Fake molecular-orbital calculations

The FAKE method of approximate molecular-orbital calculations is presented and illustrated by application to a number of molecules. The method is of the extended Huckel type but uses accurately computed kinetic-energy matrix elements and avoids scale factors of the Wolfsberg—Helmholtz type. It also includes neighbor-atom charge effects and single-center off-diagonal matrix elements. These features permit FAKE occupied-orbital energies and charge distributions to come into close agreement with corresponding ab initio quantities.     

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.     

Quantitative TOF-SIMS analysis of oligomeric degradation products at the surface of biodegradable poly(alpha-hydroxy acid)s

This paper reports the development of a new method for quantification of the hydrolytic surface degradation kinetics of biodegradable poly(alpha-hydroxy acid)s using time-of-flight secondary ion mass spectrometry (TOF-SIMS). We report results from static SIMS spectra of a series of poly(alpha-hydroxy acid)s including poly(glycolic acid), poly(L-lactic acid), and random poly(D,L-lactic acid-co-glycolic acid) hydrolyzed in various buffer systems. The distribution of the most intense peak intensities of ions generated in high mass range of the spectrum reflects the intact degradation products (oligomeric hydrolysis products) of each biodegradable polymer. First, a detailed analysis of the oligomeric ions is given based on rearrangement of the intact hydrolysis products. The pattern of ions can distinguish both degradation-generated intact oligomers and their fragment ion peaks with a variety of combinations of each repeat unit. Then, the integration and summation of the area of all ion peaks with the same number of repeat units is proposed as a measurement that provides a more accurate MW average than the typically used method which counts only the most intense peak. The multiple ion summation method described in this paper would be practical in the improvement of quantitative TOF-SIMS studies as a better data reduction method, especially in the surface degradation kinetics of biodegradable polymers.     

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.     

Edge spreading lithography and its application to the fabrication of mesoscopic gold and silver rings

A new form of edge lithography, edge spreading lithography (ESL), has been demonstrated and applied to the formation of coinage metal rings. In this process, alkanethiols are delivered from a flat PDMS stamp to the surface of a metal film through a two-dimensional array of spherical silica colloids. The thiols further spread on the metal surface, forming highly ordered SAMs in the form of a ring pattern. Following lift-off of beads, the pattern in the SAMs can be transferred into the metal film through wet chemical etching, with SAMs serving as the resist. The dimensions of the rings can be readily controlled by several parameters such as the beads diameter, the concentration of the thiol solution, and the contact time between the stamp and the silica beads.     

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.