Anthony Wickstead

Nebular and Auroral Emission Lines of [Ar IV] in the Optical Spectra of Planetary Nebulae

Recent R-matrix calculations of electron impact excitation rates in Ar IV are used to calculate the emission-line ratio: ratio diagrams (R1, R2), (R1, R3), and (R1, R4), where R1 = I(4711 ?)/I(4740 ?), R2 = I(7238 ?)/I(4711 + 4740 ?), R3 = I(7263 ?)/I(4711 + 4740 ?), and R4 = I(7171 ?)/I(4711 + 4740 ?), for a range of electron temperatures (Te = 5000-20,000 K) and electron densities (Ne = 10-106 cm-3) appropriate to gaseous nebulae. These diagrams should, in principle, allow the simultaneous determination of Te and Ne from measurements of the [Ar IV] lines in a spectrum. Plasma parameters deduced for a sample of planetary nebulae from (R1, R3) and (R1, R4), using observational date obtained with the Hamilton echelle spectrograph on the 3 m Shane Telescope at the Lick Observatory, are found to show excellent internal consistency and to be in generally good agreement with the values of Te and Ne estimated from other line ratios in the echelle spectra. These results provide observational support for the accuracy of the theoretical ratios and, hence, the atomic data adopted in their derivation. In addition, they imply that the 7171 ? line is not as seriously affected by telluric absorption as previously thought. However, the observed values of R2 are mostly larger than the theoretical high-temperature and density limit, which is due to blending of the Ar IV 7237.54 ? line with the strong C II transition at 7236 ?.

An approximate formula for the relaxation time of a single domain ferromagnetic particle with uniaxial anisotropy and collinear field

Recently there have been renewed efforts to solve the Fokker–Planck equation resulting from Brown’s model [Phys. Rev. 130, 1677 (1963)] of single domain ferromagnetic particle relaxation for the case of uniaxial anisotropy and zero external field. In particular, the usefulness of a simple analytic formula for the relaxation time based on the reciprocal of the lowest non‐zero eigenvalue resulting from this model has been stressed. Here we suggest an improved analytic formula which we extend to the case of applied collinear field and compare it with the exact numerical solution and previous analytic expressions. The results are presented in terms of the pre‐factor dependence which is commonly taken as constant. The formula exhibits good agreement with the exact results throughout the full range of anisotropy and reduced external field values.

Simple approximate formulae for the magnetic relaxation time of single domain ferromagnetic particles with uniaxial anisotropy

Abstract Approximate simple formulae valid for all barrier heights for the longitudinal relaxation time of a single domain ferromagnetic particle with uniaxial anisotropy are compared with the exact solution. It is concluded that a formula based on the application of a variational principle to the Sturn-Liouville equation associated with Browns Fokker-Planck equation yields the closest approximation to the exact solution.

When each continuous operator is regular II

The following theorem is essentially due to L.~Kantorovich and B. Vulikh and it describes one of the most important classes of Banach lattices between which each continuous operator is regular. {\bf Theorem 1.1.} {\sl Let

Positive Compact Operators on Banach Lattices: Some Loose Ends

E

Some Applications of Rademacher Sequences in Banach Lattices

be an arbitrary L-space and

The order boundedness of band preserving operators on uniformly complete vector lattices

F

Relative weak compactness of solid hulls in Banach lattices

be an arbitrary Banach lattice with Levi norm. Then

Weak and unbounded order convergence in Banach lattices

{\cal L}(E,F)={\cal L}^r(E,F),\ (\star)

[N II] AND [O III] MEAN ELECTRON TEMPERATURES IN PLANETARY NEBULAE

that is, every continuous operator from