Uniformly bounded components of normality

Suppose that $f(z)$ is a transcendental entire function and that the Fatou set $F(f)\neq\emptyset$. Set $$B_1(f):=\sup_{U}\frac{\sup_{z\in U}\log(|z|+3)}{\inf_{w\in U}\log(|w|+3)}$$ and $$B_2(f):=\sup_{U}\frac{\sup_{z\in U}\log\log(|z|+30)}{\inf_{w\in U}\log(|w|+3)},$$ where the supremum $\sup_{U}$ is taken over all components of $F(f)$. If $B_1(f)<\infty$ or $B_2(f)<\infty$, then we say $F(f)$ is strongly uniformly bounded or uniformly bounded respectively. In this article, we will show that, under some conditions, $F(f)$ is (strongly) uniformly bounded.Comment: 17 pages, a revised version, to appear in Mathematical Proceedings Cambridge Philosophical Societ

Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p

Let $F$ be a function field of characteristic $p>0$, \F/F a Galois extension with Gal(\F/F)\simeq \Z_l^d (for some prime $l\neq p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_r$ ($r$ any prime) as $L$ varies through the subextensions of \F via appropriate versions of Mazur's Control Theorem. As a consequence we prove that Sel_E(\F)_r is a cofinitely generated (in some cases cotorsion) \Z_r[[Gal(\F/F)]]-module.Comment: Final version to appear in Annales de l'Institut Fourie

Linear response conductance and magneto-resistance of ferromagnetic single-electron transistors

The current through ferromagnetic single-electron transistors (SET's) is considered. Using path integrals the linear response conductance is formulated as a function of the tunnel conductance vs. quantum conductance and the temperature vs. Coulomb charging energy. The magneto-resistance of ferromagnet-normal metal-ferromagnet (F-N-F) SET's is almost independent of the Coulomb charging energy and is only reduced when the transport dwell time is longer than the spin-flip relaxation time. In all-ferromagnetic (F-F-F) SET's with negligible spin-flip relaxation time the magneto-resistance is calculated analytically at high temperatures and numerically at low temperatures. The F-F-F magneto-resistance is enhanced by higher order tunneling processes at low temperatures in the 'off' state when the induced charges vanishes. In contrast, in the 'on' state near resonance the magneto-resistance ratio is a non-monotonic function of the inverse temperature.Comment: 10 pages, 6 figures. accepted for publication in Phys. Rev.