Volume 25, Issue 4 - 4

In this paper we determine the ranges for which the inequalities (1.2) below hold.

As for Poincaré's centre-problem Poincaré [3] himself assumed that if the origin is a centre then there is an integral (constant of motion) being analytic in a neighborhood of the origin. We are going to prove this assumption and vice versa in a concise way by using techniques developed by C. L. Siegel [6].

We estimate the growth of homomorphism numbers of a torsion-free nilpotent group G using a variant of the circle method together with the analytic continuation of ζ G ( s ) established in [4]. As an application, we obtain information on the subgroup growth of free products of nilpotent groups.

In this article we find a new class of uniqueness polynomials for entire curves into complex projective space. Moreover, we also prove that the zero set of every polynomial in the above-mentioned class is a hyperbolic hypersurface in the sense of Kobayashi and has the arithmetic finiteness property.

We study the spectrum of a biharmonic Steklov eigenvalue problem in a bounded domain of R n . We characterize it in general and give its explicit form in the case where the domain is a ball. Then, we focus our attention on the first eigenvalue of this problem. We prove some estimates and study its isoperimetric properties. By recalling a number of known results, we finally highlight the main open problems still to be solved.

This paper is a continuation of our earlier works [7] and [9] on the fractal structure of expanding and subexpanding meromorphic functions of the form F = H ° exp ° Q , where H and Q are non-constant rational maps. Under some assumptions on the forward trajectories of asymptotic values and critical values of F we construct a conformal measure m for the projection f of F onto an infinite cylinder P . Let J r f consist of all points whose trajectory returns infinitely often to some compact set whose intersection with the postsingular set is empty and h = HD( J r f ) be the Hausdorff dimension of J r f . We prove that h ∊ ( s ,2) ( s will be defined in Corollary 3.4) and an h -dimensional Hausdorff measure H h of J r f is finite. We construct a unique Borel probability f -invariant measure μ on J r f . The measure μ is ergodic and equivalent to m . We also show that h -dimensional packing measure of J r f is locally infinite at every point of this set.