Asymptotic approximations to the Hardy-Littlewood function

The function $Q(x):=\sum_{n\ge 1} (1/n) \sin(x/n)$ was introduced by Hardy and Littlewood [10] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a surprising appearance in the recent disproof by Alzer, Berg and Koumandos [1] of a conjecture by Clark and Ismail [2]. More precisely, Alzer et. al. [1] have shown that the Clark and Ismail conjecture is true if and only if $Q(x)\ge -\pi/2$ for all $x>0$. It is known that $Q(x)$ is unbounded in the domain $x \in (0,\infty)$ from above and below, which disproves the Clark and Ismail conjecture, and at the same time raises a natural question of whether we can exhibit at least one point $x$ for which $Q(x) < -\pi/2$. This turns out to be a surprisingly hard problem, which leads to an interesting and non-trivial question of how to approximate $Q(x)$ for very large values of $x$. In this paper we continue the work started by Gautschi in [7] and develop several approximations to $Q(x)$ for large values of $x$. We use these approximations to find an explicit value of $x$ for which $Q(x)<-\pi/2$.Comment: 16 pages, 3 figures, 2 table

Terminal Summation: Extending the Concept of Convergence

This paper presents an atypical method for summing divergent series, and provides a sum for the divergent series log(n). We use an idea of T.E. Phipps, called Terminal Summation, which uses asymptotic analysis to assign a value to divergent series. The method associates a series to an appropriate difference equations having boundary conditions at infinity, and solves the difference equations which then provide a value for the original series. We point out connections between Phipps\u27 method, the Euler-MacLaurin sum formula, the Ramanujan sum and other traditional methods for summing divergent series

COMPUTER TOOLS FOR SOLVING MATHEMATICAL PROBLEMS: A REVIEW

The rapid development of digital computer hardware and software has had a dramatic influence on mathematics, and contrary. The advanced hardware and modern sophistical software such as computer visualization, symbolic computation, computerassisted proofs, multi-precision arithmetic and powerful libraries, have provided resolving many open problems, a huge very difficult mathematical problems, and discovering new patterns and relationships, far beyond a human capability. In the first part of the paper we give a short review of some typical mathematical problems solved by computer tools. In the second part we present some new original contributions, such as intriguing consequence of the presence of roundoff errors, distribution of zeros of random polynomials, dynamic study of zero-finding methods, a new three-point family of methods for solving nonlinear equations and two algorithms for the inclusion of a simple complex zero of a polynomial

Oodatavate vastuste ja arvutialgebra süsteemide vastuste erinevused koolimatemaatika võrrandite puhul

Arvutialgebra süsteemidega saab lahendada erinevat tüüpi matemaatikaülesandeid, sealhulgas koolimatemaatika võrrandeid. Sageli langevad arvutialgebra süsteemide vastused kokku koolikontekstis oodatavate vastustega (koolivastustega), vahel aga mitte. Sellised ootamatud arvutialgebra süsteemide vastused on tihti küll matemaatiliselt korrektsed, aga mõne teise standardi järgi, näiteks kompleksarvude vallas. Arvutialgebra süsteemide vastuste ja koolivastuste erinevuste süstemaatiline ülevaade on kasulik arvutialgebra süsteemide arendamisel ning õppetöö planeerimisel. Käesolev dissertatsioon annab ülevaate arvutialgebra süsteemide vastuste ja koolivastuste erinevustest ning nende põhjustest koolimatemaatika võrrandite puhul. Erinevuste spektrit selgitatakse kahe võimaliku klassifikatsiooni abil. Esimese klassifikatsiooni aluseks on see, kas arvutialgebra süsteemi vastus sisaldab rohkem või vähem lahendeid kui oodatav vastus. Teine klassifikatsioon on sisupõhisem ja toob esile vastuste kuju, täielikkuse, arvuvallast sõltuvuse, harunemise ja automaatse lihtsustamise teemad. Arvuvalla ja harunemisega seotud erinevusi käsitletakse dissertatsioonis põhjalikumalt eraldi peatükkides. Koolivastuste ja arvutialgebra süsteemide vastuste erinevusi saab kasutada õpetamisel ja õppimisel. Käesolev dissertatsioon pakubki välja pedagoogilise lähenemise, mis põhineb arvutialgebra süsteemide vastuste ja õppijate endi vastuste võrdlemisel paaristööna. Lisaks õpetamisele ja õppimisele saab selle formaadiga koguda andmeid õppijate teemamõistmise kohta. Väljapakutud lähenemist kasutati tunnisituatsioonis trigonomeetriliste võrrandite käsitlemisel. Põhjalikumalt analüüsiti, kui adekvaatselt õppijad tuvastasid enda vastuse ja arvutialgebra vastuse ekvivalentsust/mitteekvivalentsust ja korrektsust. Leiti, et isegi kui õppijate lahendus paistab korrektne, võib siiski olla lünki arusaamises.It is possible to solve most mathematical problems, including equations of school mathematics, with the help of Computer Algebra Systems (CAS). The answers offered by CAS (CAS answers) often coincide with the answers that are expected in the school context (school answers), but sometimes not. Such unexpected CAS answers are often correct, but based on different standards, for example in complex domain. A systematic review of the differences between CAS answers and school answers is useful for development of CAS and organizing the teaching process. A review of the differences between CAS answers and school answers and their reasons in case of school mathematics equations is provided in this dissertation. The spectrum of differences is explained by using two possible classifications. A key criterion of the first classification is comparing whether the CAS answer includes a larger or a smaller number of solutions than the expected answer. The other classification is more content-oriented, highlighting the issues of the form, completeness, dependence on the number domain, branching and automatic simplification of answers. The differences caused by number domain and branching are discussed separately in greater depth in separate chapters. The differences between school answers and CAS answers can be used in teaching and learning. This dissertation proposes a pedagogical approach that is based on comparative discussions on students' answers and CAS answers in pairs. In addition to teaching and learning, the format is also suitable for collecting data on students' understandings and misunderstandings. The proposed approach was used in lessons on trigonometric equations. The focus was on analyzing whether students can adequately identify the equivalence/non-equivalence and correctness of their answer and CAS answer. It is found that even if a student's solution looks to be correct, students can have misunderstandings and knowledge gaps

Divergent Perturbation Series

Various perturbation series are factorially divergent. The behavior of their high-order terms can be found by Lipatov's method, according to which they are determined by the saddle-point configurations (instantons) of appropriate functional integrals. When the Lipatov asymptotics is known and several lowest order terms of the perturbation series are found by direct calculation of diagrams, one can gain insight into the behavior of the remaining terms of the series. Summing it, one can solve (in a certain approximation) various strong-coupling problems. This approach is demonstrated by determining the Gell-Mann - Low functions in \phi^4 theory, QED, and QCD for arbitrary coupling constants. An overview of the mathematical theory of divergent series is presented, and interpretation of perturbation series is discussed. Explicit derivations of the Lipatov asymptotic forms are presented for some basic problems in theoretical physics. A solution is proposed to the problem of renormalon contributions, which hampered progress in this field in the late 1970s. Practical schemes for summation of perturbation series are described for a coupling constant of order unity and in the strong-coupling limit. An interpretation of the Borel integral is given for 'non-Borel-summable' series. High-order corrections to the Lipatov asymptotics are discussed.Comment: Review article, 45 pages, PD

Métodos numérico-simbólicos para calcular soluciones liouvillianas de ecuaciones diferenciales lineales

El objetivo de esta tesis es dar un algoritmo para decidir si un sistema explicitable de ecuaciones diferenciales kJiferenciales de orden superior sobre las funciones racionales complejas, dado simbólicamente,admite !Soluciones liouvillianas no nulas, calculando una (de laforma dada por un teorema de Singer) en caso !afirmativo. mediante métodos numérico-simbólicos del tipo Introducido por van der Hoeven.donde el uso de álculo numérico no compromete la corrección simbólica. Para ello se Introduce untipo de grupos algebraicos lineales, los grupos euriméricos, y se calcula el cierre eurimérico del grupo de Galois diferencial,mediante una modificación del algoritmo de Derksen y van der Hoeven, dado por los generadores de Ramis.Departamento de Algebra, Análisis Matemático, Geometría y Topologí

Basic Analysis: Introduction to Real Analysis

This free online textbook (OER more formally) is a course in undergraduate real analysis (somewhere it is called advanced calculus ). The book is meant both for a basic course for students who do not necessarily wish to go to graduate school, but also as a more advanced course that also covers topics such as metric spaces and should prepare students for graduate study. A prerequisite for the course is a basic proof course. An advanced course could be two semesters long with some of the second-semester topics such as multivariable differential calculus, path integrals, and the multivariable integral using the second volume. There are more topics than can be covered in two semesters, and it can also be reading for beginning graduate students to refresh their analysis or fill in some of the holes

Techniques for Accelerating Iterative Methods for the Solution of Mathematical Problems

This thesis discusses the solving of mathematical problems by accelerating sequences generated by a numerically derived itertive scheme. Acceleration methods are applied to these sequences to attempt to speed up their convergence or to force the convergence of a divergent sequence. The thesis presents the derivation of some of these acceleration methods and then compares the methods theoretically and numerically. Numerical results are presented in the form of tables and/or graphs.Higher Educatio