Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

9 pages, no figures.-- MSC2000 code: 33C45.MR#: MR2431543 (2009g:41009)Zbl#: Zbl 1155.33006We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same $n$th root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e−φ(x), giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.Research by first two authors (C.D.M. and R.O.) was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grants MTM2005-08571 and MTM2007-68114. Research by third author (H.P.) was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grant MTM2006-13000-C03-02, by Comunidad de Madrid-Universidad Carlos III de Madrid, under grants CCG06-UC3M/EST-0690 and CCG07-UC3M/ESP-3339, by Centro de Investigación Matemática de Canarias (CIMAC) and by Vicerrectorado de Investigación de La Universidad de La Laguna: Convocatoria 2005 de Ayudas a Profesores Invitados.Publicad

Learning analytics visualizations of student-activity time distribution for the open Edx platform

MOOCs are one of the current trending topics in educational technology. They surged with the vision of a democratization in education worldwide by removing some access barriers. As every technology, MOOCs have promoters and detractors but truth is, they are an invaluable source of data related to student interaction with courses and their resources as has been available never before. This data is susceptible to shed light on the learning process in this online environment and potentially in uence in a positive way the learning outcomes. Students can be presented with visual, friendly information that enable them to re ect on their performance and gain awareness of their own learning style based on data beyond intuition. Teachers can be given the same metrics augmented with student aggregates for their courses. Thus, they can tune their pedagogical approach and resource quality for the better. In this context, Open edX is one of the most prominent MOOC platforms. However, its learning analytics support is low at present. This project extends the learning analytics support of the Open edX platform by adding new six visualizations related to time on video and problem modules, namely: 1) video time watched, 2) video and 3) problem time distributions, 4) video repetition pro le, 5) daily time on video and problem and 6) distribution of video events. The main technologies used have been Python, Django, MySQL, JavaScript, Google Charts and MongoDBLos MOOCs están de moda en lo que se refiere a tecnología educativa. Surgieron con la visión de remover algunas barreras de acceso en aras de la democratización de la educación en cada rincón del mundo. Como toda tecnología, tienen sus promotores y detractores, pero lo cierto es que constituyen una valiosa fuente de datos como no ha habido antes en lo que respecta a la interacción de los estudiantes con estos cursos y sus recursos. Estos datos pueden ayudarnos a entender el proceso de aprendizaje en estos entornos. Tienen además el potencial de in uir positivamente en los resultados del aprendizaje. Se puede presentar a los estudiantes una información visual fácil de entender, que les permita re exionar sobre su rendimiento y ganar conciencia de su estilo de aprendizaje a partir de los datos, más allá de lo que les pueda indicar la intuición. Las mismas métricas se pueden poner a disponibilidad de los profesores, en conjunto con valores agregados de la clase. De esta manera, los profesores pueden ajustar el enfoque pedagógico del curso y mejorar la calidad de los recursos. En este contexto, Open edX es una de las plataformas proveedoras de MOOCs más prominentes. Sin embargo, tiene todavía poco soporte para analitica del aprendizaje. Este proyecto extiende ese soporte al incorporar seis visualizaciones nuevas sobre tiempo en vídeos y problemas, especícamente: 1) tiempo visto de vídeos, distribución de tiempo en 2) vídeos y 3) problemas, 4) peril de repetición de vídeo, 5) tiempo diario en vídeos y problemas y 6) distribuci on de eventos de vídeo. Las principales tecnologías usadas son: Python, Django, MySQL, JavaScript, Google Charts y MongoDB.Ingeniería de Telecomunicació

Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights

8 pages, no figures.-- MSC2000 code: 42C05.Zbl#: Zbl 1177.42020We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form x\sp \gamma e\sp {\-varphi(x)} with γ>0, which include as particular cases the counterparts of the so-called Freud (i.e., when φ has a polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived.The research of C.D.M. and R.O. was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grants MTM2005-08571 and MTM2007-68114. The research of H.P. was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grant MTM2006-13000-C03-02, by Comunidad de Madrid- Universidad Carlos III de Madrid, under grant CCG06-UC3M/EST-0690 and by Centro de Investigación Matemática de Canarias (CIMAC).Publicad

Electrostatic models for zeros of Laguerre-Sobolev polynomials

Let {$\{S_n\}_{n\geqslant 0}$} be the sequence of orthogonal polynomials with respect to the Laguerre-Sobolev inner product $$\langle f,g\rangle_S =\!\int_{0}^{+\infty}\! f(x) g(x)x^{\alpha}e^{-x}dx+\sum_{j=1}^{N}\sum_{k=0}^{d_j}\lambda_{j,k} f^{(k)}(c_j)g^{(k)}(c_j),$$ where $\lambda_{j,k}\geqslant 0$, $\alpha >-1$ and $c_i \in (-\infty, 0)$ for $i=1,2,\dots,N$. We provide a formula that relates the Laguerre-Sobolev polynomials $S_n$ to the standard Laguerre orthogonal polynomials. We find the ladder operators for the polynomial sequence $\{S_n\}_{n\geqslant 0}$ and a second-order differential equation with polynomial coefficients for $\{S_n\}_{n\geqslant 0}$. We establish a sufficient condition for an electrostatic model of the zeros of orthogonal Laguerre-Sobolev polynomials. Some examples are given where this condition is either satisfied or not.Comment: arXiv admin note: substantial text overlap with arXiv:2308.0617

Asymptotic zero distribution for a class of extremal polynomials

We consider extremal polynomials with respect to a Sobolev-type p-norm, with 1and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved and the asymptotic critical point distribution is studied. We also find the nth root asymptotic behavior of the corresponding sequence of Sobolev extremal polynomials and their derivatives.A.D.G. was supported by the Research Fellowship Program, Ministerio de Economía, Industria y Competitividad of Spain, under grant BES-2016-076613. The authors G.L.L. and H.P.C. were supported by the Ministerio de Economía, Industria y Competitividad of Spain, under grant MTM2015-65888-C4-2-P

Polynomials of least deviation from zero in Sobolev p-Norm

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm (1<p<∞) for the case p=1. Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least n−d∗ zeros on the convex hull of the support of the measure, where d∗ denotes the number of terms in the discrete part.The research of H. Pijeira-Cabrera was partially supported by Ministry of Science, Innovation and Universities of Spain, under grant PGC2018-096504-B-C33. Funding Open Access funding provided by Universidad Carlos III de Madrid thanks to the CRUE-CSIC 2021 agreement with Springer Nature

Discrete-continuous Jacobi-Sobolev spaces and Fourier series

Let p≥1,ℓ∈N,α,β>−1 and ϖ=(ω0,ω1,…,ωℓ−1)∈Rℓ. Given a suitable function f, we define the discrete–continuous Jacobi–Sobolev norm of f as: ∥f∥s,p:=(∑k=0ℓ−1∣∣f(k)(ωk)∣∣p+∫1−1∣∣f(ℓ)(x)∣∣pdμα,β(x))1p, where dμα,β(x)=(1−x)α(1+x)βdx. Obviously, ∥⋅∥s,2=⟨⋅,⋅⟩s−−−−√, where ⟨⋅,⋅⟩s is the inner product ⟨f,g⟩s:=∑k=0ℓ−1f(k)(ωk)g(k)(ωk)+∫1−1f(ℓ)(x)g(ℓ)(x)dμα,β(x). In this paper, we summarize the main advances on the convergence of the Fourier–Sobolev series, in norms of type Lp, in the continuous and discrete cases, respectively. Additionally, we study the completeness of the Sobolev space of functions associated with the norm ∥⋅∥s,p and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in ∥⋅∥s,p norm of the partial sum of the Fourier–Sobolev series of orthogonal polynomials with respect to ⟨⋅,⋅⟩s.Authors thank the valuable comments by the referees. Their suggestions have contributed to improve the presentation of this manuscript. The research of F. Marcellán and H. Pijeira-Cabrera was partially supported by Spanish State Research Agency, under Grant PGC2018-096504-B-C33. The research of A. Díaz-González was supported by the Research Fellowship Program, Ministry of Economy and Competitiveness of Spain, under grant BES-2016-076613

Comparing the Accuracy of Automatic Scoring Solutions for a Text Comprehension Diagramming Intervention

Students typically have great difficulty monitoring their comprehension of textual materials. Completing diagrams about causal relations in expository texts has been a successful intervention to enhance the accuracy of students’ reading comprehension judgments (ie, monitoring accuracy), although there is still room for improvement. Such judgments play a role in crucial self-regulated learning decisions that students make such as allocating time and effort, selecting content for restudy, and/or consulting additional sources. The automated scoring of students’ diagram content can provide a basis for strengthening the diagramming intervention with individual and simultaneous feedback to a high number of students. Leveraging an existing human-coded (correct and incorrect) dataset of 6000+ diagram answers (completed in Dutch by 700+ secondary students), we compared different automatic scoring solutions in terms of classification accuracy. Four computational linguistic models for Dutch were identified and tested in combination with four popular machine learning classification algorithms. The best solution reached 81% accuracy (ie, four out of five answers matched the human coding). Depending on the accuracy required for different applications, these results could be used for fully-or semiautomated scorings of students’ answers to generative activities used in reading comprehension interventions