An alternating direction scheme on a nonuniform mesh for reaction-diffusion parabolic problems

In this paper we develop a numerical method for two-dimensionaltime-dependent reaction-diffusion problems. This method, whichcan immediately be generalized to higher dimensions, is shownto be uniformly convergent with respect to the diffusion problems.This method, which can immediately be generalized to higherdimensions, is shown to be uniformly convergent with respectto the diffusion parameter.     

Photoacoustic detection of metastatic melanoma cells in the human circulatory system

Detection of disseminating tumor cells among patients suffering from various types and stages of cancer can function as an early warning system, alerting the physician of the metastatic spread or recurrence of the disease. Early detection of such cells can result in preventative treatment of the disease, while late stage detection can serve as an indicator of the effectiveness of chemotherapeutics. The prognostic value of exposing disseminating tumor cells poses an urgent need for an efficient, accurate screening method for metastatic cells. We propose a system for the detection of metastatic circulating tumor cells based on the thermoelastic properties of melanoma. The method employs photoacoustic excitation coupled with a detection system capable of determining the presence of disseminating cells within the circulatory system in vitro. Detection trials consisting of tissue phantoms and a human melanoma cell line resulted in a detection threshold of the order of ten individual cells, thus validating the effectiveness of the proposed mechanism. Results imply the potential to assay simple blood draws, from healthy and metastatic patients, for the presence of cancerous melanoma providing an unprecedented method for routine cancer screening.     

Symmetry Classification Using Noncommutative Invariant Differential Operators

Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G_f or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined "defining system" of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations v_x = u, v_t = B(u) u_x - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.     

Internal energy content of n-butylbenzene, bromobenzene, iodobenzene and aniline molecular ions generated by two-photon ionization at 266 nm. A photodissociation study

A technique to investigate photodissociation kinetics on a nanosecond time scale has been devised for molecular ions generated by multiphoton ionization (MPI) using mass-analyzed ion kinetic energy spectrometry. The branching ratio or rate constant has been determined for the photodissociation of the n-butylbenzene, bromobenzene, iodobenzene, and aniline molecular ions generated by MPI at 266 nm. The ion internal energies have been estimated by comparing the measured kinetic data with the previous energy dependence data. The analysis has shown that only those molecular ions generated by two-photon ionization contribute to the photodissociation signals. Around half of the available energy has been found to remain as molecular ion internal energy in the two-photon ionization process. Copyright 1999 John Wiley & Sons, Ltd.