Quasi-exactly solvable quartic: elementary integrals and asymptotics

We study elementary eigenfunctions y=p exp(h) of operators L(y)=y"+Py, where p, h and P are polynomials in one variable. For the case when h is an odd cubic polynomial, we found an interesting identity which is used to describe the spectral locus. We also establish some asymptotic properties of the QES spectral locus.Comment: 20 pages, 1 figure. Added Introduction and several references, corrected misprint

Permutable entire functions and multiply connected wandering domains

Let f and g be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of f and g are equal; in particular, we show that J(f)=J(g) provided that neither f nor g has a simply connected wandering domain in the fast escaping set

Connectedness properties of the set where the iterates of an entire function are unbounded

We investigate the connectedness properties of the set I+(f) of points where the iterates of an entire function f are unbounded. In particular, we show that I+(f) is connected whenever iterates of the minimum modulus of f tend to ∞. For a general transcendental entire function f, we show that I+(f)∪ \{\infty\} is always connected and that, if I+(f) is disconnected, then it has uncountably many components, infinitely many of which are unbounded

Magnetoelastic Coupling and Possibility of Spintronic Electromagnetomechanical Effects

Nanoelectromangetomechanical systems (NEMMS) open up a new path for the development of high speed autonomous nanoresonators and signal generators that could be used as actuators, for information processing, as elements of quantum computers etc. Those NEMMS that include ferromagnetic layers could be controlled by the electric current due to effects related with spin transfer. In the present paper we discuss another situation when the current-controlled behaviour of nanorod that includes an antiferro- (instead of one of ferro-) magnetic layer. We argue that in this case ac spin-polarized current can also induce resonant coupled magneto-mechanical oscillations and produce an oscillating magnetization of antiferromagnetic (AFM) layer. These effects are caused by \emph{i}) spin-transfer torque exerted to AFM at the interface with nonmagnetic spacer and by \emph{ii}) the effective magnetic field produced by the spin-polarized free electrons due to $sd$-exchange.The described nanorod with an AFM layer can find an application in magnetometry and as a current-controlled high-frequency mechanical oscillator.Comment: 8 pages, 2 figures, submitted to Low Temp. Physic

Rigidity of escaping dynamics for transcendental entire functions

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that this conjugacy is essentially unique. In particular, we show that an Eremenko-Lyubich class function f has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f and g which belong to the same parameter space are conjugate on their sets of escaping points.Comment: 28 pages; 2 figures. Final version (October 2008). Various modificiations were made, including the introduction of Proposition 3.6, which was not formally stated previously, and the inclusion of a new figure. No major changes otherwis