K-invariants in the universal enveloping algebra of the desitter group

Let G be a non-compact connected semisimple Lie group with finite center and let G^K denote the centralizer of a maximal compact subgroup K of G inG, the universal enveloping algebra over of the Lie algebra of G. In [4] Lepowsky defines an injective anti-homo morphism P:G ^KK ^MA, where M is the centralizer in K of a Cartan subalgebraa of the symmetric pair (G,K),K andA are the universal enveloping algebras over corresponding to K anda, respectively, andK ^M is the centralizer of M inK. The subalgebra P(G ^K) ofK ^MA has considerable significance in the infinite dimensional representation theory of G. In this paper we explicitly compute P(G ^K) when G=S0_o(4,1), and show how this result leads to the determination of all irreducible representations of G and its universal covering group Spin(4,1).Partially supported by CONICET (Argentina) grants.     

Compact flat manifolds with holonomy group

In this paper we construct a family of compact flat manifolds, for all dimensions , with holonomy group isomorphic to and first Betti number zero.

     

Oxidation state characterization in Cr oxides by means of Cr-Kβ emission spectroscopy

High-resolution Kβ spectra of Cr oxide were measured using a non-conventional spectrometer. Theoretical spectra were obtained using the DV-Xα method in order to interpret the Kβ spectrum structures. Kβ spectrum structures were analyzed and spectral parameters show a great sensitivity to the oxidation state and to the Cr-O distance. High-purity samples of CrO₂ were obtained by means of thermal treatment at 513 °C under oxygen pressure of 200 bar. X-ray diffraction patterns show a typical rutile structure, without spurious phases. The CrO₂ data allowed to confirm the linear dependency of the Kβ_1,3 and Kβ_2,5 energy positions with the oxidation state. The energy of the Kβ_2,5 line relative to the Kβ_1,3 line seems to be a suitable parameter for characterization of the oxidation state. The relative Kβ″ transition probability per Cr-O falls exponentially with Cr-O increasing distance. This behaviour was not found in the literature for Cr oxides.     

X‐ray spectra by means of Monte Carlo simulations for imaging applications

X‐ray tubes have a broad range of applications worldwide, including several techniques for atomic physics, like X‐ray fluorescence, as well as for medical imaging, like computed tomography. The performances of X‐ray imaging detectors have shown to be significantly sensitive to the incident beam spectrum. Therefore, an accurate knowledge of the X‐ray beam becomes necessary for the emission source characterization and the whole imaging process comprehension. Direct measurements and suitable Monte Carlo simulations may be used to establish the X‐ray spectra. Dedicated Monte Carlo simulation routines, based on the PENELOPE code, have been developed to determine the Bremsstrahlung X‐ray spectra generated by conventional X‐ray tubes. The simulated spectra have been validated by comparison with the corresponding experimental data showing an overall good agreement. The incorporation of a suitably designed virtual grid allowed to assess the angular distribution of Bremsstrahlung yield, showing a remarkable anisotropy. In addition, a dedicated program has been developed for virtual imaging, which enables to perform suitable X‐ray absorption contrast images. Also, the developed program includes a user‐friendly graphic interface to allow the upload of required input parameters, which include setup arrangement, beam characteristics, sample properties and image simulation parameters (spatial resolution, tracks per run, etc.). The software includes dedicated subroutines which handle the physical process from X‐ray generation up to detector signal acquisition. The aim of the developed program is to perform virtual imaging by means of absorption contrast and using conventional X‐ray sources, which may be a useful tool for the study the X‐ray imaging techniques in several research fields as well as for educational purposes. The performed comparisons with experimental data have shown good agreement. The obtained results for X‐ray imaging may constitute useful information for the comprehension and improvement of X‐ray image quality, like absorption contrast optimization, detail visualization, definition and detectability. Copyright © 2010 John Wiley & Sons, Ltd.     

Reducibility of matrix weights

In this paper, we discuss the notion of reducibility of matrix weights and introduce a real vector space \(\mathcal C_\mathbb R\) which encodes all information about the reducibility of W. In particular, a weight W reduces if and only if there is a nonscalar matrix T such that \(TW=WT^*\). Also, we prove that reducibility can be studied by looking at the commutant of the monic orthogonal polynomials or by looking at the coefficients of the corresponding three-term recursion relation. A matrix weight may not be expressible as direct sum of irreducible weights, but it is always equivalent to a direct sum of irreducible weights. We also establish that the decompositions of two equivalent weights as sums of irreducible weights have the same number of terms and that, up to a permutation, they are equivalent. We consider the algebra of right-hand-side matrix differential operators \(\mathcal D(W)\) of a reducible weight W, giving its general structure. Finally, we make a change of emphasis by considering the reducibility of polynomials, instead of reducibility of matrix weights.