(2+2)-cycloaddition of triazolinediones to strained bicycloalkenes

Substituted norbornenes and dibenzobarrelene react under forced conditions with $\text{N}$-methyl-1,2,4-triazolin-3,5-diones to afford urazoles via (2+2)-cycloaddition.     

Sugli zeri delle funzioni zeta di Dedekind

Riassunto Scopo di questo lavoro è dare una formula asintotica per il numero degli zeri di ReF _K(λ+it) e di ImF _K(λ+it), dove eζ _K(8) è la funzione zeta di Dedekind associata al campo numericoK, con 0<t<T e λ numero reale fissato tale che 1−1/n<λ<1 doven è il grado diK.

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Summary The aim of this paper is to give an asymptotic formula for the number of zeros of ReF _K(λ+it) and ImF _K(λ+it), where andζ _K(8) is the Dedekind zeta function for a number fieldK, with 0<t<T and λ fixed real number such that 1−1/n<λ<1, wheren is the degree ofK.

     

CHIRAL VARIANTS OF ARYLALKYL SULFUR REAGENTS. (R)-CAMPHOR KETIMINES OF OGIOVANNA AMINOTHIOETHERS AND OXIDES

Abstract

Ketimines la-d derived from ortho-aminosubstituted phenylthioethers were prepared in order to determine the degree of chirality transfer from the chiral auxiliary to the sulfur atom in the formation of the sulfoxide or to the α-carbon atom in the reaction of the anion with alkyl halides or benzaldehyde. Oxidation to the sulfoxide occurred with little or no asymmetric induction. The crystalline benzyl sulfone 4c was deprotonated by alkyllithium or Grignard reagents and reacted with alkyl halides and benzaldehyde, in all cases with little to fair transfer of chirality. The major diastereoisomer from methylation of the anion of 4c with methyl iodide, was isolated, and afforded the enantiomerically pure amine 5 after removal of the chiral auxiliary. An X-ray structure determination of 4d allowed the assignment of the absolute configuration of the asymmetric carbon and revealed that the conformation of the ketimine in the crystal state is not homogeneous.     

Stereoconvergent Generation of a Contrasteric syn‐Bicyclopropylidene (=syn‐Cyclopropylidenecyclopropane) by Stille‐Like Coupling

Stereoisomerically pure endo‐ and exo‐7‐halo‐7‐(trimethylstannyl)benzonorcar‐3‐enes (=endo‐ and exo‐(1‐halo‐1a,2,7,7a‐tetrahydro‐1H‐cyclopropa[b]naphthalen‐1‐yl)trimethylstannane) 4 and 6 were selectively obtained by lithium? tin or magnesium? tin transmetalation in good yields (Scheme 2 and 3). The reaction of these compounds with copper(I) thiophene‐2‐carboxylate (CuTC) produced in both cases the corresponding C_S‐symmetric bicyclopropylidene (=cyclopropylidenecyclopropane) syn‐ 1 , a single diastereoisomer (Schemes 5 and 6). The structure of syn‐ 1 was undoubtedly elucidated by X‐ray single crystal diffraction. The coupling mechanism of the carbenoid cyclopropane is discussed (Scheme 7).     

Freie Minerals?uren neben organischen S?uren

Ohne Zusammenfassung     

Numerical detection of instability regions for delay models with delay-dependent parameters

In this paper we are interested in gaining local stability insights about the interior equilibria of delay models arising in biomathematics. The models share the property that the corresponding characteristic equations involve delay-dependent coefficients. The presence of such dependence requires the use of suitable criteria which usually makes the analytical work harder so that numerical techniques must be used. Most existing methods for studying stability switching of equilibria fail when applied to such a class of delay models. To this aim, an efficient criterion for stability switches was recently introduced in [E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002) 1144–1165] and extended [E. Beretta, Y. Tang, Extension of a geometric stability switch criterion, Funkcial Ekvac 46(3) (2003) 337–361]. We describe how to numerically detect the instability regions of positive equilibria by using such a criterion, considering both discrete and distributed delay models.