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In this paper we consider torus homeomorphisms f homotopic to Dehn twists. We prove that if the vertical rotation set of f is reduced to zero, then there exists a compact connected essential "horizontal" set K, invariant under f . In other words, if we consider the lift f of f to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of f . Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland's conjecture to this setting: If f is area preserving and has a lift f to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under f , or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.

In this article we study central configurations of the (n + 1)-body problem. For the planar (n + 1)-body problem we study central configurations performed by n ≥ 2 bodies with equal masses at the vertices of a regular n-gon and one body with null mass. We also study spatial central configurations considering n bodies with equal masses at the vertices of a regular polyhedron and one body with null mass.

Abstract-In the last years, nanostructures are widely used as dopants in liquid crystals to manipulate either their electrical or optical properties. In this work, we have analyzed the electrical response of a planar cell filled with a mixture of E7 liquid crystal doped with TiO2 nanoparticles. The effect of these dopants on the effective permittivity and conductivity of the cell has been studied in a wide frequency range at different temperatures.Nowadays, devices based on liquid crystal (LC) materials are widely used. Liquid crystal displays (LCDs) dominate the display market and other applications of LC devices (optical filters and switches, microwave and security devices) [1][2][3] have been developed in the last decades. In order to improve the electro-optical behavior of these devices, considerable attention has been paid recently to analyze the influence of incorporating nanoscale materials on an LC host. The modification of the behavior of the LC depends on the nanoparticles (NPs) used as guest dopants. Many research studies on the electro-optical response of LC-NPs mixtures using metallic [4][5], insulating [6][7] or ferroelectric [8] nanoparticles or carbon nanotubes (CNTs) [9] can be found in the literature. These works reveal that LC-NPs mixtures can exhibit enhancement of some characteristics like dielectric anisotropy, birefringence and threshold voltage, among others. In addition, LC has been also shown as an interesting active medium to govern the optical response of guest NPs, e.g. actively shifting the spectral position of scattering resonances [10]. In this work, a study of the frequency-dependent electrical properties of liquid crystal E7 doped with TiO 2 nanoparticles is presented. The temperature dependence of these properties is also analyzed, including the behavior of the mixture when the LC material E7 is in the isotropic phase. Experimental results show that doping the LC material E7 with TiO 2 nanoparticles substantially modifies its dielectric properties. A composite of E7 LC doped with TiO 2 nanoparticles of 250nm (© Sigma-Aldrich, concentration 53-57wt.% in diethylene glycol monobutyl ether/ethylene glycol) was * jctzafra@ing.uc3m.es obtained using the following procedure. Nanoparticles were mixed in pure liquid crystal with a 1:2 proportion, although only a small part of TiO 2 NPs are finally mixed. After that, the mixture was stirred for 1 min till the solution was homogeneously mixed. Thereafter, the solution was left for 6 hours at a temperature of 80°C in order to evaporate completely the organic solvent. LC cells were filled with the final mixture by capillary action, with the LC in the isotropic phase. To obtain a uniform distribution of the mixture inside the cell, the sample was cooled slowly. LC cells were composed of two glass plates, 5μm apart. Both plates were covered with transparent conductive films of indium tin oxide (ITO) as electrodes. A thin layer of polyimide SE-130 with parallel rubbing was used as an alignment layer for the LC in both substrates. Experimental ...