishes the stability of the CD complexes.     

A geometric method in nonlinear programming

A differential geometric approach to the constrained function maximization problem is presented. The continuous analogue of the Newton-Raphson method due to Branin for solving a system of nonlinear equations is extended to the case where the system is under-determined. The method is combined with the continuous analogue of the gradient-projection method to obtain a constrained maximization method with enforced constraint restoration. Detailed analysis of the global behavior of both methods is provided. It is shown that the conjugate-gradient algorithm can take advantage of the sparse structure of the problem in the computation of a vector field, which constitutes the main computational task in the methods.This is part of a paper issued as Stanford University, Computer Science Department Report No. STAN-CS-77-643 (Ref. 45), which was presented at the Gatlinburg VII Conference, Asilomar, California, 1977. This work was supported in part by NSF Grant No. NAT BUR OF ECON RES/PO No. 4369 and by Department of Energy Contract No. EY-76-C-02-0016.The main part of this work was presented at the Japan-France Seminar on Functional Analysis and Numerical Analysis, Tokyo, Japan, 1976. The paper was prepared in part while the author was a visitor at the Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 1976–77, and was completed while he was a visitor at the Computer Science Department, Stanford University, Stanford, California, 1977. He acknowledges the hospitality and stimulating environment provided by Professor G. H. Golub, Stanford University, and Professors N. J. Rose and C. D. Meyer, North Carolina State University.     

Kinetic equation approach to phase transitions

An exact mathematical discussion of the linearized Enskog-Vlasov equation is given. A criterion for the occurrence of the linear instability is related to a criterion for the occurrence of the bifurcation of the equilibrium stationary solution to the nonlinear Enskog-Vlasov equation. Mathematical results are interpreted physically in connection with phase transitions.     

An observation on the effect of fluctuations far from equilibrium: The Lotka-Volterra model

Spontaneous fluctuations in the Lotka-Volterra model of chemical reactions are known to grow in an unbounded way when species held in excess are neglected. This result is obtained in a simple way using generalized fluctuation-dissipation principles but appears to be an artifact of ignoring fluctuations in the variables that are held fixed. When fluctuations in the other concentrations are included in the model, the spontaneous fluctuations become bounded.This work was supported by the National Science Foundation through Research Grant No. MPS 74-00483 AO1.