Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems

We develop a level set method for the computation of multi-valued physical observables (density, velocity, energy, etc.) for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions. We take two approaches to derive the method.The first one starts with a weakly coupled system of an eikonal equation for phase S and a transport equation for density ρ:

The main idea is to evolve the density near the n-dimensional bi-characteristic manifold of the eikonal (Hamiltonian–Jacobi) equation, which is identified as the common zeros of n level set functions in phase space . These level set functions are generated from solving the Liouville equation with initial data chosen to embed the phase gradient. Simultaneously, we track a new quantity f = ρ(t,x,k)|det(_k)| by solving again the Liouville equation near the obtained zero level set = 0 but with initial density as initial data. The multi-valued density and higher moments are thus resolved by integrating f along the bi-characteristic manifold in the phase directions.The second one uses the high frequency limit of symmetric hyperbolic systems derived by the Wigner transform. This gives rise to Liouville equations in the phase space with measure-valued solution in its initial data. Due to the linearity of the Liouville equation we can decompose the density distribution into products of function, each of which solves the Liouville equation with L^∞ initial data on any bounded domain. It yields higher order moments such as energy and energy flux.The main advantages of these new approaches, in contrast to the standard kinetic equation approach using the Liouville equation with a Dirac measure initial data, include: (1) the Liouville equations are solved with L^∞ initial data, and a singular integral involving the Dirac-δ function is evaluated only in the post-processing step, thus avoiding oscillations and excessive numerical smearing; (2) a local level set method can be utilized to significantly reduce the computation in the phase space. These methods can be used to compute all physical observables for multi-dimensional problems.Our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix. One such example is the wave equation, which will be studied numerically in this paper.     

Coherent population trapping of electron spins in a high-purity n-type GaAs semiconductor

In high-purity n-type GaAs under a strong magnetic field, we are able to isolate a lambda system composed of two Zeeman states of neutral-donor-bound electrons and the lowest Zeeman state of bound excitons. When the two-photon detuning of this system is zero, we observe a pronounced dip in the excited-state photoluminescence, indicating the creation of the coherent population-trapped state. Our data are consistent with a steady-state three-level density-matrix model. The observation of coherent population trapping in GaAs indicates that this and similar semiconductor systems could be used for various electromagnetically induced transparency type experiments.     

Preparation and solid-state characterization of the novel mixed biradical *NSNSC-CNSSN*

Reduction of the radical-cation [*NSSNC-CNSNS][AsF(6)] with ferrocene affords the novel biradical *NSNSC-CNSSN* containing both 1,2,3,5- and 1,3,2,4-isomeric dithiadiazolyl rings. Biradicals form centrosymmetric dimers with pi(*)-pi(*) interactions between different isomeric rings. Biradical *NSNSC-CNSSN* is diamagnetic in the solid state (C = 0.00035, TIP = 6.5 x 10(-)(5) emu/Oe.mol); however, an increase in paramagnetism was observed upon grinding (C = 0.003, TIP = 4.2 x 10(-)(4) emu/Oe.mol).     

Corrigendum to “generalized martin’s axiom and souslin’s hypothesis for higher cardinals”

Correct proofs are given for Theorem 3 and the Propositions of §§5, 6 of [4]. For the latter, we must modify the principle (S)″ in a technical way. We prove a weaker version of Theorem 2, where □ is replaced by the stronger hypothesis PΓ_{N 1} ^b. Partially supported by NSF grant MCS 8301042.     

On the chromatic index of outerplanar graphs

Vizing [Diskret. Analiz3 (1964), 25–30] has shown that if ? denotes the maximum valency of a simple graph, then its chromatic index is either ? or ? + 1. The object of this paper is to show that the chromatic index of an outerplanar graph G is ? if and only if G is not an odd circuit.     

Ab initio study of catalyzed and uncatalyzed amide bond formation as a model for peptide bond formation: Ammonia-Glycine reactions

As a model reaction for peptide and bond formation, the S_N2 reactions between glycine and ammonia have been studied with and without amine catalysis: using ab initio molecular-orbital methods. For each of the catalyzed and uncatalyzed reactions, two reaction mechanisms have been examined: a two-step and a concerted mechanism. The stationary points of each reaction, including intermediate and transition states, have been identified and free energies calculated for all geometry-optimized reaction species to determine the thermodynamics and kinetics of the reaction. The calculations demonstrate that a second ammonia molecule catalyzes amide bond formation, and that the two-step mechanism is more favorable than the concerted one for the catalyzed reaction, while for the uncatalyzed reaction both mechanisms are competitive.     

Kinetics and mechanism for the iodination of methylmalonic acid

The reaction\documentclass{article}\pagestyle{empty}\begin{document}$ CH\left({CH_3 } \right)\left({COOH} \right)_2 + I_2 \rightleftharpoons CI(CH_3)\left({COOH} \right)_2 + H^ + + I^ - $\end{document} was followed spectrophotometrically at 353 nm and 470 nm at 25°C under various conditions of pH and methylmalonic acid concentration. The equilibrium constant for the reaction is 0.11 ± 0.02. An iterative technique was used to integrate postulated rate equations. Agreement between experimental and calculated absorbance versus time curves was generally better than 0.005 A (approximately 5% of maximum) at both wavelengths for a mechanism where the rate-determining step is formation of an enolate (k = 1.63 Θ 10^?4 ± 0.03 Θ 10^?4 sec^?1). The enolate may be rapidly transformed to the enol or enol carboxylate anion depending on the pH. All three forms are rapidly iodinated. The mechanism of general base catalysis is supported by rate increases proportional to base concentration in buffer solutions. The bases, acetate ion, chloracetate ion, sulfate ion, dichloracetate ion, and water, follow a Brønsted relationship with β = 0.7.