Greedy Algorithms for Reduced Bases in Banach Spaces

Given a Banach space X and one of its compact sets $\mathcal{F}$ , we consider the problem of finding a good n-dimensional space X _n ?X which can be used to approximate the elements of $\mathcal{F}$ . The best possible error we can achieve for such an approximation is given by the Kolmogorov width $d_{n}(\mathcal{F})_{X}$ . However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) in the case $X=\mathcal{H}$ is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457–1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces.     

Approaches to the enantioselective synthesis of ferrugine and its analogues

A four-step synthetic route, to ferrugine (2α-benzoyltropane), its methyl analogue (2-acetyltropane) and their N-benzyl analogues is reported. The reaction sequence uses tropinone or N-benzylnortropinone aldols as key intermediates. Reduction of aldol derived N-tosylhydrazones and oxidation of the side chain hydroxyl group followed by spontaneous diastereomer equilibration provides the final products. Relative configuration of the exo,anti N-methyl and N-benzyl aldols was retained during N-tosylhydrazone formation. The relative stereochemistry of N-tosylhydrazones was assigned by single crystal diffraction. The final products, ferrugine and its methyl analogue, were synthesized in enantiomerically pure form via asymmetric deprotonation of tropinone using chiral lithium amide/lithium chloride aggregate prepared in situ from (S,S)-N,N-bis(1-phenylethyl)amine hydrochloride.     

Synthesis and Crystal Structure of Europium(II) Tartrate Tetrahydrate, [Eu(C4H4O6)(H2O)2](H2O)2

Single crystals of [Eu(C₄H₄O₆)(H₂O)₂](H₂O)₂ were obtained from the combination of solutions of EuCl₂, previously obtained by electrolysis of an aqueous solution of EuCl₃, and tartraric acid, neutralized by LiOH. The crystal structure (orthorhombic, P2₁2₁2₁, Z = 4, a = 948.9(1), b = 954.6(1), c = 1098.4(1) pm; R(F) = 0.0242 and R_w(F²) = 0.0585 for I > 2σ(I); R(F) = 0.0256 and R_w(F²) = 0.0592 for all data) is isotypic with [Ca(C₄H₄O₆)(H₂O)₂](H₂O)₂ and [Sr(C₄H₄O₆)(H₂O)₂](H₂O)₂ exhibiting a three‐dimensional structure. The divalent cations (Eu²⁺, Ca²⁺, Sr²⁺) are eight‐coordinate by oxygen atoms that originate from carboxylate and hydroxyl groups of the tartraric dianion and two of the four water molecules.     

Non-hydrolysable Phosphinic Analogues of Dipeptides

Abstract

Peptide analogues 1 in which the phosphonamide group replaces the amide function are thought to mimic the tetrahedral intermediates and transition states in peptide hydrolysis. While pepti-des 1 are potent inhibitors of some peptidases, their unsefulness is limited by rapid hydrolysis of P-N bond in aqueous media. To circumvent the instability problem we designed peptide analogues 2 with a CH₂ group inserted between the phosphorus and nitrogen atoms. Such structures are resistant to hydrolysis but it was interesting to see if they are similar enough to pepti-des to retain any affinity to enzyme active sites.     

The Study of Derivatization Prior MALDI MSI Analysis—Charge Tagging Based on the Cholesterol and Betaine Aldehyde

Mass spectrometry imaging is a powerful tool for analyzing the different kinds of molecules in tissue sections, but some substances cannot be measured easily, due to their physicochemical properties. In such cases, chemical derivatization could be applied to introduce the charge into the molecule and facilitate its detection. Here, we study cholesterol derivatization with betaine aldehyde from tissue slices and evaluate how different sample preparation methods influence the signal from the derivatization product. In this study, we have tested different solutions for betaine aldehyde, different approaches to betaine aldehyde deposition (number of layers, deposition nozzle height), and different MALDI matrices for its analysis. As a result, we proved that the proposed approach could be used for the analysis of cholesterol in different tissues.     

Direct High-Throughput Screening Assay for mRNA Cap Guanine-N7 Methyltransferase Activity

In eukaryotes, mature mRNA is formed through modifications of precursor mRNA, one of which is 5’ cap biosynthesis, involving RNA cap guanine-N7 methyltransferase (N7-MTase). N7-MTases are also encoded by some eukaryotic viruses and facilitate their replication. N7-MTase inhibitors have therapeutic potential, but their discovery is difficult because long RNA substrates are usually required for activity. Herein, we report a universal N7-MTase activity assay based on small-molecule fluorescent probes. We synthesized 12 fluorescent substrate analogues (GpppA and GpppG derivatives) varying in the dye type, dye attachment site, and linker length. GpppA labeled with pyrene at the 3’-O position of adenosine acted as an artificial substrate with the properties of a turn-off probe for all three tested N7-MTases (human, parasite, and viral). Using this compound, a N7-MTase inhibitor assay adaptable to high-throughput screening was developed and used to screen synthetic substrate analogues and a commercial library. Several inhibitors with nanomolar activities were identified.     

Optimal Stable Nonlinear Approximation

Foundations of Computational Mathematics - While it is well-known that nonlinear methods of approximation can often perform dramatically better than linear methods, there are still questions on how...     

Cationic copolymerization of racemic‐β‐butyrolactone with L,L‐lactide: One‐pot synthesis of block copolymers

Cationic copolymerization of racemic‐β‐butyrolactone (β‐BL) with l,l ‐lactide (LA) initiated by alcohol and catalyzed by trifluoromethanesulfonic acid proceeding by activated monomer (AM) mechanism was investigated. Although both comonomers were present from the beginning in the reaction mixture, polymerization proceeded in sequential manner, with poly‐BL formed at the first stage acting as a macroinitiator for the subsequent polymerization of LA. Such course of copolymerization was confirmed by following the consumption of both comonomers throughout the process as well as by observing the changes of growing chain‐end structure using ¹H NMR. ¹³C NMR analysis and thermogravimetry revealed the block structure of resulting copolymers. The proposed mechanism of copolymerization was confirmed by the studies of changes of ¹H NMR chemical shift of acidic proton in the course of copolymerization, providing an indication that indeed protonated species and hydroxyl groups are present throughout the process, as required for AM mechanism. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2013 , 51, 4873–4884