On Hartogs Compacts of Holomorphy

Abstract

The CALPHAD (calculations of phase diagrams) method is used to examine the effects of applied magnetic fields on the α/γ phase boundary in the Fe-Si system in the paramagnetic state. The reported susceptibility data for pure Fe is first re-evaluated. The contributions to the total Gibbs energy of the ferrite (α) and austenite (γ ) from the external fields are calculated based on the Curie-Weiss law and the re-evaluated susceptibility data. The Fe-Si phase diagram on the Fe-rich side as a function of applied field is calculated using the Thermo-Calc ™ package. With increasing field strength, the γ loop shrinks monotonically; that is, the α/γ -Fe transition temperature increases while that for γ /δ-Fe transition decreases, albeit more slowly. Finally, in conformance with the existing CALPHAD databank, Redlich-Kister polynomials are proposed to account for the compositional and temperature dependence of the contribution to the total Gibbs energy from the applied field in the paramagnetic state in the range over which the Curie-Weiss law is obeyed.

Subharmonic Extensions and Approximations

“…So once again we would have the same solution for the subharmonic problem as for the harmonic problem. The proof in [26] is not quite complete; however, Shirinbekov has subsequently given a complete proof (in the continuous case) [27]. This result is also a corollary of the solution to problem 21 below.…”

“…Walsh) set if each function upper-semicontinuous on E and subharmonic on £° can be extended (resp. approximated) by a function subharmonic on a neighbourhood of E. The approximation version of this problem was considered by Shirinbekov [26] who claimed that a compact set E C R n is a subharmonic Walsh set if and only if it is a harmonic Walsh set. So once again we would have the same solution for the subharmonic problem as for the harmonic problem.…”