Bounds on linear PDEs via semidefinite optimization

Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this paper a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on linear functionals defined on solutions of linear partial differential equations. We apply the proposed method to examples of PDEs in one and two dimensions, with very encouraging results. We pay particular attention to a PDE with oblique derivative conditions, commonly arising in queueing theory. We also provide computational evidence that the semidefinite constraints are critically important in improving the quality of the bounds, that is, without them the bounds are weak. Research supported by the SMA-MI Talliance. Research partially supported by the SMA-MIT alliance and an NSF predoctoral fellowship.     

Lower bounds on the spectra of symmetric matrices with nonnegative entries

If we normalize a symmetric n × n matrix with nonnegative entriesso that its largest entry is 1, then its spectrum is bounded below by $\text{?n}\text{2}$. The lower bound is achieved in all even dimensions for (and only for) adjacency matrices of complete bipartite graphs with equal parts.     

Cooperative effects in the two atom resonance fluorescence spectrum

We consider the resonance fluorescence spectrum of two similar interacting atoms in the limit of high photon densities. The excitation spectrum of the symmetric modes contains two sidebands in addition to the usual three peaks which are analogous to those of the isolated atom. These two new sidebands are due entirely to the cooperative behaviour of the two atoms and vanish when the atoms are far apart. The energy shifts and spectral widths for these two sidebands are two and five times larger than those for the isolated atom respectively. The probability of occurrence of these sidebands depends on the parameters V_AB/Ω and γ²₀/Ω², where V_AB is the dipole-dipole interaction energy, γ₀ is the spontaneous emission probability and Ω is the Rabi frequency. The asymmetric broadening of both sidebands depends on the parameter γ₀/Ω. The possibility to measure the dipole-dipole energy through the observation of these sidebands is discussed.     

Wave fans are special

It is shown that self-similar BV solutions of genuinely nonlinear strictly hyperbolic systems of conservation laws are special functions of bounded variation, with vanishing Cantor part.     

ANNOUNCEMENT

A special issue of Acta Mathematica Scientia,to be published in 2011,will be dedicated to the 85th birthday of Peter D.Lax,who is one of the most prominent     

Interaction of the early 3d transition metals Sc, Ti, V, and Cr with N2: an ab initio study

The interaction of the early 3d transition elements M=Sc, Ti, V, and Cr with N2(X 1Sigmag+) has been studied by coupled-cluster and multiconfigurational techniques in conjunction with quantitative basis sets. We investigated both triatomic (MN2) and tetratomic (M2N2) species but focused mainly on high-spin linear and T-shaped triatomics. The lowest bound states of ScN2(4B1),TiN2(5Delta), and VN2(6Sigma+) correlate to the first excited state of the M atom, with M-N2 binding energies (De) of 24, 14, and 8 kcal/mol, respectively. In CrN2, the first bound state (7) product operator correlates to the sixth excited state of the Cr atom (7P) with De = 27 kcal/mol. The M-N2-M bond strength of high-spin linear tetratomics is twice as large the binding energy of the corresponding M-N2 linear triatomics, M = Sc, Ti, V, and Cr.     

Bosonic molecules in rotating traps

We present a variational many-body wave function for repelling bosons in rotating traps, focusing on rotational frequencies that do not lead to restriction to the lowest Landau level. This wave function incorporates correlations beyond the Gross-Pitaevskii (GP) mean-field approximation, and it describes rotating boson molecules (RBMs) made of localized bosons that form polygonal-ring-like crystalline patterns in their intrinsic frame of reference. The RBMs exhibit characteristic periodic dependencies of the ground-state angular momenta on the number of bosons in the polygonal rings. For small numbers of neutral bosons, the RBM ground-state energies are found to be always lower than those of the corresponding GP solutions, in particular, in the regime of GP vortex formation.     

Functional determinants in euclidean Yang-Mills theory

We study functional determinants entering the path integral for topologically non-trivial sectors of Euclidean SU(2) Yang-Mills theory and we derive some results in exact closed form.This work is supported through funds provided by the US Department of Energy (DOE) under Contract EY-76-C-02-3069 and by the National Science Foundation through funds provided under Contract PHY 79-16812     

Electron impact and chemical ionisation mass spectrometry in the characterisation of organometallic silicon, germanium, tin and lead compounds

The electron impact and chemical ionisation mass spectra of a number of organometallic derivatives of silicon, germanium, tin and lead, incorporating inter alia 2-bis(trimethylsilyl)methylpyridyl or cyclopentadienyl ligands have been recorded. Representative spectra are presented and illustrate the utility of these mass spectrometric techniques in the characterisation of such compounds, particularly so in light of the highly informative nature arising from their fragmentation patterns and the numerous isotopes of the nuclei involved. Copyright 2000 John Wiley & Sons, Ltd.