Linear independence of time-frequency shifts under a generalized Schrödinger representation

Let r_\mathbbR \rho_{\mathbb{R}} be the classical Schrödinger representation of the Heisenberg group and let L \Lambda be a finite subset of \mathbbR ×\mathbbR \mathbb{R} \times \mathbb{R} . The question of when the set of functions {t ? e^{2 pi y t} f(t + x) = (r_\mathbbR(x, y, 1) f)(t) : (x, y) ? L} \{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\} is linearly independent for all f ? L²(\mathbbR), f 1 0 f \in L^2(\mathbb{R}), f \neq 0 , arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset L \Lambda of G ×[^(G)] G \times \widehat{G} and r_G \rho_G the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set {r_G (x, w, 1)f : (x, w) ? L} \{\rho_G (x, w, 1)f : (x, w) \in \Lambda\} to be linearly independent for all f ? L²(G), f 1 0 f \in L^2(G), f \neq 0 .     

Effect of radioactive and non-radioactive mercury on wheat germination and the anti-toxic role of glutathione

Studies to test the noxious effect of mercury ions on wheat germination and seedling growth showed that germination rate, shoot length, and fresh weights varied as a function of their concentration in the treatment solutions. At the same concentration, the radioactive mercury proved to be more harmful to the living seeds and seedlings. The detoxification action of glutathione for both radioactive and non-radioactive mercury was also followed. After a seven-day period of germination in the presence of the investigated compounds, the wheat plantlets were cut from the seeds, and their height, weight, and residual radioactivity were measured. The shoot length decreased from 8.1 (blank) to 4.6 cm (non-radioactive mercury) or even to 2.5 cm ((203)Hg), while glutathione had both an anti-toxic and an anti-radiotoxic effect (6.4 and 6.0 cm, respectively). The root weight of the lot decreased from 1.7 to 0.7 g and 0.4 g, respectively, while glutathione showed a healing action (1.5 and 1.7 g). The radioactive ions accumulated especially in roots (35.5 %), and less in shoots (11.2 %). Results were statistically validated.     

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝ ^d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe. The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.      

Robust dimension reduction,fusion frames,and Grassmannian packings

We consider estimating a random vector from its measurements in a fusion frame, in presence of noise and subspace erasures. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first consider the linear minimum mean-squared error (LMMSE) estimation of the random vector of interest from its fusion frame measurements in the presence of additive white noise. Each fusion frame measurement is a vector whose elements are inner products of an orthogonal basis for a fusion frame subspace and the random vector of interest. We derive bounds on the mean-squared error (MSE) and show that the MSE will achieve its lower bound if the fusion frame is tight. We then analyze the robustness of the constructed LMMSE estimator to erasures of the fusion frame subspaces. We limit our erasure analysis to the class of tight fusion frames and assume that all erasures are equally important. Under these assumptions, we prove that tight fusion frames consisting of equi-dimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace among all tight fusion frames, and that the optimal subspace dimension depends on signal-to-noise ratio (SNR). We also prove that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures, among the class of equi-dimensional tight fusion frames. We call such fusion frames equi-distance tight fusion frames. We prove that the squared chordal distance between the subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for the construction of equi-distance tight fusion frames.     

Metal-Dependent and Selective Crystallization of CAU-10 and MIL-53 Frameworks through Linker Nitration

The reaction of the V-shaped linker molecule 5-hydroxyisophthalic acid (H₂L⁰), with Al or Ga nitrate under almost identical reaction conditions leads to the nitration of the linker and subsequent formation of metal–organic frameworks (MOFs) with CAU-10 or MIL-53 type structure of composition [Al(OH)(L)], denoted as Al-CAU-10-L^{0, 2, 4, 6} or [Ga(OH)(L)], denoted as Ga-MIL-53-L². The Al-MOF contains the original linker L⁰ as well as three different nitration products (L², L⁴ and L^4/6), whereas the Ga-MOF mainly incorporates the linker L². The compositions were deduced by ¹H NMR spectroscopy and confirmed by Rietveld refinement. In situ and ex situ studies were carried out to follow the nitration and crystallization, as well as the composition of the MOFs. The crystal structures were refined against powder X-ray diffraction (PXRD) data. As anticipated, the use of the V-shaped linker results in the formation of the CAU-10 type structure in the Al-MOF. Unexpectedly, the Ga-MOF crystallizes in a MIL-53 type structure, which is usually observed with linear or slightly bent linker molecules. To study the structure directing effect of the in situ nitrated linker, pure 2-nitrobenzene-1,3-dicarboxylic acid (m-H₂BDC-NO₂) was employed which exclusively led to the formation of [Ga(OH)(C₈H₃NO₆)] (Ga-MIL-53-m-BDC-NO₂), which is isoreticular to Ga-MIL-53-L². Density Functional Theory (DFT) calculations confirmed the higher stability of Ga-MIL-53-L² compared to Ga-CAU-10-L² and grand canonical Monte Carlo simulations (GCMC) are in agreement with the observed water adsorption isotherms of Ga-MIL-53-L².     

Matrix/analyte ratio influencing polymer molecular weight distribution in matrix‐assisted laser desorption/ionization time‐of‐flight mass spectrometry

Matrix‐assisted laser desorption/ionization time‐of‐flight mass spectrometry (MALDI‐TOF‐MS) has been used to characterize poly(L‐lysine) polymers and unique oligomer peptides, like 10‐, 15‐ and 25‐mer [Lys]_n oligolysine peptides. Several matrices have been tried in order to find optimal conditions, but only α‐cyano‐4‐hydroxycinnamic acid gave analytically useful spectra. The synthetic oligomers and their mixtures gave good quality spectra, showing protonated and cationized molecules, including doubly charged species. The polymers, analogously, gave a wide distribution of single‐ and double‐cationized peak series. The polymer distributions observed indicate the presence of significant suppression effects. The concentration (matrix/analyte ratio) was found to influence the results significantly; distributions shifting to higher masses when higher polymer concentrations were used. This effect was studied in detail using the synthetic (‘monodisperse’) oligolysine peptides. It was found that the relative intensities change by over an order of magnitude in the 0.1–10 pmol/µL concentration range (typical for most proteomic analyses). The results indicate that concentration effects need to be considered when MALDI‐MS is used for quantitative purposes. Copyright © 2009 John Wiley & Sons, Ltd.     

Altered Glycosylation of Human Alpha-1-Acid Glycoprotein as a Biomarker for Malignant Melanoma

A high-resolution HILIC-MS/MS method was developed to analyze anthranilic acid derivatives of N-glycans released from human serum alpha-1-acid glycoprotein (AGP). The method was applied to samples obtained from 18 patients suffering from high-risk malignant melanoma as well as 19 healthy individuals. It enabled the identification of 102 glycan isomers separating isomers that differ only in sialic acid linkage (α-2,3, α-2,6) or in fucose positions (core, antenna). Comparative assessment of the samples revealed that upregulation of certain fucosylated glycans and downregulation of their nonfucosylated counterparts occurred in cancer patients. An increased ratio of isomers with more α-2,6-linked sialic acids was also observed. Linear discriminant analysis (LDA) combining 10 variables with the highest discriminatory power was employed to categorize the samples based on their glycosylation pattern. The performance of the method was tested by cross-validation, resulting in an overall classification success rate of 96.7%. The approach presented here is significantly superior to serological marker S100B protein in terms of sensitivity and negative predictive power in the population studied. Therefore, it may effectively support the diagnosis of malignant melanoma as a biomarker.     

A qualitative uncertainty principle for functions generating a Gabor frame on LCA groups

Let G be a locally compact Abelian group. In this paper we study in which way the qualitative uncertainty principle is modified when we consider only functions f∈L²(G) which generate a Gabor frame associated with a uniform lattice K in G. This provides us with sharp lower bounds for the measure of the support of such functions and their Plancherel transforms.     

Detection of immune complexes by matrix-assisted laser desorption/ionization mass spectrometry

Matrix-assisted laser desorption/ionization mass spectrometry (MALDI-MS) was used to detect an immune complex formed between beta-lactoglobulin and polyclonal anti-beta-lactoglobulin antibody in the gas phase. The most important experimental parameters to detect such a specific antibody-antigen complex by MALDI were the use of solutions at near-neutral pH and of sinapinic acid matrix prepared by the dried-droplet method. Under such conditions, predominantly one but also two molecules of antigen protein were complexed by the antibody. Specific formation of the antibody-antigen complex was confirmed by performing competitive reactions. Addition of antibody to a 1:1 mixture of beta-lactoglobulin and one control protein resulted not only in the appearance of the expected antibody-antigen complex, but also in a strong decrease in the free beta-lactoglobulin signal, while the abundance of the control protein was not influenced.