Volume 27, Issue 1

After having explained the underlying motivations, we study the location of the zeros of the functions T ( z ) := Ae az +Be bz +Ce cz of the complex variable z with complex coefficients A , B , C and real a < b < c . As normal form of T ( z )=0 serves the equation e -pz/2 · sinh ( z /2)= P with a complex parameter P and a real p ∈ (-1,1). The problem of finding all solutions z of this equation is reduced to the calculation of the unique solution in a horizontal fundamental strip F := { z ∈ C: - π < Im( z ) ≤ π }. By detailed estimations, we find tight enclosures for the zero in F . Series expansions and algorithms to find the zero z in F are propounded. A complete stability analysis for real trinomials is given. In a discussion, the problem is set into a wider perspective.

The paper is related to the question of stability for the motionless spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. It considers a perturbation of the linearization of the governing equations of this problem, taking a step in the derivation of estimates which will allow us to prove non-linear stability of the equilibria. The perturbed operator, like the linearization considered earlier, generates an analytic semigroup, which allows us to derive asymptotic estimates as t → ∞.

In this paper, we investigate the first order neutral differential equation of Euler form with variable unbounded delay where 0 ≤ c < 1, 0 < α < 1, 0 < β i < 1 and p i > 0 are constants, i =1,2,…, n . Some necessary and sufficient as well as explicit sufficient conditions for the oscillation of all solutions are established.

The purpose of this paper is twofold. Firstly we carry out a modification of the finite volume WENO (weighted essentially non-oscillatory) scheme of Titarev and Toro [14] and [15].This modification is done by using two fluxes as building blocks in spatially fifth order WENO schemes instead of the second order TVD flux proposed by Titarev and Toro [14] and [15]. These fluxes are the second order TVD flux [19] and the third order TVD flux [20]. Secondly, we propose to use these fluxes as a building block in spatially seventh order WENO schemes. The numerical solution is advanced in time by the third order TVD Runge–Kutta method. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws, in one and two dimension is presented. Systematic assessment of the proposed schemes shows substantial gains in accuracy and better resolution of discontinuities, particularly for problems involving long time evolution containing both smooth and non-smooth features.

We present explicit expressions for the Mellin transforms of Laguerre and Hermite functions in terms of a variety of special functions. We show that many of the properties of the resulting functions, including functional equations and reciprocity laws, are direct consequences of transformation formulae of hypergeometric functions. Interest in these results is reinforced by the fact that polynomial or other factors of the Mellin transforms have zeros only on the critical line Re s = 1/2. We additionally present a simple-zero Proposition for the Mellin transform of the wavefunction of the D -dimensional hydrogenic atom. These results are of interest to several areas including quantum mechanics and analytic number theory.

The main theme of this paper is a thorough study of a parametrized family (Ω w ) w ∈D of solutions of a nondegenerate matricial Carathéodory problem. If w =0 the function Ω w coincides with the so-called central solution which has several extremal properties amongst the set of all solutions. This observation inspired us to look for corresponding extremal properties of the functions Ω w in the case of an arbitrary w ∈D. It turns out that the functions Ω w are extremal in several directions (semi-radii of Weyl matrix balls, extremal entropy etc.). Our methods are based on a combination of the theory of orthogonal matrix polynomials with respect to a nonnegative Hermitian matrix-valued measure on the unit circle developed by Delsarte, Genin, and Kamp with the authors´ former investigations on the matricial Carathéodory problem.

In this paper we consider fully nonlinear elliptic equations of the form including the Monge–Ampère, the Hessian and the Weingarten equations and give conditions which ensure that a singular set E of a solution u is removable. This is shown by proving suitable maximum principles along the lines of the Aleksandrov–Bakel´man maximum principle.