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We analyse the angular eigenfunctions - spin-weighted spheroidal harmonics-and eigenvalues of Teukolsky’s equation. This equation describes infinitesimal scalar, electromagnetic and gravitational perturbations of rotating (Kerr) black holes. We derive analytic expressions for the eigenvalues up to sixth order in the expansion parameter for low frequencies and an analogous expansion in the high-frequency limit. Spinweighted spheroidal harmonics form a complete and orthonormal set of functions on a prolate spheroid. They are, however, not the eigenfunctions of the natural Laplace operator on a spheroid and thus do not allow an obvious geometrical interpretation as the corresponding spin-weighted spherical harmonics.

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The isovector fields which generate the invariance group of a class of nonlinear reaction-diffusion equations are used to obtain a "general" similarity solution. The solution is characterized by the largest number of parameters admitted by the isogroup of the equation. For a particular example a specialization of this solution yields a one-parameter group similarity solution previously reported in the literature.

The invariance group of a class of reaction-diffusion equations, where the diffusion coefficient varies as some power of the population density, is studied. The Lie algebra of the invariance group and the isovector fields which generate the invariance group are established, and then the isovectors are used to derive a "general" invariant group similarity solution which admits the largest number of group parameters. It is also shown that some exact particular analytical solutions previously reported in the literature for special cases of the equation here considered can be rederived from the present solution.

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In this paper the streaming term of the one-speed transport equation is expressed as a duality operation of modern tensor analysis. This representation allows one to write the equation easily in any orthogonal coordinate system. An application of the present transformation technique is made to three different geometries of current interest in controlled thermonuclear reactor design, namely cylindrical and toroidal (with both circular and general elliptic cross sections) .