Quadrature formulas on the unit circle based on rational functions

AbstractQuadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szegő polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made

Convergence of modified approximants associated with orthogonal rational functions

AbstractLet {αn} be a sequence in the unit disk D = {z ∈ C: ¦z¦ < 1} consisting of a finite number of points cyclically repeated, and let L be the linear space generated by the functions Bn(z) = Πk=0n − αk(z − αk)¦αk¦(1 − αkz). Let {ϕn(z)} be orthogonal rational functions obtained from the sequence {Bn(z)} (orthogonalization with respect to a given functional on L), and let {ψn(z)} be the corresponding functions of the second kind (with superstar transforms ϕn∗(z) and ψn∗(z) respectively). Interpolation and convergence properties of the modified approximants Rn(z, un, vn) = (unψn(z) − vnψn∗(z))(unϕn(z) + vnϕn∗(z)) that satisfy ¦un¦ = ¦vn¦ are discussed

Analyzing and Predicting Purchase Intent in E-commerce: Anonymous vs. Identified Customers

The popularity of e-commerce platforms continues to grow. Being able to understand, and predict customer behavior is essential for customizing the user experience through personalized result presentations, recommendations, and special offers. Previous work has considered a broad range of prediction models as well as features inferred from clickstream data to record session characteristics, and features inferred from user data to record customer characteristics. So far, most previous work in the area of purchase prediction has focused on known customers, largely ignoring anonymous sessions, i.e., sessions initiated by a non-logged-in or unrecognized customer. However, in the de-identified data from a large European e-commerce platform available to us, more than 50% of the sessions start as anonymous sessions. In this paper, we focus on purchase prediction for both anonymous and identified sessions on an e-commerce platform. We start with a descriptive analysis of purchase vs. non-purchase sessions. This analysis informs the definition of a feature-based model for purchase prediction for anonymous sessions and identified sessions; our models consider a range of session-based features for anonymous sessions, such as the channel type, the number of visited pages, and the device type. For identified user sessions, our analysis points to customer history data as a valuable discriminator between purchase and non-purchase sessions. Based on our analysis, we build two types of predictors: (1) a predictor for anonymous that beats a production-ready predictor by over 17.54% F1; and (2) a predictor for identified customers that uses session data as well as customer history and achieves an F1 of 96.20%. Finally, we discuss the broader practical implications of our findings.Comment: 10 pages, accepted at SIGIR eCommerce 202

An Unusual Triad in Pediatric Neurology:A Case Report on Cerebral Palsy, Epilepsy, and Duchenne Muscular Dystrophy

We present a case of an unusual triad in pediatric neurology: a currently 12-year-old boy with cerebral palsy and epilepsy who was later also diagnosed with Duchenne muscular dystrophy. We describe the clinical path that resulted in this exceptional diagnosis. This case report illustrates how different neurological disorders may overshadow each other. In addition, it demonstrates that every child with cerebral palsy and either an atypical clinical course or with inexplicable laboratory values-as well as every infant boy born to a theoretical Duchenne muscular dystrophy carrier-should be subjected to additional investigations.</p

Increase in incidence of anorexia nervosa among 10- to 14-year-old girls:A nationwide study in the Netherlands over four decades

Objective: This primary care study examined time trends in the incidence of anorexia nervosa (AN) and bulimia nervosa (BN) in the Netherlands across four decades. Methods: A nationwide network of general practitioners, serving approximately 1% of the total Dutch population, recorded newly diagnosed patients with AN and BN in their practices from 1985 to 2019 (2,890,978 person-years). DSM-IV diagnostic criteria were consistently used and the same psychiatrist was responsible for the final diagnostic decision. Incidence rates (IRs) were calculated for: the total population (all ages), females overall, and females per 5-year age category. Time trends in IRs were analyzed using JoinPoint regression analyses. Results: In four decades, the incidence of AN among 10- to 14-year-old females increased significantly from 8.6 to 38.6 per 100,000 person-years (average period percentage change [APPC] = 56.7; 95% confidence interval [CI] = 6.5–130.6. The overall incidence of AN was stable, with IRs ranging from 6.0 (95% CI = 4.3–8.1) to 8.4 (95% CI = 6.4–10.8). The IR of BN decreased significantly from 8.7 (95% CI = 6.7–11.0) to 3.2 (95% CI = 2.0–4.9) in the 2000s, before leveling off in the 2010s (IR 3.2; 95% CI = 2.0–4.8). Discussion: The incidence of AN among 10- to 14-year-old girls increased significantly over four decades. Both biological and sociocultural factors, for example, early pubertal timing and the impact of social media, might explain this. In other age groups and overall, the incidence of AN remained stable. The significant decrease of the incidence of BN in the previous decades halted in the last decade. Public Significance: An important finding of the present study is that for 10- to 14-year-old girls, the risk for developing anorexia nervosa has increased significantly over 40 years. More healthcare facilities for younger people are needed, and prevention programs could include social media use. For bulimia nervosa, the general decrease in the occurrence of new cases has halted in the 2010s.</p