Properties of the conjugate-gradient and Davidon methods

Two quadratically convergent gradient methods for minimizing an unconstrained function of several variables are examined. The heart of the Fletcher and Powell reformulation of Davidon's method is a variableH-matrix. The eigenvalues and eigenvectors of this matrix for a quadratic function are explored, leading to a proof that the gradient vectors at each step are mutually orthogonal. From this, a geometric interpretation of theH-matrix in terms of the projection of the gradient into a solution subspace is derived. These properties are then used to arrive at the main result, which states that, for a quadratic function, the direction vectors generated by the Davidon algorithm and the conjugate-gradient algorithm of Hestenes and Stiefel are scalar multiples of each other, provided the initial step each takes is in the direction of steepest descent. If one assumes no round-off error and a perfect one-dimensional search, the methods generate identical steps leading to the minimum.It is also shown that, for a quadratic function, the Davidon algorithm has a simplified version which searches in the same directions. However, the unique advantage of the original scheme, that it yields the curvature of the function at the minimum, is sacrificed for simplicity.Although these results apply to the case of a quadratic function, a comparative study of the same algorithms for a general function can be found in a companion paper.This research was carried out under Contract No. NAS 9-4036, NASA-Manned Spacecraft Center, Houston, Texas. The author is indebted to Dr. H. J. Kelley, whose suggestions and encouragement provided the inspiration for this paper.     

Plutonium-244 dating VI. Initial ratios of plutonium to uranium in achondrites

Re-examination of all known xenon isotopic data for achondrites reveals that²⁴⁴Pu fission xenon can be resolved in about three-fourths of the meteorites of this class. The amounts of²⁴⁴Pu fission xenon found in these meteorites range from ca. 1–2 up to 20–40·10^–12 ccSTP/g. These meteorites started to retain their xenon some 200–500 million years later than did the carbonaceous chondrites Allende, Groznaya, Mokoia, Murchison, Murray, and Renazzo, which began to retain their xenon over 4800 million years ago.     

A computational study of the chromatographic separation of racemic mixtures

Summary A model is devised using molecular mechanics to simulate chromatographic separations of enantiomers. Theoretical results derived from this model are compared with experimental findings obtained using supercritical fluid chromatography. The model is then developed to incorporate the effects of binding the stationary phase to a matrix. Computed results show that addition of the matrix into the model has significant effects on the ability of the stationary phase to separate racemic mixtures.     

Adjustability as a requirement of ideal clocks

A coordinate frame is considered as an arrangement of clocks that meet certain criteria of synchronization. Einstein's ideal clock is compared with the behavior required for clocks to maintain synchrony in the group of coordinate frames that leaves Maxwell's equations invariant. The required clock rates differ from the rate of Einstein's clock. An ideal adjustable clock is defined as an Einstein clock augmented by variable “gearing” that can offset its rate. Ongoing adjustment of these clocks enables them to meet all synchronization criteria in the group of coordinate frames. The need for adjustment is due to the well known invariance of Maxwell's equations under a group of coordinate transformations larger than the Lorentz group, and has nothing to do with imperfections in clocks. It is shown that the adjustments needed by ideal adjustable clocks to maintain synchrony can be measurably separated from additional adjustments that may be needed to compensate for random imperfections. The necessity for adjustment brings with it the necessity for ongoing measurement of the light signals whose exchange defines synchronization. Implications are discussed, both for the interpretation of Maxwell's equations and for the role of measuring instruments.     

Determination of rhenium in molybdenite by X-ray fluorescence: A combined chemical-spectrometric technique

Rhenium in molybdenite is separated from molybdenum by distillation of rhenium heptoxide from a perchloric-sulphuric acid mixture. It is concentrated by precipitation of the sulphide and then determined by X-ray fluorescence. From 3 to 1000 microg of rhenium can be measured with a precision generally within 2%. The procedure tolerates larger amounts of molybdenum than the usual colorimetric methods.     

An XPS study of the valence bands in solid and liquid bismuth

High resolution XPS measurements on bismuth show the valence band structure of the liquid phase to be very similar to that of the solid and very different from that of free electrons. The spectra are in general accord with recent calculations of the densities of states for crystalline and amorphous group V elements.     

Resonance fluorescence spectrum of nitrogen dioxide

We report an investigation of the resonance fluorescence spectrum of NO₂ excited by several laser lines. Sixty transitions, mostly in ν₂ progressions of the ground state, have been assigned. Analysis of the spectra extend the knowledge of the ground state constants, especially of the anharmonic coefficients. It is possible to establish that transitions are occurring to a ²B₁ state and to a ²B₂ state in the same energy region.     

Some investigations of fusing and dissolving samples

When both Si-H and Si-vinyl are present, the sum of these groups can be determined bromometrically. For Si-H a gasvolumetric method is applied and Si-vinyl results from the difference. The bromometric determination is carried out by dissolving the sample in a mixture of carbon tetrachloride/acetic acid, addition of excess solution of bromine in acetic acid and back-titration of the unconsumed bromine. Mercuric chloride serves as a catalyst for the bromine addition. The Si-H determination is carried out by measuring the hydrogen gas evolved after hydrolysis of the sample in wet n-butanol in the presence of sodium n-butanolate. The bromometric method proved to be also useful for the rapid and sensitive determination of either Si-vinyl or Si-H. The time required is 45 min, the lower limit is 0.02 mmole per g.     

Microwave spectrum of silyl methyl ether

The microwave spectra of silyl methyl ether, SiH₃OCH₃, and its isotopic modifications, SiH₃OCD₃, SiD₃OCH₃, and SiD₃OCD₃, have been observed and assigned. Large splittings arising from the internal rotation of the methyl top and somewhat smaller splittings arising from the internal rotation of the silyl top are observed. The “effective barrier” to internal rotation of the methyl top is approximately 550 cal/mole. The effective barrier to internal rotation of the silyl top is approximately 1100 cal/mole. The internal rotation of the two tops is strongly coupled, but no values for the potential coupling constants have been obtained. The dipole moment has been determined to be 1.15 ± 0.02 D (|μ_a| = 0.647 ± 0.01 and |μ_b| = 0.95 ± 0.02 D) from measurements of the Stark effect.