Showing papers on "Auxiliary function published in 1979"

Linearized analysis of inhomogeneous plasma equilibria: General theory

Abstract: A generalized framework is presented for analyzing the linearized equations for perturbations of inhomogeneous plasma equilibria in which there is a collisionless species, some properties of the solutions of the linearized equations are described, and a basis is provided for numerical computations of the linearized properties of such equilibria. It is useful to expland the perturbation potentials in eigenfunctions of the field operator which appears in the linearized equations, and to define a dispersion matrix whose analytical properties determine the nature of the solutions of the initial‐value problem. It is also useful to introduce auxiliary functions to replace the usual perturbation distribution functions, and to expand the auxiliary functions in eigenfunctions of the equilibrium Liouville operators. By introducing the auxiliary functions, great freedom is achieved in the choice of the field operator which appears in the linearized equations. This freedom can be useful in some problems to define expansion functions for the potentials that are particularly suitable for studying specific normal modes.

A new method for the investigation of arithmetical properties of analytic functions

Abstract: The general scheme of the proof uses the action of the Galois group and a random walk. We consider auxiliary functions of the form Fj(z) = P(z + Xi; f(z)) for algebraic xi such that each number conjugate to Fk>)(w) for w e S has the same form Fk)(w). Then we evaluate the order ui of the zero of Fj(z) at z = w and show that uj satisfies a system of inequalities (3.14) (or (W)). This system of inequalities describes a random walk in a multidimensional cube (discrete Markov chain), and using generating functions, we show that IS I ? p. Section 1 consists of a number of auxiliary lemmas (cf. [3]-[5]). In Section 2 we construct the numbers Xj using the group ring Z[G] and in Section 3 we investigate the auxiliary functions Fj(z) and establish connections between the u4. Finally, in Section 4 it is proved that the random

A Natural Structure Theorem for Complex Fields

Abstract: This paper presents a theorem which describes the structure of algebraic relationships which must hold when a certain set of transcendental functions are algebraically dependent. The functions in the set may be logarithmic, exponential, trigonometric, hyperbolic, or indefinite integrals. This structure theorem has important applications to symbolic mathematical computation. A procedure for finding regular representations for classes of transcendental expressions based upon the structure theorem is discussed. By use of this representation procedure, it is possible to solve the equivalence problem for expressions in the class being considered.

Analysis of the Light Changes in the Frequency-Domain: Spherical Stars

Abstract: In the preceding chapter of this book we gave an outline of the iterative solutions of the light curves of eclipsing variables in the time-domain for the elements of the eclipses of the two stars. In retrospect, we note that while the problem at issue does not admit of any closed (or even algebraic) solution on account of its non-linearity, it can be solved by successive approximations (the convergence of which will be discussed more fully in Chapter VII). But whichever method we employed in our efforts to decipher the light changes of eclipsing variables in the time-domain so far, we found one auxiliary function consistently in our way: namely, the ‘geometrical depth’ p(k, α) of the eclipse as defined by Equation (0.6). It is this function — rather than the fractional loss of light α(k, p) — which played directly so fundamental a role in all methods of interpretation of the light changes caused by eclipses in the time-domain. Unfortunately, a transcendental character of its dependence on k and α has made it impossible to construct a mathematical solution of the underlying problem otherwise than by successive approximations; and no one succeeded so far to express this function analytically in an algebraic or differential form — no one has, indeed, seen it in any form other than of numerical tables!