Antonio Palomares

A bi-proportional method applied to the Spanish Congress

A bi-proportional divisor method is applied to allocate the seats of the 2004 Spanish Congress, thus achieving proportionality relative to the population counts in the fifty-two districts, as well as proportionality relative to the vote counts for the political parties. Also, advantages and disadvantages of the method are discussed.

Degressively Proportional Methods for the Allotment of the European Parliament Seats Amongst the EU Member States

In this work we present several methods for distributing the seats of the European Parliament amongst the States of the European Union, in accordance with the restrictions established in article I-20 of the projected European Constitution. The proposed methods can be applied to the current composition of the EU, but also if there is a change in the number of states or their populations. They are based on adjusting the quotas of every country so that they verify the constitutional restrictions, and so that their rounding to an integer number will constitute an allotment of the seats of the Parliament.

ISOMORPHISMS, SCHAUDER BASES IN BANACH SPACES, AND NUMERICAL SOLUTION OF INTEGRAL AND DIFFERENTIAL EQUATIONS

ABSTRACT The authors determine the inverse of an isomorphism between two Banach spaces, the first of them with a Schauder basis, by means of certain best approximation sequences. As a consequence, they state an algorithm that allows one to solve numerically some linear integral and differential equations.

A more balanced composition of the European Parliament with degressive proportionality

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 EPs internal rules of procedure and existing EU Treaties, we contend that our results are not only of interest to the academic world.

Schauder bases in banach spaces: application to numerical solutions of differential equations

Abstract In this work, we present a numerical method for approximating the solution of a linear differential problem. We formulate a general problem in a Banach space making use of a Schauder basis and the Tau method to approximate the load function and the solution of the differential problem. Finally, we offer a numerical example.

The senatorial election in Spain. Proportional Borda methods for selecting several candidates

Abstract Firstly, in this paper, we analyze how the electors vote in the Senate election in Spain. We show examples in which the election outcome is predictable and not proportional. We also show similar examples for other social choice problems, in which the strategic vote affords a great advantage to those who put it into practice. Secondly, we introduce the definition of a proportional social choice method. We then prove the main result that determines the only Borda type social choice method that has the same proportionality as a given divisor method. We include results with the vote thresholds that guarantee the candidates will be elected under Limited Approval Voting and, finally, compare these thresholds with the proportional Borda Method.

The whole and its parts: On the coherence theorem of Balinski and Young

A new proof of the Coherence Theorem of Balinski and Young is presented. The theorem elucidates the methods used to apportion parliamentary seats among political parties proportionately to their vote counts, or among geographical districts proportionately to their population figures. A proportional apportionment method is coherent when each seat apportionment among all claimants is such that every part of it is a valid solution for the subset of claimants concerned. The Coherence Theorem states that every coherent apportionment method is compatible with a divisor method.

A convergence result for a least-squares method using Schauder bases

In this work we introduce a method, by using the least-squares method and a Schauder basis, which provides a numerical solution for a wide class of linear differential or integral equations. In addition, we give a convergence result and an application.

Filling holes with geometric and volumetric constraints

Abstract We propose and analyze different methods to reconstruct a function that is defined outside a sub-domain (hole) of a given domain. The reconstructed function is a smooth Powell–Sabin spline that is defined also inside this hole, filling then this lack of information, and, at the same time, fulfills certain global geometric considerations and other local volume constraints on the hole. We give several examples and we include a technique to estimate the volume of the function inside the hole by using just the data of the function where it is known, that is, outside the hole.

Approximation of patches by Cr-finite elements of Powell–Sabin type

In this work we develop a method to extend a function that is defined in a finite set of disjoint patches to a bigger domain containing all of them. The way to extend the function is by minimizing an energy functional which controls the closeness of the extended function to the original one over the patches, as well as the smoothness of the final reconstructed function. We show the existence and uniqueness of solution of this problem and we give a convergence result as well as several graphical and numerical examples.