Decay spectrum and decay subspace of normal operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Surrogate-Based Optimization Using Multifidelity Models with Variable Parameterization and Corrected Space Mapping

Surrogate-based-optimization methods provide a means to achieve high-fidelity design optimization at reduced computational cost by using a high-fidelity model in combination with lower-fidelity models that are less expensive to evaluate. This paper presents a provably convergent trust-region model-management methodology for variableparameterization design models: that is, models for which the design parameters are defined over different spaces. Corrected space mapping is introduced as a method to map between the variable-parameterization design spaces. It is then used with a sequential-quadratic-programming-like trust-region method for two aerospace-related design optimization problems. Results for a wing design problem and a flapping-flight problem show that the method outperforms direct optimization in the high-fidelity space. On the wing design problem, the new method achieves 76% savings in high-fidelity function calls. On a bat-flight design problem, it achieves approximately 45% time savings, although it converges to a different local minimum than did the benchmark.

Effect of quantization of vibrations on the structural properties of crystals

We study the structural effects produced by the quantization of vibrational degrees of freedom in periodic crystals at zero temperature. To this end we introduce a methodology based on mapping a suitable subspace of the vibrational manifold and solving the Schrödinger equation in it. A number of increasingly accurate approximations ranging from the quasiharmonic approximation (QHA) to the vibrational self-consistent field (VSCF) method and the exact solution are described. A thorough analysis of the approximations is presented for model monatomic and hydrogen-bonded chains, and results are presented for a linear H-F chain where the potential-energy surface is obtained via first-principles electronic structure calculations. We focus on quantum nuclear effects on the lattice constant and show that the VSCF is an excellent approximation, meaning that correlation between modes is not extremely important. The QHA is excellent for covalently bonded mildly anharmonic systems, but it fails for hydrogen-bonded ones. In the latter, the zero-point energy exhibits a nonanalytic behavior at the lattice constant where the H atoms center, which leads to a spurious secondary minimum in the quantum-corrected energy curve. An inexpensive anharmonic approximation of noninteracting modes appears to produce rather good results for hydrogen-bonded chains for small system sizes. However, it converges to the incorrect QHA results for increasing size. Isotope effects are studied for the first-principles H-F chain. We show how the lattice constant and the H-F distance increase with decreasing mass and how the QHA proves to be insufficient to reproduce this behavior.

Stokes imaging, Doppler mapping and Roche tomography of the AM Herculis system V834 Cen

We report on new simultaneous phase-resolved spectroscopic and polarimetric observations of the polar (AM Herculis star) V834 Cen during a high state of accretion. Strong emission lines and high levels of variable circular and linear polarization are observed over the orbital period. The polarization data are modelled using the Stokes imaging technique of Potter et al. The spectroscopic emission lines are investigated using the Doppler tomography technique of Marsh and Horne and the Roche tomography technique of Dhillon and Watson. Up to now, all three techniques have been used separately to investigate the geometry and accretion dynamics in cataclysmic variables. For the first time, we apply all three techniques to simultaneous data for a single system. This allows us to compare and test each of the techniques against each other and hence to derive a better understanding of the geometry, dynamics and system parameters of V834 Cen.

Mapping the realm: A New Look at the Gough Map of Britain (c.1360)

The mid-fourteenth century map of Britain known as the ‘Gough’ map is the earliest extant depiction of the island in geographically recognizable form. Hitherto, however, interest in the road or route patterns marked on the map has meant that the map's extraordinarily rich settlement geography has not received the attention it merits and that, consequently, the point of the map may have been missed. The availability of a digital scan of the map coupled with the use of Geographical Information System (GIS) software provides the opportunity for a new look at the Gough map and the questions it poses. Attention in this article is directed to the settlement geography it shows, and in particular to the map's 654 cities, towns, villages, castles and monasteries. Their geographical positions as given on the manuscript are compared with their modern equivalents to shed light on some of the basic questions—the map's place of origin, the purpose or purposes for which it was made and the circumstances of its production—that have posed such a challenge for scholars. Our preliminary conclusion is that the key to understanding the original primary role of the Gough map lies in its accurate but selective depiction of the settlement geography of fourteenth-century Britain.

Application of NMR based metabolomics for mapping metabolite variation in European wheat

In this study, we report on the use of NMR-based metabolomics to access variation in low molecular weight polar metabolites between the European wheat cultivars Apache, Charger, Claire and Orvantis. Previous unassigned resonances in the published NMR spectra of wheat extracts were identified using C NMR and two dimensional proton-carbon NMR. These included a peak for trans-aconitate (d3.43) and resonances corresponding to fructose in the crowded carbohydrate region of the spectra. Large metabolite differences were observed between two different growth stages, namely the coleoptile and two week old leaf tissue extracts which were consistent across cultivars. Two week old leaf tissue extracts had higher abundances of glutamine, glutamate, sucrose and trans-aconitate and less glucose and fructose than were observed in the coleoptile extracts. Across both growth stages the cultivars Apache and Charger showed the greatest differences in metabolite profiles. Charger had higher abundances of betaine, the single most influential metabolite in the principal component analysis, in addition to fructose and sucrose. However, Charger had lower levels of aspartate, choline and glucose than Apache. These findings demonstrate the potential for a biochemical mapping approach using NMR, across European wheat germplasm, for metabolites of known importance to functional characteristics. © Springer Science+Business Media, LLC 2009.