Degree theory on oriented infinite dimensional varieties and the Morse number of minimal surfaces spanning a curve in ℝn

The algebraic number of disc minimal surfaces spanning a wire in ₃ is defined and shown to be equal to one.The author wishes to acknowledge the support of the NSF.     

Molecular structure of 1,2,4-triazole

The molecular structure of 1,2,4-triazole has been determined by gas phase electron diffraction. The intemuclear distances and bond angles were obtained by applying a least-squares analysis to the experimental intensity. The bond distances (r_g) and bond angles were N₁-N₂ = 1.380 ± 0.010 Å, N₂C₃ = 1.329 ± 0.009 Å, C₃-N₄ = 1.348 ± 0.009 Å, N₁-C₅ = 1.377 ± 0.004 Å, N₄C₅ = 1.305 Å (calculated value). N-H = 0.990 Å, C-H = 1.054 Å, ∠N₁N₂C₃ = 102.7± 0.5°, ∠N₂C₃N₄ = 113.8 ± 1.3°, ∠N₂N₁C₅ = 108.9 ± 0.8°, ∠H₁N₁N₂ = 110.9°, ∠H₂C₃N₄ = 119.2°, ∠H₃C₅N₁ = 131.0°, ∠C₃N₄C₅ = 105.7° (calculated value) and ∠N₄C₅N₁ = 108.7° (calculated value).     

An additivity principle for Goldie rank

LetA be a noetherian ring. In generalA will not admit a classical Artinian ring of quotients. Yet a problem in enveloping algebras leads one to consider the possible embedding ofA in a prime ringB which is finitely generated as a left and a rightA module. Under certain additional technical assumptions, it is shown that the setS of regular elements ofA is regular inB and is an Ore set in bothA andB withS ⁻¹ A andS ⁻¹ B Artinian. This enables one to establish the following additivity principle for Goldie rank. Let {P ₁,P ₂, …P ₁} be the set of minimal primes ofA. Then under the above conditions it is shown that there exist positive integersz ₁,z ₂, …,z, such that , where rk denotes Goldie rank. This applies to the study of primitive ideals in the enveloping algebra of a complex semisimple Lie algebra. This paper was written while the authors were guests of the Institute for Advanced Studies, The Hebrew University of Jerusalem. The first author was on leave of absence from the Centre Nationale de la Recherche Scientifique, France.     

Stereospecific elimination of acetic acid in 3- and 4-arylcyclohexyl acetates under electron impact

The elimination of CH₃COOH from the molecular ions of trans-3- and 4-arylcyclohexyl acetates takes place to a greater extent than in the cis isomers. Deuterium labelling shows that the elimination involves mainly the benzylic hydrogen in the trans-acetates, but not in the cis isomers. This behaviour is similar qualitatively to that of the corresponding alcohols and methyl ethers, but entirely different from that of t-butylcyclohexyl acetates, which do not exhibit any stereospecificity. Substituent effects on the elimination for both cis and trans isomers are discussed.     

Stereoselective H2O eliminations in 3- and 4-arcylcyclohexanols under electron impact

The elimination of H20 from the molecular ions of trans-4- and trans-3-arylcyclohexanois takes place to a greater extent than in the corresponding cis isomers. The remarkable differences in abundance taken together with substituent effects and the results of deuterium labelling, show that configuration is retained in the molecular ions which undergo the elimination, and that this process is a cis-1,4 and cis-1,3 elimination in the trans-4- and trans-3- arylcyclohexanois, respectively. Possible mechanisms for the elimination in the cis isomers are discussed.     

The synthesis of some 4,4′-disubstituted-2,2′-biquinolines

2,2′-Biquinoline dioxide and 4,4′-dichloro-2,2′-biquinoline have been used for the preparation of the following 4,4′-disubstituted-2,2′-biquinolines: dimethoxy, diethoxy, dihydroxy, dipiperidino, dipyrrolidino, dibromo, diphenoxy, dithiophenoxy, di(p-chloro-phenoxy), di(p-bromophenoxy), di(p-fluorophenoxy), di(β-naphthoxy) and the dinitro dioxide. Molar extinction coefficients have been determined for several of the copper (I) complexes of these compounds.     

Thermally reversible polymer systems by cyclopentadienylation. I. A model for termination by cyclopentadienylation of olefin polymerization

The reaction between tert-butylchloride (t-BuCl) and dimethylcyclopentadienylaluminum (Me₂AlCPD) was studied as a model for initiation by the tert-butyl cation (t-Bu^⊕) and termination by cyclopentadienylation by the Me₂Al(CPD)Cl^? counteranion of isobutylene polymerization. All reaction products formed in this model system have been identified and quantitatively determined. A comprehensive scheme that indicates pathways to these products was developed (scheme III). It is proposed that the predominant product, tert-butylcyclopentadiene (t-BuCPD), arises in the collapse of the t-Bu^⊕-Me₂Al(CPD)Cl^? ion pair, mainly by CPD^? transfer to the tert-butyl cation. The minor products are neopentane (t-BuMe) and isobutylene (i-C₄H₈), which are probably formed, respectively, by Me^? transfer to and proton loss from the t-butyl cation. Cyclopentadienylation selectivity increases by lowering the temperature and extrapolation of results suggests 100% cyclopentadienylation at ?55°C. The t-BuCl/Me₂AlCPD ratio strongly influences the overall reaction pathway. The reaction is first order in t-BuCl with ΔE_a of ca. 7 kcal/mole (1,2-dichloroethane or chlorobenzene solvents, +24 to ?29°C). Indirect evidence indicates that the kinetic product of cyclopentadienylation is 5-t-BuCPD and that this isomer cannot be tert-butylated; that is, the initiation of 5-t-BuCPD polymerization by t-Bu^⊕ is sterically unfavorable. Detailed analysis of the chemistry and kinetics of the t-BuCl/Me₂AlCPD model system holds important clues to the controlled polymerization of olefins leading to macromolecules with cyclopentadiene (CPD) termini.     

Convergence of series of translations

For a mean zero norm one sequence (f _n )L ₂[0, 1], the sequence (f _n {nx+y}) is an orthonormal sequence inL ₂([0, 1]²); so if , then converges for a.e. (x, y)[0, 1]² and has a maximal function inL ₂([0, 1]²). But for a mean zerofL ₂[0, 1], it is harder to give necessary and sufficient conditions for theL ₂-norm convergence or a.e. convergence of . Ifc _n 0 and , then this series will not converge inL ₂-norm on a denseG subset of the mean zero functions inL ₂[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c _n | = 0(n ^–) for >1/2, then converges a.e. and unconditionally inL ₂[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL _p[0, 1] with 1p<. Finally, if (f _n )L [0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x _n )[0, 1]. Moreover, iffL [0, 1] is mean zero and , then for a.e. (x _n )[0, 1], converges for a.e.y and in allL _p [0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.     

Generalized quadrangles in finite projective spaces

If a pointset of the projective spacePG(d,q), together with a lineset ofPG(d,q) form a generalized quadrangleS, thenS is of classical type. This beautiful theorem was proved by F. Buekenhout and C. Lefèvre. In this paper we give a simple proof of this theorem in the cased 4 (we suppose that the result is established ford = 3). We remark that in our proof a central role is played by the theory of subquadrangles.