How Fast can a Subharmonic Function Decay Along a Ray?

Much is known about the connection between the growth and decayof subharmonic functions. The results indicate that there isa general principle: asubharmonic function cannot decay ‘toofast’ relative to its growth.Three theorems are provedwhich, together with work previously published elsewhere, givea fairly complete account of how this principle works out fora subharmonic function having extremal decay along a ray. 1991Mathematics Subject Classification: 30D20, 31A05.     

Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of orderr is applied to a smooth system of ordinary differential equations.Given any integer m 1, the method may be shown to be of orderr + m as an approximation to a certain modified equation. Ifthe method and the system have a particular qualitative propertythen it is important to determine whether the modified equationsinherit this property. In this article, a technique is introducedfor proving that the modified equations inherit qualitativeproperties from the method and the underlying system. The techniqueuses a straightforward contradiction argument applicable toarbitrary one-step methods and does not rely on the detailedstructure of associated power series expansions. Hence the conclusionsapply, but are not restricted, to the case of Runge-Kutte methods.The new approach unifies and extends results of this type thathave been derived by other means: results are presented forintegral preservation, reversibility, inheritance of fixed points.Hamiltonian problems and volume preservation. The techniquealso applies when the system has an integral that the methodpreserves not exactly, but to order greater than r. Finally,a negative result is obtained by considering a gradient systemand gradient numerical method possessing a global property thatis not shared by the associated modified equations.     

Transitivity of Jordan Algebras of Linear Operators: On Two Questions by Grünenfelder, Omladič and Radjavi

Let be a Jordan algebra of linear operators on a vector space over a field of characteristic different from 2. In this short note, we show that (1) if is 2-transitive, then it is dense, and (2) if is n-transitive, n ≥ 1, then a nonzero Jordan ideal of is also n-transitive. These answer two questions posed by Grünenfelder, Omladič and Radjavi. The second author was partially supported by National Science Council, Taiwan, grant #095- 2811-M-006-005.     

Studying parity-nonconservation effect in the conversion transition between the levels of the (5/2)± doublet in the 229Pa nucleus

Effects of weak nucleon interaction are calculated for the ground-state doublet of 5/2^± levels of the strongly deformed nucleus ²²⁹Pa₉₁. A parity-nonconservation effect in the doublet states can be observed in the conversion spectrum for the isomeric transition between the doublet levels. By using a generalized model of the nucleus, the matrix element of the effective one-nucleon weak-interaction potential, which determines the weight of the opposite parity admixture in the doublet components is estimated in the single-particle approximation. The reduced probabilities of the E1 and M1 nuclear transitions between the doublet states are calculated within various models of the deformed nuclear potential. The effect of Coriolis forces on the dipole electric transition in question is considered. The lifetime of the upper doublet state is estimated.     

Specific features of the ionization of atmospheric air resulting from radioactive pollution

Specific features of inhomogeneous ionization resulting from radioactive pollution in the lower atmosphere are investigated. A model of track ionization of air is proposed that allows for such elementary processes as kinetics, molecular diffusion, and drift of ions in an electric field. The applicability limits of the model are determined.     

A counterexample to the Bishop-Phelps Theorem in complex spaces

The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subsetS of a real Banach spaceX is norm dense inX ^*. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.