Matrix-Valued (Θ,Θ∗)(\Theta, \Theta^*)-Gabor Frames over LCA Groups for Hyponormal Operators

G\v avruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely $K$-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator $K$. For a locally compact abelian group G and a positive integer $n$, we study frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space $L^2(G, \mathbb{C}^{n\times n})$ , where a bounded linear operator $\Theta$ on $L^2(G, \mathbb{C}^{n\times n})$ controls not only lower but also the upper frame condition. We term such frames matrix-valued $(\Theta, \Theta^*)$-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued $(\Theta, \Theta^*)$- Gabor frames in terms of hyponormal operators. It is shown that if $\Theta$ is adjointable hyponormal operator, then $L^2(G, \mathbb{C}^{n\times n})$ admits a $\lambda$-tight $(\Theta, \Theta^*)$-Gabor frame for every positive real number $\lambda$. A characterization of matrix-valued $(\Theta, \Theta^*)$-Gabor frames is given. Finally, we show that matrix-valued $(\Theta, \Theta^*)$-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study

Application of Mass Spectroscopy in Pharmaceutical and Biomedical Analysis

Mass spectrometry (MS) is a powerful analytical tool with many applications in pharmaceutical and biomedical field. The increase in sensitivity and resolution of the instrument has opened new dimensions in analysis of pharmaceuticals and complex metabolites of biological systems. Compared with other techniques, mass spectroscopy is only the technique for molecular weight determination, through which we can predict the molecular formula. It is based on the conversion of the sample into ionized state, with or without fragmentation which are then identified by their mass-to-charge ratios (m/e). Mass spectroscopy provides rich elemental information, which is an important asset to interpret complex mixture components. Thus, it is an important tool for structure elucidation of unknown compounds. Mass spectroscopy also helps in quantitative elemental analysis, that is, the intensity of a mass spectra signal is directly proportional to the percentage of corresponding element. It is also a noninvasive tool that permits in vivo studies in humans. Recent research has looked into the possible applications of mass spectrometers in biomedical field. It is also used as a sensitive detector for chromatographic techniques like LC–MS, GC–MS and LC/MS/MS. These recent hyphenated technological developments of the technique have significantly improved its applicability in pharmaceutical and biomedical analyses

Accurate calculations of interstellar lines of Mg<SUP>+</SUP> using the coupled cluster approach

One of the most successful ab initio, highly correlated all-order many-body methods, the relativistic coupled cluster theory, is employed to calculate excitation energies of the doublet states of Mg+ and allowed transitions among them that are of interest in astrophysical problems. We have also calculated oscillator strength for the 3s-4p doublet transitions, which is improved over the existing results. These transition lines have been sought after in astronomical observations because they represent the best column density identifier in the interstellar medium. Our calculated oscillator strength (9.3 &#215; 10-4) and branching ratio (1.80) of these doublet lines matches well with the recent empirical and semiempirical calculations

Relativistic coupled-cluster-based linear response theory for ionization potentials of alkali-metal and alkaline-earth-metal atoms

We have developed and applied the relativistic coupled-cluster-based linear response theory (RCCLRT) for computing the principal as well as the shake-up ionization potentials (IP's) of Li, Be, Na, and Mg where the single-particle orbitals are generated by solving the relativistic Hartree-Fock-Roothaan equations using the Gaussian basis functions on a grid. The computed principal and shake-up ionization energies by the RCCLRT approach are in favorable agreement with the experimental results. Since for the (one-valence) IP problem, there is a formal equivalence between the principal IP values as obtained from the CCLRT and those obtained as eigenvalues of the multireference coupled-cluster theory, the computed quantities are fully size extensive. The approach via the RCCLRT has the additional advantage of providing the shake-up IP's as well. These are, however, not fully size extensive, but the error scales as the number of valence excitations (2h-1p), so the inextensivity error is rather small

Core effects on ionization potentials in thallium

Ionization potentials (IP's) are evaluated for various excited states of Tl using the relativistic coupled cluster (CCCD) theory in the even-parity pair channel approximation (CCSD-EPC). An average accuracy below half a percent is reached. The effect of deep core electrons on the core-valence correlations is investigated. It is found that electrons in the third subshell (n=3) modify the IP's of the 6p orbitals by 100 cm-1. By comparison with calculations made in the linearized CCSD (LCCSD) approximation it is demonstrated that nonlinear contributions are mandatory to reach an accuracy below half a percent for the 6p&#189; orbital