Polinomios y Ecuaciones Relacionadas

Con este cursillo se pretende dar a conocer algunas propiedades de los polinomios como lo son las fórmulas de Vieta, algunas herramientas simples para encontrar raíces, las transformaciones de los polinomios y las identidades de Newton. Estas propiedades dan alternativas que generan problemas con un alto contenido de buenos hechos, los cuales ayudarán a enriquecer los conocimientos sobre polinomios y dar alternativas de solución a problemas de ecuaciones que a simple vista pueden parecer complicados

Endpoint estimates and weighted norm inequalities for commutators of fractional integrals

We prove that the commutator [b, Iα], b ∈ BMO, Iα the fractional integral operator, satisfies the sharp, modular weak-type inequality f(x) tdx, where B(t) = tlog(e + t) and Ψ(t)=[tlog(e + tα/n)]n/(n−α). These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality, M#([b, Iα]f)(x) ≤ CbBMO [Iαf(x) + Mα,Bf(x)], where M# is the sharp maximal operator, and Mα,B is a generalization of the fractional maximal operator in the scale of Orlicz spaces. Using this inequality we also prove one-weight inequalities for the commutator; to do so we prove one and two-weight norm inequalities for Mα,B which are of interest in their own right.[b, Iα]f(x

Logarithmic bump conditions for Calderón-Zygmund operators on spaces of homogeneous type

We establish two-weight norm inequalities for singular integral operators defined on spaces of homogeneous type. We do so first when the weights satisfy a double bump condition and then when the weights satisfy separated logarithmic bump conditions. Our results generalize recent work on the Euclidean case, but our proofs are simpler even in this setting. The other interesting feature of our approach is that we are able to prove the separated bump results (which always imply the corresponding double bump results) as a consequence of the double bump theorem

Norm inequalities for the minimal and maximal operator, and differentiation of the integral

We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Hölder inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability of the integral in cases when the associated maximal operator is large, and we give a new condition for this maximal operator to be weak (1,1)

Characterization of non-intentional emissions from distributed energy resources up to 500 kHz: A case study in Spain

Narrow Band Power Line Communications (NB-PLC) systems are currently used for smart metering and power quality monitoring as a part of the Smart Grid (SG) concept. However, non-intentional emissions generated by the devices connected to the grid may sometimes disturb the communications and isolate metering equipment. Though some research works have been recently developed to characterize these emissions, most of them have been limited to frequencies below 150 kHz and they are mainly focused on in-house electronic appliances and lightning devices. As NB-PLC can also be allocated in higher frequencies up to 500 kHz, there is still a lack of analysis in this frequency range, especially for emissions from Distributed Energy Resources (DERs). The identification and characterization of the emissions is essential to develop solutions that avoid a negative impact on the proper performance of NB-PLC. In this work, the non-intentional emissions of different types of DERs composing a representative microgrid have been measured in the 35–500 kHz frequency range and analyzed both in time and frequency domains. Different working conditions and coupling and commutation procedures to mains are considered in the analysis. Results are then compared to the limits recommended by regulatory bodies for spurious emissions from communication systems in this frequency band, as no specific limits for DERs have been established. Field measurements show clear differences in the characteristics of non-intentional emissions for different devices, working conditions and coupling procedures and for frequencies below and above 150 kHz. Results of this study demonstrate that a further characterization of the potential emissions from the different types of DERs connected to the grid is required in order to guarantee current and future applications based on NB-PLC.This work has been financially supported in part by the Basque Government (Elkartek program)

Boundedness of Pseudodifferential Operators on Banach Function Spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$ is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$ or to the the Miyachi class $S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$ with $0\le\delta\le\rho\le 1$, $0\le\delta0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.Comment: To appear in a special volume of Operator Theory: Advances and Applications dedicated to Ant\'onio Ferreira dos Santo