scholarly journals Jonathan Michael Borwein 1951 − 2016: Life and Legacy

2021 ◽  
Vol 1 (2) ◽  
Author(s):  
Richard Brent
Keyword(s):  
Number Theory ◽  
Special Functions ◽  
Mathematical Finance ◽  
Mathematical Education ◽  
Experimental Mathematics

Jonathan M. Borwein (1951−2016) was a prolific mathematician whose career spanned several countries(UK, Canada, USA, Australia) and whose many interests includedanalysis, optimization, number theory, special functions, experimental mathematics, mathematical finance, mathematical education,and visualization. We describe his life and legacy, and give anannotated bibliography of some of his most significant books and papers.

scholarly journals A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2294
Author(s):  
Hari Mohan Srivastava
Keyword(s):  
Number Theory ◽  
Special Functions ◽  
Applied Mathematics ◽  
Analytic Number Theory ◽  
Review Article ◽  
Mathematical Functions ◽  
Analytic Number ◽  
Recent Developments ◽  
Transcendental Functions ◽  
Mathematical Sciences

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.

scholarly journals Experimenting with Apéry Limits and WZ pairs

2021 ◽  
Vol 1 (2) ◽  
Author(s):  
Robert Dougherty-Bliss ◽  
Doron Zeilberger
Keyword(s):  
Number Theory ◽  
Infinite Family ◽  
Experimental Mathematics

This article, dedicated with admiration in memory of Jon and Peter Borwein,illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apéry limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apéry limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationalitymeasures (lower than Liouville's generic 3), and that we conjectured to go down to the optimal 2. As it turned out, as pointed out by Paul Voutier (see the postscript kindly written by him), our conjectures follow from deep results in number theory. Nevertheless we believe that further experiments with our Maple programs would lead to new and interesting results.

scholarly journals Mathematics and outdoor photography experience – exploration of an approach to mathematical education, based on the theory of Dewey’s aesthetics

2018 ◽  
Vol 6 (2) ◽  
Author(s):  
Antje Meier ◽  
Markku S. Hannula ◽  
Miika Toivanen
Keyword(s):  
Mathematics Education ◽  
Visual Attention ◽  
Positive Impact ◽  
Group Activity ◽  
Teaching Mathematics ◽  
Mathematical Education ◽  
Experimental Mathematics ◽  
Gaze Tracking ◽  
The Gaze ◽  
Teacher Students

Based on Dewey’s theory of art, aesthetics, and experiences and photographer Barnbaums’ writing about expanded perception through photography, we conducted a one-day experimental mathematics education unit. Using photography in outdoor conditions had a positive impact on teacher students’ perception of the use of photography for teaching mathematics. To study the changes in students’ visual attention deeper, we used gaze-tracking to analyse one student’s visual attention when walking outdoors after the activity. The gaze data indicated that more visual attention was given to objects he had photographed or discussed during the group activity in comparison to other objects.

scholarly journals From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results

2021 ◽  
Vol 18 (2 Jul-Dec) ◽  
pp. 020203
Author(s):  
Wolfgang Bietenholz
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Number Theory ◽  
Zeta Function ◽  
Casimir Force ◽  
Vacuum Energy ◽  
Special Functions ◽  
Divergent Series ◽  
Quantum Field ◽  
Riemann’S Zeta Function

A century ago Srinivasa Ramanujan --- the great self-taught Indian genius of mathematics --- died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, $\sum_{n \geq 1} n$ and $\sum_{n \geq 1} n^{3}$. These values are sensible, however, as analytic continuations, which correspond to Riemann's $\zeta$-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured.  We discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory.

Introduction to Experimental Mathematics

2017 ◽  
Author(s):  
Soren Eilers ◽  
Rune Johansen
Keyword(s):  
Experimental Mathematics
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