Eigenvalue perturbation theory of symplectic,orthogonal, and unitary matrices under generic structured rank one perturbations

We study the perturbation theory of structured matrices under structured rank one perturbations, with emphasis on matrices that are unitary, orthogonal, or symplectic with respect to an indefinite inner product. The rank one perturbations are not necessarily of arbitrary small size (in the sense of norm). In the case of sesquilinear forms, results on selfadjoint matrices can be applied to unitary matrices by using the Cayley transformation, but in the case of real or complex symmetric or skew-symmetric bilinear forms additional considerations are necessary. For complex symplectic matrices, it turns out that generically (with respect to the perturbations) the behavior of the Jordan form of the perturbed matrix follows the pattern established earlier for unstructured matrices and their unstructured perturbations, provided the specific properties of the Jordan form of complex symplectic matrices are accounted for. For instance, the number of Jordan blocks of fixed odd size corresponding to the eigenvalue 1 or ?1 have to be even. For complex orthogonal matrices, it is shown that the behavior of the Jordan structures corresponding to the original eigenvalues that are not moved by perturbations follows again the pattern established earlier for unstructured matrices, taking into account the specifics of Jordan forms of complex orthogonal matrices. The proofs are based on general results developed in the paper concerning Jordan forms of structured matrices (which include in particular the classes of orthogonal and symplectic matrices) under structured rank one perturbations. These results are presented and proved in the framework of real as well as of complex matrices.     

Autoignition of gasoline and its surrogates in a rapid compression machine

The analysis and interpretation of the combustion chemistry is greatly simplified by using simple mixtures of pure components, referred to as surrogates, in lieu of fully-blended transportation fuels, such as gasoline. Recognizing that the ability to model autoignition chemistry is critical to understanding the operation of Homogeneous Charged Compression Ignition engines, this work is an attempt to experimentally and computationally assess the autoignition responses of research grade gasoline and two of its proposed surrogates reported in the literature using a rapid compression machine (RCM), for the low-to-intermediate temperature range and at high pressures. The first surrogate studied is a three-component mixture of iso-octane, n-heptane, and toluene. The second is a four-component mixture that includes an olefin (2-pentene), in addition to the ones noted above. Ignition delay times of stoichiometric mixtures, for gasoline and the two surrogates in air, are measured using an RCM for pressures of 20 and 40 bar, and in the temperature range of 650–900 K. The four-component surrogate is found to emulate the ignition delay times of gasoline more closely when compared to the three-component surrogate. Additionally, the experimental data are compared against the computed results from a recently developed surrogate model for gasoline combustion. A good agreement between the experimental and computed results is observed, while discrepancies are also identified and discussed.     

Numerical methods for palindromic eigenvalue problems: Computing the anti‐triangular Schur form

We present structure‐preserving numerical methods for the eigenvalue problem of complex palindromic pencils. Such problems arise in control theory, as well as from palindromic linearizations of higher degree palindromic matrix polynomials. A key ingredient of these methods is the development of an appropriate condensed form—the anti‐triangular Schur form. Ill‐conditioned problems with eigenvalues near the unit circle, in particular near ±1, are discussed. We show how a combination of unstructured methods followed by a structured refinement can be used to solve such problems accurately. Copyright © 2008 John Wiley & Sons, Ltd.     

Self-organizing properties of natural and related synthetic glycolipids

In this article we report on the syntheses, self-organizing properties, and structures of a variety of cerebrosides and related synthetic glycolipids. The thermotropic and lyotropic liquid crystalline properties of the materials were evaluated in detail. All of the families of materials studied exhibited columnar mesophases. In the dry state the aliphatic chains were found to be located on the exterior of the columns, whereas in the wet state the reverse was the case with the polar headgroups on the exterior. Thus, the aliphatic chains act almost like hydrocarbon solvents in the dry state.     

Deuterium NMR investigation of the biaxial nematic phase in an organosiloxane tetrapode

Deuterium NMR is used to examine the molecular order exhibited by an organosiloxane tetrapode giving the first experimental evidence, using a bulk sample, for the existence of a biaxial nematic phase in this type of compounds. The temperature dependence of the averaged quadrupolar coupling constant and asymmetry parameter was determined in the compound's nematic phase. Two distinct regimes could be identified, one with a vanishing asymmetry parameter corresponding to a uniaxial nematic phase and another with a significant temperature dependent asymmetry parameter, corresponding to a biaxial nematic phase. The high values obtained for the asymmetry parameter at the lower end of the nematic range are well above experimental error and constitute a definite proof of the biaxial nature of the nematic phase exhibited by the studied compound for those temperatures.