William H. Julian

The spin state and homogeneity of Comet Halley's nucleus

Abstract We determine a model for the spin state of Comet Halleys nucleus that simultaneously satisfies imaging data from the Vega and Giotto encounters, and a wide range of ground-based data including observations of CN-jets, CN-shells, C 2 production rates, and photometric variability. The model is that of an excited, axially symmetric, rotator whose shape is taken to be that of a prolate spheroid. The motion is assumed to be unaffected by jet-induced torques. The long-axis of the nucleus is inclined to the total angular momentum vector, M, by 66°.0, and rotates around M with a period, P φ , of 3.69 days. The component of spin around the long axis has a period, P Ψ , of 7.1 days which compounds with P φ to produce a total spin period, P T = 2.84 days. The total spin vector, S, is inclined to M by 21°.4 and freely precesses around the angular momentum vector with a period of 3.69 days. The model has a ratio of maximum to minimum moments of inertia of 2.28, which, when compared to the observed linear dimensions of the nucleus, implies an approximately constant density distribution throughout its interior. When compared to the sense of orbital motion, the spin is direct and M points toward right ascension, declination (1950) = (6°·2, -60°·7). The model is characterized by five, localized, areas on the nucleus which dominate the observed activity. One of these, located near the “waist” of the nucleus, appears to be solely responsible for the initiation of CN-shell boundaries seen propagating through the coma and may also be responsible for photometric activity seen at large (≥5 AU) heliocentric distances on approach to the sun. This behavior may indicate the presence of either a large-scale chemical inhomogeneity in the nucleus or, possibly, an extensive region on the surface with unusual physical structure.

The comet halley nucleus: random jets

Abstract The Comet Halley nucleus is modeled as a homogeneous, triaxial ellipsoid precessing under the torque of nine randomly placed cometographically fixed jets. We employ the ansatz of random jets to elicit the dominant periods in the solar-heating-induced emission spectrum and to test the permanence of the spin state against disruption by the torque from the jets. The precessing nucleus is followed numerically using nonsingular, matrix differential equations. Eight models in the literature are evaluated in relation to the constraints imposed by: (i) the observed long-axis directions at the spacecraft encounters, (ii) the observed ground-based emission periods harmonically related to ≈7.4 days ( M. J. S. Belton 1990 , Icarus 86, 30–51), (iii) the need for a 2-day total spin period in models of the jet morphology, (iv) the limitation on the net rotation (roll) around the long axis between the Vega 2 and Giotto spacecraft encounters ( Smith et al. 1987 , Nature (London) 326, 573–574), (v) the resistance to spin-state change from the torque of the jets, and (vi) the return of the nucleus to the same spatial orientation roughly every 7.4 days. All eight models were constructed to satisfy spacecraft-encounter long-axis directions. The ground-based emission periods disfavor the five models (numbered 1, 2, 6, 7, and 8 below) that were derived from a 2.2-day long-axis precession. The same models fail to repeat their orientation every 7.4 days. The three remaining models (3, 4, and 5 below) proposed by Belton (1990) are based on a 3.7-day long-axis precession and his Assumption SV1G2 that the long axis is reversed at Vega 1 and reflected in the plane of the sky as seen from Giotto. Models 1, 3, and 8 fail the test of a permanent spin state. The extremely inhomogeneous LAM Model 4 satisfies the six constraints. But, the emission spectrum of Model 4 is weighted toward the fifth harmonic of 7.4 days while that of Model 5 is weighted toward the third. Models 1, 2, 5, and 7 require reexamination of the Smith et al (1987 , Nature 326, 573–574) constraint. Models 3 and 5 require reanalysis of the jet morphology with a 3-day spin period.

Precession of triaxial cometary nuclei

Abstract Comet nuclei are modeled as rotating, homogeneous, triaxial ellipsoids that precess during perihelion passage due to a subsolar dust and gas jet induced by solar heating. We develop a phase-averaged system of equations in which the torque is averaged over a rotation about the spin axis. The inertial space precession of the rotation axis is computed numerically using the phase-averaged system. The result is checked against independent calculations using the exact Euler-Poisson system of equations.

Extensions and fixed points of contractive maps in Rn

Abstract This paper, which is written within the framework of Bishops constructive mathematics, deals with the construction of the fixed point ξ of a contractive self-map f of R n, and with the rate at which the sequence (fn(x)) converges to ξ for any x in R n. It also discusses contractive extensions of contractive mappings on compact subsets of R n, and almost uniform contractions of complete metric spaces.

Bounded linear mappings of finite rank

Abstract This paper comprises a constructive investigation of the relationship between compactness, finite rank and located kernel for a bounded linear mapping into a finite-dimensional normed space. The main result is that a bounded linear mapping of a normed space into a finite-dimensional normed space is constructively compact if and only if its kernel is located. Several examples are given which highlight the constructive distinction between a mapping into a finite-dimensional space and a mapping with finite-dimensional range.

Picard’s theorem

This paper deals with the numerical content of Picards Thsorem. Two classically equivalent versions of this theorem are proved which are distinct from a computational point of view. The proofs are elementary, and constructive in the sense of Bishop. A Brouwerian counterexample is given to the original version of the theorem.

The constructive equivalence of covering and inductive dimensions

Abstract The equivalence of the notions of covering and inductive dimension of compact metric spaces is proved in a constructive setting. It is shown that, for each n , Cov X ⩽ n if and only if Ind X ⩽ n . A Brouwerian counterexample shows that Cov X > 0 does not imply Ind X > 0.

Banyan models for the parallel implementation of the fast Fourier transform algorithm

Abstract We give banyan models for the fast Fourier transform (FFT) algorithms for calculating the discrete Fourier transforms (DFT) of: (1) 2 k -point data; (2) -point; (3) two-dimensional ( n × n )-arrays of data and (4) μ-dimensional n μ -arrays of data. We discuss the parallel implementation of each of these FFT algorithms.

Multistability, chains, and cycles in optical multiwave mixing processes

We exhibit the information processing capabilities of the first few terms that arise in the amplitude expansion for resonant scattering in a medium with a delay nonlinearity (generalized volume hologram). We begin by showing how the physics of intensity dependent charge transport near a two-photon resonance gives both delayed quadratic and quartic nonlinearities. After reviewing the utility for matrix associative memories exhibited by the delayed quadratic nonlinearity (the ordinary Gabor hologram), we examine the role of the quartic nonlinearity, which is a fourth rank tensor. The symmetries of this tensor determine the information processing capabilities (via multilinear correlations) of the medium in an optical computing paradigm. We find multiple basins of stability, Jordan strings, and cycles as possible dynamic behaviors for the medium. We indicate how each corresponds to an information processing task: multiple basins to multiassociative memory, Jordan strings and cycles to chain and sequence memory and to group-invariant pattern recognition. We briefly indicate how branching processes may be implemented by the fourth rank mode-coupling tensor.