Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy

Multiple orthogonality is considered in the realm of a Gauss--Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of M-Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution, that can be obtained from the solution of a Gauss--Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with snake shape, and results like the ABC theorem or the Christoffel--Darboux formula are re-derived in this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss--Borel factorization problem, is discussed. Deformations of the weights, natural for M-Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov--Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the $\tau$-function representation of the multiple orthogonality is given.Comment: 53 pages. In this version minor revisions regarding the Christoffel-Darboux operators are performe

Review of real brain-controlled wheelchairs

This paper presents a review of the state of the art regarding wheelchairs driven by a brain-computer interface (BCI). Using a brain-controlled wheelchair (BCW), disabled users could handle a wheelchair through their brain activity, granting autonomy to move through an experimental environment. A classification is established, based on the characteristics of the BCW, such as the type of electroencephalographic (EEG) signal used, the navigation system employed by the wheelchair, the task for the participants, or the metrics used to evaluate the performance. Furthermore, these factors are compared according to the type of signal used, in order to clarify the differences among them. Finally, the trend of current research in this field is discussed, as well as the challenges that should be solved in the future

The personal income tax applied in the member states of European Union. The case of Spain.

In the Communication on "Tax policy in the European Union - Priorities for the years ahead" (COM/2001/260 of 23 May 2001), the Commission reiterated its belief that there is no need for an across the board harmonisation of Member States' direct tax systems. For tax policy, the Communication established, as a main priority, the need to address the concerns of individuals and businesses operating within the Internal Market by focusing on the elimination of tax obstacles to all forms of cross-border economic activity, in addition to continuing the fight against harmful tax competition. This approach was confirmed in the Communication "The contribution of taxation and customs policies to the Lisbon strategy" (COM/2005/532 of 25 October 2005) (European Commission, 2006). The Spanish Law No 35/2006, 28 th November of Personal Income Tax, reformed that tax. The long-term capital gains will be taxed at 18% (before 15%) ;the tax scale will be comprised of only 3 or 4 brackets (before 5) and the top marginal rate (before 45%) will be reduced; and some tax credits and allowances (acquisition of permanent home and contributions to pension funds for example) were readjusted in order to make them more accessible to low-income earners. The present paper makes a brief approach to the harmonisation in the European Union, explains the reform of the Spanish Personal Income Tax, introduces new figures and formulas never seen before at book of taxes, analyses the concept of the Spanish Personal Income Tax, studies the elements of this tax as the beneficiary, taxable person, territoriality, basis of assessment, exemptions, explains the basic mechanism of the tax, deductions, the taxable base, the tax rates, collections and examples. This paper is the result of three researches that the authors are carrying out at The Institute for Fiscal Studies, Ministry of Economy and Finance, University of CEU San Pablo, Madrid and University of Oviedo Spain from 2006 to 2008

Brain switch mode: an alternative to drive a brain-controlled wheelchair

To date, different control paradigms of low level navigation have been tested for brain-controlled wheelchairs, mainly divided into continuous or discrete control [1]. However, these paradigms have certain drawbacks such as the need to keep the mental tasks active for a long time, as in continuous mode, or the impossibility to freely choose any distance of the movement or the turn, as in the discrete mode. An alternative paradigm to solve these problems could be the use of the brain switch mode [2], which would allow a more flexible control of the distance, requiring a lower workload for the user.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Proposals of Control Paradigms Applied to a Brain-Controlled Wheelchair

Proposals of Control Paradigms Applied to a Brain-Controlled Wheelchair, Ron-Angevin R., Velasco-Álvarez F., Fernández Rodriguez A., Proceeding og the BITs 4th Annual World Congress of Smart Material 2018, Osaka (Japan), 6-8 March 2018Several of the neurological diseases that human beings can result in severe disabilities. In some cases, people who suffer from such deficiencies lose any chance of communication with their environment, being the only possible alternative to give the brain a new channel not based on muscular activity, allowing these people to send messages and commands to the external world. The systems that allows the latter is what is known as Brain-Computer Interfaces (BCI). Their common feature is to process the brain’s electrical activity for extracting information that can be used to command an external device, as for example, a wheelchair to provide them some mobility. One of the most important limitations of these brain controlled wheelchair is to guarantee that a person can, through his mental activity, safely control the variety of navigation commands that provide control of the wheelchair: advance, turn, move back, and stop. The vast majority of the mobile robot navigation applications that are controlled via a BCI demand that the user performs as many different mental tasks as there are different control commands, worsening the classification accuracy. In order to enable an effective and autonomous wheelchair navigation with a BCI system without worsening user performance, the Brain–Computer Interface (BCI) group of the University of Málaga (UMA-BCI) proposed and later developed a new paradigm based on the discrimination of only two classes (one active mental task versus any other mental activity), which enabled the selection of four commands: move forwards, turn right, move backward and turn left. The final aim of this contribution is to show how to control a robotic wheelchair through the use of only two mental tasks. The mapping of these two mental tasks into several navigation commands allows the Brain-Controlled Wheelchair to be moved and turned in order to achieve effective navigation.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech