Equilibrium measures in the presence of certain rational external fields

Equilibrium measures in the real axis in the presence of rational external fields are considered. These external fields are called rational since their derivatives are rational functions. We analyze the evolution of the equilibrium measure, and its support, when the size of the measure, $t$, or other parameters in the external field vary. Our analysis is illustrated by studying with detail the case of a generalized Gauss-Penner model, which, in addition to its mathematical relevance, has important physical applications (in the framework of random matrix models). This paper is a natural continuation of \cite{MOR2013}, where equilibrium measures in the presence of polynomial external fields are thoroughly studied

Equilibrium problems in weakly admissible external fields created by pointwise charges

The main subject of this paper is equilibrium problems on an unbounded conductor $\Sigma$ of the complex plane in the presence of a weakly admissible external field. An admissible external field $Q$ on $\Sigma$ satisfies, along with other mild conditions, the following growth property at infinity: $$\lim_{|x| \rightarrow \infty}(Q(x) - \log |x|) = +\infty.$$ This condition guarantees the existence and uniqueness of the equilibrium measure in the presence of $Q$, and the compactness of its support. In the last 10-15 years, several papers have dealt with weakly admissible external fields, in the sense that $Q$ satisfies a weaker condition at infinity, namely, $$\exists M\in(-\infty,\infty],\quad\liminf_{|x| \rightarrow \infty}(Q(x) - \log |x|) = M.$$ Under this last assumption, there still exists a unique equilibrium measure in the external field $Q$, but the support need not be a compact subset of $\Sigma$ anymore. In most examples considered in the literature the support is indeed unbounded. Our main goal in this paper is to illustrate this topic by means of a simple class of external fields on the real axis created by a pair of attractive and repellent charges in the complex plane, and to study the dynamics of the associated equilibrium measures as the strength of the charges evolves. As one of our findings, we exhibit configurations where the support of the equilibrium measure in a weakly admissible external field is a compact subset of the real axis. To achieve our goal, we extend some results from potential theory, known for admissible external fields, to the weakly admissible case. These new results may be of independent interest. Finally, the so--called signed equilibrium measure is an important tool in our analysis. Its relationship with the (positive) equilibrium measure is also explored.Comment: To appear in Journal of Approximation Theor

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect towallows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasiorthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems.We also discuss the discrete-to-continuous transition of thesemodels in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.The first author was partially supported by Simons Foundation Collaboration Grants for Mathematicians (grant 710499). He also acknowledges the support of the Spanish Government and the European RegionalDevelopment Fund (ERDF) through grant PID2021-124472NB-I00, Junta deAndalucía (research group FQM-229 and Instituto Interuniversitario Carlos I de Física Teórica y Computacional), and by the University of Almería (Campus de Excelencia Internacional del Mar CEIMAR) in the early stages of this project. The second and third authors were partially supported by Spanish Ministerio de Ciencia, Innovación y Universidades, under grant MTM2015-71352-P. The third author was additionally supported by Junta de Andalucía (research group FQM-384), the University of Granada (Research Project ERDF-UGR A-FQM-246-UGR20), and by the IMAG-Maria de Maeztu grant CEX2020-001105- M/AEI/10.13039/501100011033. Funding for open access publishing: Universidad de Granada/CBU

The urban transport companies in Spain: analysis of efficiency with data envelopment analysis.

Purpose This paper aims to define the efficiency achieved by urban transport companies in Spain concerning the resources they use, considering the type of management used for implementation, public-private, and size. Design/methodology/approach This study consisted of an analysis of the efficiency of 229 public-private urban transport operators during the period 2012–2021 using Data Envelopment Analysis, the Malmquist Index and inference estimators to determine productivity, efficiency change into Pure Technical Efficiency Change (PTECH), and scale efficiency change. Findings Based on the efficiency analysis, the authors concluded that of the 229 companies studied, more than 35 were inefficient in all analysed periods. Considering the sample used, direct management is considered significantly more efficient. It cannot be concluded that the size of these companies influences their efficiency, as the data show unequal development behaviours in the studied years. Originality/value This study provides arguments on whether there is a significant difference between the two types of management in the urban transport sector. It also includes firm size as a study variable, which has not been previously considered in other studies related to urban transport efficiency. Efficiency should be a crucial factor in determining funding allocation in this sector, as it encourages operators to optimize and improve their services

On Gegenbauer Point Processes on the Unit Interval

In this paper we compute the logarithmic energy of points in the unit interval [-1,1] chosen from different Gegenbauer Determinantal Point Processes. We check that all the different families of Gegenbauer polynomials yield the same asymptotic result to third order, we compute exactly the value for Chebyshev polynomials and we give a closed expression for the minimal possible logarithmic energy. The comparison suggests that DPPs cannot match the value of the minimum beyond the third asymptotic term.CRUE-CSIC agreementSpringer Natur

Cuatro proyectos de investigación para luchar contra el COVID-19

La aparición del COVID-19 ha supuesto un cambio radical en la forma de vivir de las personas. Hace unos años, nadie se hubiese imaginado que habría que pasar meses de confinamiento en casa, ni que ponerse una mascarilla o lavarse las manos fuese algo que permitiría salvar vidas

La salida itinerante de Geografía

La Salida Itinerante de Geografía es un proyecto educativo transversal y extracurricular, consistente en la realización de una salida de campo de varios días de duración. A pie y por un espacio de carácter natural y rural, profesores y alumnos de la titulación de Geografía de la Universidad de Sevilla, previamente organizados en grupos de trabajo, analizan las principales claves territoriales del ámbito recorrido. Hasta la fecha se han realizado nueve ediciones, una en cada curso académico desde 2008. El objetivo de esta comunicación es mostrar las principales aportaciones didácticas de esta actividad, detectar sus fortalezas y debilidades, y exponer perspectivas de mejora en el futuro

Del extractivismo cultural y lingüístico a la revitalización de las lenguas

Son bastante conocidos los efectos destructores de la conquista y la colonización de América sobre centenares de lenguas originarias en América Latina. Además, también se ha investigado acerca de la continuidad del proceso glotofágico bajo el dominio de las élites criollas, después de la Independencia. Durante el siglo XX, el indigenismo no cambió sustancialmente las relaciones interculturales (políticas, económicas y sociales) entre las culturas dominantes hispanófonas y las culturas minorizadas, aun cuando acumulara e hiciera uso de la simbología de las producciones precolombinas como seña de identidad nacional, de modo muy notorio en México. Sin embargo, las últimas décadas del siglo XX y los inicios del siglo XXI, a causa del neoliberalismo y la globalización, han multiplicado las amenazas sobre las culturas y las lenguas supervivientes, en la medida que se ha pretendido aniquilar el entorno medioambiental, los ecosistemas locales y el mundo vital, que podríamos categorizar como territorio indígena. El proyecto Ndatiaku Tu’un Savi es solo uno de los muchos gérmenes que pretenden revitalizar y revalorizar el conocimiento y el uso de las lenguas originarias, por medio de las redes sociales, entre sus propios hablantes, a la vez que confrontan las estrategias meramente forenses de la lingüístic

Análisis evolutivo económico-financiero de las empresas del servicio público de tranporte urbano en España. 2008-2013

El Observatorio de Costes y Financiación del Transporte Urbano Colectivo, desde su creación en noviembre de 2009, además de la realización de informes bianuales de la situación del sector español, ha realizado otros trabajos específicos en el sector del transporte en superficie ampliando año tras año la información a la que tiene acceso. En este sentido, y tras un acuerdo establecido con la Asociación del Transporte Urbano Colectivo, se ha elaborado un informe evolutivo del período 2008-2013. En la presente comunicación se muestran algunas de las conclusiones obtenidas al analizar una serie de elementos clave para observar las necesidades de financiación, la gestión competitiva y la correcta aplicación de los recursos por parte de las empresas gestoras de los transportes colectivos urbanos.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech