Derivation of Orthogonality Relations in Plasma Theory using Singular Integral Equations

A mathematical technique is developed whereby weighted orthogonalityrelations can be established between the eigenfunctions of thelinear integro-differential Vlasov-Maxwell operators of thekind which occur in plasma physics, so giving to these operatorsthe property of self-adjointness. The technique involves thesolution of a singular integral equation. Previous discussionof the technique has considered the case of positive and zeroindex of this equation. The remaining case of negative indexis dealt with by considering a particular operator.     

Near-best Lp Approximations by Real and Complex Chebyshev Series

The expansion of a real or complex function in a series of Chebyshevpolynomials of the first and second kinds is discussed in thecontext of near-best approximation. The discussion covers realand complex approximation on the real interval [–1, 1]as a special example of the complex elliptical contour , as well as complex approximationon an elliptical domain, an ellipse exterior, and an ellipticalannulus (including special cases in which part of the boundarycollapses into a "crack"). Two distinct types of function spacesare considered, namely appropriately weighted L_p measure spacesand analytic function spaces, and resulting approximations areshown in all cases to be near-best in the L_p norm within a relativedistance asymptotic to (4^-2 log n)^2p-1–1 for all p (1p ), where relates to the order of approximation.     

Effects of non-uniform interfacial tension in small Reynolds number flow past a spherical liquid drop

A singular perturbation solution is given for small Reynolds number flow past a spherical liquid drop. The interfacial tension required to maintain the drop in a spherical shape is calculated. When the interfacial tension gradient exceeds a critical value, a region of reversed flow occurs on the interface at the rear and the interior flow splits into two parts with reversed circulation at the rear. The magnitude of the interior fluid velocity is small, of order the Reynolds number. A thin transition layer attached to the drop at the rear occurs in the exterior flow. The effects could model the stagnant cap which forms as surfactant is added but the results apply however the variability in the interfacial tension might have been induced.     

Near-minimax Interpolation by a Polynomial in z and z-1 on a Circular Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space A(N_p) of functions f analytic in the circular annulusp^–1 < <p and continuous on itsboundaries = p^–1, p. The points ofinterpolation are chosen to be spaced at equal angles aroundthe two boundaries, with arguments on the inner boundary midwaybetween those on the outer boundary. By calculating the Lebesgueconstants numerically, is found to be close to a minimax approximation for all p 1and all degrees n in the range 1 n 15. In the limiting casesp = 1 and, it is proved that is asymptotic to 2^–1 log n. More specifically and , where _nis the Lebesgueconstant of Gronwall for equally spaced interpolation on a circleby a polynomial of degree n. It is also demonstrated that is not in general monotonic in p, and that is not everywhere differentiable in p.