Some positive cotes numbers for the chebyshev weight function

We show that, for the Chebyshev weight function (1–x ²)^–1/2, the Cotes numbers for the quadrature rule with nodes at the zeros of the ultraspherical polynomialP _n ^/() are nonnegative if and only if –1/2<1.     

Sur la forme canonique des formes extérieures

Sans résumé

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The article is the text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.     

Infrared investigation of aluminum- and chromium-substituted magnetites and of the lacunar spinels resulting from their oxidation

When Fe³⁺ ions are substituted by aluminum or chromium on magnetite octahedral sites, the ir spectrum shows the conversion of an inverse spinel to a normal spinel. Both broad bands of magnetite are gradually replaced by the four characteristic bands of normal spinels II–III. They are also observed for solid solutions, FeCr₂O₄FeAl₂O₄, with, however, a further band at 780 cm^?1 which may be assigned to Al³⁺ ions on tetrahedral sites. Low-temperature (<400°C) oxidation of these compounds whose sizes are less than 2000 Å results in lacunar spinels III–III. The ir spectrum of these solids is characterized by two absorption bands (as for inverse spinels II–III) except for compounds close to pure γFe₂O₃ in which an order of vacancies could be put in evidence.     

Theoretical Study of Weakly Bound Dimers between Hydrogen Fluoride and Some Polar Molecules

Weakly bound linear and bent dimers, FH—X (where X = CO, OC, CNH, NCH, N₂O and ON₂), are investigated using the DFT B3LYP and ab initio MP2 methods with the same basis sets (6–311++G(3df,2pd)). The strengths of the H—C or H—N H‐bonds in dimers FH—CO, FH—CNH, and FH—N₂O are compared with those of the H—O or H—N H‐bonds in dimers FH—OC, FH—NCH, and FH—ON₂. The results obtained for the H‐bond distances, the elongation effect of the HF bond, the red shift of the HF stretching frequency, and the energy difference between the dimer and the charge transfer reveal that the H‐bonds of the first group of dimers are stronger than those of the second. The Gibbs energies calculated for the six dimer formations indicate that the weakly bound dimers are unstable at room temperature (T = 298 K) (FH—X's → FH + X's, ΔG < 0).     

Quantum chemistry study on B32 and B32H2–32

The structures of B32 and B₃₂H^2–₃₂ with I_h symmetry have been investigated by means of ab initio calculations at STO-3G level. The relationship between molecular orbitals of them has been analyzed and their bonding properties have been discussed. Then the possibility of their existence, as well as the similarity and difference between B₃₂ (B₃₂H^2–₃₂) and C₆₀ (C₆₀H₆₀) have been inferred.     

KT-nearfields of rank 2

Summary Nearfields of rank 2 over their kernel which are not Dickson nearfields have recently been constructed by H. Zassenhaus. Here it is shown that KT-nearfields of rank 2 over their kernels and of characteristic 2 are necessarily Dickson nearfields which are coupled to quadratic field extensions and their Dickson group is isomorphic to ₂. Using results of Gröger one obtains all KT-nearfields of rank 2 over the field of rational numbers.     

Quadratically constrained least squares and quadratic problems

Summary We consider the following problem: Compute a vectorx such that Ax–b₂=min, subject to the constraint x₂=. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse.It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx ^T A x–2b ^T x subject to the constraint x₂=.Some numerical examples are given.This work was in part supported by the National Science Foundation under Grant DCR-8412314 and by the National Institute of Standards and Technology under Grant 60NANB9D0908.     

A high power N2 laser using water filled strip lines

A fast Blümlein circuit with a water strip line having an impedance of Z=0.078 is used to excite a TE high power nitrogen laser at =337 nm. A pseudospark switch works as a high power switch. The laser operates without any preionization and delivers an energy of 15 mJ per pulse (7.5 ns FWHM) at 10 kPa nitrogen pressure. The shot to shot reproducibility is better than 5%.     

On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations

Summary This paper is a continuation of our previous work [10] on projection methods. We study first existing higher order projection schemes in the semidiscretized form for the Navier-Stokes equations. One error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent Neumann boundary condition satisfied by the pressure approximations. We then propose a penalty-projection scheme for which we obtain improved error estimates.This work is partially supported by NSF grant MS-8802596.     

On Jensen's functional equation

Summary Letf be a map from a groupG into an abelian groupH satisfyingf(xy) + f(xy ^–1) = 2f(x), f(e) = 0, wherex, y G ande is the identity inG. A set of necessary and sufficient conditions forS(G, H) = Hom(G, H) is given whenG is abelian, whereS(G, H) denotes all the solutions of the functional equation. The case whenG is non-abelian is also discussed.