Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation

Wavefront solutions of scalar reaction-diffusion equations havebeen intensively studied for many years. There are two basiccases, typified by quadratic and cubic kinetics. An intermediatecase is considered in this paper, namely, u_l = u_xx + u²(1 –u). It is shown that there is a unique travelling-wave solution,with a speed 1/2, for which the decay to zero ahead of the waveis exponential with x. Moreover, numerical evidence is presentedwhich suggests that initial conditions with such exponentialdecay evolve to this travelling-wave solution, independentlyof the half-life of the initial decay. It is then shown thatfor all speeds greater than 1/2 there is also a travelling-wavesolution, but that these faster waves decay to zero algebraically,in proportion to 1/x. The numerical evidence suggests that initialconditions with such a decay rate evolve to one of these fasterwaves; the particular speed depends in a simple way on the detailsof the initial condition. Finally, initial conditions decayingalgebraically for a power law other than 1/x are considered.It is shown numerically that such initial conditions evolveeither to an algebraically decaying travelling wave, or in somecases to a wavefront whose shape and speed vary as a functionof time. This variation is monotonic and can be quite pronounced,and the speed is a function of u as well as of time. Using asimple linearization argument, an approximate formula is derivedfor the wave speed which compares extremely well with the numericalresults. Finally, the extension of the results to the more generalcase of u_l = u_xx + u^m(1 – u), with m > 1, is discussed.     

Hydrodynamic and hydromagnetic stability of black holes with radiative transfer

Subrahmanyan Chandrasekhar (Chandra) was just eight years old when the first astrophysical jet was discovered in M87. Since then, jets have been uncovered with a wide variety of sources including accretion disks orbiting stellar and massive black holes, neutron stars, isolated pulsars, γ-ray bursts, protostars and planetary nebulae. This talk will be primarily concerned with collimated hydromagnetic outflows associated with spinning, massive black holes in active galactic nuclei. Jets exhibit physical processes central to three of the major research themes in Chandrasekhar’s research career – radiative transfer, magnetohydrodynamics and black holes. Relativistic jets can be thought of as ‘exhausts’ from both the hole and its orbiting accretion disk, carrying away the energy liberated by the rotating spacetime and the accreting gas that is not radiated. However, no aspect of jet formation, propagation and radiation can be regarded as understood in detail. The combination of new γ-ray, radio and optical observations together with impressive advances in numerical simulation make this a good time to settle some long-standing debates.     

Biphotonic photochemistry of benzophenones in dimethylsulphoxide: a flash photolysis EPR study

Benzophenone and its derivatives in dimethylsulphoxide (DMSO) exhibit biphotonic photochemistry under 355 nm laser photolysis. Flash photolysis electron paramagnetic resonance experiments demonstrate that a single laser pulse is capable of producing and exciting benzophenone triplets, which can sensitize dimethylsulphoxide and subsequently lead to photodecomposition. In decafluorobenzophenone, electron transfer is the dominant process of the highly excited triplet state. Despite the rapid radiationless decay of 2-hydroxybenzophenone (2OHbenzophenone) in non-polar solvents, radical signals are observed from the photoexcitation of 2OHbenzophenone in DMSO. This is attributed to the sufficiently rapid excitation of the triplet state, which competes with the radiationless decay process, aided by the unique solvent properties of DMSO. It is concluded, in contrast to literature data, that the excited triplet state of DMSO is reactive, and can produce methyl radicals that show triplet mechanism polarization via the biphotonic photoexcitation of benzophenone.     

Electron affinities of cyano-substituted ethylenes

Electron affinities of ethylene and six cyano-substituted ethylenes (cyanoethylene, 1, 1-dicyanoethylene, cis-1, 2-dicyanoethylene, trans-1, 2-dicyanothylene, tricyanoethylene, and tetracyanoethylene) were determined using six different density functional or hybrid Hartree-Fock/density functional methods. Equilibrium geometries and harmonic vibrational frequencies for each species were determined with each density functional method. Experimental electron affinities exist for three of the six systems studied (cyanoethylene, trans-1, 2-dicyanoethylene, and tetracyanoethylene); for the three systems, the absolute average EA errors for the different methods are 0.10eV (BLYP), 0.19ev (BHLYP), 0.22eV (B3LYP), 0.20eV (BP86), 0.78eV (B3P86), and 0.81eV (LSDA). The electron affinities of gem-dicyanoethylene, cis-discyanoethylene, and tricyanoethylene are not known from experiment but are predicted here to be 1.23eV (gem-dicyanoethylene), 1.32eV (cis-dicyanoethylene), and 2.41eV (tricyanoethylene). Contrary to earlier suggestions, tetracyanoethylene is predicted to be planar, rather than twisted. Density functional theory predicts that the ²B_1u state of the ethylene anion lies lower than the ²B_2g state, which is reported by experimentalists as the (transient) ground state, and lower than the ²A_g state. Coupled-cluster results indicate that the ²A_g state is lower than either the ²B_2g or ²B_1u states. The energetic stabilization of cyano substitution on ethylene results from the π and π? conjugation of multiple cyano groups. The HOMO-LUMO gap in ethylene decreases with increasing cyano substitution, from 7.2eV in C₂H₄ to 3.8eV in C₂(CN)₄, explaining the extreme difference between the electron affinities of ethylene (negative) and tetracyanoethylene (～T3.0eV).     

ACS Student Affiliates at University of Michigan Use Their Passion for Chemistry to Inspire the Community and Other Students

Going from almost zero to more than sixty active members in less than a year, the University of Michigan American Chemical Society Student Affiliate chapter is a success story about revitalizing a struggling chapter. Innovative officer positions, increased delegation of authority, and courage to try new ideas generated this special chemistry. In order to encourage each member to be creative and reach his or her full chapter potential, the officers stressed, Dream of an outstanding activity, and everyone will make it happen. Strategically, this philosophy was reinforced by electing members who had spearheaded projects to the new officer positions. In the following article, we will present the events and philosophies that created the strong intra-Student Affiliate and inter-Student Affiliate/community chemical bonds.     

Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

The authors study the effect of advection on reaction-diffusionpatterns. It is shown that the addition of advection to a two-variablereaction–diffusion system with periodic boundary conditionsresults in the appearance of a phase difference between thepatterns of the two variables which depends on the differencebetween the advection coefficients. The spatial patterns movelike a travelling wave with a fixed velocity which depends onthe sum of the advection coefficients. By a suitable choiceof advection coefficients, the solution can be made stationaryin time. In the presence of advection a continuous change inthe diffusion coefficients can result in two Turing-type bifurcationsas the diffusion ratio is varied, and such a bifurcation canoccur even when the inhibitor species does not diffuse. It isalso shown that the initial mode of bifurcation for a givendomain size depends on both the advection and diffusion coefficients.These phenomena are demonstrated in the numerical solution ofa particular reaction–diffusion system, and finally apossible application of the results to pattern formation inDrosophila larvae is discussed.     

Approximating Reachable Sets for n-dimensional Linear Discrete Systems

We present a technique for approximating the reachable set fromthe origin for n-dimensional discrete-time linear systems subjectto bounded control, where the state matrix is stable and diagonalizable.The procedure is based on decomposing the system into one- andtwo-dimensional subsystems for which the reachable set of eachof the subsystems can be over-estimated by a polygon. Thesepolygons are then used to create an n-dimensional polyhedronwhich contains the reachable set of the original system     

Smoothing Data with Tolerances by Use of Linear Programming

This report describes a method for smoothing observed data whenthe values of the dependent variable are assumed to lie withinspecified tolerance of the observed value. Linear programmingis employed to obtain optimal smoothing of the (local) degreedesired within the restraints imposed by the tolerances. First-andthird-degree smoothing are discussed, but the method may beextended to higher order.     

Low field RYDMR: effects of orthogonal static and oscillating magnetic fields on radical recombination reactions

Reactions involving spin correlated radical pairs as intermediates are known to be sensitive to applied static and/or oscillating magnetic fields. In the reaction yield detected magnetic resonance (RYDMR) technique, an electromagnetic field in resonance with the electron Zeeman splitting produced by a strong static field is used to perturb the singlet ? triplet interconversion of the radical pair and so to affect the yield of geminate recombination. New experiments are described in which weak radiofrequency fields (? 300μT) in the frequency range 1–80 MHz are applied to radical ion pairs derived from pyrene and 1,3-dicyanobenzene, in the presence of a weak (? 3.0 mT) static magnetic field. Such experiments test the viability of RYDMR in low fields, provide insight into the crossover region between the zero-field and high field cases, and may give information on the distribution of radical pair lifetimes.     

Variational calculations of vibrational energy levels for XY4 molecules 1. Stretching states

A variational method for calculating excited stretching states of symmetric tetrahedral pentaatomic molecules is presented based on the use of Radau coordinates and Morse oscillator-like basis functions. Symmetry is used both to reduce the size of the secular matrix to be diag-onalized and to calculate the potential energy matrix elements over a reduced grid of quadrature points. Test results are presented for methane, silane and germane. For CH₄, stretch-bend coupling is found to be significant, whereas it is less important for the more strongly local mode SiH₄ and GeH₄ molecules. Converged results are obtained for stretching states significantly higher than considered in previous calculations. These states will be used to represent stretching motions in a fully coupled stretch-bend calculation.