Logarithmic correction to the probability of capture for dissipatively perturbed Hamiltonian systems

Hamiltonian systems are analyzed with a double homoclinic orbit connecting a saddle to itself. Competing centers exist. A small dissipative perturbation causes the stable and unstable manifolds of the saddle point to break apart. The stable manifolds of the saddle point are the boundaries of the basin of attraction for the competing attractors. With small dissipation, the boundaries of the basins of attraction are known to be tightly wound and spiral-like. Small changes in the initial condition can alter the equilibrium to which the solution is attracted. Near the unperturbed homoclinic orbit, the boundary of the basin of attraction consists of a large sequence of nearly homoclinic orbits surrounded by close approaches to the saddle point. The slow passage through an unperturbed homoclinic orbit (separatrix) is determined by the change in the value of the Hamiltonian from one saddle approach to the next. The probability of capture can be asymptotically approximated using this change in the Hamiltonian. The well-known leading-order change of the Hamiltonian from one saddle approach to the next is due to the effect of the perturbation on the homoclinic orbit. A logarithmic correction to this change of the Hamiltonian is shown to be due to the effect of the perturbation on the saddle point itself. It is shown that the probability of capture can be significantly altered from the well-known leading-order probability for Hamiltonian systems with double homoclinic orbits of the twisted type, an example of which is the Hamiltonian system corresponding to primary resonance. Numerical integration of the perturbed Hamiltonian system is used to verify the accuracy of the analytic formulas for the change in the Hamiltonian from one saddle approach to the next. (c) 1995 American Institute of Physics.     

Results and estimates on pseudopowers

Let be a positive integer. We say looks like a power of 2 modulo a prime if there exists an integer such that . First, we provide a simple proof of the fact that a positive integer which looks like a power of modulo all but finitely many primes is in fact a power of . Next, we define an -pseudopower of the base to be a positive integer that is not a power of , but looks like a power of modulo all primes . Let denote the least such . We give an unconditional upper bound on , a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that is about for a certain constant . We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than are also given.

     

Oligodeoxyfluorosides: Strong Sequence Dependence of Fluorescence Emission

We describe the properties of a series of oligomeric polyfluorophores assembled on the DNA backbone. The 11 oligomers (oligodeoxyfluorosides, ODFs), 4-7 monomers in length, were composed of only two fluorescent monomers and a spacer in varied sequences, and were designed to test how fluorescent nucleobases can interact electronically to yield complexity in fluorescence emission. The monomer fluorophores were deoxyribosides of pyrene and perylene, which emit light in violet and blue wavelengths, respectively. The experiments show that simple variation in sequence and spacing can dramatically change fluorescence, yielding emission maxima ranging from 380 to 557 nm and visible colors from violet to orange-red. Fluorescence lifetime data, excitation spectra, and absorption data point to a number of multi-fluorophore electronic interactions, including pyrene-pyrene and perylene-perylene excimers, pyrene-perylene exciplexes, as well as monomer dye emissions, contributing to the final spectral outcomes. Thus, two simple fluorophores can be readily combined to give emissions over much of the visible spectrum, all requiring only a single excitation. The results demonstrate that fluorescent nucleobases in oligomeric form can act cooperatively as electronic units, and that fluorophore sequence in such oligomers is as important as fluorophore composition in determining fluorescence properties.     

Reactions of 6-diazopenicillanates with allylic sulphides,selenides, and bromides

6-Diazopenicillanates react with allylic sulphides, selenides and bromides, to give 6,6-disubstituted penicillanates $\text{via}$ [2,3]-sigmatropic rearrangements.     

Contents of Volume 6

Volume Contents

Contents of Volume 6     

Vibrational Spectroscopic Investigations of M(Imidazole)2Ni(CN)4·2 Dioxane Clathrates

Four new clathrates of the formula M(Im)2Ni(CN)4·2·Dioxane (where M = Co, Ni, Cu, Cd; Im = Imidazole) have been prepared in powder form and their FT-IR and laser-Raman spectra are reported for the first time. These clathrates are analogues to the previously reported classical Hofmann type clathrates except for the copper clathrate. The Cu clathrate has different spectral features in comparison with its analogues due to the Jahn-Teller effect.     

Exciton dissociation dynamics in model donor-acceptor polymer heterojunctions. I. Energetics and spectra

In this paper we consider the essential electronic excited states in parallel chains of semiconducting polymers that are currently being explored for photovoltaic and light-emitting diode applications. In particular, we focus upon various type II donor-acceptor heterojunctions and explore the relation between the exciton binding energy to the band offset in determining the device characteristic of a particular type II heterojunction material. As a general rule, when the exciton binding energy is greater than the band offset at the heterojunction, the exciton will remain the lowest-energy excited state and the junction will make an efficient light-emitting diode. On the other hand, if the offset is greater than the exciton binding energy, either the electron or hole can be transferred from one chain to the other. Here we use a two-band exciton to predict the vibronic absorption and emission spectra of model polymer heterojunctions. Our results underscore the role of vibrational relaxation and suggest that intersystem crossings may play some part in the formation of charge-transfer states following photoexcitation in certain cases.     

Threshold detection using indicator-displacement assays: an application in the analysis of malate in Pinot Noir grapes

The mathematics for modeling indicator-displacement assay isotherms is presented and contrasted to the classical host-guest binding isotherm. It is shown that the signal response can be tuned to occur closer to 1 equiv of guest relative to a standard binding algorithm. This delay in response leads to a better triggering protocol for threshold detection schemes. The determination of malate in Pinot Noir must was calculated using this new mathematical model, which demonstrates how a color change can be tuned to occur near a desired concentration of analyte.