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.     

New Light on the Alchemical Writings of Michael Sendivogius (1566–1636)

The Polish alchemist Michael Sendivogius is best known for the influential Novum lumen chymicum, a work composed of three separate texts. Sendivogius's authorship was questioned in the mid-seventeenth century, and these reservations are still held by some modern historians. On the other hand, other early modern and modern readers not only accepted his authorship of all three texts, but also ascribed as many as eleven texts to him. This paper discusses the key works published under the anagrammatised name of Sendivogius with the aim of resolving the authorship question. Newly discovered evidence makes it possible to trace the circumstances leading to the publication of these works in much greater detail than previously, and to present new arguments affirming Sendivogius's authorship. In the Tractatus de sulphure, Sendivogius promised to write another two treatises, which readers sought to identify and (in some cases) to write themselves. This paper sets out, and rejects, the arguments for including them in the corpus of genuine Sendivogian writings.