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

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods

Integral inequalities of hermite-hadamard type and their applications

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 17 October 2016.The role of mathematical inequalities in the growth of different branches of mathematics as well as in other areas of science is well recognized in the past several years. The uses of contributions of Newton and Euler in mathematical analysis have resulted in a numerous applications of modern mathematics in physical sciences, engineering and other areas sciences and hence have employed a dominat effect on mathematical inequalities. Mathematical inequalities play a dynamic role in numerical analysis for approximation of errors in some quadrature rules. Speaking more specifically, the error approximation in quadrature rules such as the mid-point rule, trapezoidal rule and Simpson rule etc. have been investigated extensively and hence, a number of bounds for these quadrature rules in terms of at most second derivative are proven by a number of researchers during the past few years. The theorey of mathematical inequalities heavily based on theory of convex functions. Actually, the theory of convex functions is very old and its commencement is found to be the end of the nineteenth century. The fundamental contributions of the theory of convex functions can be found in the in the works of O. Hölder [50], O. Stolz [151] and J. Hadamard [48]. At the beginning of the last century J. L. W. V. Jensen [72] first realized the importance convex functions and commenced the symmetric study of the convex functions. In years thereafter this research resulted in the appearance of the theory of convex functions as an independent domain of mathematical analysis. Although, there are a number of results based on convex function but the most celebrated results about convex functions is the Hermite-Hadamard inequality, due to its rich geometrical significance and many applications in the theory of means and in numerical analysis. A huge number of research articles have been written during the last decade by a number of mathematicians which give new proofs, generalizations, extensions and refitments of the Hermite-Hadamard inequality. Applications of the results for these classes of functions are given. The research upshots of this thesis make significant contributions in the theory of means and the theory of inequalities.MT 201

Αριθμητική ολοκλήρωση με σημεία ρίζες πολυωνύμων του Chebyshev

Το θέμα της εργασίας, όπως δηλώνει και ο τίτλος της, είναι η αριθμητική ολοκλήρωση, δηλαδή, η προσέγγιση της τιμής ενός ορισμένου ολοκληρώματος με μια αριθμητική μέθοδο. Η αριθμητική ολοκλήρωση αποτελεί κλασσικό θέμα της αριθμητικής ανάλυσης και η χρησιμότητά της έγγυται σε δυο βασικούς λόγους: Αν f είναι η συνάρτηση που ολοκληρώνουμε, τότε μια παράγουσά της F μπορεί να προσδιορισθεί αναλυτικά μόνο σε σπάνιες περιπτώσεις, ενώ ακόμα κι όταν αυτό είναι εφικτό, ο υπολογισμός της F μπορεί να είναι ασύμφορος. Για την, κατά το δυνατόν, αρτιότερη παρουσίαση των εννοιών που μελετάμε, η εργασία διαρθρώνεται σε τέσσερα κεφάλαια, όπου: Στο πρώτο κεφάλαιο εισάγουμε την έννοια των ορθογωνίων πολυωνύμων σημειώνοντας τις βασικότερες ιδιότητές τους, όπως ο αναδρομικός τύπος τους, οι ιδιότητες των ριζών τους, η ταυτότητα Christoffel-Darboux και κατόπιν τα πολυώνυμα του Chebyshev πρώτου και δευτέρου είδους. Στο δεύτερο κεφάλαιο παρουσιάζουμε βασικά στοιχεία αριθμητικής ολοκλήρωσης: Τύπους αριθμητικής ολοκλήρωσης εκ παρεμβολής, το βαθμό ακριβείας τους, σύγκλιση αυτών των τύπων για διάφορες κλάσεις συναρτήσεων, καθώς και το σφάλμα τους, μέσω μεθόδων χώρων Hilbert, για αναλυτικές συναρτήσεις. Στο τρίτο κεφάλαιο μελετάμε τέσσερις συγκεκριμένους τύπους αριθμητικής ολοκλήρωσης εκ παρεμβολής: Τους λεγόμενους τύπους του Fejer πρώτου και δευτέρου είδους, τον τύπο του Basu και τον τύπο των Clenshaw-Curtis, εξετάζοντας για τον καθένα τα ζητήματα που αναλύσαμε στο τρίτο κεφάλαιο. Στο τέταρτο κεφάλαιο προχωρούμε σε κάποια αριθμητικά παραδείγματα. Συγκεκριμένα, υπολογίζουμε το σφάλμα των τύπων που μελετήσαμε στο τρίτο κεφάλαιο για μια σειρά από συναρτήσεις, προσεγγίζουμε το ολοκλήρωμα μιας συνάρτησης που παρουσιάζει μια ανωμαλία στο ένα άκρο του διαστήματος ολοκλήρωσης, ενώ τέλος βρίσκουμε φράγματα για το σφάλμα του τύπου του Fejer δευτέρου είδους για αναλυτικές συναρτήσεις.The theme of the work, as indicated by its title, is the numerical integration, ie, the estimated price of a certain integral with a numerical method. The numerical integration is a classic issue of numerical analysis and its utility New is two main reasons: If f is the function we conclude, then a factor of F can be determined only analytically in rare cases, and even when it is feasible, the calculation of F can be disadvantageous. For, if possible, better presentation of the concepts we are studying, the work is divided into four chapters, where: In the first chapter we introduce the concept of orthogonal polynomials noting their basic properties, such as the retro type, the properties of their roots, the Christoffel-Darboux identity and then the Chebyshev polynomials of the first and second kind. In the second chapter we present basic numerical integration elements: Types numerical integration interpolated, the degree of accuracy, convergence of these types of functions for different classes and their error, by methods Hilbert spaces, for analytic functions. In the third chapter we study four specific types of numerical integration interpolated: They called Fejer types of first and second type, the type of Basu and type of Clenshaw-Curtis, looking for everyone matters analyzed in the third chapter. In the fourth chapter we move to some numerical examples. Specifically, we estimate the error of the type studied in the third chapter of a series of functions, we approach the integral of a function that shows an anomaly at one end of the integration period, and finally we find bounds for the error type of Fejer second type for analytic functions

Spinfoams and high performance computing

Numerical methods are a powerful tool for doing calculations in spinfoam theory. We review the major frameworks available, their definition, and various applications. We start from $\texttt{sl2cfoam-next}$, the state-of-the-art library to efficiently compute EPRL spin foam amplitudes based on the booster decomposition. We also review two alternative approaches based on the integration representation of the spinfoam amplitude: Firstly, the numerical computations of the complex critical points discover the curved geometries from the spinfoam amplitude and provides important evidence of resolving the flatness problem in the spinfoam theory. Lastly, we review the numerical estimation of observable expectation values based on the Lefschetz thimble and Markov-Chain Monte Carlo method, with the EPRL spinfoam propagator as an example.Comment: 33 pages, 11 figures. Invited chapter for the book "Handbook of Quantum Gravity" (Eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer Singapore, expected in 2023

Richard von Mises’ work for ZAMM until his emigration in 1933 and glimpses of the later history of ZAMM

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