Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all the Hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields.     

Polynomial first integrals of quadratic vector fields

We classify all quadratic polynomial differential systems having a polynomial first integral, and provide explicit normal forms for such systems and for their first integrals.     

On some open problems in planar differential systems and Hilbert’s 16th problem

This review paper contains a brief summary of topics and concepts related with some open problems of planar differential systems. Most of them are related with 16th Hilbert problem which refers to the existence of a uniform upper bound on the number of limit cycles of a polynomial system in function of its degree. These open problems are proposed as open questions throughout the text. Finally, an extensive bibliography, which does not intend to be exhaustive, is also given.     

Compatibilized poly(ether imide)/LCP blends: drawing ability and mechanical properties of the ribbons

The effects of the addition of polyarylate (PAr) on the drawing ability of poly(ether imide) (PEI)/Rodrun liquid‐crystalline 5000 (Ro LC5000) (Ro) blends and on the mechanical properties of their ribbons were studied. The compatibilizing effect of the PAr led to an increase in the drawing ability of the blends, as seen by the fact that the maximum Ro content compatible with the drawing process increased from 15 to 30%. This may lead to new applications in the field of gas barrier materials. The presence of PAr also led to an increase in the adhesion between the two phases of the blends and consequent improved ductility. However, the very high modulus of elasticity and tensile strength in the direction of orientation (up to three‐fold those of the matrix) due to the less developed fibrillation were smaller than the corresponding values before compatibilization. These very large modulus of elasticity and tensile strength values and those perpendicular to the direction of orientation, which were similar to those of the matrix, led to a high anisotropy in the extruded ribbons that increased with the draw ratio (DR) and the Ro contents. The high mechanical properties of the ribbons may be used and the anisotropy diminished by layering the ribbons in, and perpendicular to, the direction of orientation, according to the expected external load. Copyright © 2002 John Wiley & Sons, Ltd.     

Riesz-Nágy singular functions revisited

In 1952 F. Riesz and Sz.-Nágy published an example of a monotonic continuous function whose derivative is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-Nágy family of functions and we relate it to a system for real number representation which we call (τ,τ−1)-expansions. With the help of these real number expansions we generalize the family. The singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.