‘Lone pair’ Electronic Structure,Conformation and Oxidation Behaviour of Diaziridines

Photoelectron spectroscopic investigations indicate that (i) monocyclic diaziridines prefer a trans-conformation of the nitrogen-lone pairs, and (ii) the interaction between these is comparable to that exhibited by open-chain analogues, i.e. alkyl-substituted hydrazines. The chemical experience concerning the particular case of ? NH? NH? →? N?N? oxidation of diaziridines can therefore not be accounted for by a prevailing unfavorable ‘lone pair’ interaction. Thermochemical as well as entropy arguments may serve as an explanation. A rather unsatisfactory agreement between predictions from various theoretical procedures and experimental results for the ionization potentials of ‘lone pair’ MO electrons is noted.     

Krümmungskennzeichnung der Ketten einer allgemeinen Geometrie Σ(ℜ,\mathfrak{L})

Lie theory renders a concept of curvature for the geometries (,) introduced by W. Benz [1]. Among regular curves in the chains of (, ) are characterized by =0.     

An analogue to a conjecture of S. Chowla and H. Walum

It is proved that (for every ε > 0)

$\text{∑}\text{n?T}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{∑n<}\text{T}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{n}{}^{\mathrm{a}}\text{m}{}^{\mathrm{b}}\text{B}{}_{\mathrm{k}}\text{({}\text{T}\text{nm}\text{}) = O(T}{}^{\mathrm{<mtext>(a+b+1)</mtext><mtext>3?</mtext>}}\text{)}$

(where {·} denotes the fractional part and B_k the Bernoulli polynomial of order k) under the suppositions that k ≥ 2 and 2a ? 1 ≥ b ≥ 1. If (1) were true for k = 1, a = b = 0, then Piltz' divisor problem (for n = 3) would be readily solved. This is an analog to a conjecture formulated by S. Chowla and H. Walum in 1963 and settled in the affirmative (under suitable suppositions) quite recently by S. Kanemitsu and R. Sita Rama Chandra Rao.     

Configurationally and conformationally homogeneous cyclicN-aryl sulfimides. II.13C-and1H NMR spectra

The configuration and (in case of mobile ring systems) the preferred conformation in a series of thiane- and ofcis-andtrans-1-thiadecalin-1-N-4-chlorophenyl imides were assigned by means of¹³C- and¹H nmr spectroscopy.¹H nmr criteria known to be valid for determination of the stereochemistry of cyclic sulfoxides may be applied (with limitations) to cyclicN-aryl sulfimides, if both isomers (S–N bond equatorial and axial, respectively) are known. The assignments are easier, and unambiguous for single isomers, by comparison of¹³C nmr chemical shifts of ring carbon atoms of sulfimides and sulfides. The influence of equatorially and axially oriented sulfimide groups on the chemical shifts of neighbouring protons, and on the carbon atoms of the heterocyclic rings are discussed in detail.

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Konfigurativ und konformationell einheitliche cyclische N-Aryl-sulfimide. II.¹³C- und¹H-NMR-Spektroskopie

Zusammenfassung Die Konfiguration und (bei beweglichen Ringsystemen) die bevorzugte Konformation einer Reihe von Thian- und voncis- undtrans-1-Thiadekalin-1-N-4-chlorophenylimiden wurde durch¹³C- und¹H-NMR-Spektroskopie bestimmt. Bekannte¹H-NMR-Kriterien zur Festlegung der Stereochemie cyclischer Sulfoxide sind (mit Einschränkungen) auch bei cyclischenN-Arylsulfimiden anwendbar, wenn beide Isomere (S–N-Bindung äquatorial bzw. axial) bekannt sind. Leichter, und auch bei Vorliegen von nur einem Isomeren eindeutig, gelingt die Zuordnung durch Vergleich der¹³C-NMR-Verschiebungen der Ringkohlenstoffatome von Sulfimiden und Sulfiden. Die Einflüsse äquatorial oder axial orientierter Sulfimidgruppen auf die chemischen Verschiebungen benachbarter Wasserstoffe und der Kohlenstoffe des Heterorings werden diskutiert.

     

Spirocyclische 3-Oxazoline durch 1,3-dipolare Cycloaddition von Benzonitrilio-2-propanid mit 1,4-Chinonen

Spiro 3-Oxazolines from the 1,3-Dipolar Cycloaddition of Benzonitrilio-2-propanide and 1,4-Quinones On irradiation with light of wavelength 290–350 nm, 2,2-dimethyl-3-phenyl-2H-azirine (1b) reacts with 1,4-naphthoquinone to give the 1H-benzo [f]isoindol-4,9-dione (11) (Scheme 3) via cycloaddition of the benzonitrilio-2-propanide (2b) onto the quinone C, C-double bond. With 2-methyl- and 2,3-dimethyl-1,4-naphthoquinone, the nitrile ylide 2b undergoes cycloaddition preferentially onto the C, O-double bond of the quinone, leading to spiro-oxazolines 12 and 14 (Scheme 4). Steric as well as electronic effects can be discussed to explain the observed site selectivity of the cycloaddition. With the 1,4-benzoquinones 15a, 15b, 15d and 15f , nitrile ylide 2b undergoes the 1,3-dipolar cycloaddition exclusively onto the C, O-double bond. The corresponding spiro-oxazolines have been isolated in 17–32% yield. This contrasts with the previously reported results with benzonitrilipo-phenylmethanide (2a) , which undergoes cycloaddition to the C, C-double bond of 1,4-benzoquinones (cf. [1]). This difference in the site selectivity of the 1,3-dipolar cycloaddition can be explained with Houk's concept of LUMO-polarization, that is, the stronger nucleophilic dipol 2b polarizes the LUMO of a α,β-unsaturated carbonyl compound more efficient than the less nucleophilic 2a. This leads to a preference of the cycloaddition to the C, O-double bond in the case of 2b. With 2,3-dimethyl- (15c) and 2,3,5,6-tetramethyl-1,4-benzoquinone (15e) , nitrile ylide 2b undergoes C, O- as well as C, C-cycloaddition (Schemes 7 and 8).     

A convenient synthesis of 4-aminoaryl substituted cyclic imides

An efficient one-step synthesis of 4-nitro-N-aryl substituted glutarimides, succinimides and maleimides in polyphosphoric acid is described together with the subsequent reduction to the corresponding anilines. The scope and limitation of this cyclocondensation are presented.     

Lattice points in a circle and divisors in arithmetic progressions

Let A(x) denote the number of lattice points in the circle u²+v²x and define as the infimum of all reals for which . The objective of this paper is to show that 35/108 which improves upon all previously known results. This estimate is an immediate consequence of a surprisingly easy generalization of KOLESNIK's work on Dirichlet's divisor problem to divisor functions with respect to arithmetic progressions.     

Die Ungleichungen von Vietoris

The inequalities $$P_{k,l} = \frac{1}{{B(l + 1,k)}}\int\limits_0^{l/(k + l)} {x^l (1 - x)^{k - 1} dx = I_{l/(k + l)} (l + 1,k)< \frac{1}{2}}$$ and $$\Phi _{k,l,\mu } = \frac{1}{{B(k,k\mu + 1)}}\int\limits_0^1 {x^{k - 1} (1 - x)^{k\mu } I_x (l + 1,l\mu )dx< \frac{1}{2}}$$ are shown to be valid for any positive real numbersk, l, μ.     

Solution manifolds and submanifolds of parametrized equations and their discretization errors

Summary The paper concerns solution manifolds of nonlinear parameterdependent equations (1)F(u, )=y₀ involving a Fredholm operatorF between (infinite-dimensional) Banach spacesX=Z× andY, and a finitedimensional parameter space . Differntial-geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds produced by numerical path-tracing procedures. Then, for arbitrary (finite) dimension of , estimates of the error between the solution manifold of (1) and its discretizations are developed. These estimates are shown to be applicable to rather general nonlinear boundaryvalue problems for partial differential equations.This work was in part supported by the U.S. Air Force Office of Scientific Research under Grant 80-0176, the National Science Foundation under Grant MCS-78-05299, and the Office of Naval Research under Contract N-00014-80-C-0455