This paper presents an analytical expression that models the tilt angle of directors in a nematic liquid crystal (LC), depending on its elastic properties, its dielectric anisotropy, and the quasi-static electric field applied. The analytical solution obtained is fast and easily computable in comparison with numerical estimations and is of special interest in radiofrequency; for instance, for the LC modeling in full-wave electromagnetic simulators in the design process of electronically tunable devices, such as microwave phase shifters or electronically steerable antennas for satellite communications. Subsequently, a comparison is made between numerical approaches (self-implemented shooting method) and the analytical formulas when varying the parameters of the LC, being demonstrated its usefulness. The average LC director is then obtained and used to form the full permittivity tensor that completely characterizes the electrical properties of the material. Finally, an electromagnetic simulation is carried out to show the capabilities of the LC as a tunable phase shifter. It is shown that only 5 cm of a commercial 200-µm LC mixture is necessary to achieve 360 • of the maximum variable phase shift at the 30-GHz band. INDEX TERMS Liquid crystal, nematic phase, analytical expression, microwave, phase shifting.

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This letter presents a closed-form integral-equation formulation that models the director tilting in nematic liquid crystals. The proposed formulation is computationally efficient compared to numerical methods and provides a physical insight on the matter. Our previous work is generalized from the one-constant approach to all the possible cases that cover the elastic constants. These constants determine in great extent the electrical properties of the liquid crystal, and subsequently, the response of the reconfigurable devices that take use of it. Therefore, a more precise but still simple modeling of their influence is pursued here. Simulations show good agreement with numerical implementations. In comparison, the error in the estimation is considerably reduced when using the integralequation formulation, specially as the polarization voltage or the dielectric anisotropy decreases.

Degressive proportionality constitutes the main conceptual criterion to determine the composition of the European Parliament (EP). In reality, however, this concept entails serious practical problems as exemplified by the current distribution of seats in the EP. This article takes up the call and presents a new method for a better balanced seat allocation in the EP after the 2014 elections. Comparing with recent methods in the field, our results reveal that the method proposed yields a more balanced and juster parliamentary seat distribution among all European member states by joining the strengths of the existing methods and avoiding their weaknesses. In light of upcoming rounds of enlargement and possible amendments to the EP's internal rules of procedure and existing EU Treaties, we contend that our results are not only of interest to the academic world.

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