Sparse fusion frames: existence and construction

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame can be regarded as a frame-like collection of subspaces in a Hilbert space, and thereby generalizes the concept of a frame for signal representation. However, when the signal and/or subspace dimensions are large, the decomposition of the signal into its fusion frame measurements through subspace projections typically requires a large number of additions and multiplications, and this makes the decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a ‘uniform basis’ over all subspaces, thereby enabling low-complexity fusion frame decompositions. We study the existence and construction of sparse fusion frames, but our focus is on developing simple algorithmic constructions that can easily be adopted in practice to produce sparse fusion frames with desired (given) operators. By a desired (or given) operator we simply mean one that has a desired (or given) set of eigenvalues for the fusion frame operator. We start by presenting a complete characterization of Parseval fusion frames in terms of the existence of special isometries defined on an encompassing Hilbert space. We then introduce two general methodologies to generate new fusion frames from existing ones, namely the Spatial Complement Method and the Naimark Complement Method, and analyze the relationship between the parameters of the original and the new fusion frame. We proceed by establishing existence conditions for 2-sparse fusion frames for any given fusion frame operator, for which the eigenvalues are greater than or equal to two. We then provide an easily implementable algorithm for computing such 2-sparse fusion frames.     

The local integrability condition for wavelet frames

In this article we study finitely generated wavelet systems with arbitrary dilation sets. In 2002 Hernández et al. gave a characterization of when such a system forms a Parseval frame, assuming that a certain hypothesis known as the local integrability condition (LIC) holds. We show that, under some mild regularity assumption on the wavelets, the LIC is solely a density condition on the dilation sets. Using this new interpretation of the LIC, we further discuss when the characterization result holds.     

Fusion frames and distributed processing

Let {W_i}_iI be a (redundant) sequence of subspaces of a Hilbert space each being endowed with a weight v_i, and let be the closed linear span of the W_is, a composite Hilbert space. {(W_i,v_i)}_iI is called a fusion frame provided it satisfies a certain property which controls the weighted overlaps of the subspaces. These systems contain conventional frames as a special case, however they reach far “beyond frame theory.” In case each subspace W_i is equipped with a spanning frame system {f_ij}_jJi, we refer to {(W_i,v_i,{f_ij}_jJi)}_iI as a fusion frame system. The focus of this article is on computational issues of fusion frame reconstructions, unique properties of fusion frames important for applications with particular focus on those superior to conventional frames, and on centralized reconstruction versus distributed reconstructions and their numerical differences. The weighted and distributed processing technique described in this article is not only a natural fit to distributed processing systems such as sensor networks, but also an efficient scheme for parallel processing of very large frame systems. Another important component of this article is an extensive study of the robustness of fusion frame systems.     

Optimum collision energies for proteomics: The impact of ion mobility separation

Ion mobility spectrometry (IMS) is a widespread separation technique used in various research fields. It can be coupled to liquid chromatography–mass spectrometry (LC–MS/MS) methods providing an additional separation dimension. During IMS, ions are subjected to multiple collisions with buffer gas, which may cause significant ion heating. The present project addresses this phenomenon from the bottom-up proteomics point of view. We performed LC–MS/MS measurements on a cyclic ion mobility mass spectrometer with varied collision energy (CE) settings both with and without IMS. We investigated the CE dependence of identification score, using Byonic search engine, for more than 1000 tryptic peptides from HeLa digest standard. We determined the optimal CE values—giving the highest identification score—for both setups (i.e., with and without IMS). Results show that lower CE is advantageous when IMS separation is applied, by 6.3 V on average. This value belongs to the one-cycle separation configuration, and multiple cycles may supposedly have even larger impact. The effect of IMS is also reflected in the trends of optimal CE values versus m/z functions. The parameters suggested by the manufacturer were found to be almost optimal for the setup without IMS; on the other hand, they are obviously too high with IMS. Practical consideration on setting up a mass spectrometric platform hyphenated to IMS is also presented. Furthermore, the two CID (collision induced dissociation) fragmentation cells of the instrument—located before and after the IMS cell—were also compared, and we found that CE adjustment is needed when the trap cell is used for activation instead of the transfer cell. Data have been deposited in the MassIVE repository (MSV000090944).     

Direct photothermal techniques for rapid quantification of total anthocyanin content in sour cherry cultivars

The analytical performance of the newly proposed laser-based photoacoustic spectroscopy (PAS) and of optothermal window (OW) method for quantification of total anthocyanin concentration (TAC) in five sour cherry varieties is compared to that of the spectrophotometry (SP). High performance liquid chromatography (HPLC) was used to identify and quantify specific anthocyanins. Both, PAS and OW are direct methods that unlike SP and HPLC obviate the need for the extraction of analyte. The outcome of the study leads to the conclusion that PAS and OW are both suitable for quick screening of TAC in sour cherries. The correlation between the two methods and SP is linear with R² = 0.9887 for PAS and R² = 0.9918 for OW, respectively. Both methods are capable of the rapid determination of TAC in sour cherries without a need for a laborious sample pretreatment.     

The azaphilic addition of organometallic reagents on tetrazines: scope and limitations

A series of tetrazines were reacted with organometallic reagents. Depending on the nature of the metal azaphilic addition, reduction of the tetrazine or simple complex formation was the predominant transformation and usually high selectivity was observed.     

Combination of solid-phase affinity capture on magnetic beads and mass spectrometry to study non-covalent interactions: example of minor groove binding drugs

A simple and novel approach was developed to detect non-covalent interactions. It is based on combination of solid-phase affinity capture with matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOFMS). One of the interacting molecules is bound to magnetic beads and is incubated with the target molecules in solution. The complex bound on the solid support is removed from the solution and transferred for MALDI analysis. Mass spectrometry is used only to detect the target compound, which is far more straightforward than detecting the intact non-covalent complex. To demonstrate the applicability of the method, an AT-rich oligonucleotide (5'-CCCCCAATTCCCCC-3') and its complementary biotinylated sequence (5'-biotin-GGGGGAATTGGGGG-3') were hybridized and immobilized to paramagnetic particles by streptavidin-biotin interaction. The immobilized duplex oligonucleotide was reacted with minor groove binding drugs, Netropsin, Distamycin A, Hoechst 33258 and 4',6-diamidino-2-phenylindole. The resulting DNA-drug complex bound to the particles was separated and analyzed by linear MALDI-TOFMS after washing. Drugs were selectively detected in the spectra. Relative binding strengths were also estimated using competitive complexation.     

Mass spectrometric and quantum-chemical study on the structure, stability, and chirality of protonated serine dimers

Stability and structure of homo- and heterochiral protonated serine (Ser) dimers were investigated by a combination of mass spectrometry and ab initio quantum chemical calculations. This established that the energy difference between the most stable homo- and heterochiral forms is very small: tandem mass spectrometry with Cooks' kinetic method yielded a negligible difference in Gibbs free energy (0.2+/-0.2 kJ mol(-1)). The various isomeric forms of (Ser)2 H+ and their energetics were determined by extensive electronic-structure calculations, which yielded homo- and heterochiral forms of the isomers with distinctly different relative energies. The most stable homochiral isomer is stabilized by two hydrogen bonds and is far more stable than any other homochiral isomer. The most stable heterochiral isomer has completely different features, and it is characterized by a salt-bridge structure. This clearly shows that salt-bridge structures do exist in the gas phase even in comparatively small molecules and in the absence of particularly basic or acidic functional groups.     

Schwermetall-π-Komplexe. X. Synthese und Kristallstruktur von {[(1,3,5-(CH3)3C6H3)2Tl][AlCl4]}2: ein arenstabilisiertes dimeres Thallium(I)-tetrachloroaluminat

π-Complexes of Heavy Metals. X. Synthesis and Crystal Structure of {[(1,3,5-(CH₃)₃C₆H₃)₂Tl][AlCl₄]}₂: an Arene Stabilized Dimeric Thallium(I) Tetrachloroaluminate From a solution of AlCl₃ and TlCl in mesitylene, the bis(arene)thallium complex {[(1,3,5-(CH₃)₃C₆H₃)₂Tl][AlCl₄]}₂ ( 1 ) (space group P2₁/c with a = 19.575(4) Å, b = 12.436(2) Å, c = 19.415(4) Å, β = 101.69(3)° at T = ?90 ± 1°C; Z = 4) will crystallize at low temperature. This compound can be described as a dimeric thallium(I) tetrachloroaluminate with a sceleton similar to that of (TeI₄)₄, shielded by four arenes, in pairs coordinated at the thallium atoms. In the solid state the complete configuration has point group symmetry 1 (C₁). Tl? Cl distances ranging from 3.292(3) to 3.679(3) Å point out an ionic bonding situation between arene₂Tl⁺ and AlCl₄^? fragments. The strengths of the η⁶ like Tl-arene interactions are characterized by distances Tl(1)–C of 3.250 Å and 3.315 Å, and Tl(2)? C of 3.285 Å and 3.328 Å and a temperature of release of all arene molecules of 61°C, which has been determined by differential thermal analysis, to yield pure thallium(I) tetrachloroaluminate.