An inequality relating the regulator and the discriminant of a number field

Let K be a number field of degree d with regulator R_K and absolute discriminant D_K. Let r(K) be the rank of the unit group in K, and let p(K) be the maximum of r(k) as k ranges over proper subfields of K. We prove

$\text{R}{}_{\mathrm{K}}\text{> c}{}_{\mathrm{d}}\text{(}\text{log}\text{γ}{}_{\mathrm{d}}\text{D}{}_{\mathrm{K}}\text{)}{}^{\mathrm{r(K)?\rho (K)}}$

for constants c_d, γ_d > 0 depending only on d.     

The molecular structure of norbornene as determined by electron diffraction and microwave spectroscopy

The molecular structure of norbornene has been investigated in the gas phase by combining electron diffraction data with microwave spectroscopic rotational constants. The interatomic distances (r_g) and bond angles were obtained by applying a least squares program to the refined experimental molecular diffraction intensities. The CC bond length was found to be 1.336 ± 0.002 Å while the

) bond length was 1. 529 ± 0.007 Å. Other bond lengths and angles included (IUPAC numbering system was used for norbornene): C₁-C₆ = 1.550 ± 0.020 Å, C₁-C₇ = 1.566± 0.005 Å, C₅-C₆ = 1.556 ± 0.005 Å, C-H_ave. = 1.103 ± 0.003 Å, ∠C₁C₂C₄ = 95.3°. The dihedral angle between planes C₁C₂C₃C₄ and C₁C₆C₅C₄ is 110.8 ± 1.5° while that between C₁C₂C₃C₄ and C₁C₇C₄ is 122.3°. The moments of inertia calculated from ED structure are in good agreement with microwave spectroscopic values.     

Recent advances in the thin-layer chromatography of pesticides: a review

Advances in the applications of thin-layer chromatography (TLC) and high-performance TLC for the separation, detection, identification, and determination of pesticides, other agrochemicals, and related compounds are reviewed for the period 2000-2002. Analyses are described for a variety of samples, such as food, biological, and environmental samples, and for residues of pesticides of various types, including insecticides, herbicides, and fungicides, belonging to different chemical classes. References are included for residue analysis, hydrophobicity studies, and the use of TLC and thin-layer radiochromatography for studies of pesticide metabolism, degradation, uptake, and related topics.     

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.