Density, Overcompleteness, and Localization of Frames. I. Theory

Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames and ( a discrete abelian group), relating the decay of the expansion of the elements of in terms of the elements of via a map . A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of , the relative measure of — both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements — and the density of the set in . Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.     

Emission characteristics of laser dyes C1 and Rh6G in cross-linked polyvinyl alcohol solutions

The lifetimes and the relative quantum yields for fluorescence of two laser dyes Coumarin 1 and Rhodamine 6G have been determined in cross-linked polyvinyl alcohol matrix. The cross-linking has been achieved using gamma radiation. The relative fluorescence quantum yields of the dyes increased with increasing cross-linking of the polymer, but the fluorescence lifetimes remained unchanged within experimental error. The results have been attributed to a reduction in the loss of the excitation energy via the internal conversion of the excited singlet state due to increase in the viscosity of the medium brought about by the increased cross-linking of the polymer.     

Molecular Imaging of Growth,Metabolism, and Antibiotic Inhibition in Bacterial Colonies by Laser Ablation Electrospray Ionization Mass Spectrometry

Metabolism in microbial colonies responds to competing species, rapidly evolving genetic makeup, and sometimes dramatic environmental changes. Conventional characterization of the existing and emerging microbial strains and their interactions with antimicrobial agents, e.g., the Kirby–Bauer susceptibility test, relies on time consuming methods with limited ability to discern the molecular mechanism and the minimum inhibitory concentration. Assessing the metabolic adaptation of microbial colonies requires their non‐targeted molecular imaging in a native environment. Laser ablation electrospray ionization (LAESI) is an ambient ionization technique that in combination with mass spectrometry (MS) enables the analysis and imaging of numerous metabolites and lipids. In this contribution, we report on the application of LAESI‐MS imaging to gain deeper molecular insight into microbe–antibiotic interactions, and enhance the quantitative nature of antibiotic susceptibility testing while significantly reducing the required incubation time.     

Expedient synthesis of sulfinamides from sulfonyl chlorides

Sulfinamides were synthesized from sulfonyl chlorides using a procedure involving in situ reduction of sulfonyl chlorides. The reaction is broad in scope and easy to perform.     

The Effects of New Zealand Grown Ginseng Fractions on Cytokine Production from Human Monocytic THP-1 Cells

Pro-inflammatory cytokines and anti-inflammatory cytokines are important mediators that regulate the inflammatory response in inflammation-related diseases. The aim of this study is to evaluate different New Zealand (NZ)-grown ginseng fractions on the productions of pro-inflammatory and anti-inflammatory cytokines in human monocytic THP-1 cells. Four NZ-grown ginseng fractions, including total ginseng extract (TGE), non-ginsenoside fraction extract (NGE), high-polar ginsenoside fraction extract (HPG), and less-polar ginsenoside fraction extract (LPG), were prepared and the ginsenoside compositions of extracts were analyzed by HPLC using 19 ginsenoside reference standards. The THP-1 cells were pre-treated with different concentrations of TGE, NGE, HPG, and LPG, and were then stimulated with lipopolysaccharide (LPS). The levels of pro-inflammatory cytokines, including tumor necrosis factor-alpha (TNF-α), interleukin-1 beta (IL-1β), interleukin-6 (IL-6), interleukin-8 (IL-8), and anti-inflammatory cytokines, such as interleukin-10 (IL-10), and transforming growth factor beta-1 (TGF-β1), were determined by enzyme-linked immunosorbent assay (ELISA). TGE at 400 µg/mL significantly inhibited LPS-induced TNF-α and IL-6 productions. NGE did not show any effects on inflammatory secretion except inhibited IL-6 production at a high dose. Furthermore, LPG displayed a stronger effect than HPG on inhibiting pro-inflammatory cytokine (TNF-α, IL-1β, and IL-6) productions. Particularly, 100 µg/mL LPG not only significantly inhibited the production of pro-inflammatory cytokines TNF-α, IL-1β, and IL-6, but also remarkably enhanced the production of anti-inflammatory cytokine IL-10. NZ-grown ginseng exhibited anti-inflammatory effects in vitro, which is mainly attributed to ginsenoside fractions (particularly less-polar ginsenosides) rather than non-saponin fractions.     

Spectroscopic Characterization of a Lead–Lead Dioxide Automobile Battery

Here we combine small angle neutron scattering measurements (SANS) with X-ray diffraction analysis (XRD) and infrared spectroscopy (IR) measurements to obtain information about nanoparticles formed in a series of lead-lead dioxide samples mixed with various CuO concentrations. New vitreous systems with the xCuO???(100?x)[4PbO₂???Pb] composition where x?=?0, 30, and 70?mol% CuO were prepared by the melt-quenching method using CuO mixed in suitable proportion with the active electrodes of a disassembled car battery as the starting materials. The X-ray diffraction patterns permit the identification of the metallic Pb phase and the presence of oxidic nanoparticles of the lead and copper ions. By doping with higher CuO contents, the SANS curves have a concave shape indicating inhomogeneities and tendency of phase separation due to formation of nanoparticles of the lead and copper ions in recycled host matrix. The studied samples can be considered as polydispersed systems. The matrix is the solvent and the soluble phase is formed from the oxidic lead and/or copper particles with sizes smaller than 69?Å dispersed either inside the host matrix grains or between the host matrix grains. The formation of the nanoparticles in the host matrix and the knowledge of the type of nanoparticles have a decisive role in applications for the construction of essential components of the automobile batteries.     

Toxicity of <Emphasis Type="Italic">Camellia sinensis</Emphasis>-Fabricated Silver Nanoparticles on Invertebrate and Vertebrate Organisms: Morphological Abnormalities and DNA Damages

Our understanding of nanoparticle toxicity and fate in the aquatic environment is still patchy. In the present study, the toxicity of silver nanoparticles coated by Camellia sinensis (Cs) leaf extract metabolites (Cs-AgNPs) was investigated in comparison with C. sinensis leaf extract and AgNO₃ on a micro-crustacean, Ceriodaphnia cornuta, and a fish Poecilia reticulata. 100% mortality of C. cornuta was observed post-exposure to AgNO₃ (40 µg/ml) if compared to the Cs leaf extract and Cs-AgNPs, showing 30 and 56% mortality at the same concentration, respectively. In P. reticulata 100% mortality was observed testing AgNO₃ and Cs-AgNPs post-exposure to 1 and 30 µg/ml, respectively. Light microscopy and CLSM images showed the accumulation of nanoparticles in the intestine of C. cornuta treated with Cs-AgNPs at 40 µg/ml. In addition, histological observations confirmed the abnormal tissue texture in nanoparticle-exposed P. reticulata, if compared to control fishes. Furthermore, C. cornuta and P. reticulata treated with Cs-AgNPs showed DNA damages compared to the control. Overall, these findings indicated relevant limits about the employ of silver-based pesticides in the environment, and also pointed out the Cs-AgNPs were less toxic to C. cornuta and P. reticulata if compared to silver ions.     

Control of the degree of surface graphitization on 3C-SiC(100)/Si(100)

The current method of growing graphene by thermal decomposition of 3C-SiC(100) on silicon substrates is technologically attractive. Here, we investigate the evolution of the surface graphitization as a function of the synthesis temperature. We establish that the carbon enrichment of the surface is characterized by a clear modulation of the surface potential and structuration. The structural properties analysis of the graphene layers by low energy electron diffraction and micro-Raman spectroscopy demonstrate a graphitization of the surface.     

Linear SPDEs Driven by Stationary Random Distributions

Using tools from the theory of stationary random distributions developed in It? (Mem. Coll. Sci., Univ. Kyoto, Ser. A: Math., 28:209?C223,?1954) and Yaglom (Theory Probab. Appl., 2:273?C320,?1957), we introduce a new class of processes which can be used as a model for the noise perturbing an SPDE. This type of noise is not necessarily Gaussian, but it includes the spatially homogeneous Gaussian noise introduced in Dalang (Electron. J. Probab. 4(6)?1999), and the fractional noise considered in Balan and Tudor (Stoch. Process. Appl., 120:2468?C2494,?2010). We derive some general conditions for the existence of a random field solution of a linear SPDE with this type of noise, under some mild conditions imposed on the Green function of the differential operator which appears in this equation. This methodology is applied to the study of the heat and wave equations (possibly replacing the Laplacian by one of its fractional powers), extending in this manner the results of Balan and Tudor (Stoch. Process. Appl., 120:2468?C2494,?2010) to the case H<1/2.