Non-ergodic maps in the tangent family

AbstractWe consider maps in the tangent family for which the asymptotic values are eventually mapped onto poles. For such functions the Julia set J(f) = C. We prove that for almost all z ∈ J(f) the limit set w(z) is the post-singular set and f is non-ergodic on J(f). We also prove that for such f does not exist a f-invariant measure absolutely continuous with respect to the Lebesgue measure finite on compact subsets of C

Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Tsuji related to the Denjoy-Carleman-Ahlfors theorem.Comment: 17 page

Experimental Modeling to Facilitate Removal of Inclusion Contaminants from the Contact Surface in Transversal Extrusion Processes

The paper presents experimental optimization of geometrical conditions regarding the shape of billet’s end pieces in the extrusion process of aluminum alloys. The experiments were performed using appropriately pre-treated plasticine as a substitute material. A special test stand was built using a Plexiglass die in order to make it possible to conduct observations and measurements of the modeled contamination surface. To find the best solution, a great number of shape variants of the billets and of their heating temperatures were experimentally analyzed. An attempt was made to find such a solution that could not only resolve the issue of the removal of contaminants from the contact surface of the pre- heated billets but also develop the least complicated technology of billet fabrication. The test results show how to develop the design process and production technology and also offer advice how to improve the quality of the produced profiles. Finally the authors recommend such a shape for the billets that will ensure effective removal of contaminants from the contact surfaces

Rapid Prototyping Technology Using for Casting Process of Mini Turbine Runner

The previous mini-hydrogenator rotors’ prototypes were made of ABS thermoplastic material with use of fused deposition modelling (FDM). However, there is a need for such rotors made of metallic materials with higher strength. In paper three dimensional printing (3DP) as rapid prototyping technology, was presented, to execute a shell mould of rotor prototype. In initial phase, the moulds were prepared which allow to cast a simple geometric solid with different alloys, for comparative reasons. The form of rotor and testing cast moulds were designed using universal CAD environment and printed composite using material zp151 (Calcium sulfate hemihydrate) with a binder zb63 (2- pyrrolidone) for preparation of colourful presentation models or casting models. The suitability of technology usage with alloys of copper , aluminium and zinc, was analyzed. The strength of the mould during casting, a material resistant to high temperature, the quality of received casting and its surface roughness were assumed as the comparative parameters which characterized method