Approximating the independence number via theϑ-function

We describe an approximation algorithm for the independence number of a graph. If a graph onn vertices has an independence numbern/k + m for some fixed integerk 3 and somem > 0, the algorithm finds, in random polynomial time, an independent set of size , improving the best known previous algorithm of Boppana and Halldorsson that finds an independent set of size (m ^1/(k–1)) in such a graph. The algorithm is based on semi-definite programming, some properties of the Lovász-function of a graph and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If the-function of ann vertex graph is at leastMn ^1–2/k for some absolute constantM, we describe another, related, efficient algorithm that finds an independent set of sizek. Several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between the-function and the independence number of a graph onn vertices. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research supported in part by a USA—Israel BSF grant and by the Fund for Basic Research administered by the Israel Academy of Sciences and by the Minkowski Minerva Center for Geometry at Tel Aviv University.This work was partly done while the author was at XEROX PARC and partly at DIMACS.     

A new strategy for the synthesis of pyrazolo[4,3-e][1,2,4]triazolo[4,3-c]pyrimidines and pyrazolo[4,3-e][1,2,4]triazolo[1,5-c]pyrimidines

A series of pyrazolo[4,3-e][1,2,4]triazolo[4,3-c]pyrimidines were prepared via oxidative cyclization of aldehyde N-(1,3-diphenylpyrazolo[3,4-d]pyrimidin-4-yl)hydrazones. Dimroth rearrangement of such a series yielded pyrazolo[4,3-e][1,2,4]triazolo[1,5-c]pyrimidines.     

Inhibitory Effects of Tunisian Marine Algal Extracts on Digestive Lipases

The lipase inhibitory activity of ethanol extracts obtained from some marine algae collected on the Tunisian coast was evaluated. Caulerpa prolifera extract markedly reduced both dog gastric (DGL) and human pancreatic lipase (HPL) activities. Generally, the inhibition reached 100% after 40 to 60 min of incubation depending on lipase types and on substrates used. Moreover, the inhibitory effect of C. prolifera extract on lipases appeared to be accelerated by adding bile salts, which likely modified the interface and allowed the inhibitory compound to inactivate the lipase. The separation of C. prolifera extract by thin-layer chromatography (TLC) resulted in eight fractions showing efficient inhibition rate against DGL, compared to the crude extract. In the case of HPL, TLC fractionation reduced the inhibitory rates, suggesting that the effect of algal extract on lipases may be caused by a synergetic action of several compounds within the extract. High-performance liquid chromatograph separation resulted in isolation of a major compound displaying high inhibition capacity of HPL activity. Caulerpa prolifera extract may therefore be useful in developing antiobesity drugs.     

Front Cover: Katritzky Salts for the Synthesis of Unnatural Amino Acids and Late-Stage Functionalization of Peptides (Eur. J. Org. Chem. 12/2023)

Peptide drug discovery often benefits from the large structural diversity permitted by unnatural amino acids (UAAs). Indeed, numerous approved peptide drugs include UAAs in their sequences. Therefore, innovative chemical approaches either to synthesize UAAs or to allow late-stage functionalization of peptides are emerging themes in peptide drug discovery. Thanks to the recent advances in deaminative strategies using alkylpyridiniums salts, often referred to as Katritzky salts, a variety of radical alkylation methods have been developed. In recent years the use of Katritzky salts have become popular in peptide chemistry due to their ease of preparation from a primary amine, which is a predominant functional group in amino acids. This review highlights the progress that has been made by using Katritzky salts in the synthesis of UAAs, late-stage peptide functionalization, and peptide macrocyclization.     

Vehicle routing problems with multiple trips

This paper presents a survey on the multi-trip vehicle routing problem (MTVRP) and on related routing problems where vehicles are allowed to perform multiple trips. The first part of the paper focuses on the MTVRP. It gives an unified view on mathematical formulations and surveys exact and heuristic approaches. The paper continues with variants of the MTVRP and other families of routing problems where multiple trips are sometimes allowed. For the latter, it specially insists on the motivations for having multiple trips and the algorithmic consequences. The expected contribution of the survey is to give a comprehensive overview on a structural property of routing problems that has seen a strongly growing interest in the last few years and that has been investigated in very different areas of the routing literature.     

Structural and Mössbauer studies of nanocrystalline Mn<Superscript>4+</Superscript>-doped Li<Subscript>0.5</Subscript>Fe<Subscript>2.5</Subscript>O<Subscript>4</Subscript> particles prepared by mechanical milling

The structure and magnetic properties of spinel-related Mn⁴⁺-doped Li_0.5Fe_2.5O₄ nanocrystalline particles of the composition Li_0.5Fe_2.25Mn_0.1875O₄, prepared by milling a pristine sample for different times, were investigated. The average crystallite and particle size, respectively, decreased form ～40 nm to ～10 nm and ～2.5 μm to ～10 nm with increasing milling time from 0 h to 70 h. Rietveld refinement of the XRD data of the non-milled sample show the Mn⁴⁺ dopant ions to substitute for Fe³⁺ at the octahedral B-sites of the spinel-related structure. The Mössbauer spectra of the milled ferrites indicate that more particles turn superparamagnetic with increasing milling time. The Mössbauer data collected at 78 K suggest that while in the non-milled sample the Mn⁴⁺ ions substitute for Fe³⁺ at the octahedral B-sites, this is reversed as milling proceeds with doped Mn⁴⁺ ions, balancing Fe³⁺ vacancies and possibly Li⁺ ions progressively migrate to the tetrahedral A-sites. This is supported by the slight increase observed in the magnetization of the milled samples relative to that of the non-milled one. The magnetic data suggest that in addition to the increasing superparamagentic component of the milled particles, thermal spin reversal and/or spin canting effects are possible at the surface layers of the nanoparticles.     

Biphasic Oxidation of cis‐Diaquabis(1,10‐phenanthroline)chromium(III) by N‐Bromosuccinimide and the Formation of Chromium(IV) and Chromium(V)

The oxidation of cis‐diaquabis(1,10‐phenanthroline)chromium(III) [cis‐Cr^III(phen)₂(H₂O)₂]³⁺ by ‐bromosuccinimide (NBS) to yield cis‐dioxobis(1,10‐phenanthroline)chromium(V) has been studied spectrophotometrically in the pH 1.57–3.56 and 5.68–6.68 ranges at 25.0°C. The reaction displayed biphasic kinetics at pH < 4.0 and a simple first order at the pH > 5.0. In the low pH range, the reaction proceeds by two successive steps; the first faster step corresponds to the oxidation of Cr(III) to Cr(IV), and the second slower one corresponds to the oxidation of Cr(IV) to Cr(V), the final product of the reaction. The formation of both Cr(IV) and Cr(V) has been detected by electron spin resonance (ESR). The ESR clearly showed the formation and decay of Cr(IV) as well as the formation of Cr(V). Each oxidation process exhibited a first‐order dependence on the initial [Cr(III)]. The pseudo–first‐order rate constants k₃₄ and k₄₅, for the faster and slower steps, respectively, were obtained by a computer program using Origin7.0. Both rate constants showed first‐order dependence on [NBS] and increased with increasing pH.     

Officinalioside,a new lignan glucoside from Borago officinalis L.

A new lignan glucoside, officinalioside (1), was isolated from n-BuOH fraction of the aerial parts of Borago officinalis L., together with four known compounds: actinidioionoside (2), roseoside (3), crotalionoside C (4) and kaempferol 3-O-β-D-galactopyranoside (5). The structure of the new compound was established by means of spectroscopic and chemical analyses. Compounds 1 and 2 showed a moderate DPPH radical scavenging activity (IC₅₀: 52.6 ± 1.70 and 41.3 ± 0.25 μM, respectively) comparable with that of the standard trolox (16.6 ± 2.2 μM) without any significant cytotoxicity towards human cell line A549 (IC₅₀ > 100 μM).     

An efficient synthesis of enantiomerically pure (R)-pipecolic acid, (S)-proline, and their N-alkylated derivatives

Enantiomerically pure (R)-(+)-pipecolic acid was synthesized in four steps and 42% overall yield starting from dihydropyran and (R)-alpha-methylbenzylamine. A general short strategy is also described for preparing (S)-proline (47.5% overall yield) and derivatives.     

Regioselective C-H Functionalization of the Six-Membered Ring of the 6,5-Fused Heterocyclic Systems: An Overview

The regioselective C-H functionalization of the five-membered ring of the 6,5-fused heterocyclic systems is nowadays well documented due to its high reactivity compared to the six-membered ring. So, developing new procedures of C-H functionalization of the six-membered ring “by thinking out of the box” is extremely challenging, which explains the limited number of reports published to date. This review paper aims to highlight advances achieved in this emerging chemistry research and discusses recently reported methods.