Coverings of random ellipsoids,and invertibility of matrices with i.i.d. heavy-tailed entries

Let A = (a_ij) be an n × n random matrix with i.i.d. entries such that Ea₁₁ = 0 and Ea ₁₁ ² = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B ₂ ⁿ of cardinality at most exp(δn) such that with probability very close to one we have

$$A\left( {B_2^n} \right)\subset\mathop \cup \limits_{y \in A\left( \mathcal{N} \right)} \left( {y + L\sqrt n B_2^n} \right)$$

. In fact, a stronger statement holds true. As an application, we show that for some L' > 0 and u ∈ [0, 1) depending only on the distribution law of a11, the smallest singular value sn of the matrix A satisfies

$$\mathbb{P}\left\{ {{s_n}\left( A \right) \leq \varepsilon {n^{ - 1/2}}} \right\} \leq L'\varepsilon + {u^n}$$

for all ε > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.

     

Regularities of the formation of iron(III) nanocrystalline particles during the oxidation of iron(II) compounds in a neutral medium

The regularities of the formation of iron(III) oxide hydroxides as nanocrystalline particles via oxidation of iron(II) compounds in a near-neutral pH region were studied by potentiometric titration, electron microscopy, chemical analysis, and X-ray diffraction. The oxidation process comprises two steps. The first step produces Fe(II)-Fe(III) hydroxo salts having a “green rust” structure in the form of nanocrystalline particles shaped as hexagons. The second step produces anisotropic nanocrystalline particles of iron(III) oxide hydroxides via the dissolution-oxidation-precipitation mechanism and via solid-phase oxidation. The oxidation of chlorine-containing suspensions helps the formation of single-phase nanocrystalline lepidocrocite, while oxidation in the presence of sulfate ions yields nanocrystalline goethite.     

A new method for pyrrole peri-annulation: synthesis of 1H-1,5,7-triazacyclopenta[c,d]phenalenes from 1H-perimidines

New methods for the synthesis of 1H-1,5,7-triazacyclopenta[c,d]phenalenes have been developed based on a sequence involving Schmidt reaction of keto-perimidines and acylation of the intermediate amides with 1,3,5-triazines or carboxylic acids. The same synthetic sequence starting from the corresponding alkylperimidine includes acylation with acetic acid as the first step.     

Key structures in the synthesis of steroid antitumor agents. Methods for upbuilding the 17β-pregnane side chain of 3-methoxy-19-norpregna-1,3,5(10)-trien-20-ones with and without an additional 16α,17α-carbocycle

Russian Chemical Bulletin - Methods for extending the side chain of 20-oxopregnanes were summarized. The possibilities of alkylation of 3-methoxy-19-norpregna-1,3,5(10)-trien-20-one and its...     

Tetra-<Emphasis Type="Italic">O</Emphasis>-acetyl-tetra-<Emphasis Type="Italic">O</Emphasis>-phosphoryl-tetra-<Emphasis Type="Italic">C</Emphasis>-naphthyl-resorcin[4]arene. Synthesis and receptor properties toward organic amines

Directed modification of rctt-tetra-C-naphthyl resorcin[4]arene afforded a heterofunctionalized derivative containing specifically oriented phosphoryl and acetyl groups, and its ability to form complexes with organic amines was studied. The structure and stereochemistry of the resulting complexes depended on the amine nature.     

The features of the synthesis and chemical behavior of some β-cyclodextrin silyl derivatives

Conditions are found for the regioselective silylation of the β-cyclodextrin primary hydroxy groups by diphenylmethylsilyl chloride and trimethylsilyl chloride. It is shown that the position of silyl substituents at the primary or secondary hydroxyl can be determined using ²⁹Si NMR spectroscopy. In the case of acetic and phosphorous acid chlorides, the subsequent functionalization of the secondary hydroxyls occurs with a significant removal of the protective silyl groups.     

Synthesis of pyrrolo[2,3-c]2,7-naphthyridine derivatives by cascade heterocyclization reaction of 2-amino-4-cyanomethyl-6-dialkylamino-3,5-pyridinedicarbonitriles

Alkylation of the title pyridinedicarbonitriles with N-substituted chloroacetamides was found to give 5,6-diamino-8-dialkylamino-2,3-dihydro-2-oxo-1H-pyrrolo[2,3-c]2,7-naphthyridine-9-carbonitriles. The structure of obtained compounds was unambiguously confirmed by X-ray crystallographic study. The heterocyclization reaction proceeded regioselectively involving 3-CN group of the starting pyridines without participation of 5-CN. The reasons of the selectivity were discussed. An interaction of prepared naphthyridine derivatives with acetic acid anhydride and cyclohexanone yielded 2-dialkylamino-6,8,9,10-tetrahydro-5-methyl-9-oxopyrimido[4,5,6-ij]pyrrolo[2,3-c]2,7-naphthyridine-1-carbonitriles and 2-dialkylamino-4,5,6,8,9,10-hexahydro-9-oxospiro{pyrimido[4,5,6-ij]pyrrolo[2,3-c]2,7-naphthyridine-5,1′-cyclohexane}-1-carbonitriles, respectively. All fused 2,7-naphthyridines obtained were derivatives of novel heterocyclic systems.     

Interaction of 1,2,5-chalcogenadiazole derivatives with thiophenolate: hypercoordination with formation of interchalcogen bond versus reduction to radical anion

According to the DFT calculations, [1,2,5]thiadiazolo[3,4-c][1,2,5]thiadiazole (4), [1,2,5]selenadiazolo[3,4-c][1,2,5]thiadiazole (5), 3,4-dicyano-1,2,5-thiadiazole (6), and 3,4-dicyano-1,2,5-selenadiazole (7) have nearly the same positive electron affinity (EA). Under the CV conditions they readily produce long-lived π-delocalized radical anions (π-RAs) characterized by EPR. Whereas 4 and 5 were chemically reduced into the π-RAs with thiophenolate (PhS(-)), 6 did not react and 7 formed a product of hypercoordination at the Se center (9) isolated in the form of the thermally stable salt [K(18-crown-6)][9] (10). The latter type of reactivity has never been observed previously for any 1,2,5-chalcogenadiazole derivatives. The X-ray structure of salt 10 revealed that the Se-S distance in the anion 9 (2.722 ?) is ca. 0.5 ? longer than the sum of the covalent radii of these atoms but ca. 1 ? shorter than the sum of their van der Waals radii. According to the QTAIM and NBO analysis, the Se-S bond in 9 can be considered a donor-acceptor bond whose formation leads to transfer of ca. 40% of negative charge from PhS(-) onto the heterocycle. For various PhS(-)/1,2,5-chalcogenadiazole reaction systems, thermodynamics and kinetics were theoretically studied to rationalize the interchalcogen hypercoordination vs reduction to π-RA dichotomy. It is predicted that interaction between PhS(-) and 3,4-dicyano-1,2,5-telluradiazole (12), whose EA slightly exceeds that of 6 and 7, will lead to hypercoordinate anion (17) with the interchalcogen Te-S bond being stronger than the Se-S bond observed in anion 9.     

New molecular markers for prostate tumor imaging: a study on 2-methylene substituted fatty acids as new AMACR inhibitors

The development of prostate carcinoma is associated with alterations in fatty acid metabolism. α‐Methylacyl‐CoA racemase (AMACR) is a peroxisomal and mitochondrial enzyme that catalyses interconversion between the (S)/(R)‐isomers of a range of α‐methylacyl‐CoA thioesters. AMACR is involved in the β‐oxidation of the dietary branched‐chain fatty acids and bile acid intermediates. It is highly expressed in prostate (more than 95 %), colon (92 %), and breast cancers (44 %) but not in the respective normal or hyperplastic tissues. Thus, targeting of AMACR could be a new strategy for molecular imaging and therapy of prostate and some other cancers. Unlabeled 2‐methylenacyl‐CoA thioesters ( 12 a – c ) were designed as AMACR binding ligands. The thioesters were tested for their ability to inhibit the AMACR‐mediated epimerization of (25R)‐THC‐CoA and were found to be strong AMACR inhibitors. Radioiodinated (E)‐¹³¹I‐13‐iodo‐2‐methylentridec‐12‐enoic acid (¹³¹I‐ 7 c ) demonstrated preferential retention in AMACR‐positive prostate tumor cells (LNCaP, LNCaP C4‐2wt and DU145) compared with both AMACR‐knockout LNCaP C4‐2 AMACR‐siRNA and benign BPH1 prostate cell lines. A significant protein‐bound radioactive fraction with main bands at 47 (sum of molecular weights of AMACR plus 12 c ), 70, and 75 kDa was detected in LNCaP C4‐2 wt cells. In contrast, only negligible amounts of protein‐bound radioactivity were found in LNCaP C4‐2 AMACR‐siRNA cells.