Matrix extensions and eigenvalue completions, the generic case

In this paper we provide new necessary and sufficient conditions for the so-called eigenvalue completion problem.

     

Linear matrix inequality representation of sets

This article concerns the question, Which subsets of ?^m can be represented with linear matrix inequalities (LMIs)? This gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory. Also, before having much hope of representing engineering problems as LMIs by automatic methods, one needs a good idea of which problems can and cannot be represented by LMIs. Little is currently known about such problems. In this article we give a necessary condition that we call “rigid convexity,” which must hold for a set ?? ? ?^m in order for ?? to have an LMI representation. Rigid convexity is proved to be necessary and sufficient when m = 2. This settles a question formally stated by Pablo Parrilo and Berndt Sturmfels in [15]. As shown by Lewis, Parillo, and Ramana [11], our main result also establishes (in the case of three variables) a 1958 conjecture by Peter Lax on hyperbolic polynomials. © 2006 Wiley Periodicals, Inc.     

Tracking poles and representing Hankel operators directly from data

Summary We propose and analyse a method of estimating the poles near the unit circleT of a functionG whose values are given at a grid of points onT: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value ofG onT, and then to estimate the modulus by seeking a bestL ² fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL ² which is well suited to the numerical representation of the Hankel operator with symbolG and thereby to the numerical solution of the Nehari problem (computing the bestH , analytic, approximation toG relative to theL norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.Partially supported by grants from the AFOSR and NSF     

Control of oxo-molybdenum reduction and ionization potentials by dithiolate donors

The compounds (L-N3)MoO(qdt) and (L-N3)MoO(tdt) [(L-N3) = hydrotris(3,5-dimethyl-1-pyrazolyl)borate; tdt = toluene-3,4-dithiolate; qdt = quinoxaline-2,3-dithiolate] have been studied by cyclic voltammetry and photoelectron, magnetic circular dichroism, and electronic absorption spectroscopies, and the experimental data have been interpreted in the context of ab initio molecular orbital calculations on a variety of dithiolate dianion ligands. The PES data reveal very substantial differences between (L-N3)MoO(qdt) and (L-N3)MoO(tdt) in that the first ionization (originating from the Mo dxy orbital) for (L-N3)MoO(qdt) is about 0.8 eV to deeper binding energy than that of (L-N3)MoO(tdt). This stabilizing effect is also reflected in the solution reduction potentials, where (L-N3)MoO(qdt) is approximately 220 mV easier to reduce than (L-N3)MoO(tdt). A direct correlation between the relative donating ability of a given dithiolate ligand and the reduction potential of the (L-N3)MoO(dithiolate) complex has been observed, and a linear relationship exists between the calculated Mulliken charge on the S atoms of the dithiolate dianion and the Mo reduction potential. The study confirms previously communicated work (Helton, M. E.; Kirk, M. L. Inorg. Chem. 1999, 38, 4384-4385) that suggests that anisotropic covalency contributions involving only the out-of-plane S orbitals of the coordinated dithiolate control the Mo reduction potential by modulating the effective nuclear charge of the metal, and this has direct relevance to understanding the mechanism of ferricyanide inhibition in sulfite oxidase. Furthermore, these results indicate that partially oxidized pyranopterins may play a role in facilitating electron and/or atom transfer in certain pyranopterin tungsten enzymes which catalyze formal oxygen atom transfer reactions at considerably lower potentials.     

Nature of the oxomolybdenum-thiolate pi-bond: implications for Mo-S bonding in sulfite oxidase and xanthine oxidase

The electronic structure of cis,trans-(L-N(2)S(2))MoO(X) (where L-N(2)S(2) = N,N'-dimethyl-N,N'-bis(2-mercaptophenyl)ethylenediamine and X = Cl, SCH(2)C(6)H(5), SC(6)H(4)-OCH(3), or SC(6)H(4)CF(3)) has been probed by electronic absorption, magnetic circular dichroism, and resonance Raman spectroscopies to determine the nature of oxomolybdenum-thiolate bonding in complexes possessing three equatorial sulfur ligands. One of the phenyl mercaptide sulfur donors of the tetradentate L-N(2)S(2) chelating ligand, denoted S(180), coordinates to molybdenum in the equatorial plane such that the OMo-S(180)-C(phenyl) dihedral angle is approximately 180 degrees, resulting in a highly covalent pi-bonding interaction between an S(180) p orbital and the molybdenum d(xy) orbital. This highly covalent bonding scheme is the origin of an intense low-energy S --> Mo d(xy) bonding-to-antibonding LMCT transition (E(max) approximately 16000 cm(-)(1), epsilon approximately 4000 M(-)(1) cm(-)(1)). Spectroscopically calibrated bonding calculations performed at the DFT level of theory reveal that S(180) contributes approximately 22% to the HOMO, which is predominantly a pi antibonding molecular orbital between Mo d(xy) and the S(180) p orbital oriented in the same plane. The second sulfur donor of the L-N(2)S(2) ligand is essentially nonbonding with Mo d(xy) due to an OMo-S-C(phenyl) dihedral angle of approximately 90 degrees. Because the formal Mo d(xy) orbital is the electroactive or redox orbital, these Mo d(xy)-S 3p interactions are important with respect to defining key covalency contributions to the reduction potential in monooxomolybdenum thiolates, including the one- and two-electron reduced forms of sulfite oxidase. Interestingly, the highly covalent Mo-S(180) pi bonding interaction observed in these complexes is analogous to the well-known Cu-S(Cys) pi bond in type 1 blue copper proteins, which display electronic absorption and resonance Raman spectra that are remarkably similar to these monooxomolybdenum thiolate complexes. Finally, the presence of a covalent Mo-S pi interaction oriented orthogonal to the MOO bond is discussed with respect to electron-transfer regeneration in sulfite oxidase and Mo=S(sulfido) bonding in xanthine oxidase.     

Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less

We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each other in a very explicit way. In particular, the averaging operator appears to be closely related to the solutions of the associated wave equation. The machinery used allows one to study a class of infinite graphs without assumption on the local finiteness.