From Analysis to Visualization

Our friend and colleague Jonathan Michael Borwein excelled in an incredibly vast range of fields: his work spans from pure analysis, functional analysis and maximal monotone operators, applied optimization, multiobjective optimisation, numerical and computational analysis, mathematics for high performance computing, probability theory, and many more. Jon’s mathematical breadth and his naturally inquisitive energy makes a unique, fascinating picture, that extends far beyond traditional areas such as functional analysis and number theory.

We consider conditions ensuring the monotonicity of right and left Riemann sums of a function \(f:[0,1]\rightarrow \mathbb {R}\) with respect to uniform partitions. Experimentation suggests that symmetrization may be important and leads us to results such as: if f is decreasing on [0, 1] and its symmetrization, \(F(x) := \frac{1}{2}\left( f(x) + f(1-x)\right) \) , is concave then its right Riemann sums increase monotonically with partition size. Applying our results to functions such as \(f(x) = 1/\left( 1+x^2\right) \) also leads to a nice application of Descartes’ rule of signs.

Measures of risk have grown in importance in expressing preferences between different manifestations of uncertain cost or loss in finance and engineering, but utility functions and expected utility have had a more traditional role. This article surveys how risk and utility are in fact more closely related than may have been appreciated by practitioners. The tools of convex analysis, including conjugate duality, are able to bring this out.

We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail.

We focus on the convergence analysis of averaged relaxations of cutters, specifically for variants that—depending upon how parameters are chosen—resemble alternating projections, the Douglas–Rachford method, relaxed reflect-reflect, or the Peaceman–Rachford method. Such methods are frequently used to solve convex feasibility problems. The standard convergence analysis of projection algorithms is based on the firm nonexpansivity property of the relevant operators. However, if the projections onto the constraint sets are replaced by cutters (which may be thought of as maps that project onto separating hyperplanes), the firm nonexpansivity is lost. We provide a proof of convergence for a family of related averaged relaxed cutter methods under reasonable assumptions, relying on a simple geometric argument. This allows us to clarify fine details related to the allowable choice of the relaxation parameters, highlighting the distinction between the exact (firmly nonexpansive) and approximate (strongly quasinonexpansive) settings. We provide illustrative examples and discuss practical implementations of the method.

Jon was a passionate advocate for mathematics research and education. It is a sense of joviality and inquiry that sustained his dedication to both. His active engagement with the communication and growth of mathematics research and pedagogy crossed popular and academic lines, and spanned from primary to tertiary platforms. The papers contained in this volume showcase his dynamic set of research interests, which is equally mirrored in the education-themed section of the September 2017 Jonathan Borwein Commemorative Conference, and its satellite meetings on Indigenising mathematics curricula, entitled “Mathematics and Education: Spirit, Culture and Community”. As a somatic approach, haptics and heuristics became integral to Jon’s model of mathematical education, with long-reaching ramifications, which this section explores through topical considerations.

This chapter describes possible legacies in mathematical culture and education of the work and influence of Jonathan M. Borwein. Three lenses are applied: views expressed by members of the community in a panel session on “Maths, Education, Research and Culture” at the Jonathan M Borwein Commemorative Conference, September 2017; scholarship of schisms, divisions and disciplinarity; and Jonathan’s own words describing his practice of experimental mathematics and a philosophy of mathematical humanism. His work has evident potential for lasting transformational effect for the good of mathematics education, which emerges from each of our three different lenses. This potential is complex and rich. Jonathan’s defining impact on experimental mathematics as a research approach is entwined with his view of experimental mathematics as a teaching methodology, and provides the ground for one such legacy. In both research and educational uses of experimental “mathodology”, Jonathan’s aim was the creation of insight and intuition, and in this context his work and writing draws our attention to the significance of tools in enabling and extending human capacity. Another facet of legacy is the spanning of math/education schisms that is exemplified by Jonathan as an interdisciplinary practitioner whose work was influenced by his early love of history and his profound appreciation of music, art and philosophy. Jonathan’s writing extended beyond mathematics into various areas including educational scholarship and the philosophy of mathematics. His role as a public intellectual fostered another potential legacy: modelling a newly vigorous stewardship of mathematics by the collective custodians of the discipline.

A project conducted in six Australian universities was designed to foster interdisciplinary collaboration between mathematicians and mathematics educators in pre-service teacher education programmes. This paper reports on two approaches to integrating mathematics content and pedagogy arising from these collaborations: co-developed and co-taught courses, and communities of practice linking pre-service and graduate teachers. The findings illustrate ways of increasing interdisciplinary understanding in preparing future teachers of mathematics.

In the spirit of Jonathan Borwein’s opportunistic and inventive use of computers in the development of the field of Experimental Mathematics, this paper recommends a computational ‘turn’ in school mathematics. At a time when students are increasingly moving away from mathematics in the senior years of schooling, we need to reconsider the relevance of current mathematics curricula and traditional approaches to mathematics pedagogy. Computational applications are transforming the world that we live in and just as Experimental Mathematics challenged the foundations of the discipline of mathematics, computational approaches are also changing almost all traditional fields of study. By persisting with the teaching of manual computation in school mathematics, we are denying the current and future worlds of our students. By doing so, we risk increasing our students’ lack of interest in mathematics as it will progressively be seen as an historical curiosity of little relevance. This paper proposes some key questions for mathematics educators to consider in order to make mathematics more relevant and of interest to today’s students.

The development, implementation and assessment of a Mathematics Workshop for Indigenous school students is described and analysed. The Workshop formed part of an intensive week-long programme conducted by the Faculty of Engineering and Information Technologies at the University of Sydney in September 2017 called the STEM Spring Workshop. The curriculum and teaching schedule development incorporated an assessment of the students’ backgrounds and a presurvey, and includes an extended description of the development of two extracurricular topics on Cryptography and the Rubik’s cube. The interactions during the classes are then described, with an emphasis on the ongoing evolution and adaptation of the teaching modes in response to real-time student feedback. The assessments of the classes were highly positive with many students indicating a desire for more time on the mathematical activities. The principle of cultural plasticity (being receptive to, learning from and adapting to the cultural perspectives of others) as used in the Mathematics Workshop is described.

How to engage Indigenous students in their education particularly to mathematics is a question being regularly discussed within schools and academic forums. This paper proposes embracing the Indigenous methodology of narrative and combining it with existing origami narrative methodologies to engage Indigenous students to mathematics principles at many different cognitive stages. Origami, the art of folding paper or other materials, is an ancient art form that has occurred across multiple societies including Indigenous and non-Indigenous populations where the folding of paper or fabrics has presented cultural knowledge from within that society. This paper reviews the connections between origami and mathematics education as well as their use in an Indigenous setting by incorporating Indigenous narrative aspects into the origami art form. We present theory in support of the use of storigami as an educational tool in Indigenous mathematics classrooms, and we provide original examples of the adaptation of origami techniques for representing and engaging with Australian Indigenous art and culture while simultaneously engaging the students mathematically.

We provide dynamic visual models of the following facts established by ancient mathematicians:

We contrast the clarity of the models by outlining formal mathematical proofs based on those timeless ideas. We also reflect about the place that proofs play in the calculus classroom.

We describe our adventures in creating a new first-year course in Experimental Mathematics that uses active learning. We used a state-of-the-art facility, called The Western Active Learning Space, and got the students to “drive the spaceship” (at least a little bit). This paper describes some of our techniques for pedagogy, some of the vignettes of experimental mathematics that we used, and some of the outcomes. EYSC was a student in the simultaneously-taught senior sister course “Open Problems in Experimental Mathematics” the first time it was taught and an unofficial co-instructor the second time. Jon Borwein attended the Project Presentation Day (the second time) and gave thoughtful feedback to each student. This paper is dedicated to his memory.

It is well-known that Jonathan Borwein had numerous wide-ranging interests and intellectual persuasions. Moreover, he never stopped only at purely musing a subject, but often concerned himself with its impact on science and society. Jon’s involvement in financial mathematics research is an excellent illustration of these multidisciplinary interests.

It is well documented that academics and practitioners focus on statistical significance (typically represented by P tests) and statistical hypothesis testing to determine if their non-statistical analytical hypothesis is correct or likely to be correct. Moreover, statistical significance is relied upon to determine which variables should be used in their models or analyses. In spite of ongoing criticism, this practice continues to the detriment of robust scientific analysis. I discuss the significant limitations of statistical significance in scientific analysis, irrespective of discipline, with some focus on economics. I place statistical significance in a broader analytical context, discussing other analytical procedures that need to be followed and emphasized for one’s analysis to be scientifically robust. This relates the development of models, assumptions underlying the models, the data collected and constructed, the relationship between statistical significance and causality and the importance of non-statistical theory to the identification of pertinent modelling variables. Analytical significance (size effects and variability) is core to any robust scientific analysis, but only in the context of all of the other prior steps in the applied research project being in place. In this broader analytical framework, the statistical significance becomes relatively insignificant. I also address why statistical significance and statistical hypothesis testing dominates the applied analytical landscape even though this dominance is not best practice. Of critical importance are the mental models of best practice and the worldview and preferences of decision makers determining what gets published and who is successful in securing grants and employment.

Many investors rely on market experts and forecasters when making investment decisions, such as when to buy or sell securities. Ranking and grading market forecasters provide investors with metrics on which they may choose forecasters with the best record of accuracy for their particular market exposure. This study develops a novel ranking methodology to rank the market forecaster. In particular, we distinguish forecasts by their specificity, rather than considering all predictions and forecasts equally important, and we also analyze the impact of the number of forecasts made by a particular forecaster. We have applied our methodology on a dataset including 6,627 forecasts made by 68 forecasters.

We highlight the role of entropy maximization in several fundamental results in financial mathematics. They are the two fund theorem for Markowitz efficient portfolios, the existence and uniqueness of a market portfolio in the capital asset pricing model, the fundamental theorem of asset pricing, the selection of a martingale measure for pricing contingent claims in an incomplete market and the calculation of super/sub-hedging bounds and portfolios. The connection of diverse important results in finance with the method of entropy maximization indicates the significant influence of methodology of physical science in financial research.

At the Jonathan Borwein Commemorative Conference (JBCC), the talks were divided into several themes, one of which was “Number Theory, Special Functions and \(\pi \)”. Here we summarise the eight contributions to that theme included in the Proceedings. There is a considerable overlap with other themes, and with Experimental Mathematics. The eight contributions described here are as follows:

We show that the Mahler measure of every Borwein polynomial—a polynomial with coefficients in \( \{-1,0,1 \}\) having non-zero constant term—can be expressed as a maximal Lyapunov exponent of a matrix cocycle that arises in the spectral theory of binary constant-length substitutions. In this way, Lehmer’s problem for height-one polynomials having minimal Mahler measure becomes equivalent to a natural question from the spectral theory of binary constant-length substitutions. This supports another connection between Mahler measures and dynamics, beyond the well-known appearance of Mahler measures as entropies in algebraic dynamics.

We consider some of Jonathan and Peter Borweins’ contributions to the high-precision computation of \(\pi \) and the elementary functions, with particular reference to their book Pi and the AGM (Wiley, 1987). Here “AGM” is the arithmetic–geometric mean of Gauss and Legendre. Because the AGM converges quadratically, it can be combined with fast multiplication algorithms to give fast algorithms for the n-bit computation of \(\pi \), and more generally the elementary functions. These algorithms run in “almost linear” time \(O(M(n)\log n)\), where M(n) is the time for n-bit multiplication. We outline some of the results and algorithms given in Pi and the AGM, and present some related (but new) results. In particular, we improve the published error bounds for some quadratically and quartically convergent algorithms for \(\pi \), such as the Gauss–Legendre algorithm. We show that an iteration of the Borwein-Borwein quartic algorithm for \(\pi \) is equivalent to two iterations of the Gauss–Legendre quadratic algorithm for \(\pi \), in the sense that they produce exactly the same sequence of approximations to \(\pi \) if performed using exact arithmetic.

We present an idiosyncratic view of the race for quantum computational supremacy. Google’s approach and IBM challenge are examined. An unexpected side effect of the race is the significant progress in designing fast classical algorithms. Quantum supremacy, if achieved, won’t make classical computing obsolete.

It is shown that a certain nonlinear expression for Bernoulli polynomials, related to higher-order convolutions, can be evaluated as a product of simple linear polynomials with integer coefficients. The proof involves higher-order Bernoulli polynomials. A similar result for Euler polynomials is also obtained, and identities for Bernoulli and Euler numbers follow as special cases.

In this paper, the metric theory of Diophantine approximation associated with mixed type small linear forms is investigated. We prove Khintchine–Groshev type theorems for both the real and complex number systems. The motivation for these metrical results comes from their applications in signal processing. One such application is discussed explicitly.

Brun’s constant is \(B=\sum _{p \in P_{2}} p^{-1} + (p+2)^{-1}\), where the summation is over all twin primes. We improve the unconditional bounds on Brun’s constant to \(1.840503< B < 2.288490\), which are about 13% tighter.

The pslq algorithm computes integer relations for real numbers and Gaussian integer relations for complex numbers. We endeavour to extend pslq to find integer relations consisting of algebraic integers from some quadratic extension fields (in both the real and complex cases). We outline the algorithm, discuss the required modifications for handling algebraic integers, problems that have arisen, experimental results, and challenges to further work.