Retaining positive definiteness in thresholded matrices

Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of complete components. This result rigorously demonstrates that, except in special cases, positive definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions.     

Exploring the Limits of π-Acid Catalysis Using Strongly Electrophilic Main Group Metal Complexes: The Case of Zinc and Aluminium

The catalytic activity of cationic NHC-Zn^II and NHC-Al^III (NHC=N-heterocyclic carbene) complexes in reactions that require the electrophilic activation of soft C−C π bonds has been studied. The former proved able to act as a soft π-Lewis acid in a variety of transformations. The benefit of the bulky IPr NHC ligand was demonstrated by comparison with simple ZnX₂ salts. The tested NHC-Al^III catalyst is not able to activate C−C π bonds but simple AlX₂⁺ ions were found potent in some cases.     

Chemical potential of triatomic polar liquids; a computer simulation study

The chemical potential of liquid sulphur dioxide and liquid water is investigated by molecular dynamics simulation. The calculation is based upon the Widom test-particle method. An empirical procedure is applied in order to reduce drastically the computing time needed to sample the computer-generated configurations. The influence of the choice of the effective pair potential is discussed.     

Copolymeres du chlorure de vinyle et de l'acrylate de glycidyle

A study of the radical copolymerization of vinyl chloride (C) and glycidyl acrylate (A) at 60° in dichloroethane solution leads to the following reactivity ratios r_c = 0·14 ± 0·02 and r_A = 7·4 ± 0·3According to the nature of the solvent, the reaction of protonic acids may cause either cationic polymerization of epoxy groups or addition of HCl or water onto the same groups. The HCl addition is observed during the thermal degradation of the copolymers so that HCl evolution is greatly delayed.     

Synthesis,Biological Evaluation,and Crystallographic Studies of Extended Guanine‐Based (lin‐Benzoguanine) Inhibitors for tRNA‐Guanine Transglycosylase (TGT)

This paper describes the rational design, synthesis, and biological evaluation of a new generation of inhibitors of the bacterial enzyme tRNA‐guanine transglycosylase (TGT), which has been identified as a new target in the fight against bacillary dysentery (Shigellosis). The enzyme catalyzes the exchange of guanine in the anticodon wobble position of tRNA by the modified base preQ₁, a guanine derivative, according to a ping‐pong mechanism involving a covalent TGT‐tRNA intermediate (Fig. 2). Based on computer modeling (Fig. 3), lin‐benzoguanine (6‐aminoimidazol[4,5‐g]quinazolin‐8(7H)‐one ( 2 )) was selected as an extended central scaffold, to form up to seven in‐plane intermolecular H‐bonds with the protein while sandwiching between Tyr106 and Met260. Versatile synthetic protocols were developed for the synthesis of 2 , and derivatives with phenyl, benzyl, and 2‐phenylethyl side chains (i.e., 16, 17a , and 12a, 12b, 13, 17 , resp.) to reach into the lipophilic pocket lined by Val282, Val45, and Leu68 (Schemes 1–3). To account for the limited solubility of the new ligands and in consequence of a recently developed detailed understanding of the mechanism of TGT catalysis (Fig. 2), the enzyme kinetic assay was completely redesigned, providing competitive (K_ic) and uncompetitive (K_iu) inhibition constants with respect to tRNA binding by TGT. The modifications of the various parameters in the new assay are described in detail. Binding affinities of the new inhibitors were found to be in the single‐digit micromolar range (K_ic values, Fig. 8). Decoration of the lin‐benzoguanine scaffold with lipophilic residues only gave a modest improvement in biological activity which was explained on structural grounds with the help of four crystal structures (Fig. 10) obtained by soaking the protein with inhibitors 2 and 12a – 12c . Both biochemical and biostructural analyses reported in this paper provide a fertile basis for the development of more potent future generations of TGT inhibitors.     

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