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

The Szeg\"o curve and Laguerre polynomials with large negative parameters

We study the asymptotic zero distribution of the rescaled Laguerre polynomials, $\displaystyle L_n^{(\alpha_n)}(nz)$, with the parameter $\alpha_n$ varying in such a way that $\displaystyle \lim_{n\rightarrow \infty}\alpha_n/n=-1$. The connection with the so-called Szeg\"{o} curve will be showed

El Chato vitoriano

La raza porcina Chato Vitoriano procede del Tronco céltico. Su área de distribución abarcó la vertiente cantábrica y zonas de Navarra, La Rioja y Castilla y León, suponiendo más del 5% del censo reproductor existente en el Estado en el año 1955. Fue un cerdo transformador de los productos agrarios de la explotación en carne y grasa destinados al consumo familiar, obteniendo unos buenos índices en las pruebas realizadas de cebamiento y de rendimiento cárnico. Su total extinción, ocurrida en la década de los años 1960-70, fue debida al cambio de las costumbres alimenticias dándose preferencia al consumo de carnes magras, que supuso la introducción de otras razas que absorbieron a nuestro Chato Vitoriano.Gasteizko Txato izeneko zerri arraza Enbor Zeltatik dator. Haren eremuak Kantauri isurialdea eta Nafarroa, Errioxa eta Gaztela eta Leongo zenbait eskualde hartu zituen, 1955eko ugaltzaileen zentsuan %5 baino gehiago zirelarik. Zerri horrek nekazaritza ustiatzetik ateratako produktuak haragi eta gantza bilakatzen zituen familia barneko kontsumorako, bai eta indize onak lortu ere gizentze eta haragi etekinei buruz egindako probetan. Erabat desagertu zen 60-70. urteetan, elikadura ohituren aldaketaren ondorioz; izan ere, haragi gihartsuei lehentasuna ematen hasi zen eta, joera horren ondorioz, gure Gasteizko Txatoa irentsiko zuten beste arraza batzuk ekarri zituzten.La race porcine Chato de Vitoria provient du Tronc celtique. Sa zone de distribution comprenait le versant cantabrique et des zones de Navarre, de La Rioja et de Castilla y Leon, ce qui suppose plus du 5% du recensement reproducteur existant au niveau national en 1955. Ce cochon transforma des produits agraires de l'exploitation en viande et graisse destinées à la consommation familiale, obtenant, lors des essais, de bons coefficients d'engraissement et de rendement de viande. Son extinction totale au cours des années 1960-70, fut la conséquence du changement des habitudes alimentaires qui donna la préférence à la consommation de viande maigre, ce qui favorisa l'introduction d'autres races qui absorbèrent notre Chato de Vitoria.The Chato vitoriano race of pigs proceeds from the Celtic Trunk. Its area of distribution encompasses the Cantabrian area and zones of Navarre, La Rioja and Castile and Leon, this meaning more than 5% of the breeding census existing in the State in 1955. The Chato vitoriano was a pig which transformed the agrarian products that resulted from the exploitation of meat and fats destined for family consumption, obtaining some good indices in the fattening and meat yield tests. Their total extinction, occurred in the sixties,was due to the change of the nutritional mores, when preference was given to lean meat consumption, which meant the introduction of other races that absorbed our Chato vitoriano

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

An experimental publish-subscribe monitoring assessment to Beyond 5G networks

Collection: Wireless Technologies for the Connectivity of the Future.The fifth generation (5G) of mobile networks is designed to accommodate different types of use cases, each of them with different and stringent requirements and key performance indicators (KPIs). To support the optimization of the network performance and validation of the KPIs, there exist the necessity of a flexible and efficient monitoring system and capable of realizing multi-site and multi-stakeholder scenarios. Nevertheless, for the evolution from 5G to 6G, the network is envisioned as a user-driven, distributed Cloud computing system where the resource pool is foreseen to integrate the participating users. In this paper, we present a distributed monitoring architecture for Beyond 5G multi-site platforms, where different stakeholders share the resource pool in a distributed environment. Taking advantage of the usage of publish-subscribe mechanisms adapted to the Edge, the developed lightweight monitoring solution can manage large amounts of real-time traffic generated by the applications located in the resource pool. We assess the performance of the implemented paradigm, revealing some interesting insights about the platform, such as the effect caused by the throughput of monitoring data in performance parameters such as the latency and packet loss, or the presence of a saturation effect due to software limitations that impacts in the performance of the system under specific conditions. In the end, the performance evaluation process has confirmed that the monitoring platform suits the requirements of the proposed scenarios, being capable of handling similar workloads in real 5G and Beyond 5G scenarios, then discussing how the architecture could be mapped to these real scenarios.This work was partly funded by the European Commission under the European Union's Horizon 2020 program-Grant Agreement Number 815074 (5G EVE project). This work was also partly funded by the Community of Madrid, under the grant approved in the "Convocatoria de 2017 de Ayudas para la Realización de Doctorados Industriales en la Comunidad de Madrid (Orden 3109/2017, de 29 de agosto)", Grant Agreement Number IND2017/TIC-7732. The paper solely reflects the views of the authors. Neither the European Commission nor the Community of Madrid are responsible for the contents of this paper or any use made thereof

Monitoring Platform Evolution towards Serverless Computing for 5G and Beyond Systems

Fifth generation (5G) and beyond systems require flexible and efficient monitoring platforms to guarantee optimal key performance indicators (KPIs) in various scenarios. Their applicability in Edge computing environments requires lightweight monitoring solutions. This work evaluates different candidate technologies to implement a monitoring platform for 5G and beyond systems in these environments. For monitoring data plane technologies, we evaluate different virtualization technologies, including bare metal servers, virtual machines, and orchestrated containers. We show that containers not only offer superior flexibility and deployment agility, but also allow obtaining better throughput and latency. In addition, we explore the suitability of the Function-as-a-Service (FaaS) serverless paradigm for deploying the functions used to manage the monitoring platform. This is motivated by the event oriented nature of those functions, designed to set up the monitoring infrastructure for newly created services. When the FaaS warm start mode is used, the platform gives users the perception of resources that are always available. When a cold start mode is used, containers running the application"s modules are automatically destroyed when the application is not in use. Our analysis compares both of them with the standard deployment of microservices. The experimental results show that the cold start mode produces a significant latency increase, along with potential instabilities. For this reason, its usage is not recommended despite the potential savings of computing resources. Conversely, when the warm start mode is used for executing configuration tasks of monitoring infrastructure, it can provide similar execution times to a microservice-based deployment. In addition, the FaaS approach significantly simplifies the code logic in comparison with microservices, reducing lines of code to less than 38%, thus reducing development time. Thus, FaaS in warm start mode represents the best candidate technology to implements such management functions.This work has been supported by EC H2020 5GPPP projects 5G-EVE and 5GROWTH under grant agreements No. 815974 and 856709, respectively

Self-organization in the olfactory system: one shot odor recognition in insects

We show in a model of spiking neurons that synaptic plasticity in the mushroom bodies in combination with the general fan-in, fan-out properties of the early processing layers of the olfactory system might be sufficient to account for its efficient recognition of odors. For a large variety of initial conditions the model system consistently finds a working solution without any fine-tuning, and is, therefore, inherently robust. We demonstrate that gain control through the known feedforward inhibition of lateral horn interneurons increases the capacity of the system but is not essential for its general function. We also predict an upper limit for the number of odor classes Drosophila can discriminate based on the number and connectivity of its olfactory neurons

Gain control network conditions in early sensory coding

Gain control is essential for the proper function of any sensory system. However, the precise mechanisms for achieving effective gain control in the brain are unknown. Based on our understanding of the existence and strength of connections in the insect olfactory system, we analyze the conditions that lead to controlled gain in a randomly connected network of excitatory and inhibitory neurons. We consider two scenarios for the variation of input into the system. In the first case, the intensity of the sensory input controls the input currents to a fixed proportion of neurons of the excitatory and inhibitory populations. In the second case, increasing intensity of the sensory stimulus will both, recruit an increasing number of neurons that receive input and change the input current that they receive. Using a mean field approximation for the network activity we derive relationships between the parameters of the network that ensure that the overall level of activity of the excitatory population remains unchanged for increasing intensity of the external stimulation. We find that, first, the main parameters that regulate network gain are the probabilities of connections from the inhibitory population to the excitatory population and of the connections within the inhibitory population. Second, we show that strict gain control is not achievable in a random network in the second case, when the input recruits an increasing number of neurons. Finally, we confirm that the gain control conditions derived from the mean field approximation are valid in simulations of firing rate models and Hodgkin-Huxley conductance based models