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Laureate Professor Jon Borwein

Laureate Professor

School of Mathematical and Physical Sciences (Mathematics)

The Life of Dr π

With a reputation as the world's most renowned π expert, University of Newcastle Laureate Professor Jon Borwein has spent more than a quarter of a century pioneering the field of experimental mathematics.

Now he is helping industries like medicine and music unearth further value from his mathematical breakthroughs.

Founder and head of the University of Newcastle's Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA), Prof Borwein has used the computer as a laboratory for more than a quarter of a century, with just some of his feats including producing the world's largest mathematical picture (of π) and proving a 60+ year-old conjecture incorrect.

When asked what brought him to Newcastle, Prof Borwein jokes that "CARMA brought me here", adding that, "the University of Newcastle is a world-class mathematical hotspot. If you want an interesting life, then you need to follow the interesting work".

The life of Doctor π - a title he has gained for his fascination and breakthroughs in understanding the irrational number calculated by dividing a circle's circumference by its diameter - has certainly been an interesting one.

Laureate Professor Jon Borwein

A finger in every π

A Rhodes Scholar and Oxford Doctor of Philosophy, Prof Borwein has held professorships as a pure mathematician, an applied mathematician, an operations researcher and a computer scientist.

Working with a new method of visualising large strings of numbers through random walks – a path described by a sequence of randomly generated points in the plane – Prof Borwein and his colleagues produce papers and digital images using computers to show information at a glance that would take many pages to set out in written numerical form.

Some of the algorithms visual resemblance to genomes and other structures seen in nature are not lost on Prof Borwein and his team. They are working with a number of other leading researchers and industry to see the potential applications and learnings from CARMA's breakthroughs, in fields as diverse as genetics, cognitive science, the fine arts, operations research and theoretical computer science.

For example, Prof Borwein is collaborating with radiology at the University of Newcastle to explore mathematical equations in spectral analysis in functional MRI. On another project, he is working with the University of Newcastle's Conservatorium to showcase the mathematical patterns of what constitutes beautiful music.

In celebration of π Day on March 14, 2013, Prof Borwein together with a team of researchers produced the world's largest mathematical picture, which was of π. The number of digits for π are infinite and seem random, however as computers have become more powerful we now know many more of the digits than ever before.

Borwein and his team had each digit describe the direction and distance being moved and used colour to designate time. The colours followed the RYPGPOB sequence, with red and yellow being used earlier on. The further the colours went along the spectrum, the later it was in the sequence.

"Since we could not see some of the colours in the image, it raised questions we would not have normally asked in a maths laboratory, like 'Where is the yellow?'. This led us to realise that some `normal' numbers, but not π, actually reroute over the same pathway several times," Prof Borwein said.

Maths in the genes

When you look at Borwein's genetics, the probability that he would end up in mathematics seems almost inevitable. His family tree reads like a who's who of contemporary mathematics - with his father, brother and son-in law all renowned mathematicians in their own right.

Prof Borwein collaborated with his brother, Peter Borwein, together with David H. Bailey and Simon Plouffe, in developing the famous Bailey-Borwein-Plouffe (BBP) formula. The BBP formula calculates the value of π and allows the binary digits of π and other constants to be determined without knowing the previous digits. With IBM, they are also calculating digits of π squared beginning at the 10th trillionth place in both base three and base two.

Each year since 1987 he has also completed at least one paper with his now 89-year old father, David Borwein.

As frequent collaborator, David H. Bailey, describes the Borweins in the foreword of one of two books on selected writings by the father-son duo, "The Borweins are inarguably the world's leading exponents of utilising state-of-the-art computer technology to discover and prove new and fundamental mathematical results."

"As a direct result of the Borwein's influence, hundreds of researchers worldwide are now engaged in "experimental" and computationally-assisted mathematics, and the pace of mathematics has measurably quickened," Bailey said.

Writing in the numbers

So prolific is Prof Borwein's output, Microsoft Academic has listed him as number 21 on their list of top mathematics authors of all time. He has also produced over a dozen books, including a monograph on convex functions which was a Choice 2011 Outstanding Academic Book, and over 450 refereed articles and proceedings.

His latest book, Lattice Sums Then and Now published by Cambridge Press, sits on the boundary between mathematical physics and number theory, and as with many of his written subjects surrounding mathematics, the book surprisingly begins with an unexpected connection – in this case a quote from Olivia Newton-John.

"Olivia Newton-John's grandfather was Nobel prize winner Max Born, an expert in lattice theory. Not many people realise this and it's a great way to engage people and make them think about mathematics in a different way. Her dad, Brin Newton-John, was actually the University of Newcastle's first Pro Vice-Chancellor!"

Changing the perspective on mathematics to include a physics-like inductive component rather than being exclusively a deductive science is a strong passion of Borwein's. 

Case in point is Borwein and his students' use of the computer as a laboratory to prove that the 60 year-old Guiga Conjecture about prime numbers can only fail for numbers with more than 20,000 decimal digits.

On a tour of the CARMA lab, there are clocks for New York, San Francisco, London and more lining the wall, that speak of the laboratory's extensive collaborations with leading mathematics institutions around the world. The group boasts an outstanding international profile in number theory and currently has 36 members, eight associates and seven students.

"The University of Newcastle is the only University to receive a rating of '5' for above world-standard research in the ERA rankings twice in Applied Mathematics," Prof Borwein said.

"In one minute we can now achieve calculations that would have been the subject of an applied maths PhD 40 years ago. I'm conducting intensive computer-assisted research on work that I once thought I had completed 15 years ago and which is providing new insights."

"One of the greatest attractions of science is that you don't know how ignorant you are until you acquire more knowledge. As has been said before, the best is the enemy of the better, we are constantly looking for new breakthroughs on things we previously thought we had solved or explored."

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The Life of Dr π

With a reputation as the world's most renowned π expert, University of Newcastle Laureate Professor Jon Borwein has spent more than a quarter of a century pioneering the field of experimental mathematics.Now he is helping industries like medicine and music…

Read more

Delving into the unknown

When people ask this world-renowned mathematician what brought him to Newcastle, he tells them it was CARMA.

Laureate Professor Jon Borwein sitting in front of his computer 

"It was once said that the only things a mathematician needed for research were a pad, pencil and a wastepaper basket," Laureate Professor Jon Borwein says. "That was never completely true – but it is even less so now."

Borwein has come a long way from using notepads and blackboards in the 40-plus years that he has been grappling with complex mathematical equations. These days the director of the University's Centre for Computer-Assisted Research Mathematics and its Applications (CARMA) works with vivid digital images that can portray at a glance information that would take many pages of figures to set out in written form.

"I have not found a piece of mathematics that I cannot learn more effectively and quickly with computer assistance," he says.

Mathematical visualisation through the use of graphics is one of the focuses of CARMA, which seeks to harness developments in information and communication technology to further the exploration of maths and undertake ever more challenging calculations.

Borwein is a pioneer in what is known as experimental maths, in which computers are used to run computations to look for patterns, identify sequences and gather evidence in support of specific mathematical assertions. His work at Newcastle has put the University at the cutting edge of this field.

His textbooks, papers and blogs are widely read and discussed by the international mathematics community. In recognition of his contributions to research, he was recently awarded a Fellowship of the Australian Academy of Science, the country's most elite group of scientists.

Borwein has had a lifelong fascination with Pi, the irrational number calculated by dividing a circle's circumference by its diameter. This fascination led to the development of the famous Borwein Algorithm, a formula he devised with his brother Peter for calculating the value of 1/Pi.

To celebrate Pi Day on March 14 (3.14), Borwein demonstrated his mathematical prowess by teaming with several other masterminds to complete what he believes is the largest single dedicated computation ever done: calculating digits of Pi squared beginning at the 10 trillionth place.

Borwein estimates it would have taken more than 1,380 years to calculate on a single PC what he and the team achieved with exclusive access to IBM's Blue Gene/P computer system, one of the world's most powerful platforms.

Where does this sort of research lead? Borwein says there are abundant applications– from the burgeoning field of random number generation widely used in banking, to testing the integrity of computer programs. For example, a 29 million digit calculation of Pi at NASA just after the 1986 shuttle disaster uncovered hardware and software faults in the space agency's supercomputer.

But to a large extent experimental maths is an exhilarating journey towards new frontiers.

"In one minute we can now achieve calculations that would have been the subject of an applied maths PhD 40 years ago," Borwein says. "At the same time, I am conducting intensive computer-assisted research on work I once thought I had completed 15 years ago.

"One of the greatest attractions of science is that you do not know how ignorant you are until you acquire more knowledge."

Visit the Centre for Computer-Assisted Research Mathematics and its Applications website

Laureate Professor Jon Borwein sitting in front of a computer

Delving into the unknown

Laureate Professor John Borwein grapples with complex mathematical equations using Computer-Assisted Research Mathematics

Read more

Career Summary

Qualifications

  • Doctor of Philosophy, University of Oxford - UK
  • Bachelor of Arts (Mathematics), University of Western Ontario - Canada
  • Master of Science (Accounting), University of Oxford - UK

Keywords

  • Classical Analysis
  • Computer Algebra
  • Convex Analysis
  • Entropy Optimization
  • Functional Analysis
  • Mathematical Programming
  • Nonsmooth Analysis
  • Nonsmooth Optimization
  • Numerical Analysis

Fields of Research

CodeDescriptionPercentage
010108Operator Algebras and Functional Analysis75
010299Applied Mathematics not elsewhere classified15
010399Numerical and Computational Mathematics not elsewhere classified10

Professional Experience

UON Appointment

DatesTitleOrganisation / Department
10/04/2015 - Laureate ProfessorUniversity of Newcastle
School of Mathematical and Physical Sciences
Australia
15/08/2008 - 19/12/2008Casual AcademicUniversity of Newcastle
School of Mathematical and Physical Sciences
Australia

Academic appointment

DatesTitleOrganisation / Department
25/06/2015 - FellowAustralian Academy of Science
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Publications

For publications that are currently unpublished or in-press, details are shown in italics.


Book (18 outputs)

YearCitationAltmetricsLink
2014Borwein J, Poorten AVD, Shallit J, Zudilin W, Neverending Fractions, Cambridge University Press, Cambridge, U.K., 224 (2014) [A2]
Co-authorsWadim Zudilin
2014Borwein J, Poorten AVD, Shallit J, Zudilin W, Neverending Fractions, Cambridge University Press, Cambridge, U.K., 224 (2014) [A2]
Co-authorsWadim Zudilin
2013Borwein JM, Glasser ML, McPhedran RC, Wan JG, Zucker IJ, Lattice sums then and now, Cambridge University Press, Cambridge, 368 (2013) [A2]
2013Borwein JM, Shparlinski I, Zudilin W, Number Theory and Related Fields: In Memory of Alf van der Poorten, Springer New York LLC (2013) [A3]
DOI10.1007/978-1-4614-6642-0
Co-authorsWadim Zudilin
2012Borwein JM, Skerritt MP, An Introduction to Modern Mathematical Computing with Mathematica, Springer, New York, 224 (2012) [A2]
2012Bailey DH, Borwein JM, Exploratory Experimentation in Mathematics: Selected Works, Perfectly Scientific Press, Portland, OR, 364 (2012) [A2]
2011Borwein JM, Skerritt MP, An Introduction to Modern Mathematical Computing: With Maple, Springer, New York, 216 (2011) [A2]
2011Borwein JM, Devlin K, Experimentelle Mathematik: Eine beispielorientierte Einfuhrung, Spektrum Akademischer Verlag, Heidelberg, DE, 158 (2011) [A4]
DOI10.1007/978-3-8274-2662-8
2010Borwein JM, Vanderwerff JD, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, Cambridge, 521 (2010) [A1]
DOI10.1017/CBO9781139087322
2009Borwein JM, Devlin K, The Computer as Crucible: An Introduction to Experimental Mathematics, A K Peters, Natick, 158 (2009) [A1]
2008Borwein J, David B, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, Massachusetts, United States, 288 (2008) [A1]
2008Borwein JM, Rocha EM, Rodrigues JF, Communicating Mathematics in the Digital Era, A. K. Peters, Wellesley, MA, 325 (2008) [A3]
2007Bailey DH, Borwein JM, Calkin NJ, Girgensohn R, Luke DR, Moll V, Experimental mathematics in action, A K Peters, Natick, Massachusetts, United States, 322 (2007) [A1]
2006Borwein JM, Lewis AS, Convex analysis and nonlinear optimization : theory and examples, Springer, Berlin, Germany, 310 (2006) [A1]
2005Borwein JM, Zhu QJ, Techniques of variational analysis, Springer, Berlin, Germany, 366 (2005) [A1]
DOI10.1007/0-387-28271-8
2004Berggren L, Borwein J, Borwein P, Pi: a Source Book, Springer, New York, United States, 797 (2004) [A1]
2004Borwein JM, Bailey DH, Girgensohn R, Experimentation in mathematics : computational paths to discovery, A K Peters Ltd., Natick, Massachusetts, United States, 357 (2004) [A1]
2002Borwein J, Morales MH, Polthier K, Multimedia tools for communicating mathematics, Springer, Berlin, Germany, 314 (2002) [A3]
Show 15 more books

Chapter (6 outputs)

YearCitationAltmetricsLink
2013Bailey DH, Borwein JM, 'Normal Numbers and Pseudorandom Generators', Computational and Analytical Mathematics, Springer, New York 1-18 (2013) [B1]
DOI10.1007/978-1-4614-7621-4_1
CitationsScopus - 1
2013Borwein JM, Yao L, Burachik RS, 'Monotone Operators Without Enlargements', Computational and Analytical Mathematics, Springer, New York 79-103 (2013) [B1]
DOI10.1007/978-1-4614-7621-4_5
2012Borwein JM, 'Exploratory experimentation: Digitally-assisted discovery and proof', Proof and Proving in Mathematics Education: The 19th ICMI Study, Springer, Dordrecht 69-96 (2012) [B1]
DOI10.1007/978-94-007-2129-6_4
2011Borwein JM, Luke DR, 'Entropic Regularization of the l(0) Function', Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, New York 65-92 (2011) [B1]
DOI10.1007/978-1-4419-9569-8_5
CitationsWeb of Science - 3
2011Borwein JM, Sims B, 'The Douglas-Rachford algorithm in the absence of convexity', Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, New York 93-109 (2011) [B1]
DOI10.1007/978-1-4419-9569-8_6
CitationsWeb of Science - 7
Co-authorsBrailey Sims
2008Borwein JM, Jungic V, Langstroth D, Macklem M, Wilson S, 'Coast-to-coast (C2C) seminar: Background, history, and practice', Communicating Mathematics in the Digital Era, A K Peters, Natick 127-140 (2008) [B1]
Show 3 more chapters

Journal article (364 outputs)

YearCitationAltmetricsLink
2015Borwein JM, Sims B, Tam MK, 'Norm convergence of realistic projection and reflection methods', OPTIMIZATION, 64 161-178 (2015)
DOI10.1080/02331934.2014.947499Author URL
2015Borwein JM, Sims B, Tam MK, 'Norm convergence of realistic projection and reflection methods', Optimization, 64 161-178 (2015)
DOI10.1080/02331934.2014.947499
2015Borwein JM, Zhu QJ, 'A Variational Approach to Lagrange Multipliers', Journal of Optimization Theory and Applications, (2015)
DOI10.1007/s10957-015-0756-2
2015Borwein JM, Tam MK, 'The cyclic Douglas-Rachford method for inconsistent feasibility problems', Journal of Nonlinear and Convex Analysis, 16 537-584 (2015)
2015Bailey DH, Borwein JM, 'Computation and theory of Mordell-Tornheim-Witten sums II', Journal of Approximation Theory, 197 115-140 (2015)

In Bailey etal. [8] the current authors, along with the late and much-missed Richard Crandall (1947-2012), considered generalized Mordell-Tornheim-Witten (MTW) zeta-function values along with their derivatives, and explored connections with multiple-zeta values (MZVs). This entailed use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory. The original motivation was to represent objects such as Eulerian log-gamma integrals; and all such integrals were expressed in terms of a MTW basis. Herein, we extend the research envisaged in Bailey etal. [8] by analyzing the relations between a significantly more general class of MTW sums. This has required significantly more subtle scientific computation and concomitant special function theory.

DOI10.1016/j.jat.2014.10.004
2014Artacho FJA, Borwein JM, Tam MK, 'Recent Results on Douglas-Rachford Methods for Combinatorial Optimization Problems', JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 163 1-30 (2014) [C1]
DOI10.1007/s10957-013-0488-0Author URL
CitationsScopus - 2Web of Science - 2
2014Borwein D, Borwein JM, Sims B, 'On the Solution of Linear Mean Recurrences', AMERICAN MATHEMATICAL MONTHLY, 121 486-498 (2014) [C1]
DOI10.4169/amer.math.monthly.121.06.486Author URL
CitationsWeb of Science - 1
Co-authorsBrailey Sims
2014Bailey DH, Borwein J, 'Pi day is upon us again and we still do not know if Pi is normal', American Mathematical Monthly, 121 191-206 (2014) [C1]
DOI10.4169/amer.math.monthly.121.03.191
2014Bailey DH, Borwein JM, de Prado ML, Zhu QJ, 'Pseudo-mathematics and financial charlatanism: The effects of backtest overfitting on out-of-sample performance', Notices of the American Mathematical Society, 61 458-471 (2014) [C1]
DOI10.1090/noti1105
2014Borwein D, Borwein JM, Straub A, 'On lattice sums and Wigner limits', Journal of Mathematical Analysis and Applications, 414 489-513 (2014) [C1]

Wigner limits are given formally as the difference between a lattice sum, associated to a positive definite quadratic form, and a corresponding multiple integral. To define these limits, which arose in work of Wigner on the energy of static electron lattices, in a mathematically rigorous way one commonly truncates the lattice sum and the corresponding integral and takes the limit along expanding hypercubes or other regular geometric shapes. We generalize the known mathematically rigorous two- and three-dimensional results regarding Wigner limits, as laid down in [3], to integer lattices of arbitrary dimension. In doing so, we also resolve a problem posed in [6, Chapter 7]. For the sake of clarity, we begin by considering the simpler case of cubic lattice sums first, before treating the case of arbitrary quadratic forms. We also consider limits taken along expanding hyperballs with respect to general norms, and connect with classical topics such as Gauss's circle problem. Appendix A is included to recall certain properties of Epstein zeta functions that are either used in the paper or serve to provide perspective. © 2014 Elsevier Inc.

DOI10.1016/j.jmaa.2014.01.008
CitationsScopus - 1
2014Borwein JM, Tam MK, 'A Cyclic Douglas-Rachford Iteration Scheme', JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 160 1-29 (2014) [C1]
DOI10.1007/s10957-013-0381-xAuthor URL
CitationsScopus - 3Web of Science - 5
2014Aragón Artacho FJ, Borwein JM, Martín-Márquez V, Yao L, 'Applications of convex analysis within mathematics', Mathematical Programming, 148 49-88 (2014) [C1]
DOI10.1007/s10107-013-0707-3
2014Bailey DH, Borwein JM, Crandall RE, 'COMPUTATION AND THEORY OF EXTENDED MORDELL-TORNHEIM-WITTEN SUMS', MATHEMATICS OF COMPUTATION, 83 1795-1821 (2014) [C1]
Author URL
2014Bailey DH, Borwein JM, Kaiser AD, 'Automated simplification of large symbolic expressions', JOURNAL OF SYMBOLIC COMPUTATION, 60 120-136 (2014) [C1]
DOI10.1016/j.jsc.2013.09.001Author URL
2014Borwein JM, Yao L, 'Legendre-Type Integrands and Convex Integral Functions', JOURNAL OF CONVEX ANALYSIS, 21 261-288 (2014) [C1]
Author URL
2014Borwein JM, Li G, Yao L, 'ANALYSIS OF THE CONVERGENCE RATE FOR THE CYCLIC PROJECTION ALGORITHM APPLIED TO BASIC SEMIALGEBRAIC CONVEX SETS', SIAM JOURNAL ON OPTIMIZATION, 24 498-527 (2014) [C1]
DOI10.1137/130919052Author URL
2014Borwein J, Howlett P, Piantadosi J, 'Modelling and Simulation of Seasonal Rainfall Using the Principle of Maximum Entropy', ENTROPY, 16 747-769 (2014) [C1]
DOI10.3390/e16020747Author URL
CitationsScopus - 4Web of Science - 1
2014Borwein JM, Burachik RS, Yao L, 'Conditions for zero duality gap in convex programming', Journal of Nonlinear and Convex Analysis, 15 167-190 (2014) [C1]

We introduce and study a new dual condition which characterizes zero duality gap in nonsmooth convex optimization. We prove that our condition is less restrictive than all existing constraint qualifications, including the closed epigraph condition. Our dual condition was inspired by, and is less restrictive than, the so-called Bertsekas' condition for monotropic programming problems. We give several corollaries of our result and special cases as applications. We pay special attention to the polyhedral and sublinear cases, and their implications in convex optimization. ©2014.

2014Artacho FJA, Borwein JM, Tam MK, 'Douglas-rachford feasibility methods for matrix completion problems', ANZIAM Journal, 55 299-326 (2014) [C1]
DOI10.1017/S1446181114000145
2014BORWEIN JM, YAO L, 'SUM THEOREMS FOR MAXIMALLY MONOTONE OPERATORS OF TYPE (FPV)', Journal of the Australian Mathematical Society, (2014) [C1]
DOI10.1017/S1446788714000056
2014Borwein JM, Yao L, 'Some results on the convexity of the closure of the domain of a maximally monotone operator', OPTIMIZATION LETTERS, 8 237-246 (2014) [C1]
DOI10.1007/s11590-012-0564-7Author URL
CitationsScopus - 1Web of Science - 1
2013Borwein JM, Crandall RE, 'Closed Forms: What they are and why we care', Notices of the American Mathematical Society, 60 50-65 (2013) [C1]
DOI10.1090/noti936
CitationsScopus - 13
2013Bailey DH, Borwein JM, Mattingly A, Wightwick G, 'The computation of previously inaccessible digits of p2 and Catalan's constant', Notices of the American Mathematical Society, 60 844-854 (2013) [C1]
DOI10.1090/noti1015
CitationsScopus - 2
2013Borwein JM, Straub A, 'Relations for Nielsen polylogarithms', Journal of Approximation Theory, (2013)

Polylogarithms appear in many diverse fields of mathematics. Herein, we investigate relations amongst the restricted class of Nielsen-type (essentially, height one) polylogarithms, both generic and at special arguments including the sixth roots of unity. Numerical computations suggest that the collected relations, partially motivated by a previous study of the authors on log-sine integrals, are complete except in the case when the argument is the fundamental sixth root of unity. For use in other applications, all our results are implemented and accessible for use in symbolic computation or to facilitate numeric computation. In particular, the relations are explicitly exhibited in the case of low weights. © 2013 Elsevier Inc. All rights reserved.

DOI10.1016/j.jat.2013.07.003
2013Aragón Artacho FJ, Borwein JM, Tam MK, 'Recent Results on Douglas-Rachford Methods for Combinatorial Optimization Problems', Journal of Optimization Theory and Applications, 1-30 (2013) [C1]
DOI10.1007/s10957-013-0488-0
CitationsScopus - 2Web of Science - 2
2013Borwein JM, Yao L, 'Structure Theory for Maximally Monotone Operators with Points of Continuity', Journal of Optimization Theory and Applications, 157 1-24 (2013) [C1]
DOI10.1007/s10957-012-0162-yAuthor URL
CitationsScopus - 3Web of Science - 3
2013Aragón Artacho FJ, Borwein JM, Martín-Márquez V, Yao L, 'Applications of convex analysis within mathematics', Mathematical Programming, 148 49-88 (2013)

In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator¿ Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis.

DOI10.1007/s10107-013-0707-3
2013Bailey DH, Borwein JM, Crandall RE, Rose MG, 'Expectations on fractal sets', APPLIED MATHEMATICS AND COMPUTATION, 220 695-721 (2013) [C1]
DOI10.1016/j.amc.2013.06.078Author URL
CitationsScopus - 2Web of Science - 1
2013Borwein JM, Straub A, 'Mahler measures, short walks and log-sine integrals', THEORETICAL COMPUTER SCIENCE, 479 4-21 (2013) [C1]
DOI10.1016/j.tcs.2012.10.025Author URL
CitationsScopus - 2Web of Science - 2
2013Artacho FJA, Bailey DH, Borwein JM, Borwein PB, 'Walking on Real Numbers', MATHEMATICAL INTELLIGENCER, 35 42-60 (2013) [C1]
DOI10.1007/s00283-012-9340-xAuthor URL
CitationsScopus - 4Web of Science - 2
2013Artacho FJA, Borwein JM, 'Global convergence of a non-convex Douglas-Rachford iteration', JOURNAL OF GLOBAL OPTIMIZATION, 57 753-769 (2013) [C1]
DOI10.1007/s10898-012-9958-4Author URL
CitationsScopus - 3Web of Science - 3
2013Bauschke HH, Borwein JM, Wang X, Yao L, 'Monotone Operators and "Bigger Conjugate" Functions', JOURNAL OF CONVEX ANALYSIS, 20 143-155 (2013) [C1]
Author URL
CitationsScopus - 1
2013Borwein JM, Straub A, Wan J, 'Three-Step and Four-Step Random Walk Integrals', Experimental Mathematics, 22 1-14 (2013) [C1]
DOI10.1080/10586458.2013.748379Author URL
CitationsScopus - 10Web of Science - 4
2013Aragón Artacho FJ, Borwein JM, Tam MK, 'Recent Results on Douglas-Rachford Methods', Serdica Mathematical Journal, 39 313-330 (2013) [C1]
2013Bailey DH, Borwein D, Borwein JM, 'On Eulerian log-gamma integrals and Tornheim¿Witten zeta functions', Ramanujan Journal, 36 43-68 (2013)

Stimulated by earlier work by Moll and his coworkers (Amdeberhan et al., Proc. Am. Math. Soc., 139(2):535¿545, 2010), we evaluate various basic log Gamma integrals in terms of partial derivatives of Tornheim¿Witten zeta functions and their extensions arising from evaluations of Fourier series. In particular, we fully evaluate Formula presented for 1 = n = 4 and make some comments regarding the general case. The subsidiary computational challenges are substantial, interesting and significant in their own right.

DOI10.1007/s11139-012-9427-1
CitationsScopus - 1
2013Bailey DH, Borwein JM, 'Compressed lattice sums arising from the Poisson equation', BOUNDARY VALUE PROBLEMS, (2013) [C1]
DOI10.1186/1687-2770-2013-75Author URL
2013Bailey DH, Borwein JM, Crandall RE, Zucker IJ, 'Lattice sums arising from the Poisson equation', Journal of Physics A: Mathematical and Theoretical, 46 (2013) [C1]
DOI10.1088/1751-8113/46/11/115201Author URL
CitationsScopus - 3Web of Science - 2
2013Borwein JM, Crandall RE, 'Closed Forms: What they are and why we care', Notices of the American Mathematical Society, 60 50-65 (2013)
DOI10.1090/noti936
CitationsScopus - 15
2013Borwein JM, Yao L, 'Maximality of the Sum of a Maximally Monotone Linear Relation and a Maximally Monotone Operator', SET-VALUED AND VARIATIONAL ANALYSIS, 21 603-616 (2013) [C1]
DOI10.1007/s11228-013-0259-yAuthor URL
CitationsScopus - 1Web of Science - 1
2013Borwein JM, Zhu QJ, 'Variational methods in the presence of symmetry', Advances in Nonlinear Analysis, 2 (2013) [C1]
DOI10.1515/anona-2013-1001
2012Borwein D, Borwein JM, Straub A, 'A sinc that sank', American Mathematical Monthly, 119 535-549 (2012) [C1]
DOI10.4169/amer.math.monthly.119.07.535
CitationsWeb of Science - 1
2012Bailey DH, Borwein JM, 'Ancient Indian square roots: An exercise in forensic paleo-mathematics', American Mathematical Monthly, 119 646-657 (2012) [C1]
DOI10.4169/amer.math.monthly.119.08.646
CitationsScopus - 3Web of Science - 1
2012Borwein JM, 'Ramanujan and Pi', American Mathematical Society. Notices, 59 1534-1537 (2012) [C1]
2012Borwein JM, Jungic V, 'Organic mathematics then and now', Notices of the AMS, 59 416-419 (2012) [C2]
2012Borwein JM, Straub A, Wan G, Zudilin V, 'Densities of short uniform random walks', Canadian Journal of Mathematics, 64 961-990 (2012) [C1]
DOI10.4153/CJM-2011-079-2
CitationsScopus - 10Web of Science - 9
Co-authorsWadim Zudilin
2012Borwein JM, Vanderwerff J, 'Constructions of uniformly convex functions', Canadian Mathematical Bulletin, 55 697-707 (2012) [C1]
2012Bauschke HH, Borwein JM, Wang X, Yao L, 'The Brezis-Browder Theorem in a general Banach space', Journal of Functional Analysis, 262 4948-4971 (2012) [C1]
CitationsScopus - 4Web of Science - 4
2012Amdeberhan T, Borwein D, Borwein JM, Straub A, 'On formulas for Pi experimentally conjectured by Jauregui-Tsallis', Journal of Mathematical Physics, 53 (2012) [C1]
2012Bailey DH, Barrio R, Borwein JM, 'High-precision computation: Mathematical physics and dynamics', Applied Mathematics and Computation, 218 10106-10121 (2012) [C1]
CitationsScopus - 19Web of Science - 12
2012Borwein JM, 'Maximum entropy and feasibility methods for convex and nonconvex inverse problems', Optimization, 61 1-33 (2012) [C1]
CitationsScopus - 5Web of Science - 5
2012Bailey DH, Borwein JM, Calude CS, Dinneen MJ, Dumitrescu M, Yee A, 'An empirical approach to the normality of p', Experimental Mathematics, 21 375-384 (2012) [C1]
CitationsScopus - 4Web of Science - 3
2012Borwein JM, Borwein P, Jungic V, 'Remote collaboration: Six years of the Coast-to-Coast Seminar Series', Science Communication, 34 419-428 (2012) [C3]
2012Bailey DH, Borwein D, Borwein JM, 'On Eulerian log-gamma integrals and Tornheim-Witten zeta functions', Ramanujan Journal, 1-26 (2012) [C1]

Stimulated by earlier work by Moll and his coworkers (Amdeberhan et al., Proc. Am. Math. Soc., 139(2):535-545, 2010), we evaluate various basic log Gamma integrals in terms of¿partial derivatives of¿Tornheim-Witten zeta functions and their extensions arising from evaluations of¿Fourier series. In particular, we fully evaluate {Mathematical expression} for 1=n=4 and make some comments regarding the general case. The subsidiary computational challenges are substantial, interesting and significant in their own right. © 2012 Springer Science+Business Media, LLC.

DOI10.1007/s11139-012-9427-1
2012Bailey DH, Borwein JM, 'Nonnormality of Stoneham constants', Ramanujan Journal, 29 409-422 (2012) [C1]
DOI10.1007/s11139-012-9417-3
CitationsScopus - 2Web of Science - 1
2012Alladi K, Andrews GE, Borwein JM, 'Preface. Ramanujan's 125th Anniversary Special Volume', Ramanujan Journal, 29 1-2 (2012) [C3]
CitationsWeb of Science - 7
2012Borwein JM, Osborn J-AH, 'Response to 'Experimental Approaches to Theoretical Thinking: Artefacts and Proofs Proof and Proving in Mathematics Education'', Proof and Proving in Mathematics Education: The 19th ICMI Study, 15 138-143 (2012) [C3]
DOI10.1007/978-94-007-2129-6
2012Borwein JM, Straub A, 'Log-sine evaluations of Mahler measures', Journal of the Australian Mathematical Society, 92 15-36 (2012) [C1]
DOI10.1017/S1446788712000067
CitationsScopus - 1Web of Science - 1
2012Janssen H, Craxi A, 'Preface', BEST PRACTICE & RESEARCH IN CLINICAL GASTROENTEROLOGY, 26 369-369 (2012)
DOI10.1016/j.bpg.2012.11.009Author URL
2012Piantadosi J, Howlett P, Borwein JM, 'Copulas with maximum entropy', Optimization Letters, 6 99-125 (2012) [C1]
CitationsScopus - 7Web of Science - 7
2012Bauschke HH, Borwein JM, Wang X, Yao L, 'Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type', Optimization Letters, 6 1875-1881 (2012) [C1]
CitationsScopus - 2Web of Science - 2
2012Borwein JM, Osborn J-AH, 'Loving and Hating Mathematics by Reuben Hersh and Vera John-Steiner [Book Review]', The Mathematical Intelligencer, 33 63-69 (2012) [C3]
2012Borwein D, Borwein JM, Straub A, Wan G, 'Log-sine evaluations of Mahler measures, II', Integers, 12 1179-1212 (2012) [C1]
2012Wang XF, Yao L, Bauschke HH, Borwein JM, 'Construction of pathological maximally monotone operators on non-reflexive Banach spaces', Set-Valued and Variational Analysis, 20 387-415 (2012) [C1]
DOI10.1007/s11228-012-0209-0
CitationsScopus - 4Web of Science - 3
2012Bailey DH, Borwein JM, 'Hand-to-hand combat with thousand-digit integrals', Journal of Computational Science, 3 77-86 (2012) [C1]
CitationsScopus - 6Web of Science - 5
2012Piantadosi J, Howlett P, Borwein J, Henstridge J, 'Maximum entropy methods for generating simulated rainfall', Numerical Algebra, Control and Optimization, 2 233-256 (2012) [C1]

We desire to generate monthly rainfall totals for a particular loca- tion in such a way that the statistics for the simulated data match the statistics for the observed data. We are especially interested in the accumulated rainfall totals over several months. We propose two different ways to construct a joint rainfall probability distribution that matches the observed grade correlation coefficients and preserves the prescribed marginal distributions. Both methods use multi-dimensional checkerboard copulas. In the first case we use the the- ory of Fenchel duality to construct a copula of maximum entropy and in the second case we use a copula derived from a multi-variate normal distribution. Finally we simulate monthly rainfall totals at a particular location using each method and analyse the statistical behaviour of the corresponding quarterly accumulations.

DOI10.3934/naco.2012.2.233
CitationsScopus - 3
2011Borwein JM, Straub A, 'Special Values of Generalized Log-sine Integrals', CoRR, abs/1103.4298 (2011)
2011Bailey DH, Borwein JM, 'Exploratory experimentation and computation', Notices of the American Mathematical Society, 58 1410-1419 (2011) [C1]
CitationsScopus - 7
2011Borwein D, Borwein JM, Glasser ML, Wan G, 'Moments of Ramanujan's generalized elliptic integrals and extensions of Catalan's constant', Journal of Mathematical Analysis and Applications, 384 478-496 (2011) [C1]
DOI10.1016/j.jmaa.2011.06.001
CitationsScopus - 2Web of Science - 2
2011Bacak M, Borwein JM, 'On difference convexity of locally Lipschitz functions', Optimization, 60 961-978 (2011) [C1]
CitationsScopus - 3Web of Science - 5
2011Borwein JM, Osborn J-A, 'Review: Hersh, R. and John-Steiner, V. Loving and Hating Mathematics', Mathematical Intelligencer, 33 63-69 (2011) [C3]
DOI10.1007/s00283-011-9260-1
2011Bailey DH, Borwein JM, 'High-precision numerical integration: Progress and challenges', Journal of Symbolic Computation, 46 741-754 (2011) [C1]
DOI10.1016/j.jsc.2010.08.010
CitationsScopus - 8Web of Science - 7
2011Borwein JM, 'Review: Dontchev, A.L. and Rockafellar, R.T, Implicit Functions and Solution Mappings', IEEE Control Systems, 31 74-77 (2011) [C3]
2011Borwein JM, Reich S, Sabach S, 'A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept', Journal of Nonlinear and Convex Analysis, 12 161-184 (2011) [C1]
CitationsScopus - 15Web of Science - 11
2011Borwein JM, Nuyens D, Straub A, Wan G, 'Some arithmetic properties of short random walk integrals', Ramanujan Journal, 26 109-132 (2011) [C1]
DOI10.1007/s11139-011-9325-y
CitationsScopus - 9Web of Science - 10
2010Borwein JM, Chan O-Y, 'Duality in tails of multiple-zeta values', International Journal of Number Theory, 6 501-514 (2010) [C1]
DOI10.1142/s1793042110003058
2010Borwein JM, 'Fifty years of maximal monotonicity', Optimization Letters, 4 473-490 (2010) [C1]
DOI10.1007/s11590-010-0178-x
CitationsScopus - 21Web of Science - 20
2010Borwein JM, Morrs WB, 'Stability of closedness of convex cones under linear mappings II', Journal of Nonlinear Analysis and Optimization: Theory and Applications, 1 1-7 (2010) [C1]
2010Borwein D, Borwein JM, Leonard IE, 'Lp norms and the sine function', American Mathematical Monthly, 117 528-539 (2010) [C1]
DOI10.4169/000298910X492817
CitationsScopus - 3Web of Science - 3
2010Bailey DH, Borwein JM, Crandall RE, 'Advances in the theory of box integrals', Mathematics of Computation, 79 1839-1866 (2010) [C1]
DOI10.1090/s0025-5718-10-02338-0
CitationsScopus - 10Web of Science - 6
2010Bailey DH, Borwein JM, Broadhurst D, Zudilin V, 'Experimental mathematics and mathematical physics', Contemporary Mathematics, 517 41-58 (2010) [C1]
CitationsWeb of Science - 3
Co-authorsWadim Zudilin
2010Bailey DH, Borwein JM, 'Experimental computation with oscillatory integrals', Contemporary Mathematics, 517 25-40 (2010) [C1]
CitationsWeb of Science - 1
2010Bacak M, Borwein JM, Eberhard A, Mordukhovich BS, 'Infimal convolutions and lipschitzian properties of subdifferentials for prox-regular functions in hilbert spaces', Journal of Convex Analysis, 17 737-763 (2010) [C1]
CitationsScopus - 7Web of Science - 5
2010Borwein JM, Vanderwerff J, 'Frechet-legendre functions and reflexive banach spaces', Journal of Convex Analysis, 17 915-924 (2010) [C1]
CitationsScopus - 2Web of Science - 2
2010Borwein JM, Chan O-Y, Crandall RE, 'Higher-dimensional box integrals', Experimental Mathematics, 19 431-446 (2010) [C1]
DOI10.1080/10586458.2010.10390634
CitationsScopus - 7Web of Science - 5
2009Bailey DH, Borwein JM, 'Experimental mathematics and computational statistics', WIREs Computational Statistics, 1 12-24 (2009) [C1]
DOI10.1002/wics.1
CitationsScopus - 2
2009Borwein JM, Calkin NJ, Manna D, 'Euler-Boole Summation Revisited', AMERICAN MATHEMATICAL MONTHLY, 116 387-412 (2009) [C1]
Author URL
CitationsScopus - 2Web of Science - 3
2009Borwein JM, Guirao AJ, Hajek P, Vanderwerff J, 'Uniformly convex functions on Banach Spaces', Proceedings of the American Mathematical Society, 137 1081-1091 (2009) [C1]
DOI10.1090/s0002-9939-08-09630-5
CitationsScopus - 7Web of Science - 5
2009Borwein JM, 'The Princeton companion to mathematics', SIAM Review, 51 790-794 (2009) [C3]
2009Borwein JM, Vanderwerff J, 'Differentiability of conjugate functions and perturbed minimization principles', Journal of Convex Analysis, 16 707-711 (2009) [C1]
CitationsScopus - 4Web of Science - 4
2009Borwein JM, Moors WB, 'Stability of closedness of convex cones under linear mappings', Journal of Convex Analysis, 16 699-705 (2009) [C1]
CitationsScopus - 2Web of Science - 1
2009Bailey DH, Borwein JM, Crandall RE, 'Resolution of the Quinn-Rand-Strogatz Constant of nonlinear physics', Experimental Mathematics, 18 107-116 (2009) [C1]
DOI10.1080/10586458.2009.10128885
CitationsScopus - 2
2009Borwein JM, Chan O-Y, 'UNIFORM BOUNDS FOR THE COMPLEMENTARY INCOMPLETE GAMMA FUNCTION', MATHEMATICAL INEQUALITIES & APPLICATIONS, 12 115-121 (2009) [C1]
Author URL
CitationsScopus - 2Web of Science - 2
2008Borwein JM, Salvy B, 'A Proof of a Recurrence for Bessel Moments.', Experimental Mathematics, 17 223-230 (2008) [C1]
DOI10.1080/10586458.2008.10129032
2008Borwein JM, Salvy B, 'A Proof of a Recurrence for Bessel Moments', Experimental Mathematics, 17 223-230 (2008) [C1]
DOI10.1080/10586458.2008.10129032
CitationsScopus - 5Web of Science - 8
2008Borwein JM, Luke DR, 'Dynamics of a Ramanujan-type Continued Fraction with Cyclic Coefficients', The Ramanujan Journal: an international journal devoted to areas of mathematics influenced by Ramanu, 16 285-304 (2008) [C1]
DOI10.1007/s11139-007-9096-7
CitationsWeb of Science - 1
2008Borwein JM, Zucker IJ, Boersma J, 'The evaluation of character Euler double sums', The Ramanujan Journal: an international journal devoted to areas of mathematics influenced by Ramanu, 15 377-405 (2008) [C1]
DOI10.1007/s11139-007-9083-z
CitationsScopus - 13Web of Science - 9
2008Bailey DH, Borwein JM, Broadhurst DM, Glasser L, 'Elliptic integral evaluations of Bessel moments and applications', Journal of Physics A: Mathematical and Theoretical, 41 5203-5231 (2008) [C1]
DOI10.1088/1751-8113/41/20/205203
CitationsScopus - 19Web of Science - 24
2008Baillie R, Borwein D, Borwein JM, 'Surprising sinc sums and integrals', American Mathematical Monthly, 115 888-901 (2008) [C1]
CitationsScopus - 7Web of Science - 5
2008Borwein JM, 'Hilbert's Inequality and Witten's Zeta-Function', American Mathematical Monthly, 115 125-137 (2008) [C1]
CitationsScopus - 4Web of Science - 3
2008Borwein D, Borwein JM, Chan O-Y, 'The Evaluation of Bessel Functions via Exp-Arc Integrals', Journal of Mathematical Analysis and Applications, 341 478-500 (2008) [C1]
DOI10.1016/j.jmaa.2007.10.003
CitationsScopus - 4Web of Science - 4
2008Borwein D, Borwein JM, Crandall RE, 'Effective Laguerre Asymptotics', SIAM Journal on Numerical Analysis, 46 3285-3312 (2008) [C1]
DOI10.1137/07068031X
CitationsScopus - 11Web of Science - 9
2008Eberhard AC, Borwein JM, 'Second Order Cones for Maximal Monotone Operators via Representative Functions', Set-Valued and Variational Analysis: an international journal devoted to the theory of multifunctions, 16 157-184 (2008) [C1]
DOI10.1007/s11228-008-0075-y
CitationsScopus - 2Web of Science - 2
2007Bartz S, Bauschke HH, Borwein JM, Reich S, Wang X, 'Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 66 1198-1223 (2007) [C1]
DOI10.1016/j.na.2006.01.013Author URL
CitationsScopus - 28Web of Science - 30
2007Bailey DH, Borwein JM, Crandall RE, 'Box integrals', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 206 196-208 (2007) [C1]
DOI10.1016/j.cam.2006.06.010Author URL
CitationsScopus - 16Web of Science - 13
2007Bauschke HH, Borwein JM, Wang X, 'Fitzpatrick functions and continuous linear monotone operators', SIAM JOURNAL ON OPTIMIZATION, 18 789-809 (2007) [C1]
DOI10.1137/060655468Author URL
CitationsScopus - 21Web of Science - 23
2007Borwein J, Wiersma H, 'Asplund decomposition of monotone operators', SIAM JOURNAL ON OPTIMIZATION, 18 946-960 (2007) [C1]
DOI10.1137/060658357Author URL
CitationsWeb of Science - 4
2007Bailey DH, Borwein D, Borwein JM, Crandall RE, 'Hypergeometric forms for ising-class integrals', EXPERIMENTAL MATHEMATICS, 16 257-276 (2007) [C1]
DOI10.1080/10586458.2007.10129003Author URL
CitationsScopus - 11Web of Science - 9
2007Borwein D, Borwein J, Crandall R, Mayer R, 'On the dynamics of certain recurrence relations', RAMANUJAN JOURNAL, 13 63-101 (2007) [C1]
DOI10.1007/s11139-006-0243-3Author URL
CitationsWeb of Science - 3
2007Borwein JM, 'Proximality and Chebyshev sets', OPTIMIZATION LETTERS, 1 21-32 (2007) [C1]
DOI10.1007/s11590-006-0014-5Author URL
CitationsScopus - 11Web of Science - 11
2007Borwein JM, Moors WB, 'Non-smooth Analysis, Optimisation Theory and Banach Space Theory 549-559 (2007)

This chapter discusses about nonsmooth analysis, optimization theory, and Banach space theory. A Banach space X is called a weak Asplund space [Gâteaux differentiability space] if each continuous convex function defined on it is Gâteaux differentiable at the points of a residual subset (that is, a subset that contains the intersection of countably many dense open subsets of X) [dense subset] of its domain. For a Banach space (X, ¿ ·. ¿), with closed unit ball BX, the Bishop-Phelps set is the set of all linear functionals in the dual X* that attain their maximum value over BX; that is, the set {. x*¿. X*: x*(x) = ¿. x*¿ for some x ¿. BX}. The Bishop-Phelps Theorem says that the Bishop-Phelps set is always dense in X*. A Banach space X has the attainable approximation property (AAP) if the set of support functionals for any closed bounded convex subset W ¿. X is norm dense in X*. The concepts related to the Bishop-Phelps problem and the complex Bishop-Phelps property are also discussed. Concepts of biorthogonal sequences and support points are also elaborated. © 2007 Elsevier B.V. All rights reserved.

DOI10.1016/B978-044452208-5/50050-8
CitationsScopus - 3
2007Borwein JM, Langstroth D, Macklem M, Wilson S, Jungic V, 'The coast-to-coast seminar and remote mathematical collaboration', Proceedings - International Symposium on High Performance Computing Systems and Applications, (2007)
DOI10.1109/HPCS.2007.30
2007Borwein JM, 'A class of Dirichlet series integrals', AMERICAN MATHEMATICAL MONTHLY, 114 70-76 (2007) [C1]
Author URL
CitationsScopus - 1Web of Science - 1
2007Bailey D, Borwein J, Bradley D, 'A hypergeometric identity', AMERICAN MATHEMATICAL MONTHLY, 114 643-645 (2007)
Author URL
2007Borwein JM, 'Maximality of sums of two maximal monotone operators in general banach space', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 135 3917-3924 (2007) [C1]
Author URL
CitationsScopus - 31Web of Science - 31
2007Borwein D, Borwein J, 'van der pol expansions of L-series', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 50 11-23 (2007) [C1]
DOI10.4153/CMB-2007-002-0Author URL
2007Borwein JM, Chamberland M, 'Integer powers of Arcsin', International Journal of Mathematics and Mathematical Sciences, 2007 1-10 (2007) [C1]
DOI10.1155/2007/19381
2006Borwein D, Borwein JM, Bradley DM, 'Parametric Euler sum identities', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 316 328-338 (2006) [C1]
DOI10.1016/j.jmaa.2005.04.040Author URL
CitationsScopus - 7Web of Science - 7
2006Bailey DH, Borwein JM, Crandall RE, 'Integrals of the Ising class', JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 39 12271-12302 (2006) [C1]
DOI10.1088/0305-4470/39/40/001Author URL
CitationsScopus - 23Web of Science - 24
2006Borwein JM, Zhu QJJ, 'Variational methods in convex analysis', JOURNAL OF GLOBAL OPTIMIZATION, 35 197-213 (2006) [C1]
DOI10.1007/s10898-005-3835-3Author URL
CitationsScopus - 6Web of Science - 3
2006Borwein J, 'Simon Fitzpatrick memorial volume', JOURNAL OF CONVEX ANALYSIS, 13 463-476 (2006)
Author URL
CitationsWeb of Science - 2
2006Borwein JM, 'Maximal monotonicity via convex analysis', JOURNAL OF CONVEX ANALYSIS, 13 561-586 (2006) [C1]
Author URL
CitationsScopus - 55Web of Science - 51
2006Borwein J, Montesinos V, Vanderwerff J, 'Boundedness, differentiability and extensions of convex functions', JOURNAL OF CONVEX ANALYSIS, 13 587-602 (2006) [C1]
Author URL
CitationsWeb of Science - 4
2006Borwein JM, 'The Life of Simon Fitzpatrick, 1953-2004', Journal of Convex Analysis, 13 463-476 (2006) [C1]
2006Borwein JM, Bradley DM, 'Thirty-two goldbach variations', INTERNATIONAL JOURNAL OF NUMBER THEORY, 2 65-103 (2006) [C1]
DOI10.1142/S1793042106000383Author URL
CitationsWeb of Science - 17
2006Bailey DH, Borwein JM, 'Effective bounds in Euler-Maclaurin-based quadrature (summary for HPCS06)', 20th International Symposium on High-Performance Computing in an Advanced Collaborative Environment, 2006. HPCS 2006, 34-34 (2006)

We analyze the behavior of Euler-Maclaurin-based integration schemes with the intention of deriving accurate and economic estimations of the error. These schemes typically provide very high-precision results (hundreds or thousands of digits), in reasonable run time, even in cases where the integrand function has a blow-up singularity or infinite derivative at an endpoint. Heretofore, researchers using these schemes have relied mostly on ad hoc error estimation schemes to project the estimated error of the present iteration. In this paper, we seek to develop some more rigorous, yet highly usable schemes to estimate these errors. © 2006 IEEE.

DOI10.1109/HPCS.2006.22
2006Bailey DH, Borwein JM, Bradley DM, 'Experimental determination of Apery-like identities for zeta(2n+2)', EXPERIMENTAL MATHEMATICS, 15 281-289 (2006) [C1]
DOI10.1080/10586458.2006.10128968Author URL
CitationsScopus - 10Web of Science - 13
2006Bailey DH, Borwein JM, Kapoor V, Weisstein EW, 'Ten problems in experimental mathematics', AMERICAN MATHEMATICAL MONTHLY, 113 481-509 (2006) [C1]
Author URL
CitationsScopus - 12Web of Science - 9
2006Borwein JM, 'Maximality of sums of two maximal monotone operators', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 134 2951-2955 (2006) [C1]
DOI10.1090/S0002-9939-06-08323-7Author URL
CitationsScopus - 7Web of Science - 8
2005Borwein JM, Girgensohn R, 'Evaluations of binomial series', Aequationes Mathematicae, 70 25-36 (2005) [C1]
DOI10.1007/s00010-005-2774-x
CitationsScopus - 15
2005Borwein JM, Choi KKS, Pigulla W, 'Continued fractions of tails of hypergeometric series', AMERICAN MATHEMATICAL MONTHLY, 112 493-501 (2005) [C1]
Author URL
CitationsScopus - 1Web of Science - 1
2005Bailey DH, Borwein JM, 'Experimental mathematics: Examples, methods and implications', Notices of the American Mathematical Society, 52 502-514 (2005) [C1]
CitationsScopus - 20
2005Borwein JM, Wang XF, 'Lipschitz functions with maximal Clarke subdifferentials are staunch', BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 72 491-496 (2005) [C1]
Author URL
CitationsScopus - 3Web of Science - 4
2005Borwein JM, Wang XF, 'Cone-monotone functions: Differentiability and continuity', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 57 961-982 (2005) [C1]
DOI10.4153/CJM-2005-037-5Author URL
CitationsScopus - 6Web of Science - 6
2005Borwein J, Stanway T, 'Knowledge and community in mathematics', MATHEMATICAL INTELLIGENCER, 27 7-16 (2005) [C1]
DOI10.1007/BF02985788Author URL
CitationsWeb of Science - 1
2005Borwein JM, 'The SIAM 100 Digits Challenge', The Mathematical Intelligencer, 27 40-48 (2005) [C3]
2005Borwein JM, Luke DR, 'Dynamics of a continued fraction of Ramanujan with random coefficients', ABSTRACT AND APPLIED ANALYSIS, 449-467 (2005) [C1]
DOI10.1155/AAA.2005.449Author URL
CitationsScopus - 3Web of Science - 2
2005Borwein JM, 'The experimental mathematician: The pleasure of discovery and the role of proof', International Journal of Computers for Mathematical Learning, 10 75-108 (2005) [C1]

The emergence of powerful mathematical computing environments, the growing availability of correspondingly powerful (multi-processor) computers and the pervasive presence of the internet allow for mathematicians, students and teachers, to proceed heuristically and 'quasi-inductively'. We may increasingly use symbolic and numeric computation, visualization tools, simulation and data mining. The unique features of our discipline make this both more problematic and more challenging. For example, there is still no truly satisfactory way of displaying mathematical notation on the web; and we care more about the reliability of our literature than does any other science. The traditional role of proof in mathematics is arguably under siege - for reasons both good and bad. © Springer 2005.

DOI10.1007/s10758-005-5216-x
CitationsScopus - 5
2004Borwein JM, Burke JV, Lewis AS, 'Differentiability of cone-monotone functions on separable Banach space', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 132 1067-1076 (2004) [C1]
DOI10.1090/S0002-9939-03-07149-1Author URL
CitationsScopus - 7Web of Science - 9
2004Borwein JM, Borwein D, Galway WF, 'Finding and excluding b-ary Machin-type individual digit formulae', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 56 897-925 (2004) [C1]
DOI10.4153/CJM-2004-041-2Author URL
CitationsScopus - 5Web of Science - 4
2004Borwein J, Cheng LX, Fabian M, Revalski JP, 'A one perturbation variational principle and applications', SET-VALUED ANALYSIS, 12 49-60 (2004) [C1]
DOI10.1023/B:SVAN.0000023400.92518.cbAuthor URL
CitationsWeb of Science - 3
2004Borwein JM, Vanderwerff JD, 'Constructible convex sets', SET-VALUED ANALYSIS, 12 61-77 (2004) [C1]
DOI10.1023/B:SVAN.0000023393.75251.05Author URL
CitationsScopus - 5Web of Science - 4
2004Borwein J, Crandall R, Fee G, 'On the Ramanujan AGM fraction, I: The real-parameter case', EXPERIMENTAL MATHEMATICS, 13 275-285 (2004) [C1]
Author URL
CitationsWeb of Science - 8
2004Borwein J, Crandall R, 'On the Ramanujan AGM fraction, II: The complex- parameter case', EXPERIMENTAL MATHEMATICS, 13 287-295 (2004) [C1]
Author URL
CitationsWeb of Science - 8
2004Bailey D, Borwein JM, Crandall R, Pomerance C, 'On the binary expansions of algebraic numbers', Journal de Theorie des Nombres de Bordeaux, 16 487-518 (2004) [C1]
2003Borwein J, Girgensohn R, 'A class of exponential inequalities', MATHEMATICAL INEQUALITIES & APPLICATIONS, 6 397-411 (2003) [C1]
Author URL
CitationsWeb of Science - 1
2003Borwein JM, Choi KKS, 'On Dirichlet series for sums of squares', RAMANUJAN JOURNAL, 7 95-127 (2003) [C1]
DOI10.1023/A:1026230709945Author URL
CitationsScopus - 6Web of Science - 6
2003Kellar M, MacKay B, Zhang R, Watters C, Kaufman D, Borwein J, 'Dynamic composition of math lessons', EDUCATIONAL TECHNOLOGY & SOCIETY, 6 100-111 (2003) [C1]
Author URL
CitationsWeb of Science - 1
2003Borwein D, Borwein JM, Rooin J, 'Problem 11009.', The American Mathematical Monthly, 110 341-341 (2003)
2003Benoist J, Borwein JM, Popovici N, 'A characterization of quasiconvex vector-valued functions', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 131 1109-1113 (2003) [C1]
DOI10.1090/S0002-9939-02-06761-8Author URL
CitationsScopus - 24Web of Science - 25
2003Borwein JM, Dontchev AL, 'On the Bartle-Graves theorem', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 131 2553-2560 (2003) [C1]
DOI10.1090/S0002-9939-03-07229-0Author URL
CitationsScopus - 6Web of Science - 7
2003Borwein J, FItzpatrick S, Girgensohn R, 'Subdifferentials whose graphs are not normxweak* closed', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 46 538-545 (2003) [C1]
DOI10.4153/CMB-2003-051-5Author URL
CitationsWeb of Science - 7
2003Bauschke HH, Borwein JM, Combettes PL, 'Bregman monotone optimization algorithms', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 42 596-636 (2003) [C1]
DOI10.1137/S0363012902407120Author URL
CitationsScopus - 72Web of Science - 79
2003Borwein JM, Vanderwerff J, Wang XF, 'Local Lipschitz-constant functions and maximal subdifferentials', SET-VALUED ANALYSIS, 11 37-67 (2003) [C1]
DOI10.1023/A:1021975622768Author URL
CitationsScopus - 2Web of Science - 2
2003Borwein J, Choksi R, Marechal P, 'Probability distributions of assets inferred from option prices via the principle of maximum entropy', SIAM JOURNAL ON OPTIMIZATION, 14 464-478 (2003) [C1]
DOI10.1137/S1052623401400324Author URL
CitationsWeb of Science - 8
2003Borwein JM, Goebel R, 'Notions of Relative Interior in Banach spaces', Journal of Mathematical Sciences: a translation of selected Russian-language serial publications in mathematics, 115 2542-2553 (2003) [C1]
DOI10.1023/A:1022988116044
2002Borwein JM, Fabian M, Loewen PD, 'The range of the gradient of a Lipschitz C-1-smooth bump in infinite dimensions', ISRAEL JOURNAL OF MATHEMATICS, 132 239-251 (2002)
DOI10.1007/BF02784514Author URL
CitationsScopus - 8Web of Science - 9
2002Borwein JM, Jimenez-Sevilla M, Moreno JP, 'Antiproximinal norms in Banach spaces', JOURNAL OF APPROXIMATION THEORY, 114 57-69 (2002)
DOI10.1006/jath.2001.3636Author URL
CitationsScopus - 3Web of Science - 1
2002Borwein J, Giles J, Vanderwerff J, 'Rotund norms, Clarke subdifferentials and extensions of Lipschitz functions', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 48 287-301 (2002)
DOI10.1016/S0362-546X(00)00187-5Author URL
CitationsScopus - 4Web of Science - 2
2002Borwein JM, Zhu QJ, 'A survey of subdifferential calculus with applications (vol 38, pg 687, 1999)', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 49 295-296 (2002)
DOI10.1016/S0362-546X(01)00099-2Author URL
CitationsScopus - 3Web of Science - 2
2002Borwein D, Borwein JM, Mares BA, 'Multi-variable sinc integrals and volumes of polyhedra', RAMANUJAN JOURNAL, 6 189-208 (2002)
DOI10.1023/A:1015727317007Author URL
CitationsScopus - 5Web of Science - 5
2002Borwein JM, Vanderwerff JD, 'On the continuity of biconjugate convex functions', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 130 1797-1803 (2002)
DOI10.1090/S0002-9939-01-06249-9Author URL
CitationsScopus - 2Web of Science - 2
2001Borwein JM, 'Generalized trigonometric functions', AMERICAN MATHEMATICAL MONTHLY, 108 473-474 (2001)
Author URL
2001Borwein JM, Bradley DM, Broadhurst DJ, Lisonek P, 'Special values of multiple polylogarithms', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 353 907-941 (2001)
DOI10.1090/S0002-9947-00-02616-7Author URL
CitationsScopus - 117Web of Science - 126
2001Borwein JM, Moors WB, Wang XF, 'Generalized subdifferentials: A Baire categorical approach', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 353 3875-3893 (2001)
DOI10.1090/S0002-9947-01-02820-3Author URL
CitationsScopus - 10Web of Science - 6
2001Bauschke HH, Borwein JM, Combettes PL, 'Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces', COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 3 615-647 (2001)
DOI10.1142/S0219199701000524Author URL
CitationsScopus - 77Web of Science - 83
2001Borwein JM, Fitzpatrick SP, 'Duality inequalities and sandwiched functions', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 46 365-380 (2001)
DOI10.1016/S0362-546X(00)00129-2Author URL
CitationsScopus - 1Web of Science - 1
2001Borwein J, Kortezov I, 'Some generic results on nonattaining functionals', SET-VALUED ANALYSIS, 9 35-47 (2001)
DOI10.1023/A:1011246414064Author URL
CitationsWeb of Science - 2
2001Borwein JM, Vanderwerff JD, 'Convex functions of Legendre type in general Banach spaces', JOURNAL OF CONVEX ANALYSIS, 8 569-581 (2001)
Author URL
CitationsScopus - 5Web of Science - 6
2001Borwein JM, Broadhurst DJ, Kamnitzer J, 'Central binomial sums, multiple Clausen values, and zeta values', EXPERIMENTAL MATHEMATICS, 10 25-34 (2001)
Author URL
CitationsScopus - 31Web of Science - 34
2001Borwein D, Borwein J, Fee G, Girgensohn R, 'Refined convexity and special cases of the Blaschke-Santalo inequality', MATHEMATICAL INEQUALITIES & APPLICATIONS, 4 631-638 (2001)
Author URL
CitationsWeb of Science - 10
2001Borwein D, Borwein JM, 'Some remarkable properties of sinc and related integrals', RAMANUJAN JOURNAL, 5 73-89 (2001)
DOI10.1023/A:1011497229317Author URL
CitationsScopus - 13Web of Science - 11
2001Borwein JM, Borwein PB, 'Challenges in mathematical computing', COMPUTING IN SCIENCE & ENGINEERING, 3 48-53 (2001)
DOI10.1109/5992.919266Author URL
CitationsScopus - 2Web of Science - 4
2001Bauschke HH, Borwein JM, 'Joint and separate convexity of the bregman distance', Studies in Computational Mathematics, 8 23-36 (2001)

Algorithms involving Bregman projections for solving optimization problems have beenreceiving much attention lately. Several of these methods rely crucially on the joint convexity of the Bregman distance. In this note, we study joint and separate convexity of Bregman distances. To bring out the main ideas more clearly, we consider first functions defined on an open interval. Our main result states that the Bregman distance of a given function is jointly convex if and only if the reciprocal of its second derivative is concave. We observe that Bregman distances induced by the two most popular choices-the energy and the Boltzmann-Shannon entropy-are limiting cases in a profound sense. This result is generalized by weakening assumptions on differentiability and strict convexity. We then consider general, not necessarily separable, convex functions. The characterization of joint convexity has a natural and beautiful analog. Finally, we discuss spectral functions, where the situation is less clear. Throughout, we provide numerous examples to illustrate our results. © 2001 Elsevier B.V. All rights reserved.

DOI10.1016/S1570-579X(01)80004-5
CitationsScopus - 5
2000Borwein JM, Bradley DM, Crandall RE, 'Computational strategies for the Riemann zeta function', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 121 247-296 (2000)
DOI10.1016/S0377-0427(00)00336-8Author URL
CitationsScopus - 56Web of Science - 53
2000Borwein J, Lucet Y, Mordukhovich B, 'Compactly epi-Lipschitzian convex sets and functions in normed spaces', JOURNAL OF CONVEX ANALYSIS, 7 375-393 (2000)
Author URL
CitationsWeb of Science - 25
2000Bauschke HH, Borwein JM, Tseng P, 'Bounded linear regularity, strong CHIP, and CHIP are distinct properties', JOURNAL OF CONVEX ANALYSIS, 7 395-412 (2000)
Author URL
CitationsScopus - 19Web of Science - 20
2000Borwein J, Choi KKS, 'On the representations of xy plus yz plus zx', EXPERIMENTAL MATHEMATICS, 9 153-158 (2000)
Author URL
CitationsWeb of Science - 5
2000Borwein JM, Marechal P, Naugler D, 'A convex dual approach to the computation of NMR complex spectra', MATHEMATICAL METHODS OF OPERATIONS RESEARCH, 51 91-102 (2000)
Author URL
CitationsScopus - 6Web of Science - 5
2000Borwein D, Borwein JM, Marechal P, 'Surprise maximization', AMERICAN MATHEMATICAL MONTHLY, 107 517-527 (2000)
DOI10.2307/2589347Author URL
CitationsScopus - 3Web of Science - 2
2000Borwein JM, 'Moments of the Poisson distribution', AMERICAN MATHEMATICAL MONTHLY, 107 659-659 (2000)
Author URL
CitationsWeb of Science - 1
2000Borwein JM, Moors WB, 'Separable determination of integrability and minimality of the Clarke subdifferential mapping', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 128 215-221 (2000)
DOI10.1090/S0002-9939-99-05001-7Author URL
CitationsScopus - 10Web of Science - 6
2000Borwein JM, Wang XF, 'Lipschitz functions with maximal Clarke subdifferentials are generic', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 128 3221-3229 (2000)
DOI10.1090/S0002-9939-00-05914-1Author URL
CitationsScopus - 6Web of Science - 8
1999Borwein JM, Corless RM, 'Emerging tools for experimental mathematics', AMERICAN MATHEMATICAL MONTHLY, 106 889-909 (1999)
DOI10.2307/2589743Author URL
CitationsScopus - 13Web of Science - 12
1999Marechal P, Togane D, Celler A, Borwein JM, 'Computation and stability analysis for regularized tomographic reconstructions', IEEE TRANSACTIONS ON NUCLEAR SCIENCE, 46 2177-2184 (1999)
DOI10.1109/23.819301Author URL
CitationsScopus - 3Web of Science - 3
1999Borwein JM, Mordukhovich BS, Shao YH, 'On the equivalence of some basic principles in variational analysis', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 229 228-257 (1999)
DOI10.1006/jmaa.1998.6157Author URL
CitationsScopus - 9Web of Science - 9
1999Bauschke HH, Borwein JM, Li W, 'Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization', MATHEMATICAL PROGRAMMING, 86 135-160 (1999)
DOI10.1007/s101070050083Author URL
CitationsScopus - 63Web of Science - 65
1999Bauschke HH, Borwein JM, 'Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators', PACIFIC JOURNAL OF MATHEMATICS, 189 1-20 (1999)
Author URL
CitationsScopus - 19Web of Science - 23
1999Bauschke HH, Noll D, Celler A, Borwein JM, 'An EM algorithm for dynamic SPECT', IEEE TRANSACTIONS ON MEDICAL IMAGING, 18 252-261 (1999)
DOI10.1109/42.764899Author URL
CitationsWeb of Science - 23
1999Borwein J, Moors WB, Shao Y, 'Subgradient representation of multifunctions', JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES B-APPLIED MATHEMATICS, 40 301-313 (1999)
Author URL
1999Borwein JM, Treiman JS, Zhu QJJ, 'Partially smooth variational principles and applications', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 35 1031-1059 (1999)
DOI10.1016/S0362-546X(98)00113-8Author URL
CitationsScopus - 5Web of Science - 4
1999Borwein JM, Zhu QJJ, 'A survey of subdifferential calculus with applications', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 38 687-773 (1999)
DOI10.1016/S0362-546X(98)00142-4Author URL
CitationsScopus - 61Web of Science - 68
1999Marechal P, Togane D, Celler A, Borwein JM, 'Assessment of the performance of reconstruction processes for computed tomography', IEEE Nuclear Science Symposium and Medical Imaging Conference, 2 1353-1357 (1999)

This paper deals with the assessment of the performance of reconstruction methods for computed tomography, including Filtered Back Projection and 'entropy-like' methods. For each of these methods, the influence of errors in the measured data on the reconstructed image is analyzed. A small perturbation of the data vector induces a perturbation of the reconstructed object which can be computed by means of the sensitivity matrix. Using appropriate matrix computation techniques, an upper bound on the size of the reconstruction error is determined, as well as the pattern of noise in the sinogram that will result in the largest reconstruction error. Simulations will illustrate our analysis and demonstrate its utility in the interpretation of computed images and in the selection of reconstruction parameters.

1998Borwein JM, Wang XF, 'The converse of the mean value theorem may fail generically', AMERICAN MATHEMATICAL MONTHLY, 105 847-848 (1998)
DOI10.2307/2589216Author URL
CitationsScopus - 1Web of Science - 1
1998Borwein JM, Treiman JS, Zhu QJ, 'Necessary conditions for constrained optimization problems with semicontinuous and continuous data', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 350 2409-2429 (1998)
DOI10.1090/S0002-9947-98-01984-9Author URL
CitationsScopus - 26Web of Science - 29
1998Borwein D, Borwein JM, Pinner C, 'Convergence of Madelung-like lattice sums', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 350 3131-3167 (1998)
DOI10.1090/S0002-9947-98-01983-7Author URL
CitationsScopus - 8Web of Science - 9
1998Borwein JM, Girgensohn R, Wang XF, 'On the construction of holder and proximal subderivatives', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 41 497-507 (1998)
DOI10.4153/CMB-1998-065-9Author URL
CitationsScopus - 3Web of Science - 3
1998Borwein JM, 'Brouwer-Heyting sequences converge', MATHEMATICAL INTELLIGENCER, 20 14-15 (1998)
Author URL
CitationsScopus - 3Web of Science - 3
1998Borwein JM, Zhu QJ, 'Limiting convex examples for nonconvex subdifferential calculus', Journal of Convex Analysis, 5 221-235 (1998)
CitationsScopus - 1
1998Borwein JM, Moors WB, 'A chain rule for essentially smooth Lipschitz functions', SIAM JOURNAL ON OPTIMIZATION, 8 300-308 (1998)
DOI10.1137/S1052623496297838Author URL
CitationsScopus - 7Web of Science - 8
1998Borwein JM, Moors WB, 'Null sets and essentially smooth Lipschitz functions', SIAM JOURNAL ON OPTIMIZATION, 8 309-323 (1998)
DOI10.1137/S1052623496305213Author URL
CitationsScopus - 16Web of Science - 16
1998Borwein JM, Bradley DM, Broadhurst DJ, Lisonek P, 'Combinatorial aspects of multiple zeta values', Electronic Journal of Combinatorics, 5 (1998)

Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally, we present extensive computational evidence supporting an in finite family of conjectured MZV identities that simultaneously generalize the Zagier identity.

CitationsScopus - 39
1998Borwein JM, 'A generalization of Young's l(p) equality', MATHEMATICAL INEQUALITIES & APPLICATIONS, 1 131-136 (1998)
Author URL
CitationsScopus - 4Web of Science - 4
1998Borwein JM, Jofre A, 'A nonconvex separation property in Banach spaces', MATHEMATICAL METHODS OF OPERATIONS RESEARCH, 48 169-179 (1998)
DOI10.1007/s001860050019Author URL
CitationsScopus - 15Web of Science - 15
1997Borwein J, Bradley D, 'Empirically Determined Apéry-Like Formulae for ¿(4n+3)', Experimental Mathematics, 6 181-194 (1997)
DOI10.1080/10586458.1997.10504608
1997Borwein J, Monagan M, 'A note from the editors - Winter 1997, Vol 4, No 1', MAPLETECH, 4 1-1 (1997)
Author URL
1997Borwein JM, Lisonek P, Macdonald JA, 'Arithmetic-geometric means revisited', MAPLETECH, 4 20-27 (1997)
Author URL
1997Borwein J, Monagan M, 'Differential equations in Maple: A short tutorial', MAPLETECH, 4 55-58 (1997)
Author URL
1997Borwein JM, Bradley DM, Broadhurst DJ, 'Evaluations of k-fold Euler/Zagier sums: A compendium of results for arbitrary k', Electronic Journal of Combinatorics, 4 1-21 (1997)

Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble results for Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of arbitrary depth, including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations, we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking, we have sketched derivations of all results that have so far been proved.

CitationsScopus - 74
1997Borwein JM, Wang XF, 'Distinct differentiable functions may share the same Clarke subdifferential at all points', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 125 807-813 (1997)
DOI10.1090/S0002-9939-97-03654-XAuthor URL
CitationsScopus - 3Web of Science - 2
1997Borwein J, Vanderwerff J, 'Convex functions on Banach spaces not containing l(1)', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 40 10-18 (1997)
DOI10.4153/CMB-1997-002-1Author URL
CitationsWeb of Science - 3
1997Borwein JM, Moors WB, 'Essentially smooth Lipschitz functions', JOURNAL OF FUNCTIONAL ANALYSIS, 149 305-351 (1997)
DOI10.1006/jfan.1997.3101Author URL
CitationsScopus - 36Web of Science - 35
1997Borwein JM, Sun W, 'The stability analysis of dynamic SPECT systems', NUMERISCHE MATHEMATIK, 77 283-298 (1997)
DOI10.1007/s002110050287Author URL
CitationsWeb of Science - 2
1997Bailey DH, Borwein JM, Crandall RE, 'On the Khintchine constant', Mathematics of Computation, 66 417-431 (1997)
CitationsScopus - 10Web of Science - 12
1997Bauschke HH, Borwein JM, 'Legendre functions and the method of random bregman projections', Journal of Convex Analysis, 4 27-67 (1997)
CitationsScopus - 106
1997Bailey DH, Borwein JM, Borwein PB, Plouffe S, 'The quest for Pi', MATHEMATICAL INTELLIGENCER, 19 50-57 (1997)
Author URL
CitationsScopus - 26Web of Science - 26
1997Borwein JM, Zhu QJJ, 'Variational analysis in nonreflexive spaces and applications to control problems with L(1) perturbations', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 28 889-915 (1997)
DOI10.1016/0362-546X(95)00186-YAuthor URL
CitationsScopus - 6Web of Science - 7
1997Borwein JM, Moors WB, Wang XF, 'Lipschitz functions with prescribed derivatives and subderivatives', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 29 53-63 (1997)
DOI10.1016/S0362-546X(96)00050-8Author URL
CitationsScopus - 9Web of Science - 11
1996Borwein D, Borwein JM, Borwein PB, Girgensohn R, 'Giuga's conjecture on primality', AMERICAN MATHEMATICAL MONTHLY, 103 40-50 (1996)
DOI10.2307/2975213Author URL
CitationsScopus - 9Web of Science - 5
1996Borwein JM, Vanderwerff JD, 'Banach spaces that admit support sets', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 124 751-755 (1996)
DOI10.1090/S0002-9939-96-03122-XAuthor URL
CitationsScopus - 12Web of Science - 12
1996Borwein JM, Vanderwerff JD, 'Epigraphical and uniform convergence of convex functions', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 348 1617-1631 (1996)
DOI10.1090/S0002-9947-96-01581-4Author URL
CitationsScopus - 13Web of Science - 12
1996Bauschke HH, Borwein JM, 'Projection algorithms for solving convex feasibility problems', SIAM REVIEW, 38 367-426 (1996)
DOI10.1137/S0036144593251710Author URL
CitationsScopus - 605Web of Science - 602
1996Borwein J, Borwein P, Girgensohn R, Parnes S, 'Making sense of experimental mathematics', MATHEMATICAL INTELLIGENCER, 18 12-18 (1996)
Author URL
CitationsWeb of Science - 9
1996Borwein JM, Zhu QJ, 'Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 34 1568-1591 (1996)
DOI10.1137/S0363012994268801Author URL
CitationsScopus - 42Web of Science - 39
1996Borwein JM, Lewis AS, Noll D, 'Maximum entropy reconstruction using derivative information .1. Fisher information and convex duality', MATHEMATICS OF OPERATIONS RESEARCH, 21 442-468 (1996)
DOI10.1287/moor.21.2.442Author URL
CitationsScopus - 15Web of Science - 16
1996Borwein JM, Ioffe A, 'Proximal analysis in smooth spaces', SET-VALUED ANALYSIS, 4 1-24 (1996)
DOI10.1007/BF00419371Author URL
CitationsScopus - 22Web of Science - 33
1996Borwein D, Borwein JM, Wang XF, 'Approximate subgradients and coderivatives in R(n)', SET-VALUED ANALYSIS, 4 375-398 (1996)
DOI10.1007/BF00436112Author URL
CitationsScopus - 2Web of Science - 3
1996Borwein JM, Girgensohn R, 'Evaluation of triple Euler sums', Electronic Journal of Combinatorics, 3 1-27 (1996)

Let a, b, c be positive integers and define the so-called triple, double and single Euler sums by ¿(a, b, c) := ¿sum;sum; x=1 y=1 z=18; x=1 y=11/xay bzc, ¿(a, b) := ¿¿x=1 y=18 x=11/xayb and ¿(a) := ¿x=181/xa. Extending earlier work about double sums, we prove that whenever a + b + c is even or less than 10, then ¿(a, b, c) can be expressed as a rational linear combination of products of double and single Euler sums. The proof involves finding and solving linear equations which relate the different types of sums to each other. We also sketch some applications of these results in Theoretical Physics.

CitationsScopus - 41
1995BORWEIN JM, HUANG WZ, 'A FAST HEURISTIC METHOD FOR POLYNOMIAL MOMENT PROBLEMS WITH BOLTZMANN-SHANNON ENTROPY', SIAM JOURNAL ON OPTIMIZATION, 5 68-99 (1995)
DOI10.1137/0805004Author URL
CitationsWeb of Science - 7
1995BORWEIN D, BORWEIN JM, 'ON AN INTRIGUING INTEGRAL AND SOME SERIES RELATED TO ZETA(4)', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 123 1191-1198 (1995)
DOI10.2307/2160718Author URL
CitationsWeb of Science - 33
1995BORWEIN JM, GIRGENSOHN R, 'ADDITION THEOREMS AND BINARY EXPANSIONS', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 47 262-273 (1995)
DOI10.4153/CJM-1995-013-4Author URL
CitationsWeb of Science - 1
1995BORWEIN D, BORWEIN JM, GIRGENSOHN R, 'EXPLICIT EVALUATION OF EULER SUMS', PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 38 277-294 (1995)
Author URL
CitationsWeb of Science - 55
1995Limber MA, Limber MN, Celler A, Barney JS, Borwein JM, 'Direct reconstruction of functional parameters for dynamic SPECT', IEEE Transactions on Nuclear Science, 42 1249-1256 (1995)

We have developed a dynamic SPECT method which allows the prevalent rotating camera systems to be used. In this method, the parameters of the time-activity curve in each pixel are recovered directly from the projection data, without first reconstructing a series of static images. Preliminary test results show that we can accurately recover time-activity curves for a simple model based on activity values expected for myocardial viability studies with fatty acid tracers.

CitationsScopus - 33
1995BORWEIN JM, VANDERWERFF JD, 'CONVERGENCE OF LIPSCHITZ REGULARIZATIONS OF CONVEX-FUNCTIONS', JOURNAL OF FUNCTIONAL ANALYSIS, 128 139-162 (1995)
DOI10.1006/jfan.1995.1026Author URL
CitationsScopus - 2Web of Science - 2
1995BORWEIN JM, LEWIS AS, LIMBER MN, NOLL D, 'MAXIMUM-ENTROPY RECONSTRUCTION USING DERIVATIVE INFORMATION .2. COMPUTATIONAL RESULTS', NUMERISCHE MATHEMATIK, 69 243-256 (1995)
DOI10.1007/s002110050090Author URL
CitationsWeb of Science - 8
1994BORWEIN JM, HSU LC, MABRY R, NEU K, ROPPERT J, TYLER DB, DEWEGER BMM, 'NEAREST INTEGER ZETA-FUNCTIONS', AMERICAN MATHEMATICAL MONTHLY, 101 579-580 (1994)
Author URL
1994BORWEIN JM, NOLL D, '2ND-ORDER DIFFERENTIABILITY OF CONVEX-FUNCTIONS IN BANACH-SPACES', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 342 43-81 (1994)
DOI10.2307/2154684Author URL
CitationsWeb of Science - 34
1994BORWEIN JM, BORWEIN PB, GARVAN FG, 'SOME CUBIC MODULAR IDENTITIES OF RAMANUJAN', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 343 35-47 (1994)
DOI10.2307/2154520Author URL
CitationsWeb of Science - 58
1994BORWEIN JM, LEWIS AS, LIMBER MA, 'ENTROPY MINIMIZATION WITH LATTICE BOUNDS', JOURNAL OF APPROXIMATION THEORY, 79 1-16 (1994)
DOI10.1006/jath.1994.1110Author URL
CitationsScopus - 1Web of Science - 2
1994Bauschke HH, Borwein JM, 'Dykstra's Alternating Projection Algorithm for Two Sets', Journal of Approximation Theory, 79 418-443 (1994)
DOI10.1006/jath.1994.1136
CitationsScopus - 64Web of Science - 60
1994BORWEIN JM, LEWIS AS, NUSSBAUM RD, 'ENTROPY MINIMIZATION, DAD PROBLEMS, AND DOUBLY STOCHASTIC KERNELS', JOURNAL OF FUNCTIONAL ANALYSIS, 123 264-307 (1994)
DOI10.1006/jfan.1994.1089Author URL
CitationsScopus - 16Web of Science - 18
1994BORWEIN JM, FABIAN M, 'A NOTE ON REGULARITY OF SETS AND OF DISTANCE FUNCTIONS IN BANACH-SPACE', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 182 566-570 (1994)
DOI10.1006/jmaa.1994.1104Author URL
CitationsScopus - 11Web of Science - 14
1994BORWEIN D, BORWEIN JM, 'ON SOME TRIGONOMETRIC AND EXPONENTIAL LATTICE SUMS', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 188 209-218 (1994)
DOI10.1006/jmaa.1994.1422Author URL
CitationsScopus - 5Web of Science - 8
1994BORWEIN J, VANDERWERFF J, 'DUAL KADEC-KLEE NORMS AND THE RELATIONSHIPS BETWEEN WIJSMAN, SLICE, AND MOSCO CONVERGENCE', MICHIGAN MATHEMATICAL JOURNAL, 41 371-387 (1994)
Author URL
CitationsWeb of Science - 6
1994Borwein JM, Huang W, 'Uniform convergence for moment problems with Fermi-Dirac type entropies', ZOR Zeitschrift für Operations Research Mathematical Methods of Operations Research, 40 239-252 (1994)
DOI10.1007/BF01432968
CitationsScopus - 1
1994BORWEIN JM, LEWIS AS, 'STRONG ROTUNDITY AND OPTIMIZATION', SIAM JOURNAL ON OPTIMIZATION, 4 146-158 (1994)
DOI10.1137/0804008Author URL
CitationsWeb of Science - 9
1994Bailey DH, Borwein JM, Girgensohn R, 'Experimental Evaluation of Euler Sums', Experimental Mathematics, 3 17-30 (1994)
DOI10.1080/10586458.1994.10504573
1993BORWEIN JM, ZHUANG D, 'SUPER EFFICIENCY IN VECTOR OPTIMIZATION', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 338 105-122 (1993)
DOI10.2307/2154446Author URL
CitationsWeb of Science - 106
1993HIRSCHHORN M, GARVAN F, BORWEIN J, 'CUBIC ANALOGS OF THE JACOBIAN THETA-FUNCTION THETA(Z,Q)', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 45 673-694 (1993)
DOI10.4153/CJM-1993-038-2Author URL
CitationsWeb of Science - 37
1993BORWEIN JM, FABIAN M, 'ON CONVEX-FUNCTIONS HAVING POINTS OF GATEAUX DIFFERENTIABILITY WHICH ARE NOT POINTS OF FRECHET DIFFERENTIABILITY', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 45 1121-1134 (1993)
DOI10.4153/CJM-1993-062-8Author URL
CitationsWeb of Science - 19
1993BORWEIN JM, FITZPATRICK S, 'A WEAK HADAMARD SMOOTH RENORMING OF L(1)(NU,MU)', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 36 407-413 (1993)
DOI10.4153/CMB-1993-055-5Author URL
CitationsWeb of Science - 14
1993BORWEIN JM, BORWEIN PB, 'INEQUALITIES FOR COMPOUND MEAN ITERATIONS WITH LOGARITHMIC ASYMPTOTES', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 177 572-582 (1993)
DOI10.1006/jmaa.1993.1278Author URL
CitationsScopus - 17Web of Science - 19
1993Beer G, Borwein JM, 'Mosco and Slice Convergence of Level Sets and Graphs of Linear Functionals', Journal of Mathematical Analysis and Applications, 175 53-67 (1993)
DOI10.1006/jmaa.1993.1151
CitationsScopus - 4Web of Science - 13
1993BORWEIN JM, BORWEIN PB, 'ON THE GENERATING FUNCTION OF THE INTEGER PART - [N-ALPHA+GAMMA]', JOURNAL OF NUMBER THEORY, 43 293-318 (1993)
DOI10.1006/jnth.1993.1023Author URL
CitationsScopus - 13Web of Science - 17
1993BORWEIN JM, 'ON THE FAILURE OF MAXIMUM-ENTROPY RECONSTRUCTION FOR FREDHOLM EQUATIONS AND OTHER INFINITE SYSTEMS', MATHEMATICAL PROGRAMMING, 61 251-261 (1993)
DOI10.1007/BF01582150Author URL
CitationsScopus - 5Web of Science - 5
1993BORWEIN J, BORWEIN P, 'PI IN THE SKY - COUNTING, THINKING, AND BEING - BARROW,JD', SCIENCE, 259 1928-1930 (1993)
DOI10.1126/science.259.5103.1928Author URL
1993BORWEIN J, BORWEIN P, GARVAN F, 'HYPERGEOMETRIC ANALOGS OF THE ARITHMETIC GEOMETRIC MEAN ITERATION', CONSTRUCTIVE APPROXIMATION, 9 509-523 (1993)
DOI10.1007/BF01204654Author URL
CitationsWeb of Science - 7
1993BORWEIN JM, FABIAN M, 'ON GENERIC 2ND-ORDER GATEAUX DIFFERENTIABILITY', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 20 1373-1382 (1993)
DOI10.1016/0362-546X(93)90166-PAuthor URL
CitationsScopus - 2Web of Science - 2
1993BORWEIN JM, BORWEIN PB, 'CLASS NUMBER 3 RAMANUJAN TYPE SERIES FOR 1/PI', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 46 281-290 (1993)
DOI10.1016/0377-0427(93)90302-RAuthor URL
CitationsScopus - 17Web of Science - 15
1993Bauschke HH, Borwein JM, 'On the convergence of von Neumann's alternating projection algorithm for two sets', Set-Valued Analysis, 1 185-212 (1993)
DOI10.1007/BF01027691
CitationsScopus - 65
1993Borwein JM, Lewis AS, 'Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces', Set-Valued Analysis, 1 355-363 (1993)
DOI10.1007/BF01027825
CitationsScopus - 4
1992BORWEIN JM, BORWEIN PB, 'STRANGE SERIES AND HIGH-PRECISION FRAUD', AMERICAN MATHEMATICAL MONTHLY, 99 622-640 (1992)
DOI10.2307/2324993Author URL
CitationsWeb of Science - 4
1992BORWEIN JM, LEWIS AS, 'DECOMPOSITION OF MULTIVARIATE FUNCTIONS', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 44 463-482 (1992)
DOI10.4153/CJM-1992-030-9Author URL
CitationsWeb of Science - 6
1992BORWEIN JM, THERA M, 'SANDWICH THEOREMS FOR SEMICONTINUOUS OPERATORS', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 35 463-474 (1992)
DOI10.4153/CMB-1992-061-7Author URL
CitationsWeb of Science - 12
1992BORWEIN J, REICH S, SHAFRIR I, 'KRASNOSELSKI-MANN ITERATIONS IN NORMED SPACES', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 35 21-28 (1992)
DOI10.4153/CMB-1992-003-0Author URL
CitationsWeb of Science - 49
1992BORWEIN J, LOU S, 'ASYMPTOTICS OF A SEQUENCE OF WITT VECTORS', JOURNAL OF APPROXIMATION THEORY, 69 326-337 (1992)
DOI10.1016/0021-9045(92)90006-AAuthor URL
CitationsWeb of Science - 3
1992BORWEIN JM, LEWIS AS, 'PARTIALLY FINITE CONVEX-PROGRAMMING .1. QUASI RELATIVE INTERIORS AND DUALITY-THEORY', MATHEMATICAL PROGRAMMING, 57 15-48 (1992)
DOI10.1007/BF01581072Author URL
CitationsScopus - 111Web of Science - 134
1992BORWEIN JM, LEWIS AS, 'PARTIALLY FINITE CONVEX-PROGRAMMING .2. EXPLICIT LATTICE MODELS', MATHEMATICAL PROGRAMMING, 57 49-83 (1992)
DOI10.1007/BF01581073Author URL
CitationsScopus - 17Web of Science - 22
1992BORWEIN JM, ZUCKER IJ, 'FAST EVALUATION OF THE GAMMA FUNCTION FOR SMALL RATIONAL FRACTIONS USING COMPLETE ELLIPTIC INTEGRALS OF THE 1ST KIND', IMA JOURNAL OF NUMERICAL ANALYSIS, 12 519-526 (1992)
DOI10.1093/imanum/12.4.519Author URL
CitationsScopus - 25Web of Science - 34
1991BORWEIN JM, DEBARRA G, 'NESTED RADICALS', AMERICAN MATHEMATICAL MONTHLY, 98 735-739 (1991)
DOI10.2307/2324426Author URL
CitationsWeb of Science - 6
1991BORWEIN JM, BORWEIN PB, 'A CUBIC COUNTERPART OF JACOBIS IDENTITY AND THE AGM', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 323 691-701 (1991)
DOI10.2307/2001551Author URL
CitationsWeb of Science - 109
1991BORWEIN JM, LEWIS AS, 'ON THE CONVERGENCE OF MOMENT PROBLEMS', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 325 249-271 (1991)
DOI10.2307/2001670Author URL
CitationsWeb of Science - 35
1991BORWEIN J, FITZPATRICK S, KENDEROV P, 'MINIMAL CONVEX USCOS AND MONOTONE-OPERATORS ON SMALL SETS', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 43 461-476 (1991)
DOI10.4153/CJM-1991-028-5Author URL
CitationsWeb of Science - 22
1991BORWEIN D, BORWEIN J, 'FIXED-POINT ITERATIONS FOR REAL FUNCTIONS', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 157 112-126 (1991)
DOI10.1016/0022-247X(91)90139-QAuthor URL
CitationsWeb of Science - 26
1991BORWEIN JM, BORWEIN PB, 'THE MAN WHO KNEW INFINITY - A LIFE OF THE INDIAN GENIUS RAMANUJAN - KANIGEL,R', SCIENCE, 253 334-335 (1991)
DOI10.1126/science.253.5017.334Author URL
1991Borwein JM, Zhuang DM, 'Super efficiency in convex vector optimization', ZOR Zeitschrift für Operations Research Methods and Models of Operations Research, 35 175-184 (1991)
DOI10.1007/BF01415905
CitationsScopus - 28
1991BORWEIN JM, LEWIS AS, 'DUALITY RELATIONSHIPS FOR ENTROPY-LIKE MINIMIZATION PROBLEMS', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 29 325-338 (1991)
DOI10.1137/0329017Author URL
CitationsScopus - 56Web of Science - 73
1990BEER G, BORWEIN JM, 'MOSCO CONVERGENCE AND REFLEXIVITY', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 109 427-436 (1990)
DOI10.2307/2048005Author URL
CitationsWeb of Science - 30
1990BORWEIN JM, BORWEIN PB, 'A REMARKABLE CUBIC MEAN ITERATION', LECTURE NOTES IN MATHEMATICS, 1435 27-31 (1990)
Author URL
CitationsWeb of Science - 6
1989BORWEIN JM, BORWEIN PB, DILCHER K, 'PI, EULER NUMBERS, AND ASYMPTOTIC EXPANSIONS', AMERICAN MATHEMATICAL MONTHLY, 96 681-687 (1989)
DOI10.2307/2324715Author URL
CitationsWeb of Science - 13
1989BORWEIN JM, BORWEIN PB, BAILEY DH, 'RAMANUJAN, MODULAR EQUATIONS, AND APPROXIMATIONS TO PI OR HOW TO COMPUTE ONE BILLION DIGITS OF PI', AMERICAN MATHEMATICAL MONTHLY, 96 201-219 (1989)
DOI10.2307/2325206Author URL
CitationsWeb of Science - 31
1989BORWEIN JM, FITZPATRICK S, 'MOSCO CONVERGENCE AND THE KADEC PROPERTY', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 106 843-851 (1989)
DOI10.2307/2047444Author URL
CitationsWeb of Science - 29
1989BORWEIN J, FITZPATRICK S, 'LOCAL BOUNDEDNESS OF MONOTONE-OPERATORS UNDER MINIMAL HYPOTHESES', BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 39 439-441 (1989)
Author URL
CitationsWeb of Science - 8
1989BORWEIN JM, FITZPATRICK S, 'EXISTENCE OF NEAREST POINTS IN BANACH-SPACES', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 41 702-720 (1989)
DOI10.4153/CJM-1989-032-7Author URL
CitationsWeb of Science - 50
1989BORWEIN D, BORWEIN JM, SHAIL R, 'ANALYSIS OF CERTAIN LATTICE SUMS', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 143 126-137 (1989)
DOI10.1016/0022-247X(89)90032-2Author URL
CitationsScopus - 12Web of Science - 19
1989BORWEIN JM, BORWEIN PB, 'ON THE MEAN ITERATION (A,B)[-(A+3B/4 SQUARE ROOT AB+B/2)', MATHEMATICS OF COMPUTATION, 53 311-326 (1989)
DOI10.2307/2008364Author URL
CitationsWeb of Science - 7
1989BORWEIN JM, BORWEIN PB, 'APPROXIMATING PI WITH RAMANUJAN MODULAR EQUATIONS', ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, 19 93-102 (1989)
Author URL
CitationsScopus - 2Web of Science - 2
1989BORWEIN JM, STROJWAS HM, 'THE HYPERTANGENT CONE', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 13 125-144 (1989)
DOI10.1016/0362-546X(89)90039-4Author URL
CitationsScopus - 4Web of Science - 3
1989BORWEIN JM, DEMPSTER MAH, 'THE LINEAR ORDER COMPLEMENTARITY-PROBLEM', MATHEMATICS OF OPERATIONS RESEARCH, 14 534-558 (1989)
DOI10.1287/moor.14.3.534Author URL
CitationsWeb of Science - 22
1988BORWEIN JM, 'A NATURAL LIMIT', AMERICAN MATHEMATICAL MONTHLY, 95 557-558 (1988)
Author URL
1988BORWEIN JM, ZHUANG DM, 'VERIFIABLE NECESSARY AND SUFFICIENT CONDITIONS FOR OPENNESS AND REGULARITY OF SET-VALUED AND SINGLE-VALUED MAPS', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 134 441-459 (1988)
DOI10.1016/0022-247X(88)90034-0Author URL
CitationsScopus - 63Web of Science - 77
1988BORWEIN JM, BORWEIN PB, 'ON THE COMPLEXITY OF FAMILIAR FUNCTIONS AND NUMBERS', SIAM REVIEW, 30 589-601 (1988)
DOI10.1137/1030134Author URL
CitationsScopus - 11Web of Science - 12
1988BORWEIN JM, BORWEIN PB, 'RAMANUJAN AND PI', SCIENTIFIC AMERICAN, 258 112-& (1988)
Author URL
CitationsWeb of Science - 8
1988BARZILAI J, BORWEIN JM, '2-POINT STEP SIZE GRADIENT METHODS', IMA JOURNAL OF NUMERICAL ANALYSIS, 8 141-148 (1988)
DOI10.1093/imanum/8.1.141Author URL
CitationsScopus - 582Web of Science - 514
1988BORWEIN D, BORWEIN JM, SHAIL R, ZUCKER IJ, 'ENERGY OF STATIC ELECTRON LATTICES', JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 21 1519-1531 (1988)
DOI10.1088/0305-4470/21/7/015Author URL
CitationsScopus - 13Web of Science - 22
1988BENTAL A, BORWEIN JM, TEBOULLE M, 'A DUAL APPROACH TO MULTIDIMENSIONAL LP SPECTRAL ESTIMATION PROBLEMS', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 26 985-996 (1988)
DOI10.1137/0326053Author URL
CitationsScopus - 4Web of Science - 16
1987BORWEIN JM, BORWEIN PB, 'THE WAY OF ALL MEANS', AMERICAN MATHEMATICAL MONTHLY, 94 519-522 (1987)
DOI10.2307/2322842Author URL
CitationsWeb of Science - 9
1987BORWEIN JM, 'AUTOMATIC-CONTINUITY AND OPENNESS OF CONVEX RELATIONS', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 99 49-55 (1987)
DOI10.2307/2046269Author URL
CitationsWeb of Science - 9
1987BORWEIN JM, GILES J, 'THE PROXIMAL NORMAL FORMULA IN BANACH-SPACE', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 302 371-381 (1987)
DOI10.2307/2000915Author URL
CitationsWeb of Science - 43
1987BORWEIN JM, PREISS D, 'A SMOOTH VARIATIONAL PRINCIPLE WITH APPLICATIONS TO SUBDIFFERENTIABILITY AND TO DIFFERENTIABILITY OF CONVEX-FUNCTIONS', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 303 517-527 (1987)
DOI10.2307/2000681Author URL
CitationsWeb of Science - 154
1987BORWEIN JM, STROJWAS HM, 'PROXIMAL ANALYSIS AND BOUNDARIES OF CLOSED-SETS IN BANACH-SPACE .2. APPLICATIONS', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 39 428-472 (1987)
DOI10.4153/CJM-1987-019-4Author URL
CitationsWeb of Science - 39
1987BORWEIN JM, BORWEIN PB, 'EXPLICIT RAMANUJAN-TYPE APPROXIMATIONS TO PI OF HIGH-ORDER', PROCEEDINGS OF THE INDIAN ACADEMY OF SCIENCES-MATHEMATICAL SCIENCES, 97 53-59 (1987)
DOI10.1007/BF02837813Author URL
CitationsScopus - 2Web of Science - 2
1987BORWEIN JM, 'EPI-LIPSCHITZ-LIKE SETS IN BANACH-SPACE - THEOREMS AND EXAMPLES', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 11 1207-1217 (1987)
DOI10.1016/0362-546X(87)90008-3Author URL
CitationsScopus - 22Web of Science - 26
1987WARD DE, BORWEIN JM, 'NONSMOOTH CALCULUS IN FINITE DIMENSIONS', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 25 1312-1340 (1987)
DOI10.1137/0325072Author URL
CitationsScopus - 17Web of Science - 40
1987Giles JR, 'The differentiability of real functions on normed linear space using generalized subgradients', J.Math.Anal. and Appl., 128 512-534 (1987)
CitationsScopus - 35Web of Science - 40
1986JAGERS AA, BORWEIN JM, 'REGULAR N AND 2K-N-GONS INSCRIBED IN A UNIT-CIRCLE', AMERICAN MATHEMATICAL MONTHLY, 93 304-304 (1986)
Author URL
1986BORWEIN D, BORWEIN JM, 'A NOTE ON ALTERNATING SERIES IN SEVERAL DIMENSIONS', AMERICAN MATHEMATICAL MONTHLY, 93 531-539 (1986)
DOI10.2307/2323026Author URL
CitationsWeb of Science - 2
1986BORWEIN JM, BORWEIN PB, 'AN EXPLICIT CUBIC ITERATION ITERATION FOR PI', BIT, 26 123-126 (1986)
DOI10.1007/BF01939368Author URL
CitationsScopus - 2Web of Science - 3
1986BORWEIN JM, STROJWAS HM, 'PROXIMAL ANALYSIS AND BOUNDARIES OF CLOSED-SETS IN BANACH-SPACE, .1. THEORY', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 38 431-452 (1986)
DOI10.4153/CJM-1986-022-4Author URL
CitationsWeb of Science - 58
1986BORWEIN JM, 'PARTIALLY MONOTONE-OPERATORS AND THE GENERIC DIFFERENTIABILITY OF CONVEX CONCAVE AND BICONVEX MAPPINGS', ISRAEL JOURNAL OF MATHEMATICS, 54 42-50 (1986)
DOI10.1007/BF02764875Author URL
CitationsScopus - 4Web of Science - 4
1986Borwein JM, 'Norm duality for convex processes and applications', Journal of Optimization Theory and Applications, 48 53-64 (1986)
DOI10.1007/BF00938589
CitationsScopus - 17Web of Science - 17
1986Borwein JM, 'Stability and regular points of inequality systems', Journal of Optimization Theory and Applications, 48 9-52 (1986)
DOI10.1007/BF00938588
CitationsScopus - 70Web of Science - 105
1986BORWEIN JM, ZHUANG D, 'ON FAN MINIMAX THEOREM', MATHEMATICAL PROGRAMMING, 34 232-234 (1986)
DOI10.1007/BF01580587Author URL
CitationsScopus - 12Web of Science - 11
1986BORWEIN JM, WOLKOWICZ H, 'A SIMPLE CONSTRAINT QUALIFICATION IN INFINITE DIMENSIONAL PROGRAMMING', MATHEMATICAL PROGRAMMING, 35 83-96 (1986)
DOI10.1007/BF01589443Author URL
CitationsScopus - 13Web of Science - 23
1986BORWEIN JM, BORWEIN PB, 'MORE QUADRATICALLY CONVERGING ALGORITHMS FOR PI', MATHEMATICS OF COMPUTATION, 46 247-253 (1986)
DOI10.2307/2008229Author URL
CitationsWeb of Science - 3
1986BORWEIN JM, 'GENERIC DIFFERENTIABILITY OF ORDER-BOUNDED CONVEX OPERATORS', JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES B-APPLIED MATHEMATICS, 28 22-29 (1986)
Author URL
CitationsWeb of Science - 3
1985BORWEIN J, 'AN EXTENSION OF HADAMARD INEQUALITY', AMERICAN MATHEMATICAL MONTHLY, 92 148-149 (1985)
DOI10.2307/2322656Author URL
1985BORWEIN JM, TINGLEY DW, 'ON SUPPORTLESS CONVEX-SETS', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 94 471-476 (1985)
DOI10.2307/2045238Author URL
CitationsWeb of Science - 9
1985BORWEIN J, 'SOME MODULAR IDENTITIES OF RAMANUJAN USEFUL IN APPROXIMATING PI', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 95 365-371 (1985)
DOI10.2307/2045804Author URL
CitationsWeb of Science - 2
1985BORWEIN D, BORWEIN JM, TAYLOR KF, 'CONVERGENCE OF LATTICE SUMS AND MADELUNG CONSTANT', JOURNAL OF MATHEMATICAL PHYSICS, 26 2999-3009 (1985)
DOI10.1063/1.526675Author URL
CitationsScopus - 40Web of Science - 49
1985BORWEIN JM, 'ALTERNATIVE THEOREMS FOR GENERAL COMPLEMENTARITY-PROBLEMS', LECTURE NOTES IN ECONOMICS AND MATHEMATICAL SYSTEMS, 259 194-203 (1985)
Author URL
CitationsWeb of Science - 6
1985BORWEIN JM, STROJWAS HM, 'TANGENTIAL APPROXIMATIONS', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 9 1347-1366 (1985)
DOI10.1016/0362-546X(85)90095-1Author URL
CitationsScopus - 70Web of Science - 79
1984BORWEIN JM, SIMS B, 'NON-EXPANSIVE MAPPINGS ON BANACH-LATTICES AND RELATED TOPICS', HOUSTON JOURNAL OF MATHEMATICS, 10 339-356 (1984)
Author URL
CitationsWeb of Science - 26
Co-authorsBrailey Sims
1984BORWEIN JM, 'AN INTEGRAL CHARACTERIZATION OF EUCLIDEAN-SPACE', BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 29 357-364 (1984)
Author URL
CitationsWeb of Science - 1
1984BORWEIN JM, STROJWAS HM, 'DIRECTIONALLY LIPSCHITZIAN MAPPINGS ON BAIRE SPACES', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 36 95-130 (1984)
DOI10.4153/CJM-1984-008-7Author URL
CitationsWeb of Science - 13
1984BORWEIN JM, RICHMOND B, 'HOW MANY MATRICES HAVE ROOTS', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 36 286-299 (1984)
DOI10.4153/CJM-1984-018-0Author URL
CitationsWeb of Science - 1
1984BORWEIN JM, BORWEIN PB, 'CUBIC AND HIGHER-ORDER ALGORITHMS FOR PI', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 27 436-443 (1984)
DOI10.4153/CMB-1984-067-7Author URL
CitationsWeb of Science - 3
1984BORWEIN JM, YOST DT, 'ABSOLUTE NORMS ON VECTOR LATTICES', PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 27 215-222 (1984)
Author URL
CitationsWeb of Science - 7
1984Borwein JM, Penot JP, Thera M, 'Conjugate convex operators', Journal of Mathematical Analysis and Applications, 102 399-414 (1984)
DOI10.1016/0022-247X(84)90180-X
CitationsScopus - 22Web of Science - 24
1984BORWEIN JM, 'GENERALIZED LINEAR COMPLEMENTARITY-PROBLEMS TREATED WITHOUT FIXED-POINT THEORY', JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 43 343-356 (1984)
DOI10.1007/BF00934459Author URL
CitationsScopus - 10Web of Science - 17
1984BORWEIN JM, NIEUWENHUIS JW, '2 KINDS OF NORMALITY IN VECTOR OPTIMIZATION', MATHEMATICAL PROGRAMMING, 28 185-191 (1984)
DOI10.1007/BF02612358Author URL
CitationsScopus - 2Web of Science - 4
1984BORWEIN JM, BORWEIN PB, 'THE ARITHMETIC-GEOMETRIC MEAN AND FAST COMPUTATION OF ELEMENTARY-FUNCTIONS', SIAM REVIEW, 26 351-366 (1984)
DOI10.1137/1026073Author URL
CitationsScopus - 24Web of Science - 43
1983BORWEIN JM, 'SEMI-INFINITE PROGRAMMING DUALITY - HOW SPECIAL IS IT', LECTURE NOTES IN ECONOMICS AND MATHEMATICAL SYSTEMS, 215 10-36 (1983)
Author URL
CitationsWeb of Science - 5
1983BORWEIN JM, 'COMPLETENESS AND THE CONTRACTION PRINCIPLE', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 87 246-250 (1983)
DOI10.2307/2043697Author URL
CitationsWeb of Science - 4
1983Borwein JM, 'ADJOINT PROCESS DUALITY.', Mathematics of Operations Research, 8 403-434 (1983)
CitationsScopus - 25Web of Science - 41
1983Borwein JM, 'ON THE EXISTENCE OF PARETO EFFICIENT POINTS.', Mathematics of Operations Research, 8 64-73 (1983)
CitationsScopus - 47Web of Science - 69
1983Borwein JM, Borwein PB, 'A very rapidly convergent product expansion for p', BIT, 23 538-540 (1983)
DOI10.1007/BF01933626
CitationsScopus - 1Web of Science - 3
1982BORWEIN JM, 'ON THE HAHN-BANACH EXTENSION PROPERTY', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 86 42-46 (1982)
DOI10.2307/2044393Author URL
CitationsWeb of Science - 7
1982BORWEIN JM, 'CONTINUITY AND DIFFERENTIABILITY PROPERTIES OF CONVEX OPERATORS', PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 44 420-444 (1982)
Author URL
CitationsWeb of Science - 66
1982BORWEIN JM, 'A NOTE ON THE EXISTENCE OF SUB-GRADIENTS', MATHEMATICAL PROGRAMMING, 24 225-228 (1982)
DOI10.1007/BF01585105Author URL
CitationsScopus - 1Web of Science - 1
1982BORWEIN JM, 'A NOTE ON EPSILON-SUBGRADIENTS AND MAXIMAL MONOTONICITY', PACIFIC JOURNAL OF MATHEMATICS, 103 307-314 (1982)
Author URL
CitationsWeb of Science - 29
1982BORWEIN JM, STYAN GPH, FOSTER LV, WOLKOWICZ H, 'SOME INEQUALITIES INVOLVING STATISTICAL EXPRESSIONS', SIAM REVIEW, 24 340-342 (1982)
DOI10.1137/1024072Author URL
CitationsWeb of Science - 4
1982BORWEIN JM, 'NECESSARY AND SUFFICIENT CONDITIONS FOR QUADRATIC MINIMALITY', NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, 5 127-140 (1982)
DOI10.1080/01630568208816135Author URL
CitationsScopus - 14Web of Science - 17
1982BORWEIN JM, WOLKOWICZ H, 'CHARACTERIZATIONS OF OPTIMALITY WITHOUT CONSTRAINT QUALIFICATION FOR THE ABSTRACT CONVEX PROGRAM', MATHEMATICAL PROGRAMMING STUDY, 19 77-100 (1982)
Author URL
CitationsWeb of Science - 24
1982Borwein JM, 'NOTE ON THE EXISTENCE OF SUBGRADIENTS.', Mathematical Programming, 24 225-228 (1982)
CitationsScopus - 3
1981BORWEIN J, WOLKOWICZ H, 'REGULARIZING THE ABSTRACT CONVEX PROGRAM', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 83 495-530 (1981)
DOI10.1016/0022-247X(81)90138-4Author URL
CitationsWeb of Science - 27
1981BORWEIN JM, 'THE LIMITING LAGRANGIAN AS A CONSEQUENCE OF HELLY THEOREM', JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 33 497-513 (1981)
DOI10.1007/BF00935755Author URL
CitationsScopus - 4Web of Science - 9
1981BORWEIN JM, 'A LAGRANGE MULTIPLIER THEOREM AND A SANDWICH THEOREM FOR CONVEX RELATIONS', MATHEMATICA SCANDINAVICA, 48 189-204 (1981)
Author URL
CitationsWeb of Science - 45
1981BORWEIN JM, 'DIRECT THEOREMS IN SEMI-INFINITE CONVEX-PROGRAMMING', MATHEMATICAL PROGRAMMING, 21 301-318 (1981)
DOI10.1007/BF01584251Author URL
CitationsScopus - 5Web of Science - 22
1981BORWEIN JM, WOLKOWICZ H, 'FACIAL REDUCTION FOR A CONE-CONVEX PROGRAMMING PROBLEM', JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES A-PURE MATHEMATICS AND STATISTICS, 30 369-380 (1981)
Author URL
CitationsWeb of Science - 10
1981BORWEIN JM, WOLKOWICZ H, 'CHARACTERIZATION OF OPTIMALITY FOR THE ABSTRACT CONVEX PROGRAM WITH FINITE DIMENSIONAL RANGE', JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES A-PURE MATHEMATICS AND STATISTICS, 30 390-411 (1981)
Author URL
CitationsWeb of Science - 17
1980GRIPENBERG G, WILANSKY A, JOYNER D, ROUSH F, WALSH B, BORWEIN J, 'RANGES IN BANACH-SPACES', AMERICAN MATHEMATICAL MONTHLY, 87 68-68 (1980)
Author URL
1980BORWEIN J, KEENER L, 'THE HAUSDORFF METRIC AND CEBYSEV CENTERS', JOURNAL OF APPROXIMATION THEORY, 28 366-376 (1980)
DOI10.1016/0021-9045(80)90071-4Author URL
CitationsWeb of Science - 9
1980BORWEIN JM, 'LEXICOGRAPHIC MULTIPLIERS', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 78 309-327 (1980)
DOI10.1016/0022-247X(80)90228-0Author URL
CitationsWeb of Science - 2
1980BORWEIN JM, 'A STRONG DUALITY THEOREM FOR THE MINIMUM OF A FAMILY OF CONVEX-PROGRAMS', JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 31 453-472 (1980)
DOI10.1007/BF00934472Author URL
CitationsScopus - 2Web of Science - 7
1980BORWEIN JM, 'A NOTE ON PERFECT DUALITY AND LIMITING LAGRANGEANS', MATHEMATICAL PROGRAMMING, 18 330-337 (1980)
DOI10.1007/BF01588327Author URL
CitationsScopus - 1Web of Science - 15
1979BORWEIN JM, 'WEAK LOCAL SUPPORTABILITY AND APPLICATIONS TO APPROXIMATION', PACIFIC JOURNAL OF MATHEMATICS, 82 323-338 (1979)
Author URL
CitationsWeb of Science - 11
1979BORWEIN JM, 'MULTIVALUED APPROACH TO THE FARKAS-LEMMA', MATHEMATICAL PROGRAMMING STUDY, 10 42-47 (1979)
Author URL
CitationsWeb of Science - 2
1978BORWEIN JM, 'STRONGLY EXPOSING FUNCTIONALS', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 69 46-48 (1978)
DOI10.2307/2043185Author URL
CitationsWeb of Science - 2
1978BORWEIN J, 'WEAK TANGENT CONES AND OPTIMIZATION IN A BANACH-SPACE', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 16 512-522 (1978)
DOI10.1137/0316034Author URL
CitationsWeb of Science - 19
1977BORWEIN J, EDELSTEIN M, OBRIEN R, 'SOME REMARKS ON VISIBILITY AND STARSHAPE', JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 15 342-344 (1977)
Author URL
1977BORWEIN J, 'MULTIVALUED CONVEXITY AND OPTIMIZATION - UNIFIED APPROACH TO INEQUALITY AND EQUALITY CONSTRAINTS', MATHEMATICAL PROGRAMMING, 13 183-199 (1977)
DOI10.1007/BF01584336Author URL
CitationsWeb of Science - 40
1977BORWEIN J, 'PROPER EFFICIENT POINTS FOR MAXIMIZATIONS WITH RESPECT TO CONES', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 15 57-63 (1977)
DOI10.1137/0315004Author URL
CitationsWeb of Science - 173
1976BORWEIN J, EDELSTEIN M, OBRIEN R, 'VISIBILITY AND STARSHAPE', JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 14 313-318 (1976)
Author URL
CitationsWeb of Science - 2
1976BORWEIN JM, 'FRACTIONAL PROGRAMMING WITHOUT DIFFERENTIABILITY', MATHEMATICAL PROGRAMMING, 11 283-290 (1976)
DOI10.1007/BF01580396Author URL
CitationsScopus - 2Web of Science - 9
Show 361 more journal articles

Conference (29 outputs)

YearCitationAltmetricsLink
2014Borwein JM, 'Experimental computation and visual theorems', Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (2014) [E2]

Long before current graphic, visualisation and geometric tools were available, John E. Littlewood (1885-1977) wrote in his delightful Miscellany: A heavy warning used to be given [by lecturers] that pictures are not rigorous; this has never had its bluff called and has permanently frightened its victims into playing for safety. Some pictures, of course, are not rigorous, but I should say most are (and I use them whenever possible myself). [p. 53] Over the past five years, the role of visual computing in my own research has expanded dramatically. In part this was made possible by the increasing speed and storage capabilities-and the growing ease of programming-of modern multi-core computing environments. But, at least as much, it has been driven by my group's paying more active attention to the possibilities for graphing, animating or simulating most mathematical research activities. © 2014 Springer-Verlag.

DOI10.1007/978-3-662-44199-2_1
2013Borwein JM, 'Keynote Lecture: Entropy and Projection Methods for (Non-)Convex Programming', MaxEnt 2013: 33rd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering Abstracts, Canberra (2013) [E3]
2013Borwein JM, Rose M, Bailey D, Crandall R, 'Expectations on fractal sets', Proceedings of the 49th ANZIAM Conference, Newcastle, NSW (2013) [E3]
2011Henstridge J, Piantadosi J, Howlett P, Borwein JM, 'Generation of simulated rainfall data at different time-scales', MODSIM 2011: 19th International Congress on Modelling and Simulation Proceedings, Perth, WA (2011) [E1]
2011Borwein JM, Straub A, 'Special values of generalized log-sine integrals', Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011, San Jose, CA (2011) [E1]
DOI10.1145/1993886.1993899
CitationsScopus - 2Web of Science - 3
2010Borwein JM, Nuyens D, Straub A, Wan J, 'Random walks in the plane', FPSAC'10 - 22nd International Conference on Formal Power Series and Algebraic Combinatorics, San Francisco, CA, United States (2010) [E1]
CitationsScopus - 1
2010Borwein JM, 'Future Challenges for Variational Analysis', VARIATIONAL ANALYSIS AND GENERALIZED DIFFERENTIATION IN OPTIMIZATION AND CONTROL: IN HONOR OF BORIS S. MORDUKHOVICH, Natl Sun Yat-Sen Univ, Dept Appl Math, Kaohsiung, TAIWAN (2010) [E1]
DOI10.1007/978-1-4419-0437-9_4Author URL
CitationsWeb of Science - 1
2010Borwein JM, Sciffer SD, 'An explicit non-expansive function whose subdifferential is the entire dual ball', Nonlinear Analysis and Optimization : A Conference in Celebration of Alex Ioffe's 70th and Simeon Reich's 60th birthdays, June 18-24, 2008, Haifa, Israel, Haifa, Israel (2010) [E1]
DOI10.1090/conm/514
CitationsWeb of Science - 1
2009Borwein JM, 'Digitally-assisted discovery and proof', Proof and Proving in Mathematics Education: ICMI Study 19 Conference Proceedings, Taipei, Taiwan (2009) [E1]
2009Borwein JM, Hamilton CH, 'Symbolic fenchel conjugation', Mathematical Programming, Univ Chile, Santiago, CHILE (2009) [E1]
DOI10.1007/s10107-007-0134-4
CitationsScopus - 6Web of Science - 5
2008Bailey DH, Borwein JM, 'Computer-assisted discovery and proof', TAPAS IN EXPERIMENTAL MATHEMATICS, New Orleans, LA (2008) [E1]
Author URL
CitationsWeb of Science - 4
2006Borwein JM, 'Aesthetics for the working mathematician', Mathematics and the Aesthetic: New Approaches to and Ancient Affinity, Queens Univ, Kingston, CANADA (2006)
Author URL
CitationsWeb of Science - 1
2006Borwein JM, Hamilton CH, 'Symbolic computation of multidimensional fenchel conjugates', Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC (2006) [E1]

Conveac optimization is a branch of mathematics dealing with non-linear optimization problems with additional geometric structure. This area has been the focus of considerable recent research due to the fact that convex optimization problems are scalable and can be efficiently solved by interior-point methods. Over the last ten years or so, convex optimization has found new applications in many areas including control theory, signal processing, communications and networks, circuit design, data analysis and finance. Of key importance in convex optimization is the notion of duality, and in particular that of Fenchel duality. This work explores algorithms for calculating symbolic Fenchel conjugates of a class of real-valued functions defined on Rn, extending earlier work to the non-separable multi-dimensional case. It also explores the potential application of the developed algorithms to automatic inequality proving. Copyright 2006 ACM.

CitationsScopus - 3
2005Borwein JM, Karamanos K, 'Algebraic dynamics of certain gamma function values', GENERALIZED CONVEXITY, GENERALIZED MONOTONICITY AND APPLICATIONS, Hanoi, VIETNAM (2005)
Author URL
CitationsWeb of Science - 6
2004Borwein JM, Choi KKS, 'Dirichet series for squares of sums of squares: A summary', NUMBER THEORY, Univ Montreal, Ctr Rech Math, Montreal, CANADA (2004)
Author URL
2003Kellar M, Mackay B, Zhang R, Watters C, Kaufman D, Borwein JM, 'Architecture to support dynamic composition of math lesson plans', Proceedings of the 2003 ACM symposium on Applied computing, Melbourne, Australia (2003) [E1]
2003Borwein J, Stanway T, 'Managing digital mathematical discourse', MATHEMATICAL KNOWLEDGE MANAGEMENT, PROCEEDINGS, BERTINORO, ITALY (2003) [E1]
Author URL
2002Borwein JM, 'The experimental mathematician: A computational guide to the mathematical unknown', MATHEMATICAL SOFTWARE, PROCEEDINGS, Beijing, PEOPLES R CHINA (2002)
DOI10.1142/9789812777171_0001Author URL
2000Borwein JM, Lisonek P, 'Applications of integer relation algorithms', DISCRETE MATHEMATICS, UNIV VIENNA, VIENNA, AUSTRIA (2000)
DOI10.1016/S0012-365X(99)00256-3Author URL
CitationsScopus - 6Web of Science - 5
1999Borwein JM, Zhu QJ, 'Multifunctional and functional analytic techniques in nonsmooth analysis', NONLINEAR ANALYSIS, DIFFERENTIAL EQUATIONS AND CONTROL, MONTREAL, CANADA (1999)
Author URL
CitationsWeb of Science - 7
1995BAILEY DH, BORWEIN JM, GIRGENSOHN R, 'FINDING NEW MATHEMATICAL IDENTITIES WITH SUPERCOMPUTERS', PROCEEDINGS OF THE SEVENTH SIAM CONFERENCE ON PARALLEL PROCESSING FOR SCIENTIFIC COMPUTING, SAN FRANCISCO, CA (1995)
Author URL
1995Borwein JM, 'A survey of examples of convex functions and classifications of normed spaces', RECENT DEVELOPMENTS IN OPTIMIZATION, DIJON, FRANCE (1995)
Author URL
1995LIMBER MA, LIMBER MN, CELLER A, BARNEY JS, BORWEIN JM, 'DIRECT RECONSTRUCTION OF FUNCTIONAL PARAMETERS FOR DYNAMIC SPECT', IEEE TRANSACTIONS ON NUCLEAR SCIENCE, NORFOLK, VA (1995)
DOI10.1109/23.467872Author URL
CitationsWeb of Science - 40
1994BORWEIN JM, ZHU QJ, 'CONTROL PROBLEMS WITH L(1) PERTURBATIONS AND VARIATIONAL ANALYSIS IN NON-REFLEXIVE SPACES', PROCEEDINGS OF THE 33RD IEEE CONFERENCE ON DECISION AND CONTROL, VOLS 1-4, LAKE BUENA VISTA, FL (1994)
Author URL
1994LIMBER MA, LIMBER MN, CELLER A, BARNEY JS, BORWEIN JM, 'Direct reconstruction of functional parameters for dynamic SPECT', NUCLEAR SCIENCE SYMPOSIUM & MEDICAL IMAGING CONFERENCE - 1994 IEEE CONFERENCE RECORD, VOLS 1-4, NORFOLK, VA (1994)
Author URL
1993BORWEIN JM, LEWIS AS, 'A SURVEY OF CONVERGENCE RESULTS FOR MAXIMUM-ENTROPY METHODS', MAXIMUM ENTROPY AND BAYESIAN METHODS, CNRS, PARIS, FRANCE (1993)
Author URL
CitationsWeb of Science - 5
1991BORWEIN JM, 'MINIMAL CUSCOS AND SUBGRADIENTS OF LIPSCHITZ FUNCTIONS', FIXED POINT THEORY AND APPLICATIONS, CTR INT RECONTRES MATH, MARSEILLE, FRANCE (1991)
Author URL
CitationsWeb of Science - 23
1991BORWEIN JM, LEWIS AS, 'PRACTICAL CONDITIONS FOR FENCHEL DUALITY IN INFINITE DIMENSIONS', FIXED POINT THEORY AND APPLICATIONS, CTR INT RECONTRES MATH, MARSEILLE, FRANCE (1991)
Author URL
CitationsWeb of Science - 2
1990BORWEIN JM, BORWEIN PB, 'A REMARKABLE CUBIC MEAN ITERATION', COMPUTATIONAL METHODS AND FUNCTION THEORY, VALPARAISO, CHILE (1990)
Author URL
CitationsWeb of Science - 6
Show 26 more conferences
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Grants and Funding

Summary

Number of grants11
Total funding$2,256,625

Click on a grant title below to expand the full details for that specific grant.


20151 grants / $10,000

2015 International Visitor from Carnegie Mellon University, Pittsburgh Pennsylvania USA$10,000

Funding body: University of Newcastle

Funding bodyUniversity of Newcastle
Project TeamLaureate Professor Jon Borwein, Professor Jeremy Avigad
SchemeInternational Research Visiting Fellowship
RoleLead
Funding Start2015
Funding Finish2015
GNoG1401314
Type Of FundingInternal
CategoryINTE
UONY

20141 grants / $415,000

Exploratory Experimentation and Computation in the Mathematical Sciences: Theory and Practice$415,000

Funding body: ARC (Australian Research Council)

Funding bodyARC (Australian Research Council)
Project TeamLaureate Professor Jon Borwein, Conjoint Professor Richard Brent, Dr David Bailey
SchemeDiscovery Projects
RoleLead
Funding Start2014
Funding Finish2014
GNoG1300158
Type Of FundingAust Competitive - Commonwealth
Category1CS
UONY

20123 grants / $657,664

Structured barrier and penalty functions in infinite dimensional optimization and analysis$390,000

Funding body: ARC (Australian Research Council)

Funding bodyARC (Australian Research Council)
Project TeamLaureate Professor Jon Borwein, Professor Andrew Eberhard, Professor Adrian Lewis, Associate Professor Brailey Sims, Professor Qiji Zhu
SchemeDiscovery Projects
RoleLead
Funding Start2012
Funding Finish2012
GNoG1100082
Type Of FundingAust Competitive - Commonwealth
Category1CS
UONY

Faculty Visiting Fellowship 2012$4,664

Funding body: University of Newcastle - Faculty of Science & IT

Funding bodyUniversity of Newcastle - Faculty of Science & IT
Project TeamLaureate Professor Jon Borwein
SchemeVisiting Fellowship
RoleLead
Funding Start2012
Funding Finish2012
GNoG1401121
Type Of FundingInternal
CategoryINTE
UONY

20112 grants / $85,000

Flexible architecture High Performance Computing facility for the Intersect consortium of NSW$45,000

Funding body: University of Newcastle

Funding bodyUniversity of Newcastle
Project TeamProfessor Leo Radom, Associate Professor Frans Henskens, Laureate Professor Jon Borwein
SchemeEquipment Grant
RoleInvestigator
Funding Start2011
Funding Finish2011
GNoG1000459
Type Of FundingInternal
CategoryINTE
UONY

DVCR Special Grant 2011$40,000

Funding body: University of Newcastle

Funding bodyUniversity of Newcastle
Project TeamLaureate Professor Jon Borwein
SchemeSpecial Project Grant
RoleLead
Funding Start2011
Funding Finish2011
GNoG1101012
Type Of FundingInternal
CategoryINTE
UONY

20101 grants / $555,000

Computer Assisted Research Mathematics and its Applications$555,000

Funding body: ARC (Australian Research Council)

Funding bodyARC (Australian Research Council)
Project TeamLaureate Professor Jon Borwein
SchemeDiscovery Projects
RoleLead
Funding Start2010
Funding Finish2010
GNoG0190001
Type Of FundingAust Competitive - Commonwealth
Category1CS
UONY

20093 grants / $533,961

Computer Assisted Research Mathematics and its Applications$267,361

Funding body: University of Newcastle

Funding bodyUniversity of Newcastle
Project TeamLaureate Professor Jon Borwein, Conjoint Professor Natashia Boland, Associate Professor Brailey Sims, Professor George Willis
SchemePriority Research Centre
RoleLead
Funding Start2009
Funding Finish2009
GNoG0189866
Type Of FundingInternal
CategoryINTE
UONY

New Theory and Algorithms for Nonsmooth Optimization with Application to Integer Programming 18/5/2015 - Final report has been approved by ARC - Bec$249,000

Funding body: ARC (Australian Research Council)

Funding bodyARC (Australian Research Council)
Project TeamConjoint Professor Natashia Boland, Laureate Professor Jon Borwein, Professor Andrew Eberhard
SchemeDiscovery Projects
RoleInvestigator
Funding Start2009
Funding Finish2009
GNoG0188718
Type Of FundingAust Competitive - Commonwealth
Category1CS
UONY

DVCR contribution to Borwien Exec assist. upgrade$17,600

Funding body: University of Newcastle

Funding bodyUniversity of Newcastle
Project TeamLaureate Professor Jon Borwein
SchemeSpecial Project Grant
RoleLead
Funding Start2009
Funding Finish2009
GNoG0190019
Type Of FundingInternal
CategoryINTE
UONY
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Research Supervision

Current Supervision

CommencedResearch Title / Program / Supervisor Type
2014Conditional Resolvability of Graphs
Mathematics, Faculty of Science and Information Technology
Co-Supervisor
2014Parallelisation, Precision, and Reproducibility in Pure Mathematical Computation
Mathematics, Faculty of Science and Information Technology
Principal Supervisor
2013Nonlinear Analysis for Optimisation
Mathematics, Faculty of Science and Information Technology
Principal Supervisor
2012Expectations over Deterministic Fractal Sets
Mathematics, Faculty of Science and Information Technology
Principal Supervisor
2012Reflection and Projection Methods for non-convex inverse problems
Mathematics, Faculty of Science and Information Technology
Principal Supervisor

Past Supervision

YearResearch Title / Program / Supervisor Type
2015Arithmetic Applications of Hankel Determinants
Mathematics, Faculty of Science and Information Technology
Co-Supervisor
2014Tools for Teaching Computational Mathematics
Mathematics, Faculty of Science and Information Technology
Principal Supervisor
2013Random Walks, Elliptic Integrals and Related Constants
Mathematics, Faculty of Science and Information Technology
Principal Supervisor
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News

It's all random

December 19, 2013

Tipsy tottering, sunlight and the smell of coffee: it's all random

Jon Borwein The Conversation

Australia needs fundamental research to build a great country

November 15, 2013

By Jonathan Borwein (Jon), University of Newcastle

Like many scientists, I was apprehensive in advance about the Abbott government's approach to science policy. Would it be pragmatic but fact-based or would it be ideological and politically driven?

Laureate Professor Jon Borwein

Position

Laureate Professor
School of Mathematical and Physical Sciences
Faculty of Science and Information Technology

Focus area

Mathematics

Contact Details

Emailjonathan.borwein@newcastle.edu.au
Phone(02) 4921 5535
Fax(02) 4921 6898

Office

RoomV120
BuildingMathematics
LocationCallaghan
University Drive
Callaghan, NSW 2308
Australia
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