Numerical computations of integrals over paths on Riemann surfaces of genusN

This paper is a continuation of work by Forest and Lee [1,2]. In [1,2] it was proved that the function theory of periodic soliton solutions occurs on the Riemann surfaces ? of genusN, where the integrals over paths on ? play the most fundamental role. In this paper a numerical method is developed to evaluate these integrals. Predisely, the aim is to develop a computational code for integrals of the form $$\int\limits_\gamma {f(z)\frac{{dz}}{{R(z)}}, or} \int\limits_\gamma {f(z)R(z)dz,} $$ wheref(z) is any single-valued analytic function on the complex planeC, andR(z) is a two-valued function onC of the form $$R^2 (z) = \prod\limits_{k = 1}^{2N + \delta } {(z - z_0 (k)), \delta = 0 or 1,} $$ where {z ₀(k),1≤k≤2N+δ} are distinct complex numbers which play the role of the branch points of the Riemann surface ? = {(z, R(z))} of genusN?1+δ. The integral path γ is continuous on ?. The numerical code is developed in “Mathematica” [3].     

Dynamics of photodissociation of 3,3,3-d(3)-propene at 157 nm: site effect and hydrogen migration

In a preceding paper [Lee et al., J. Chem. Phys. 119, 827 (2003)], we measured the kinetic-energy distributions P(E(t)) and branching ratios of products from photolysis of propene at 157 nm using time-of-flight spectroscopy combined with photoionization. In the present work, hydrogen migration before fragmentation and a site effect on P(E(t)) and branching ratios were revealed from the photodissociation of CD(3)CHCH(2). Labeling of the methyl group with deuterium enabled us to differentiate between elimination of atomic and molecular hydrogen from the vinyl moiety and from the methyl moiety; the P(E(t)) and relative yields for the formation of H, D, H(2), HD, and D(2) were measured. Deuterium labeling allowed us to also differentiate the fragmentation after hydrogen transfer from that before hydrogen migration. The observation of isotopic variants of CD(3) and C(2)H(3) radicals in the C-C bond cleavage provides evidence for hydrogen transfer of propene because of site specificity. The fraction of fragmentation after hydrogen transfer is estimated to be 25%. The isotope-specific branching ratios for five dissociation pathways of CD(3)CHCH(2) were evaluated.