Separable solutions of the sine-Gordon equation in terms of a Painlevé transcendent

We show that, when expressed in characteristic coordinates, the two-dimensional sine-Gordon equation may be reduced to a separable form via the use of a fifth Painlevé transcendent.     

A direct error analysis for least squares

A direct error analysis is given for orthogonal factorization methods for calculating the least squares solution of an overdetermined system of linear equations. The direct method has the interesting advantage in that it permits the separation of errors occurring in the transformation and back-substitution phases of solution. This shows the partial elimination of potentially significant terms occurring in different stages of the algorithm. Presumably it is prudent to minimize the error at each stage of the algorithm, so it is significant that numerical evidence shows column pivoting can reduce the magnitude of these terms. This is offered as an explanation for the common observation that column pivoting is beneficial in least squares calculations. We also summarize the corresponding error analysis for the calculation of the minimum norm solution of an underdetermined system using orthogonal transformations.     

Discrete,nonlinear approximation problems in polyhedral norms

Summary A Levenberg-like algorithm is given for the solution of discrete nonlinear approximation problems in polyhedral norms. Convergence and rate of convergence results are discussed and some numerical results presented.     

On Nordsieck's method for the numerical solution of ordinary differential equations

The connection between the class of methods suggested by Nordsieck and the class of linear multi-step methods is examined. It is shown that the starting procedure suggested by Nordsieck is specially suited to the Adams method.     

Charged ρ photoproduction in the energy range 2.8–4.8 GeV

Measurements of the photoproduction processes γρ→ρ⁺ n and γρ→ρ^-Δ⁺⁺ (1236) are reported in the energy range 2.8 to 4.8 GeV. The data show shrinkage of the differential cross section in this energy region for the process γρ→ρ^-Δ⁺⁺ (1236); no shrinkage is observed for the ρ⁺ n process. The energy dependences of the ρ⁺ n and ρ^-Δ⁺⁺ (1236) total cross sections are much steeper than current model prediction. The ρ spin density matrices for each process are also presented.     

Counterintuitive kinetics in Tsuji-Trost allylation: ion-pair partitioning and implications for asymmetric catalysis

The kinetics of Pd-catalyzed Tsuji-Trost allylation employing simple phosphine ligands (L = Ar3P, etc.) are consistent with turnover-limiting nucleophilic attack of an electrophilic [L2Pd(allyl)]+ catalytic intermediate. Counter-intuitively, when L is made more electron donating, which renders [L2Pd(allyl)]+ less electrophilic (by up to an order of magnitude), higher rates of turnover are observed. In the presence of catalytic NaBAr'F, large rate differentials arise by attenuation of ion-pair return (via generation of [L2Pd(allyl)]+ [BAr'F]-) a process that also increases the asymmetric induction from 28 to 78% ee in an archetypal asymmetric allylation employing BINAP (L*) as ligand. There is substantial potential for analogous application of [M]n+([BAr'F]-)n cocatalysis in other transition metal catalyzed processes involving an ionic reactant or reagent and an ionogenic catalytic cycle.     

Dissociative electron-ion recombination of the interstellar species protonated glycolaldehyde, acetic acid, and methyl formate

Recently, methyl formate, glycolaldehyde, and acetic acid have been detected in the Interstellar Medium, ISM. The rate constants, α(e), for dissociative electron-ion recombination of protonated gycolaldehyde, (HOCH(2)CHO)H(+), and protonated methyl formate, (HCOOCH(3))H(+), have been determined at 300 K in a variable temperature flowing afterglow using a Langmuir probe to obtain the electron density. The recombination rate constants at 300 K are 3.2 × 10(-7) cm(3) s(-1) for protonated methyl formate and 7.5 × 10(-7) cm(3) s(-1) for protonated glycolaldehyde. The recombination rate constant of protonated acetic acid could not be directly measured, but it appears to have a rate constant, α(e), on the 10(-7) cm(3) s(-1) scale. Several high- and low-temperature measurements for protonated methyl formate were made. In addition, an α(e) measurement at 220 K for protonated glycolaldehyde was performed. The astrochemical implications of the rates of recombination, α(e), and protonation routes are discussed.     

Propagation of a solitary fission wave

Reaction-diffusion phenomena are encountered in an astonishing array of natural systems. Under the right conditions, self stabilizing reaction waves can arise that will propagate at constant velocity. Numerical studies have shown that fission waves of this type are also possible and that they exhibit soliton like properties. Here, we derive the conditions required for a solitary fission wave to propagate at constant velocity. The results place strict conditions on the shapes of the flux, diffusive, and reactive profiles that would be required for such a phenomenon to persist, and this condition would apply to other reaction diffusion phenomena as well. Numerical simulations are used to confirm the results and show that solitary fission waves fall into a bistable class of reaction diffusion phenomena.