Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

Унутрашњост скраћених усредњених гаусовских квадратура и оцена грешке Гаус-Кронродових квадратура

Онда када функција није позната аналитички, већ су познате само њене вредности добијене експериментално у неким тачкама, или пак њен интеграл није елементарно израчунљив, на располагању су различите методе квадратуре, тј. нумеричке интеграције. Многе од ових метода одређују приближну вредност интеграла коришћењем вредности функције у појединим тачкама - чворовима. Проблеми који се при томе јављају су разноврсни: од проналажења оптималних чворова и тежинских коефицијената (који су често такође нумерички одређени), преко што тачнијег израчунавања вредности функције у тим чворовима (ако је у њима функција уопште дефинисана), до процене грешке квадратурне формуле (која зависи како од квадратурне формуле, тако и од природе функције). Једна од метода процене квадратурне грешке користи Гаус-Кронродова раширења Гаусових квадратурних формула. У овој дисертацији је разматрана једна модификација Гаусових квадратура, у виду тзв. уопштене усредњене гаусовске квадратуре, која може послужити као замена онда када Гаус-Кронродова квадратура не постоји или није практична. За ове квадратуре дати су неки услови под којима су сви њихови чворови унутар интервала интеграције. Такође су посматране Гаус-Кронродове формуле за тзв. Бернштајн-Сегеове тежинске функције и у тим случајевима су дате експлицитне оцене квадратурне грешке.When a function is not known analytically but only by a set of sampled values, or its integral is not an elementary function, various methods for quadrature, i.e. numerical integration can be used instead. Many of these methods use the values of the function at a finite set of points (nodes) to compute the approximate value of the integral. A variety of problems can arise throughout the process. These include finding optimal nodes and weights (often numerically, as well), evaluating the function accurately enough at the nodes (provided that these nodes are actually in its domain), or finding effective bounds for the quadrature error (which clearly depends on natures of both the quadrature and the function itself). One useful method of estimating the quadrature error involves Gauss-Kronrod extensions of Gaussian quadrature formulae. This thesis discusses a modification of Gaussian quadratures, known as generalized averaged Gaussian quadratures, which may serve as a substitute when Gauss-Kronrod quadratures are not available. For these quadratures, some conditions are given under which all their nodes lie inside the domain of integration. Also, the thesis studies Gauss-Kronrod quadrature formulae in the case of Bernstein-Szeg˝o weight functions and gives explicit bounds for the quadrature error

Truncated generalized averaged Gauss quadrature rules

Generalized averaged Gaussian quadrature formulas may yield higher accuracy than Gauss quadrature formulas that use the same moment information. This makes them attractive to use when moments or modified moments are cumbersome to evaluate. However, generalized averaged Gaussian quadrature formulas may have nodes outside the convex hull of the support of the measure defining the associated Gauss rules. It may therefore not be possible to use generalized averaged Gaussian quadrature formulas with integrands that only are defined on the convex hull of the support of the measure. Generalized averaged Gaussian quadrature formulas are determined by symmetric tridiagonal matrices. This paper investigates whether removing some of the last rows and columns of these matrices gives quadrature rules whose nodes live in the convex hull of the support of the measure

Fixed Points of Geraghty-Type Mappings in Various Generalized Metric Spaces

Fixed point theorems for mappings satisfying Geraghty-type contractive conditions are proved in the frame of partial metric spaces, ordered partial metric spaces, and metric-type spaces. Examples are given showing that these results are proper extensions of the existing ones

Convergence of iterates with errors of uniformly quasi-Lipschitzian mappings in cone metric spaces

The aim of this paper is to consider an Ishikawa type iteration process with errors to approximate the fixed point of two uniformly quasi-Lipschitzian mappings in cone metric spaces. We also extend some fixed point results of these mappings from complete generalized convex metric spaces to cone metric spaces. Our results extend and generalize many known results

Truncated generalized averaged Gauss quadrature rules

Generalized averaged Gaussian quadrature formulas may yield higher accuracy than Gauss quadrature formulas that use the same moment information. This makes them attractive to use when moments or modified moments are cumbersome to evaluate. However, generalized averaged Gaussian quadrature formulas may have nodes outside the convex hull of the support of the measure defining the associated Gauss rules. It may therefore not be possible to use generalized averaged Gaussian quadrature formulas with integrands that only are defined on the convex hull of the support of the measure. Generalized averaged Gaussian quadrature formulas are determined by symmetric tridiagonal matrices. This paper investigates whether removing some of the last rows and columns of these matrices gives quadrature rules whose nodes live in the convex hull of the support of the measure

Convergence of iterates with errors of uniformly quasi-Lipschitzian mappings in cone metric spaces

The aim of this paper is to consider an Ishikawa type iteration process with errors to approximate the fixed point of two uniformly quasi-Lipschitzian mappings in cone metric spaces. We also extend some fixed point results of these mappings from complete generalized convex metric spaces to cone metric spaces. Our results extend and generalize many known results

SOCIAL MEDIA ALGORITHMS AND DATA MANAGEMENT

Although the audience in the digital media space has more power than in the traditional media environment, as indicated by their ability to create, reshape and share content, media users’ behavior is shaped by the use of algorithms and big data management. Taking into consideration the fact that students use the internet and social media platforms daily, this paper aims to examine their perceptions and viewpoints on the operation of algorithms and data management on the Internet. According to a survey conducted by the authors, which consists of 200 respondents, two-thirds of students notice the results of the algorithmic personalization, filtered selection of content and news, and the customized display of content on social media.  Even though 70% of them realize that user activities are continually monitored and that control over personal data online is taken over by large companies and/or a third party, most respondents express only moderate concern for their data online (82%), which further confirms the fact that only a small percentage of students (18%) almost always read the terms of use on a website, application, or internet service. Keywords: social media, big data, algorithms, student