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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."

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

Biography

Research Expertise
Pure Mathematics - Analysis, Functional Analysis Applied Mathematics - Optimisation Computational mathematics - Numerical and computational analysis High performance computing Collaborative mathematics

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

Code Description Percentage
010108 Operator Algebras and Functional Analysis 75
010299 Applied Mathematics not elsewhere classified 15
010399 Numerical and Computational Mathematics not elsewhere classified 10

Professional Experience

UON Appointment

Title Organisation / Department
Laureate Professor University of Newcastle
School of Mathematical and Physical Sciences
Australia

Academic appointment

Dates Title Organisation / Department
25/06/2015 -  Fellow Australian 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)

Year Citation Altmetrics Link
2014 Borwein J, Poorten AVD, Shallit J, Zudilin W, Neverending Fractions, Cambridge University Press, Cambridge, U.K., 224 (2014) [A2]
Co-authors Wadim Zudilin
2014 Borwein J, Poorten AVD, Shallit J, Zudilin W, Neverending Fractions, Cambridge University Press, Cambridge, U.K., 224 (2014) [A2]
Co-authors Wadim Zudilin
2013 Borwein JM, Glasser ML, McPhedran RC, Wan JG, Zucker IJ, Lattice sums then and now, Cambridge University Press, Cambridge, 368 (2013) [A2]
2013 Borwein JM, Shparlinski I, Zudilin W, Number Theory and Related Fields: In Memory of Alf van der Poorten, Springer New York LLC (2013) [A3]
DOI 10.1007/978-1-4614-6642-0
Co-authors Wadim Zudilin
2012 Borwein JM, Skerritt MP, An Introduction to Modern Mathematical Computing with Mathematica, Springer, New York, 224 (2012) [A2]
2012 Bailey DH, Borwein JM, Exploratory Experimentation in Mathematics: Selected Works, Perfectly Scientific Press, Portland, OR, 364 (2012) [A2]
2011 Borwein JM, Skerritt MP, An Introduction to Modern Mathematical Computing: With Maple, Springer, New York, 216 (2011) [A2]
2011 Borwein JM, Devlin K, Experimentelle Mathematik: Eine beispielorientierte Einfuhrung, Spektrum Akademischer Verlag, Heidelberg, DE, 158 (2011) [A4]
DOI 10.1007/978-3-8274-2662-8
2010 Borwein JM, Vanderwerff JD, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, Cambridge, 521 (2010) [A1]
DOI 10.1017/CBO9781139087322
2009 Borwein JM, Devlin K, The Computer as Crucible: An Introduction to Experimental Mathematics, A K Peters, Natick, 158 (2009) [A1]
2008 Borwein J, David B, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, Massachusetts, United States, 288 (2008) [A1]
2008 Borwein JM, Rocha EM, Rodrigues JF, Communicating Mathematics in the Digital Era, A. K. Peters, Wellesley, MA, 325 (2008) [A3]
2007 Bailey 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]
2006 Borwein JM, Lewis AS, Convex analysis and nonlinear optimization : theory and examples, Springer, Berlin, Germany, 310 (2006) [A1]
2005 Borwein JM, Zhu QJ, Techniques of variational analysis, Springer, Berlin, Germany, 366 (2005) [A1]
DOI 10.1007/0-387-28271-8
2004 Berggren L, Borwein J, Borwein P, Pi: a Source Book, Springer, New York, United States, 797 (2004) [A1]
2004 Borwein JM, Bailey DH, Girgensohn R, Experimentation in mathematics : computational paths to discovery, A K Peters Ltd., Natick, Massachusetts, United States, 357 (2004) [A1]
2002 Borwein J, Morales MH, Polthier K, Multimedia tools for communicating mathematics, Springer, Berlin, Germany, 314 (2002) [A3]
Show 15 more books

Chapter (6 outputs)

Year Citation Altmetrics Link
2013 Bailey DH, Borwein JM, 'Normal Numbers and Pseudorandom Generators', Computational and Analytical Mathematics, Springer, New York 1-18 (2013) [B1]
DOI 10.1007/978-1-4614-7621-4_1
Citations Scopus - 1
2013 Borwein JM, Yao L, Burachik RS, 'Monotone Operators Without Enlargements', Computational and Analytical Mathematics, Springer, New York 79-103 (2013) [B1]
DOI 10.1007/978-1-4614-7621-4_5
2012 Borwein JM, 'Exploratory experimentation: Digitally-assisted discovery and proof', Proof and Proving in Mathematics Education: The 19th ICMI Study, Springer, Dordrecht 69-96 (2012) [B1]
DOI 10.1007/978-94-007-2129-6_4
2011 Borwein 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]
DOI 10.1007/978-1-4419-9569-8_5
Citations Web of Science - 3
2011 Borwein 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]
DOI 10.1007/978-1-4419-9569-8_6
Citations Web of Science - 8
Co-authors Brailey Sims
2008 Borwein 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)

Year Citation Altmetrics Link
2015 Borwein JM, Sims B, Tam MK, 'Norm convergence of realistic projection and reflection methods', OPTIMIZATION, 64 161-178 (2015)
DOI 10.1080/02331934.2014.947499
Co-authors Brailey Sims
2015 Borwein JM, Sims B, Tam MK, 'Norm convergence of realistic projection and reflection methods', Optimization, 64 161-178 (2015)
DOI 10.1080/02331934.2014.947499
2015 Borwein JM, Zhu QJ, 'A Variational Approach to Lagrange Multipliers', Journal of Optimization Theory and Applications, (2015)
DOI 10.1007/s10957-015-0756-2
2015 Borwein JM, Tam MK, 'The cyclic Douglas-Rachford method for inconsistent feasibility problems', Journal of Nonlinear and Convex Analysis, 16 537-584 (2015)
2015 Bailey 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 value... [more]

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.

DOI 10.1016/j.jat.2014.10.004
2014 Artacho 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]
DOI 10.1007/s10957-013-0488-0
Citations Scopus - 2Web of Science - 3
2014 Borwein D, Borwein JM, Sims B, 'On the Solution of Linear Mean Recurrences', AMERICAN MATHEMATICAL MONTHLY, 121 486-498 (2014) [C1]
DOI 10.4169/amer.math.monthly.121.06.486
Citations Web of Science - 1
Co-authors Brailey Sims
2014 Bailey 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]
DOI 10.4169/amer.math.monthly.121.03.191
2014 Bailey 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]
DOI 10.1090/noti1105
2014 Borwein 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 ... [more]

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.

DOI 10.1016/j.jmaa.2014.01.008
Citations Scopus - 1
2014 Borwein JM, Tam MK, 'A Cyclic Douglas-Rachford Iteration Scheme', JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 160 1-29 (2014) [C1]
DOI 10.1007/s10957-013-0381-x
Citations Scopus - 3Web of Science - 5
2014 Aragón Artacho FJ, Borwein JM, Martín-Márquez V, Yao L, 'Applications of convex analysis within mathematics', Mathematical Programming, 148 49-88 (2014) [C1]
DOI 10.1007/s10107-013-0707-3
2014 Bailey DH, Borwein JM, Crandall RE, 'COMPUTATION AND THEORY OF EXTENDED MORDELL-TORNHEIM-WITTEN SUMS', MATHEMATICS OF COMPUTATION, 83 1795-1821 (2014) [C1]
2014 Bailey DH, Borwein JM, Kaiser AD, 'Automated simplification of large symbolic expressions', JOURNAL OF SYMBOLIC COMPUTATION, 60 120-136 (2014) [C1]
DOI 10.1016/j.jsc.2013.09.001
2014 Borwein JM, Yao L, 'Legendre-Type Integrands and Convex Integral Functions', JOURNAL OF CONVEX ANALYSIS, 21 261-288 (2014) [C1]
2014 Borwein 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]
DOI 10.1137/130919052
2014 Borwein J, Howlett P, Piantadosi J, 'Modelling and Simulation of Seasonal Rainfall Using the Principle of Maximum Entropy', ENTROPY, 16 747-769 (2014) [C1]
DOI 10.3390/e16020747
Citations Scopus - 4Web of Science - 2
2014 Borwein 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 existin... [more]

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.

2014 Artacho FJA, Borwein JM, Tam MK, 'Douglas-rachford feasibility methods for matrix completion problems', ANZIAM Journal, 55 299-326 (2014) [C1]
DOI 10.1017/S1446181114000145
2014 BORWEIN JM, YAO L, 'SUM THEOREMS FOR MAXIMALLY MONOTONE OPERATORS OF TYPE (FPV)', Journal of the Australian Mathematical Society, (2014) [C1]
DOI 10.1017/S1446788714000056
2014 Borwein 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]
DOI 10.1007/s11590-012-0564-7
Citations Scopus - 1Web of Science - 1
2013 Borwein JM, Crandall RE, 'Closed Forms: What they are and why we care', Notices of the American Mathematical Society, 60 50-65 (2013) [C1]
DOI 10.1090/noti936
Citations Scopus - 13
2013 Bailey 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]
DOI 10.1090/noti1015
Citations Scopus - 2
2013 Borwein 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... [more]

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.

DOI 10.1016/j.jat.2013.07.003
2013 Aragón 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]
DOI 10.1007/s10957-013-0488-0
Citations Scopus - 2Web of Science - 2
2013 Borwein JM, Yao L, 'Structure Theory for Maximally Monotone Operators with Points of Continuity', Journal of Optimization Theory and Applications, 157 1-24 (2013) [C1]
DOI 10.1007/s10957-012-0162-y
Citations Scopus - 3Web of Science - 3
2013 Aragón 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 d... [more]

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.

DOI 10.1007/s10107-013-0707-3
2013 Bailey DH, Borwein JM, Crandall RE, Rose MG, 'Expectations on fractal sets', APPLIED MATHEMATICS AND COMPUTATION, 220 695-721 (2013) [C1]
DOI 10.1016/j.amc.2013.06.078
Citations Scopus - 2Web of Science - 1
2013 Borwein JM, Straub A, 'Mahler measures, short walks and log-sine integrals', THEORETICAL COMPUTER SCIENCE, 479 4-21 (2013) [C1]
DOI 10.1016/j.tcs.2012.10.025
Citations Scopus - 2Web of Science - 2
2013 Artacho FJA, Bailey DH, Borwein JM, Borwein PB, 'Walking on Real Numbers', MATHEMATICAL INTELLIGENCER, 35 42-60 (2013) [C1]
DOI 10.1007/s00283-012-9340-x
Citations Scopus - 4Web of Science - 2
2013 Artacho FJA, Borwein JM, 'Global convergence of a non-convex Douglas-Rachford iteration', JOURNAL OF GLOBAL OPTIMIZATION, 57 753-769 (2013) [C1]
DOI 10.1007/s10898-012-9958-4
Citations Scopus - 3Web of Science - 3
2013 Bauschke HH, Borwein JM, Wang X, Yao L, 'Monotone Operators and "Bigger Conjugate" Functions', JOURNAL OF CONVEX ANALYSIS, 20 143-155 (2013) [C1]
Citations Scopus - 1
2013 Borwein JM, Straub A, Wan J, 'Three-Step and Four-Step Random Walk Integrals', Experimental Mathematics, 22 1-14 (2013) [C1]
DOI 10.1080/10586458.2013.748379
Citations Scopus - 10Web of Science - 4
2013 Aragón Artacho FJ, Borwein JM, Tam MK, 'Recent Results on Douglas-Rachford Methods', Serdica Mathematical Journal, 39 313-330 (2013) [C1]
2013 Bailey 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 pa... [more]

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.

DOI 10.1007/s11139-012-9427-1
Citations Scopus - 1
2013 Bailey DH, Borwein JM, 'Compressed lattice sums arising from the Poisson equation', BOUNDARY VALUE PROBLEMS, (2013) [C1]
DOI 10.1186/1687-2770-2013-75
2013 Bailey DH, Borwein JM, Crandall RE, Zucker IJ, 'Lattice sums arising from the Poisson equation', Journal of Physics A: Mathematical and Theoretical, 46 (2013) [C1]
DOI 10.1088/1751-8113/46/11/115201
Citations Scopus - 3Web of Science - 2
2013 Borwein JM, Crandall RE, 'Closed Forms: What they are and why we care', Notices of the American Mathematical Society, 60 50-65 (2013)
DOI 10.1090/noti936
Citations Scopus - 15
2013 Borwein 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]
DOI 10.1007/s11228-013-0259-y
Citations Scopus - 1Web of Science - 1
2013 Borwein JM, Zhu QJ, 'Variational methods in the presence of symmetry', Advances in Nonlinear Analysis, 2 (2013) [C1]
DOI 10.1515/anona-2013-1001
2012 Borwein D, Borwein JM, Straub A, 'A sinc that sank', American Mathematical Monthly, 119 535-549 (2012) [C1]
DOI 10.4169/amer.math.monthly.119.07.535
Citations Web of Science - 1
2012 Bailey DH, Borwein JM, 'Ancient Indian square roots: An exercise in forensic paleo-mathematics', American Mathematical Monthly, 119 646-657 (2012) [C1]
DOI 10.4169/amer.math.monthly.119.08.646
Citations Scopus - 3Web of Science - 1
2012 Borwein JM, 'Ramanujan and Pi', American Mathematical Society. Notices, 59 1534-1537 (2012) [C1]
2012 Borwein JM, Jungic V, 'Organic mathematics then and now', Notices of the AMS, 59 416-419 (2012) [C2]
2012 Borwein JM, Straub A, Wan G, Zudilin V, 'Densities of short uniform random walks', Canadian Journal of Mathematics, 64 961-990 (2012) [C1]
DOI 10.4153/CJM-2011-079-2
Citations Scopus - 10Web of Science - 9
Co-authors Wadim Zudilin
2012 Borwein JM, Vanderwerff J, 'Constructions of uniformly convex functions', Canadian Mathematical Bulletin, 55 697-707 (2012) [C1]
2012 Bauschke 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]
Citations Scopus - 4Web of Science - 4
2012 Amdeberhan T, Borwein D, Borwein JM, Straub A, 'On formulas for Pi experimentally conjectured by Jauregui-Tsallis', Journal of Mathematical Physics, 53 (2012) [C1]
2012 Bailey DH, Barrio R, Borwein JM, 'High-precision computation: Mathematical physics and dynamics', Applied Mathematics and Computation, 218 10106-10121 (2012) [C1]
Citations Scopus - 22Web of Science - 14
2012 Borwein JM, 'Maximum entropy and feasibility methods for convex and nonconvex inverse problems', Optimization, 61 1-33 (2012) [C1]
Citations Scopus - 6Web of Science - 5
2012 Bailey 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]
Citations Scopus - 4Web of Science - 3
2012 Borwein JM, Borwein P, Jungic V, 'Remote collaboration: Six years of the Coast-to-Coast Seminar Series', Science Communication, 34 419-428 (2012) [C3]
2012 Bailey 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¿pa... [more]

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.

DOI 10.1007/s11139-012-9427-1
2012 Bailey DH, Borwein JM, 'Nonnormality of Stoneham constants', Ramanujan Journal, 29 409-422 (2012) [C1]
DOI 10.1007/s11139-012-9417-3
Citations Scopus - 2Web of Science - 1
2012 Alladi K, Andrews GE, Borwein JM, 'Preface. Ramanujan's 125th Anniversary Special Volume', Ramanujan Journal, 29 1-2 (2012) [C3]
Citations Web of Science - 7
2012 Borwein 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]
DOI 10.1007/978-94-007-2129-6
2012 Borwein JM, Straub A, 'Log-sine evaluations of Mahler measures', Journal of the Australian Mathematical Society, 92 15-36 (2012) [C1]
DOI 10.1017/S1446788712000067
Citations Scopus - 1Web of Science - 1
2012 Janssen H, Craxi A, 'Preface', BEST PRACTICE & RESEARCH IN CLINICAL GASTROENTEROLOGY, 26 369-369 (2012)
DOI 10.1016/j.bpg.2012.11.009
2012 Piantadosi J, Howlett P, Borwein JM, 'Copulas with maximum entropy', Optimization Letters, 6 99-125 (2012) [C1]
Citations Scopus - 7Web of Science - 7
2012 Bauschke 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]
Citations Scopus - 2Web of Science - 2
2012 Borwein 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]
2012 Borwein D, Borwein JM, Straub A, Wan G, 'Log-sine evaluations of Mahler measures, II', Integers, 12 1179-1212 (2012) [C1]
2012 Wang 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]
DOI 10.1007/s11228-012-0209-0
Citations Scopus - 4Web of Science - 3
2012 Bailey DH, Borwein JM, 'Hand-to-hand combat with thousand-digit integrals', Journal of Computational Science, 3 77-86 (2012) [C1]
Citations Scopus - 7Web of Science - 5
2012 Piantadosi 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 a... [more]

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.

DOI 10.3934/naco.2012.2.233
Citations Scopus - 3
2011 Borwein JM, Straub A, 'Special Values of Generalized Log-sine Integrals', CoRR, abs/1103.4298 (2011)
2011 Bailey DH, Borwein JM, 'Exploratory experimentation and computation', Notices of the American Mathematical Society, 58 1410-1419 (2011) [C1]
Citations Scopus - 7
2011 Borwein 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]
DOI 10.1016/j.jmaa.2011.06.001
Citations Scopus - 2Web of Science - 2
2011 Bacak M, Borwein JM, 'On difference convexity of locally Lipschitz functions', Optimization, 60 961-978 (2011) [C1]
Citations Scopus - 5Web of Science - 6
2011 Borwein JM, Osborn J-A, 'Review: Hersh, R. and John-Steiner, V. Loving and Hating Mathematics', Mathematical Intelligencer, 33 63-69 (2011) [C3]
DOI 10.1007/s00283-011-9260-1
2011 Bailey DH, Borwein JM, 'High-precision numerical integration: Progress and challenges', Journal of Symbolic Computation, 46 741-754 (2011) [C1]
DOI 10.1016/j.jsc.2010.08.010
Citations Scopus - 8Web of Science - 7
2011 Borwein JM, 'Review: Dontchev, A.L. and Rockafellar, R.T, Implicit Functions and Solution Mappings', IEEE Control Systems, 31 74-77 (2011) [C3]
2011 Borwein 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]
Citations Scopus - 16Web of Science - 11
2011 Borwein JM, Nuyens D, Straub A, Wan G, 'Some arithmetic properties of short random walk integrals', Ramanujan Journal, 26 109-132 (2011) [C1]
DOI 10.1007/s11139-011-9325-y
Citations Scopus - 9Web of Science - 10
2010 Borwein JM, Chan O-Y, 'Duality in tails of multiple-zeta values', International Journal of Number Theory, 6 501-514 (2010) [C1]
DOI 10.1142/s1793042110003058
2010 Borwein JM, 'Fifty years of maximal monotonicity', Optimization Letters, 4 473-490 (2010) [C1]
DOI 10.1007/s11590-010-0178-x
Citations Scopus - 21Web of Science - 20
2010 Borwein 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]
2010 Borwein D, Borwein JM, Leonard IE, 'Lp norms and the sine function', American Mathematical Monthly, 117 528-539 (2010) [C1]
DOI 10.4169/000298910X492817
Citations Scopus - 4Web of Science - 4
2010 Bailey DH, Borwein JM, Crandall RE, 'Advances in the theory of box integrals', Mathematics of Computation, 79 1839-1866 (2010) [C1]
DOI 10.1090/s0025-5718-10-02338-0
Citations Scopus - 10Web of Science - 6
2010 Bailey DH, Borwein JM, Broadhurst D, Zudilin V, 'Experimental mathematics and mathematical physics', Contemporary Mathematics, 517 41-58 (2010) [C1]
Citations Web of Science - 3
Co-authors Wadim Zudilin
2010 Bailey DH, Borwein JM, 'Experimental computation with oscillatory integrals', Contemporary Mathematics, 517 25-40 (2010) [C1]
Citations Web of Science - 1
2010 Bacak 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]
Citations Scopus - 8Web of Science - 5
2010 Borwein JM, Vanderwerff J, 'Frechet-legendre functions and reflexive banach spaces', Journal of Convex Analysis, 17 915-924 (2010) [C1]
Citations Scopus - 2Web of Science - 2
2010 Borwein JM, Chan O-Y, Crandall RE, 'Higher-dimensional box integrals', Experimental Mathematics, 19 431-446 (2010) [C1]
DOI 10.1080/10586458.2010.10390634
Citations Scopus - 7Web of Science - 5
2009 Bailey DH, Borwein JM, 'Experimental mathematics and computational statistics', WIREs Computational Statistics, 1 12-24 (2009) [C1]
DOI 10.1002/wics.1
Citations Scopus - 2
2009 Borwein JM, Calkin NJ, Manna D, 'Euler-Boole Summation Revisited', AMERICAN MATHEMATICAL MONTHLY, 116 387-412 (2009) [C1]
Citations Scopus - 3Web of Science - 3
2009 Borwein JM, Guirao AJ, Hajek P, Vanderwerff J, 'Uniformly convex functions on Banach Spaces', Proceedings of the American Mathematical Society, 137 1081-1091 (2009) [C1]
DOI 10.1090/s0002-9939-08-09630-5
Citations Scopus - 7Web of Science - 5
2009 Borwein JM, 'The Princeton companion to mathematics', SIAM Review, 51 790-794 (2009) [C3]
2009 Borwein JM, Vanderwerff J, 'Differentiability of conjugate functions and perturbed minimization principles', Journal of Convex Analysis, 16 707-711 (2009) [C1]
Citations Scopus - 4Web of Science - 4
2009 Borwein JM, Moors WB, 'Stability of closedness of convex cones under linear mappings', Journal of Convex Analysis, 16 699-705 (2009) [C1]
Citations Scopus - 2Web of Science - 1
2009 Bailey DH, Borwein JM, Crandall RE, 'Resolution of the Quinn-Rand-Strogatz Constant of nonlinear physics', Experimental Mathematics, 18 107-116 (2009) [C1]
DOI 10.1080/10586458.2009.10128885
Citations Scopus - 2
2009 Borwein JM, Chan O-Y, 'UNIFORM BOUNDS FOR THE COMPLEMENTARY INCOMPLETE GAMMA FUNCTION', MATHEMATICAL INEQUALITIES & APPLICATIONS, 12 115-121 (2009) [C1]
Citations Scopus - 2Web of Science - 2
2008 Borwein JM, Salvy B, 'A Proof of a Recurrence for Bessel Moments.', Experimental Mathematics, 17 223-230 (2008) [C1]
DOI 10.1080/10586458.2008.10129032
2008 Borwein JM, Salvy B, 'A Proof of a Recurrence for Bessel Moments', Experimental Mathematics, 17 223-230 (2008) [C1]
DOI 10.1080/10586458.2008.10129032
Citations Scopus - 5Web of Science - 8
2008 Borwein 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]
DOI 10.1007/s11139-007-9096-7
Citations Web of Science - 1
2008 Borwein 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]
DOI 10.1007/s11139-007-9083-z
Citations Scopus - 13Web of Science - 10
2008 Bailey 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]
DOI 10.1088/1751-8113/41/20/205203
Citations Scopus - 20Web of Science - 24
2008 Baillie R, Borwein D, Borwein JM, 'Surprising sinc sums and integrals', American Mathematical Monthly, 115 888-901 (2008) [C1]
Citations Scopus - 7Web of Science - 5
2008 Borwein JM, 'Hilbert's Inequality and Witten's Zeta-Function', American Mathematical Monthly, 115 125-137 (2008) [C1]
Citations Scopus - 4Web of Science - 4
2008 Borwein 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]
DOI 10.1016/j.jmaa.2007.10.003
Citations Scopus - 4Web of Science - 4
2008 Borwein D, Borwein JM, Crandall RE, 'Effective Laguerre Asymptotics', SIAM Journal on Numerical Analysis, 46 3285-3312 (2008) [C1]
DOI 10.1137/07068031X
Citations Scopus - 12Web of Science - 10
2008 Eberhard 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]
DOI 10.1007/s11228-008-0075-y
Citations Scopus - 2Web of Science - 2
2007 Bartz 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]
DOI 10.1016/j.na.2006.01.013
Citations Scopus - 28Web of Science - 30
2007 Bailey DH, Borwein JM, Crandall RE, 'Box integrals', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 206 196-208 (2007) [C1]
DOI 10.1016/j.cam.2006.06.010
Citations Scopus - 16Web of Science - 13
2007 Bauschke HH, Borwein JM, Wang X, 'Fitzpatrick functions and continuous linear monotone operators', SIAM JOURNAL ON OPTIMIZATION, 18 789-809 (2007) [C1]
DOI 10.1137/060655468
Citations Scopus - 21Web of Science - 23
2007 Borwein J, Wiersma H, 'Asplund decomposition of monotone operators', SIAM JOURNAL ON OPTIMIZATION, 18 946-960 (2007) [C1]
DOI 10.1137/060658357
Citations Web of Science - 4
2007 Bailey DH, Borwein D, Borwein JM, Crandall RE, 'Hypergeometric forms for ising-class integrals', EXPERIMENTAL MATHEMATICS, 16 257-276 (2007) [C1]
DOI 10.1080/10586458.2007.10129003
Citations Scopus - 11Web of Science - 9
2007 Borwein D, Borwein J, Crandall R, Mayer R, 'On the dynamics of certain recurrence relations', RAMANUJAN JOURNAL, 13 63-101 (2007) [C1]
DOI 10.1007/s11139-006-0243-3
Citations Web of Science - 3
2007 Borwein JM, 'Proximality and Chebyshev sets', OPTIMIZATION LETTERS, 1 21-32 (2007) [C1]
DOI 10.1007/s11590-006-0014-5
Citations Scopus - 12Web of Science - 11
2007 Borwein 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] i... [more]

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.

DOI 10.1016/B978-044452208-5/50050-8
Citations Scopus - 3
2007 Borwein 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)
DOI 10.1109/HPCS.2007.30
2007 Borwein JM, 'A class of Dirichlet series integrals', AMERICAN MATHEMATICAL MONTHLY, 114 70-76 (2007) [C1]
Citations Scopus - 1Web of Science - 1
2007 Bailey D, Borwein J, Bradley D, 'A hypergeometric identity', AMERICAN MATHEMATICAL MONTHLY, 114 643-645 (2007)
2007 Borwein JM, 'Maximality of sums of two maximal monotone operators in general banach space', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 135 3917-3924 (2007) [C1]
Citations Scopus - 31Web of Science - 31
2007 Borwein D, Borwein J, 'van der pol expansions of L-series', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 50 11-23 (2007) [C1]
DOI 10.4153/CMB-2007-002-0
2007 Borwein JM, Chamberland M, 'Integer powers of Arcsin', International Journal of Mathematics and Mathematical Sciences, 2007 1-10 (2007) [C1]
DOI 10.1155/2007/19381
2006 Borwein D, Borwein JM, Bradley DM, 'Parametric Euler sum identities', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 316 328-338 (2006) [C1]
DOI 10.1016/j.jmaa.2005.04.040
Citations Scopus - 7Web of Science - 7
2006 Bailey DH, Borwein JM, Crandall RE, 'Integrals of the Ising class', JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 39 12271-12302 (2006) [C1]
DOI 10.1088/0305-4470/39/40/001
Citations Scopus - 23Web of Science - 24
2006 Borwein JM, Zhu QJJ, 'Variational methods in convex analysis', JOURNAL OF GLOBAL OPTIMIZATION, 35 197-213 (2006) [C1]
DOI 10.1007/s10898-005-3835-3
Citations Scopus - 6Web of Science - 3
2006 Borwein J, 'Simon Fitzpatrick memorial volume', JOURNAL OF CONVEX ANALYSIS, 13 463-476 (2006)
Citations Web of Science - 2
2006 Borwein JM, 'Maximal monotonicity via convex analysis', JOURNAL OF CONVEX ANALYSIS, 13 561-586 (2006) [C1]
Citations Scopus - 56Web of Science - 51
2006 Borwein J, Montesinos V, Vanderwerff J, 'Boundedness, differentiability and extensions of convex functions', JOURNAL OF CONVEX ANALYSIS, 13 587-602 (2006) [C1]
Citations Web of Science - 5
2006 Borwein JM, 'The Life of Simon Fitzpatrick, 1953-2004', Journal of Convex Analysis, 13 463-476 (2006) [C1]
2006 Borwein JM, Bradley DM, 'Thirty-two goldbach variations', INTERNATIONAL JOURNAL OF NUMBER THEORY, 2 65-103 (2006) [C1]
DOI 10.1142/S1793042106000383
Citations Web of Science - 17
2006 Bailey 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... [more]

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.

DOI 10.1109/HPCS.2006.22
2006 Bailey DH, Borwein JM, Bradley DM, 'Experimental determination of Apery-like identities for zeta(2n+2)', EXPERIMENTAL MATHEMATICS, 15 281-289 (2006) [C1]
DOI 10.1080/10586458.2006.10128968
Citations Scopus - 10Web of Science - 13
2006 Bailey DH, Borwein JM, Kapoor V, Weisstein EW, 'Ten problems in experimental mathematics', AMERICAN MATHEMATICAL MONTHLY, 113 481-509 (2006) [C1]
Citations Scopus - 12Web of Science - 9
2006 Borwein JM, 'Maximality of sums of two maximal monotone operators', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 134 2951-2955 (2006) [C1]
DOI 10.1090/S0002-9939-06-08323-7
Citations Scopus - 7Web of Science - 8
2005 Borwein JM, Girgensohn R, 'Evaluations of binomial series', Aequationes Mathematicae, 70 25-36 (2005) [C1]
DOI 10.1007/s00010-005-2774-x
Citations Scopus - 15
2005 Borwein JM, Choi KKS, Pigulla W, 'Continued fractions of tails of hypergeometric series', AMERICAN MATHEMATICAL MONTHLY, 112 493-501 (2005) [C1]
Citations Scopus - 1Web of Science - 1
2005 Bailey DH, Borwein JM, 'Experimental mathematics: Examples, methods and implications', Notices of the American Mathematical Society, 52 502-514 (2005) [C1]
Citations Scopus - 22
2005 Borwein JM, Wang XF, 'Lipschitz functions with maximal Clarke subdifferentials are staunch', BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 72 491-496 (2005) [C1]
Citations Scopus - 3Web of Science - 4
2005 Borwein JM, Wang XF, 'Cone-monotone functions: Differentiability and continuity', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 57 961-982 (2005) [C1]
DOI 10.4153/CJM-2005-037-5
Citations Scopus - 7Web of Science - 7
2005 Borwein J, Stanway T, 'Knowledge and community in mathematics', MATHEMATICAL INTELLIGENCER, 27 7-16 (2005) [C1]
DOI 10.1007/BF02985788
Citations Web of Science - 1
2005 Borwein JM, 'The SIAM 100 Digits Challenge', The Mathematical Intelligencer, 27 40-48 (2005) [C3]
2005 Borwein JM, Luke DR, 'Dynamics of a continued fraction of Ramanujan with random coefficients', ABSTRACT AND APPLIED ANALYSIS, 449-467 (2005) [C1]
DOI 10.1155/AAA.2005.449
Citations Scopus - 3Web of Science - 2
2005 Borwein 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 i... [more]

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.

DOI 10.1007/s10758-005-5216-x
Citations Scopus - 6
2004 Borwein 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]
DOI 10.1090/S0002-9939-03-07149-1
Citations Scopus - 8Web of Science - 10
2004 Borwein 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]
DOI 10.4153/CJM-2004-041-2
Citations Scopus - 5Web of Science - 4
2004 Borwein J, Cheng LX, Fabian M, Revalski JP, 'A one perturbation variational principle and applications', SET-VALUED ANALYSIS, 12 49-60 (2004) [C1]
DOI 10.1023/B:SVAN.0000023400.92518.cb
Citations Web of Science - 3
2004 Borwein JM, Vanderwerff JD, 'Constructible convex sets', SET-VALUED ANALYSIS, 12 61-77 (2004) [C1]
DOI 10.1023/B:SVAN.0000023393.75251.05
Citations Scopus - 5Web of Science - 4
2004 Borwein J, Crandall R, Fee G, 'On the Ramanujan AGM fraction, I: The real-parameter case', EXPERIMENTAL MATHEMATICS, 13 275-285 (2004) [C1]
Citations Web of Science - 8
2004 Borwein J, Crandall R, 'On the Ramanujan AGM fraction, II: The complex- parameter case', EXPERIMENTAL MATHEMATICS, 13 287-295 (2004) [C1]
Citations Web of Science - 8
2004 Bailey 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]
2003 Borwein J, Girgensohn R, 'A class of exponential inequalities', MATHEMATICAL INEQUALITIES & APPLICATIONS, 6 397-411 (2003) [C1]
Citations Web of Science - 1
2003 Borwein JM, Choi KKS, 'On Dirichlet series for sums of squares', RAMANUJAN JOURNAL, 7 95-127 (2003) [C1]
DOI 10.1023/A:1026230709945
Citations Scopus - 6Web of Science - 6
2003 Kellar M, MacKay B, Zhang R, Watters C, Kaufman D, Borwein J, 'Dynamic composition of math lessons', EDUCATIONAL TECHNOLOGY & SOCIETY, 6 100-111 (2003) [C1]
Citations Web of Science - 1
2003 Borwein D, Borwein JM, Rooin J, 'Problem 11009.', The American Mathematical Monthly, 110 341-341 (2003)
2003 Benoist J, Borwein JM, Popovici N, 'A characterization of quasiconvex vector-valued functions', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 131 1109-1113 (2003) [C1]
DOI 10.1090/S0002-9939-02-06761-8
Citations Scopus - 24Web of Science - 25
2003 Borwein JM, Dontchev AL, 'On the Bartle-Graves theorem', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 131 2553-2560 (2003) [C1]
DOI 10.1090/S0002-9939-03-07229-0
Citations Scopus - 7Web of Science - 7
2003 Borwein J, FItzpatrick S, Girgensohn R, 'Subdifferentials whose graphs are not normxweak* closed', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 46 538-545 (2003) [C1]
DOI 10.4153/CMB-2003-051-5
Citations Web of Science - 7
2003 Bauschke HH, Borwein JM, Combettes PL, 'Bregman monotone optimization algorithms', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 42 596-636 (2003) [C1]
DOI 10.1137/S0363012902407120
Citations Scopus - 75Web of Science - 79
2003 Borwein JM, Vanderwerff J, Wang XF, 'Local Lipschitz-constant functions and maximal subdifferentials', SET-VALUED ANALYSIS, 11 37-67 (2003) [C1]
DOI 10.1023/A:1021975622768
Citations Scopus - 2Web of Science - 2
2003 Borwein 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]
DOI 10.1137/S1052623401400324
Citations Web of Science - 8
2003 Borwein 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]
DOI 10.1023/A:1022988116044
2002 Borwein 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)
DOI 10.1007/BF02784514
Citations Scopus - 8Web of Science - 9
2002 Borwein JM, Jimenez-Sevilla M, Moreno JP, 'Antiproximinal norms in Banach spaces', JOURNAL OF APPROXIMATION THEORY, 114 57-69 (2002)
DOI 10.1006/jath.2001.3636
Citations Scopus - 4Web of Science - 1
2002 Borwein J, Giles J, Vanderwerff J, 'Rotund norms, Clarke subdifferentials and extensions of Lipschitz functions', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 48 287-301 (2002)
DOI 10.1016/S0362-546X(00)00187-5
Citations Scopus - 4Web of Science - 2
2002 Borwein JM, Zhu QJ, 'A survey of subdifferential calculus with applications (vol 38, pg 687, 1999)', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 49 295-296 (2002)
DOI 10.1016/S0362-546X(01)00099-2
Citations Scopus - 3Web of Science - 2
2002 Borwein D, Borwein JM, Mares BA, 'Multi-variable sinc integrals and volumes of polyhedra', RAMANUJAN JOURNAL, 6 189-208 (2002)
DOI 10.1023/A:1015727317007
Citations Scopus - 5Web of Science - 5
2002 Borwein JM, Vanderwerff JD, 'On the continuity of biconjugate convex functions', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 130 1797-1803 (2002)
DOI 10.1090/S0002-9939-01-06249-9
Citations Scopus - 2Web of Science - 3
2001 Borwein JM, 'Generalized trigonometric functions', AMERICAN MATHEMATICAL MONTHLY, 108 473-474 (2001)
2001 Borwein JM, Bradley DM, Broadhurst DJ, Lisonek P, 'Special values of multiple polylogarithms', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 353 907-941 (2001)
DOI 10.1090/S0002-9947-00-02616-7
Citations Scopus - 118Web of Science - 126
2001 Borwein JM, Moors WB, Wang XF, 'Generalized subdifferentials: A Baire categorical approach', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 353 3875-3893 (2001)
DOI 10.1090/S0002-9947-01-02820-3
Citations Scopus - 10Web of Science - 6
2001 Bauschke HH, Borwein JM, Combettes PL, 'Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces', COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 3 615-647 (2001)
DOI 10.1142/S0219199701000524
Citations Scopus - 85Web of Science - 86
2001 Borwein JM, Fitzpatrick SP, 'Duality inequalities and sandwiched functions', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 46 365-380 (2001)
DOI 10.1016/S0362-546X(00)00129-2
Citations Scopus - 1Web of Science - 1
2001 Borwein J, Kortezov I, 'Some generic results on nonattaining functionals', SET-VALUED ANALYSIS, 9 35-47 (2001)
DOI 10.1023/A:1011246414064
Citations Web of Science - 2
2001 Borwein JM, Vanderwerff JD, 'Convex functions of Legendre type in general Banach spaces', JOURNAL OF CONVEX ANALYSIS, 8 569-581 (2001)
Citations Scopus - 5Web of Science - 6
2001 Borwein JM, Broadhurst DJ, Kamnitzer J, 'Central binomial sums, multiple Clausen values, and zeta values', EXPERIMENTAL MATHEMATICS, 10 25-34 (2001)
Citations Scopus - 31Web of Science - 34
2001 Borwein D, Borwein J, Fee G, Girgensohn R, 'Refined convexity and special cases of the Blaschke-Santalo inequality', MATHEMATICAL INEQUALITIES & APPLICATIONS, 4 631-638 (2001)
Citations Web of Science - 10
2001 Borwein D, Borwein JM, 'Some remarkable properties of sinc and related integrals', RAMANUJAN JOURNAL, 5 73-89 (2001)
DOI 10.1023/A:1011497229317
Citations Scopus - 13Web of Science - 11
2001 Borwein JM, Borwein PB, 'Challenges in mathematical computing', COMPUTING IN SCIENCE & ENGINEERING, 3 48-53 (2001)
DOI 10.1109/5992.919266
Citations Scopus - 2Web of Science - 4
2001 Bauschke 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... [more]

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.

DOI 10.1016/S1570-579X(01)80004-5
Citations Scopus - 5
2000 Borwein JM, Bradley DM, Crandall RE, 'Computational strategies for the Riemann zeta function', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 121 247-296 (2000)
DOI 10.1016/S0377-0427(00)00336-8
Citations Scopus - 57Web of Science - 55
2000 Borwein J, Lucet Y, Mordukhovich B, 'Compactly epi-Lipschitzian convex sets and functions in normed spaces', JOURNAL OF CONVEX ANALYSIS, 7 375-393 (2000)
Citations Web of Science - 25
2000 Bauschke HH, Borwein JM, Tseng P, 'Bounded linear regularity, strong CHIP, and CHIP are distinct properties', JOURNAL OF CONVEX ANALYSIS, 7 395-412 (2000)
Citations Scopus - 19Web of Science - 20
2000 Borwein J, Choi KKS, 'On the representations of xy plus yz plus zx', EXPERIMENTAL MATHEMATICS, 9 153-158 (2000)
Citations Web of Science - 5
2000 Borwein 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)
Citations Scopus - 6Web of Science - 5
2000 Borwein D, Borwein JM, Marechal P, 'Surprise maximization', AMERICAN MATHEMATICAL MONTHLY, 107 517-527 (2000)
DOI 10.2307/2589347
Citations Scopus - 3Web of Science - 2
2000 Borwein JM, 'Moments of the Poisson distribution', AMERICAN MATHEMATICAL MONTHLY, 107 659-659 (2000)
Citations Web of Science - 1
2000 Borwein JM, Moors WB, 'Separable determination of integrability and minimality of the Clarke subdifferential mapping', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 128 215-221 (2000)
DOI 10.1090/S0002-9939-99-05001-7
Citations Scopus - 10Web of Science - 6
2000 Borwein JM, Wang XF, 'Lipschitz functions with maximal Clarke subdifferentials are generic', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 128 3221-3229 (2000)
DOI 10.1090/S0002-9939-00-05914-1
Citations Scopus - 6Web of Science - 8
1999 Borwein JM, Corless RM, 'Emerging tools for experimental mathematics', AMERICAN MATHEMATICAL MONTHLY, 106 889-909 (1999)
DOI 10.2307/2589743
Citations Scopus - 13Web of Science - 12
1999 Marechal P, Togane D, Celler A, Borwein JM, 'Computation and stability analysis for regularized tomographic reconstructions', IEEE TRANSACTIONS ON NUCLEAR SCIENCE, 46 2177-2184 (1999)
DOI 10.1109/23.819301
Citations Scopus - 3Web of Science - 3
1999 Borwein 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)
DOI 10.1006/jmaa.1998.6157
Citations Scopus - 9Web of Science - 9
1999 Bauschke 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)
DOI 10.1007/s101070050083
Citations Scopus - 63Web of Science - 65
1999 Bauschke 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)
Citations Scopus - 19Web of Science - 23
1999 Bauschke HH, Noll D, Celler A, Borwein JM, 'An EM algorithm for dynamic SPECT', IEEE TRANSACTIONS ON MEDICAL IMAGING, 18 252-261 (1999)
DOI 10.1109/42.764899
Citations Web of Science - 23
1999 Borwein J, Moors WB, Shao Y, 'Subgradient representation of multifunctions', JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES B-APPLIED MATHEMATICS, 40 301-313 (1999)
1999 Borwein JM, Treiman JS, Zhu QJJ, 'Partially smooth variational principles and applications', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 35 1031-1059 (1999)
DOI 10.1016/S0362-546X(98)00113-8
Citations Scopus - 5Web of Science - 4
1999 Borwein JM, Zhu QJJ, 'A survey of subdifferential calculus with applications', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 38 687-773 (1999)
DOI 10.1016/S0362-546X(98)00142-4
Citations Scopus - 62Web of Science - 69
1999 Marechal 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 ... [more]

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.

1998 Borwein JM, Wang XF, 'The converse of the mean value theorem may fail generically', AMERICAN MATHEMATICAL MONTHLY, 105 847-848 (1998)
DOI 10.2307/2589216
Citations Scopus - 1Web of Science - 1
1998 Borwein 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)
DOI 10.1090/S0002-9947-98-01984-9
Citations Scopus - 26Web of Science - 29
1998 Borwein D, Borwein JM, Pinner C, 'Convergence of Madelung-like lattice sums', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 350 3131-3167 (1998)
DOI 10.1090/S0002-9947-98-01983-7
Citations Scopus - 8Web of Science - 9
1998 Borwein JM, Girgensohn R, Wang XF, 'On the construction of holder and proximal subderivatives', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 41 497-507 (1998)
DOI 10.4153/CMB-1998-065-9
Citations Scopus - 3Web of Science - 3
1998 Borwein JM, 'Brouwer-Heyting sequences converge', MATHEMATICAL INTELLIGENCER, 20 14-15 (1998)
Citations Scopus - 3Web of Science - 3
1998 Borwein JM, Zhu QJ, 'Limiting convex examples for nonconvex subdifferential calculus', Journal of Convex Analysis, 5 221-235 (1998)
Citations Scopus - 1
1998 Borwein JM, Moors WB, 'A chain rule for essentially smooth Lipschitz functions', SIAM JOURNAL ON OPTIMIZATION, 8 300-308 (1998)
DOI 10.1137/S1052623496297838
Citations Scopus - 7Web of Science - 8
1998 Borwein JM, Moors WB, 'Null sets and essentially smooth Lipschitz functions', SIAM JOURNAL ON OPTIMIZATION, 8 309-323 (1998)
DOI 10.1137/S1052623496305213
Citations Scopus - 16Web of Science - 16
1998 Borwein 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 ... [more]

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.

Citations Scopus - 39
1998 Borwein JM, 'A generalization of Young's l(p) equality', MATHEMATICAL INEQUALITIES & APPLICATIONS, 1 131-136 (1998)
Citations Scopus - 4Web of Science - 4
1998 Borwein JM, Jofre A, 'A nonconvex separation property in Banach spaces', MATHEMATICAL METHODS OF OPERATIONS RESEARCH, 48 169-179 (1998)
DOI 10.1007/s001860050019
Citations Scopus - 15Web of Science - 15
1997 Borwein J, Bradley D, 'Empirically Determined Apéry-Like Formulae for ¿(4n+3)', Experimental Mathematics, 6 181-194 (1997)
DOI 10.1080/10586458.1997.10504608
1997 Borwein J, Monagan M, 'A note from the editors - Winter 1997, Vol 4, No 1', MAPLETECH, 4 1-1 (1997)
1997 Borwein JM, Lisonek P, Macdonald JA, 'Arithmetic-geometric means revisited', MAPLETECH, 4 20-27 (1997)
1997 Borwein J, Monagan M, 'Differential equations in Maple: A short tutorial', MAPLETECH, 4 55-58 (1997)
1997 Borwein 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 ... [more]

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.

Citations Scopus - 74
1997 Borwein 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)
DOI 10.1090/S0002-9939-97-03654-X
Citations Scopus - 3Web of Science - 2
1997 Borwein J, Vanderwerff J, 'Convex functions on Banach spaces not containing l(1)', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 40 10-18 (1997)
DOI 10.4153/CMB-1997-002-1
Citations Web of Science - 3
1997 Borwein JM, Moors WB, 'Essentially smooth Lipschitz functions', JOURNAL OF FUNCTIONAL ANALYSIS, 149 305-351 (1997)
DOI 10.1006/jfan.1997.3101
Citations Scopus - 36Web of Science - 35
1997 Borwein JM, Sun W, 'The stability analysis of dynamic SPECT systems', NUMERISCHE MATHEMATIK, 77 283-298 (1997)
DOI 10.1007/s002110050287
Citations Web of Science - 2
1997 Bailey DH, Borwein JM, Crandall RE, 'On the Khintchine constant', Mathematics of Computation, 66 417-431 (1997)
Citations Scopus - 10Web of Science - 13
1997 Bauschke HH, Borwein JM, 'Legendre functions and the method of random bregman projections', Journal of Convex Analysis, 4 27-67 (1997)
Citations Scopus - 111
1997 Bailey DH, Borwein JM, Borwein PB, Plouffe S, 'The quest for Pi', MATHEMATICAL INTELLIGENCER, 19 50-57 (1997)
Citations Scopus - 27Web of Science - 27
1997 Borwein 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)
DOI 10.1016/0362-546X(95)00186-Y
Citations Scopus - 6Web of Science - 7
1997 Borwein JM, Moors WB, Wang XF, 'Lipschitz functions with prescribed derivatives and subderivatives', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 29 53-63 (1997)
DOI 10.1016/S0362-546X(96)00050-8
Citations Scopus - 9Web of Science - 11
1996 Borwein D, Borwein JM, Borwein PB, Girgensohn R, 'Giuga's conjecture on primality', AMERICAN MATHEMATICAL MONTHLY, 103 40-50 (1996)
DOI 10.2307/2975213
Citations Scopus - 10Web of Science - 5
1996 Borwein JM, Vanderwerff JD, 'Banach spaces that admit support sets', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 124 751-755 (1996)
DOI 10.1090/S0002-9939-96-03122-X
Citations Scopus - 12Web of Science - 12
1996 Borwein JM, Vanderwerff JD, 'Epigraphical and uniform convergence of convex functions', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 348 1617-1631 (1996)
DOI 10.1090/S0002-9947-96-01581-4
Citations Scopus - 13Web of Science - 12
1996 Bauschke HH, Borwein JM, 'Projection algorithms for solving convex feasibility problems', SIAM REVIEW, 38 367-426 (1996)
DOI 10.1137/S0036144593251710
Citations Scopus - 624Web of Science - 613
1996 Borwein J, Borwein P, Girgensohn R, Parnes S, 'Making sense of experimental mathematics', MATHEMATICAL INTELLIGENCER, 18 12-18 (1996)
Citations Web of Science - 9
1996 Borwein 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)
DOI 10.1137/S0363012994268801
Citations Scopus - 43Web of Science - 39
1996 Borwein 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)
DOI 10.1287/moor.21.2.442
Citations Scopus - 15Web of Science - 16
1996 Borwein JM, Ioffe A, 'Proximal analysis in smooth spaces', SET-VALUED ANALYSIS, 4 1-24 (1996)
DOI 10.1007/BF00419371
Citations Scopus - 22Web of Science - 33
1996 Borwein D, Borwein JM, Wang XF, 'Approximate subgradients and coderivatives in R(n)', SET-VALUED ANALYSIS, 4 375-398 (1996)
DOI 10.1007/BF00436112
Citations Scopus - 2Web of Science - 3
1996 Borwein 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=1... [more]

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.

Citations Scopus - 41
1995 BORWEIN JM, HUANG WZ, 'A FAST HEURISTIC METHOD FOR POLYNOMIAL MOMENT PROBLEMS WITH BOLTZMANN-SHANNON ENTROPY', SIAM JOURNAL ON OPTIMIZATION, 5 68-99 (1995)
DOI 10.1137/0805004
Citations Web of Science - 7
1995 BORWEIN D, BORWEIN JM, 'ON AN INTRIGUING INTEGRAL AND SOME SERIES RELATED TO ZETA(4)', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 123 1191-1198 (1995)
DOI 10.2307/2160718
Citations Web of Science - 33
1995 BORWEIN JM, GIRGENSOHN R, 'ADDITION THEOREMS AND BINARY EXPANSIONS', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 47 262-273 (1995)
DOI 10.4153/CJM-1995-013-4
Citations Web of Science - 1
1995 BORWEIN D, BORWEIN JM, GIRGENSOHN R, 'EXPLICIT EVALUATION OF EULER SUMS', PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 38 277-294 (1995)
Citations Web of Science - 55
1995 Limber 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... [more]

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. © 1995 IEEE.

DOI 10.1109/23.467872
Citations Scopus - 35
1995 BORWEIN JM, VANDERWERFF JD, 'CONVERGENCE OF LIPSCHITZ REGULARIZATIONS OF CONVEX-FUNCTIONS', JOURNAL OF FUNCTIONAL ANALYSIS, 128 139-162 (1995)
DOI 10.1006/jfan.1995.1026
Citations Scopus - 2Web of Science - 2
1995 BORWEIN JM, LEWIS AS, LIMBER MN, NOLL D, 'MAXIMUM-ENTROPY RECONSTRUCTION USING DERIVATIVE INFORMATION .2. COMPUTATIONAL RESULTS', NUMERISCHE MATHEMATIK, 69 243-256 (1995)
DOI 10.1007/s002110050090
Citations Web of Science - 8
1994 BORWEIN JM, HSU LC, MABRY R, NEU K, ROPPERT J, TYLER DB, DEWEGER BMM, 'NEAREST INTEGER ZETA-FUNCTIONS', AMERICAN MATHEMATICAL MONTHLY, 101 579-580 (1994)
1994 BORWEIN JM, NOLL D, '2ND-ORDER DIFFERENTIABILITY OF CONVEX-FUNCTIONS IN BANACH-SPACES', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 342 43-81 (1994)
DOI 10.2307/2154684
Citations Web of Science - 34
1994 BORWEIN JM, BORWEIN PB, GARVAN FG, 'SOME CUBIC MODULAR IDENTITIES OF RAMANUJAN', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 343 35-47 (1994)
DOI 10.2307/2154520
Citations Web of Science - 59
1994 BORWEIN JM, LEWIS AS, LIMBER MA, 'ENTROPY MINIMIZATION WITH LATTICE BOUNDS', JOURNAL OF APPROXIMATION THEORY, 79 1-16 (1994)
DOI 10.1006/jath.1994.1110
Citations Scopus - 1Web of Science - 2
1994 Bauschke HH, Borwein JM, 'Dykstra's Alternating Projection Algorithm for Two Sets', Journal of Approximation Theory, 79 418-443 (1994)
DOI 10.1006/jath.1994.1136
Citations Scopus - 67Web of Science - 61
1994 BORWEIN JM, LEWIS AS, NUSSBAUM RD, 'ENTROPY MINIMIZATION, DAD PROBLEMS, AND DOUBLY STOCHASTIC KERNELS', JOURNAL OF FUNCTIONAL ANALYSIS, 123 264-307 (1994)
DOI 10.1006/jfan.1994.1089
Citations Scopus - 16Web of Science - 18
1994 BORWEIN 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)
DOI 10.1006/jmaa.1994.1104
Citations Scopus - 11Web of Science - 14
1994 BORWEIN D, BORWEIN JM, 'ON SOME TRIGONOMETRIC AND EXPONENTIAL LATTICE SUMS', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 188 209-218 (1994)
DOI 10.1006/jmaa.1994.1422
Citations Scopus - 5Web of Science - 8
1994 BORWEIN J, VANDERWERFF J, 'DUAL KADEC-KLEE NORMS AND THE RELATIONSHIPS BETWEEN WIJSMAN, SLICE, AND MOSCO CONVERGENCE', MICHIGAN MATHEMATICAL JOURNAL, 41 371-387 (1994)
Citations Web of Science - 6
1994 Borwein 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)
DOI 10.1007/BF01432968
Citations Scopus - 1
1994 BORWEIN JM, LEWIS AS, 'STRONG ROTUNDITY AND OPTIMIZATION', SIAM JOURNAL ON OPTIMIZATION, 4 146-158 (1994)
DOI 10.1137/0804008
Citations Web of Science - 9
1994 Bailey DH, Borwein JM, Girgensohn R, 'Experimental Evaluation of Euler Sums', Experimental Mathematics, 3 17-30 (1994)
DOI 10.1080/10586458.1994.10504573
1993 BORWEIN JM, ZHUANG D, 'SUPER EFFICIENCY IN VECTOR OPTIMIZATION', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 338 105-122 (1993)
DOI 10.2307/2154446
Citations Web of Science - 107
1993 HIRSCHHORN 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)
DOI 10.4153/CJM-1993-038-2
Citations Web of Science - 37
1993 BORWEIN 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)
DOI 10.4153/CJM-1993-062-8
Citations Web of Science - 20
1993 BORWEIN JM, FITZPATRICK S, 'A WEAK HADAMARD SMOOTH RENORMING OF L(1)(NU,MU)', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 36 407-413 (1993)
DOI 10.4153/CMB-1993-055-5
Citations Web of Science - 14
1993 BORWEIN JM, BORWEIN PB, 'INEQUALITIES FOR COMPOUND MEAN ITERATIONS WITH LOGARITHMIC ASYMPTOTES', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 177 572-582 (1993)
DOI 10.1006/jmaa.1993.1278
Citations Scopus - 17Web of Science - 19
1993 Beer 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)
DOI 10.1006/jmaa.1993.1151
Citations Scopus - 7Web of Science - 13
1993 BORWEIN JM, BORWEIN PB, 'ON THE GENERATING FUNCTION OF THE INTEGER PART - [N-ALPHA+GAMMA]', JOURNAL OF NUMBER THEORY, 43 293-318 (1993)
DOI 10.1006/jnth.1993.1023
Citations Scopus - 13Web of Science - 17
1993 BORWEIN JM, 'ON THE FAILURE OF MAXIMUM-ENTROPY RECONSTRUCTION FOR FREDHOLM EQUATIONS AND OTHER INFINITE SYSTEMS', MATHEMATICAL PROGRAMMING, 61 251-261 (1993)
DOI 10.1007/BF01582150
Citations Scopus - 5Web of Science - 5
1993 BORWEIN J, BORWEIN P, 'PI IN THE SKY - COUNTING, THINKING, AND BEING - BARROW,JD', SCIENCE, 259 1928-1930 (1993)
DOI 10.1126/science.259.5103.1928
1993 BORWEIN J, BORWEIN P, GARVAN F, 'HYPERGEOMETRIC ANALOGS OF THE ARITHMETIC GEOMETRIC MEAN ITERATION', CONSTRUCTIVE APPROXIMATION, 9 509-523 (1993)
DOI 10.1007/BF01204654
Citations Web of Science - 7
1993 BORWEIN JM, FABIAN M, 'ON GENERIC 2ND-ORDER GATEAUX DIFFERENTIABILITY', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 20 1373-1382 (1993)
DOI 10.1016/0362-546X(93)90166-P
Citations Scopus - 2Web of Science - 2
1993 BORWEIN JM, BORWEIN PB, 'CLASS NUMBER 3 RAMANUJAN TYPE SERIES FOR 1/PI', JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 46 281-290 (1993)
DOI 10.1016/0377-0427(93)90302-R
Citations Scopus - 17Web of Science - 15
1993 Bauschke HH, Borwein JM, 'On the convergence of von Neumann's alternating projection algorithm for two sets', Set-Valued Analysis, 1 185-212 (1993)
DOI 10.1007/BF01027691
Citations Scopus - 69
1993 Borwein JM, Lewis AS, 'Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces', Set-Valued Analysis, 1 355-363 (1993)
DOI 10.1007/BF01027825
Citations Scopus - 5
1992 BORWEIN JM, BORWEIN PB, 'STRANGE SERIES AND HIGH-PRECISION FRAUD', AMERICAN MATHEMATICAL MONTHLY, 99 622-640 (1992)
DOI 10.2307/2324993
Citations Web of Science - 4
1992 BORWEIN JM, LEWIS AS, 'DECOMPOSITION OF MULTIVARIATE FUNCTIONS', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 44 463-482 (1992)
DOI 10.4153/CJM-1992-030-9
Citations Web of Science - 6
1992 BORWEIN JM, THERA M, 'SANDWICH THEOREMS FOR SEMICONTINUOUS OPERATORS', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 35 463-474 (1992)
DOI 10.4153/CMB-1992-061-7
Citations Web of Science - 12
1992 BORWEIN J, REICH S, SHAFRIR I, 'KRASNOSELSKI-MANN ITERATIONS IN NORMED SPACES', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 35 21-28 (1992)
DOI 10.4153/CMB-1992-003-0
Citations Web of Science - 49
1992 BORWEIN J, LOU S, 'ASYMPTOTICS OF A SEQUENCE OF WITT VECTORS', JOURNAL OF APPROXIMATION THEORY, 69 326-337 (1992)
DOI 10.1016/0021-9045(92)90006-A
Citations Web of Science - 3
1992 BORWEIN JM, LEWIS AS, 'PARTIALLY FINITE CONVEX-PROGRAMMING .1. QUASI RELATIVE INTERIORS AND DUALITY-THEORY', MATHEMATICAL PROGRAMMING, 57 15-48 (1992)
DOI 10.1007/BF01581072
Citations Scopus - 121Web of Science - 135
1992 BORWEIN JM, LEWIS AS, 'PARTIALLY FINITE CONVEX-PROGRAMMING .2. EXPLICIT LATTICE MODELS', MATHEMATICAL PROGRAMMING, 57 49-83 (1992)
DOI 10.1007/BF01581073
Citations Scopus - 19Web of Science - 22
1992 BORWEIN 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)
DOI 10.1093/imanum/12.4.519
Citations Scopus - 25Web of Science - 34
1991 BORWEIN JM, DEBARRA G, 'NESTED RADICALS', AMERICAN MATHEMATICAL MONTHLY, 98 735-739 (1991)
DOI 10.2307/2324426
Citations Web of Science - 6
1991 BORWEIN JM, BORWEIN PB, 'A CUBIC COUNTERPART OF JACOBIS IDENTITY AND THE AGM', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 323 691-701 (1991)
DOI 10.2307/2001551
Citations Web of Science - 111
1991 BORWEIN JM, LEWIS AS, 'ON THE CONVERGENCE OF MOMENT PROBLEMS', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 325 249-271 (1991)
DOI 10.2307/2001670
Citations Web of Science - 35
1991 BORWEIN 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)
DOI 10.4153/CJM-1991-028-5
Citations Web of Science - 22
1991 BORWEIN D, BORWEIN J, 'FIXED-POINT ITERATIONS FOR REAL FUNCTIONS', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 157 112-126 (1991)
DOI 10.1016/0022-247X(91)90139-Q
Citations Web of Science - 27
1991 BORWEIN JM, BORWEIN PB, 'THE MAN WHO KNEW INFINITY - A LIFE OF THE INDIAN GENIUS RAMANUJAN - KANIGEL,R', SCIENCE, 253 334-335 (1991)
DOI 10.1126/science.253.5017.334
1991 Borwein JM, Zhuang DM, 'Super efficiency in convex vector optimization', ZOR Zeitschrift für Operations Research Methods and Models of Operations Research, 35 175-184 (1991)
DOI 10.1007/BF01415905
Citations Scopus - 28
1991 BORWEIN JM, LEWIS AS, 'DUALITY RELATIONSHIPS FOR ENTROPY-LIKE MINIMIZATION PROBLEMS', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 29 325-338 (1991)
DOI 10.1137/0329017
Citations Scopus - 64Web of Science - 74
1990 BEER G, BORWEIN JM, 'MOSCO CONVERGENCE AND REFLEXIVITY', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 109 427-436 (1990)
DOI 10.2307/2048005
Citations Web of Science - 30
1990 BORWEIN JM, BORWEIN PB, 'A REMARKABLE CUBIC MEAN ITERATION', LECTURE NOTES IN MATHEMATICS, 1435 27-31 (1990)
Citations Web of Science - 6
1989 BORWEIN JM, BORWEIN PB, DILCHER K, 'PI, EULER NUMBERS, AND ASYMPTOTIC EXPANSIONS', AMERICAN MATHEMATICAL MONTHLY, 96 681-687 (1989)
DOI 10.2307/2324715
Citations Web of Science - 13
1989 BORWEIN 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)
DOI 10.2307/2325206
Citations Web of Science - 31
1989 BORWEIN JM, FITZPATRICK S, 'MOSCO CONVERGENCE AND THE KADEC PROPERTY', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 106 843-851 (1989)
DOI 10.2307/2047444
Citations Web of Science - 29
1989 BORWEIN J, FITZPATRICK S, 'LOCAL BOUNDEDNESS OF MONOTONE-OPERATORS UNDER MINIMAL HYPOTHESES', BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 39 439-441 (1989)
Citations Web of Science - 8
1989 BORWEIN JM, FITZPATRICK S, 'EXISTENCE OF NEAREST POINTS IN BANACH-SPACES', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 41 702-720 (1989)
DOI 10.4153/CJM-1989-032-7
Citations Web of Science - 50
1989 BORWEIN D, BORWEIN JM, SHAIL R, 'ANALYSIS OF CERTAIN LATTICE SUMS', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 143 126-137 (1989)
DOI 10.1016/0022-247X(89)90032-2
Citations Scopus - 12Web of Science - 19
1989 BORWEIN JM, BORWEIN PB, 'ON THE MEAN ITERATION (A,B)[-(A+3B/4 SQUARE ROOT AB+B/2)', MATHEMATICS OF COMPUTATION, 53 311-326 (1989)
DOI 10.2307/2008364
Citations Web of Science - 7
1989 BORWEIN JM, BORWEIN PB, 'APPROXIMATING PI WITH RAMANUJAN MODULAR EQUATIONS', ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, 19 93-102 (1989)
Citations Scopus - 2Web of Science - 2
1989 BORWEIN JM, STROJWAS HM, 'THE HYPERTANGENT CONE', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 13 125-144 (1989)
DOI 10.1016/0362-546X(89)90039-4
Citations Scopus - 5Web of Science - 3
1989 BORWEIN JM, DEMPSTER MAH, 'THE LINEAR ORDER COMPLEMENTARITY-PROBLEM', MATHEMATICS OF OPERATIONS RESEARCH, 14 534-558 (1989)
DOI 10.1287/moor.14.3.534
Citations Web of Science - 24
1988 BORWEIN JM, 'A NATURAL LIMIT', AMERICAN MATHEMATICAL MONTHLY, 95 557-558 (1988)
1988 BORWEIN 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)
DOI 10.1016/0022-247X(88)90034-0
Citations Scopus - 67Web of Science - 77
1988 BORWEIN JM, BORWEIN PB, 'ON THE COMPLEXITY OF FAMILIAR FUNCTIONS AND NUMBERS', SIAM REVIEW, 30 589-601 (1988)
DOI 10.1137/1030134
Citations Scopus - 12Web of Science - 12
1988 BORWEIN JM, BORWEIN PB, 'RAMANUJAN AND PI', SCIENTIFIC AMERICAN, 258 112-& (1988)
Citations Web of Science - 8
1988 BARZILAI J, BORWEIN JM, '2-POINT STEP SIZE GRADIENT METHODS', IMA JOURNAL OF NUMERICAL ANALYSIS, 8 141-148 (1988)
DOI 10.1093/imanum/8.1.141
Citations Scopus - 593Web of Science - 523
1988 BORWEIN D, BORWEIN JM, SHAIL R, ZUCKER IJ, 'ENERGY OF STATIC ELECTRON LATTICES', JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 21 1519-1531 (1988)
DOI 10.1088/0305-4470/21/7/015
Citations Scopus - 13Web of Science - 22
1988 BENTAL A, BORWEIN JM, TEBOULLE M, 'A DUAL APPROACH TO MULTIDIMENSIONAL LP SPECTRAL ESTIMATION PROBLEMS', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 26 985-996 (1988)
DOI 10.1137/0326053
Citations Scopus - 8Web of Science - 16
1987 BORWEIN JM, BORWEIN PB, 'THE WAY OF ALL MEANS', AMERICAN MATHEMATICAL MONTHLY, 94 519-522 (1987)
DOI 10.2307/2322842
Citations Web of Science - 9
1987 BORWEIN JM, 'AUTOMATIC-CONTINUITY AND OPENNESS OF CONVEX RELATIONS', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 99 49-55 (1987)
DOI 10.2307/2046269
Citations Web of Science - 9
1987 BORWEIN JM, GILES J, 'THE PROXIMAL NORMAL FORMULA IN BANACH-SPACE', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 302 371-381 (1987)
DOI 10.2307/2000915
Citations Web of Science - 43
1987 BORWEIN 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)
DOI 10.2307/2000681
Citations Web of Science - 155
1987 BORWEIN 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)
DOI 10.4153/CJM-1987-019-4
Citations Web of Science - 40
1987 BORWEIN 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)
DOI 10.1007/BF02837813
Citations Scopus - 2Web of Science - 2
1987 BORWEIN JM, 'EPI-LIPSCHITZ-LIKE SETS IN BANACH-SPACE - THEOREMS AND EXAMPLES', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 11 1207-1217 (1987)
DOI 10.1016/0362-546X(87)90008-3
Citations Scopus - 25Web of Science - 26
1987 WARD DE, BORWEIN JM, 'NONSMOOTH CALCULUS IN FINITE DIMENSIONS', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 25 1312-1340 (1987)
DOI 10.1137/0325072
Citations Scopus - 17Web of Science - 40
1987 Giles JR, 'The differentiability of real functions on normed linear space using generalized subgradients', J.Math.Anal. and Appl., 128 512-534 (1987)
Citations Scopus - 35Web of Science - 40
1986 JAGERS AA, BORWEIN JM, 'REGULAR N AND 2K-N-GONS INSCRIBED IN A UNIT-CIRCLE', AMERICAN MATHEMATICAL MONTHLY, 93 304-304 (1986)
1986 BORWEIN D, BORWEIN JM, 'A NOTE ON ALTERNATING SERIES IN SEVERAL DIMENSIONS', AMERICAN MATHEMATICAL MONTHLY, 93 531-539 (1986)
DOI 10.2307/2323026
Citations Web of Science - 2
1986 BORWEIN JM, BORWEIN PB, 'AN EXPLICIT CUBIC ITERATION ITERATION FOR PI', BIT, 26 123-126 (1986)
DOI 10.1007/BF01939368
Citations Scopus - 2Web of Science - 3
1986 BORWEIN 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)
DOI 10.4153/CJM-1986-022-4
Citations Web of Science - 58
1986 BORWEIN JM, 'PARTIALLY MONOTONE-OPERATORS AND THE GENERIC DIFFERENTIABILITY OF CONVEX CONCAVE AND BICONVEX MAPPINGS', ISRAEL JOURNAL OF MATHEMATICS, 54 42-50 (1986)
DOI 10.1007/BF02764875
Citations Scopus - 4Web of Science - 4
1986 Borwein JM, 'Norm duality for convex processes and applications', Journal of Optimization Theory and Applications, 48 53-64 (1986)
DOI 10.1007/BF00938589
Citations Scopus - 17Web of Science - 17
1986 Borwein JM, 'Stability and regular points of inequality systems', Journal of Optimization Theory and Applications, 48 9-52 (1986)
DOI 10.1007/BF00938588
Citations Scopus - 85Web of Science - 105
1986 BORWEIN JM, ZHUANG D, 'ON FAN MINIMAX THEOREM', MATHEMATICAL PROGRAMMING, 34 232-234 (1986)
DOI 10.1007/BF01580587
Citations Scopus - 12Web of Science - 11
1986 BORWEIN JM, WOLKOWICZ H, 'A SIMPLE CONSTRAINT QUALIFICATION IN INFINITE DIMENSIONAL PROGRAMMING', MATHEMATICAL PROGRAMMING, 35 83-96 (1986)
DOI 10.1007/BF01589443
Citations Scopus - 19Web of Science - 23
1986 BORWEIN JM, BORWEIN PB, 'MORE QUADRATICALLY CONVERGING ALGORITHMS FOR PI', MATHEMATICS OF COMPUTATION, 46 247-253 (1986)
DOI 10.2307/2008229
Citations Web of Science - 3
1986 BORWEIN JM, 'GENERIC DIFFERENTIABILITY OF ORDER-BOUNDED CONVEX OPERATORS', JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES B-APPLIED MATHEMATICS, 28 22-29 (1986)
Citations Web of Science - 3
1985 BORWEIN J, 'AN EXTENSION OF HADAMARD INEQUALITY', AMERICAN MATHEMATICAL MONTHLY, 92 148-149 (1985)
DOI 10.2307/2322656
1985 BORWEIN JM, TINGLEY DW, 'ON SUPPORTLESS CONVEX-SETS', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 94 471-476 (1985)
DOI 10.2307/2045238
Citations Web of Science - 9
1985 BORWEIN J, 'SOME MODULAR IDENTITIES OF RAMANUJAN USEFUL IN APPROXIMATING PI', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 95 365-371 (1985)
DOI 10.2307/2045804
Citations Web of Science - 2
1985 BORWEIN D, BORWEIN JM, TAYLOR KF, 'CONVERGENCE OF LATTICE SUMS AND MADELUNG CONSTANT', JOURNAL OF MATHEMATICAL PHYSICS, 26 2999-3009 (1985)
DOI 10.1063/1.526675
Citations Scopus - 40Web of Science - 49
1985 BORWEIN JM, 'ALTERNATIVE THEOREMS FOR GENERAL COMPLEMENTARITY-PROBLEMS', LECTURE NOTES IN ECONOMICS AND MATHEMATICAL SYSTEMS, 259 194-203 (1985)
Citations Web of Science - 6
1985 BORWEIN JM, STROJWAS HM, 'TANGENTIAL APPROXIMATIONS', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 9 1347-1366 (1985)
DOI 10.1016/0362-546X(85)90095-1
Citations Scopus - 71Web of Science - 79
1984 BORWEIN JM, SIMS B, 'NON-EXPANSIVE MAPPINGS ON BANACH-LATTICES AND RELATED TOPICS', HOUSTON JOURNAL OF MATHEMATICS, 10 339-356 (1984)
Citations Web of Science - 27
Co-authors Brailey Sims
1984 BORWEIN JM, 'AN INTEGRAL CHARACTERIZATION OF EUCLIDEAN-SPACE', BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 29 357-364 (1984)
Citations Web of Science - 1
1984 BORWEIN JM, STROJWAS HM, 'DIRECTIONALLY LIPSCHITZIAN MAPPINGS ON BAIRE SPACES', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 36 95-130 (1984)
DOI 10.4153/CJM-1984-008-7
Citations Web of Science - 13
1984 BORWEIN JM, RICHMOND B, 'HOW MANY MATRICES HAVE ROOTS', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 36 286-299 (1984)
DOI 10.4153/CJM-1984-018-0
Citations Web of Science - 1
1984 BORWEIN JM, BORWEIN PB, 'CUBIC AND HIGHER-ORDER ALGORITHMS FOR PI', CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 27 436-443 (1984)
DOI 10.4153/CMB-1984-067-7
Citations Web of Science - 3
1984 BORWEIN JM, YOST DT, 'ABSOLUTE NORMS ON VECTOR LATTICES', PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 27 215-222 (1984)
Citations Web of Science - 7
1984 Borwein JM, Penot JP, Thera M, 'Conjugate convex operators', Journal of Mathematical Analysis and Applications, 102 399-414 (1984)
DOI 10.1016/0022-247X(84)90180-X
Citations Scopus - 22Web of Science - 24
1984 BORWEIN JM, 'GENERALIZED LINEAR COMPLEMENTARITY-PROBLEMS TREATED WITHOUT FIXED-POINT THEORY', JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 43 343-356 (1984)
DOI 10.1007/BF00934459
Citations Scopus - 13Web of Science - 17
1984 BORWEIN JM, NIEUWENHUIS JW, '2 KINDS OF NORMALITY IN VECTOR OPTIMIZATION', MATHEMATICAL PROGRAMMING, 28 185-191 (1984)
DOI 10.1007/BF02612358
Citations Scopus - 3Web of Science - 4
1984 BORWEIN JM, BORWEIN PB, 'THE ARITHMETIC-GEOMETRIC MEAN AND FAST COMPUTATION OF ELEMENTARY-FUNCTIONS', SIAM REVIEW, 26 351-366 (1984)
DOI 10.1137/1026073
Citations Scopus - 25Web of Science - 43
1983 BORWEIN JM, 'SEMI-INFINITE PROGRAMMING DUALITY - HOW SPECIAL IS IT', LECTURE NOTES IN ECONOMICS AND MATHEMATICAL SYSTEMS, 215 10-36 (1983)
Citations Web of Science - 5
1983 BORWEIN JM, 'COMPLETENESS AND THE CONTRACTION PRINCIPLE', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 87 246-250 (1983)
DOI 10.2307/2043697
Citations Web of Science - 4
1983 Borwein JM, 'ADJOINT PROCESS DUALITY.', Mathematics of Operations Research, 8 403-434 (1983)
Citations Scopus - 29Web of Science - 41
1983 Borwein JM, 'ON THE EXISTENCE OF PARETO EFFICIENT POINTS.', Mathematics of Operations Research, 8 64-73 (1983)
Citations Scopus - 62Web of Science - 69
1983 Borwein JM, Borwein PB, 'A very rapidly convergent product expansion for p', BIT, 23 538-540 (1983)
DOI 10.1007/BF01933626
Citations Scopus - 1Web of Science - 3
1982 BORWEIN JM, 'ON THE HAHN-BANACH EXTENSION PROPERTY', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 86 42-46 (1982)
DOI 10.2307/2044393
Citations Web of Science - 7
1982 BORWEIN JM, 'CONTINUITY AND DIFFERENTIABILITY PROPERTIES OF CONVEX OPERATORS', PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 44 420-444 (1982)
Citations Web of Science - 66
1982 BORWEIN JM, 'A NOTE ON THE EXISTENCE OF SUB-GRADIENTS', MATHEMATICAL PROGRAMMING, 24 225-228 (1982)
DOI 10.1007/BF01585105
Citations Scopus - 1Web of Science - 1
1982 BORWEIN JM, 'A NOTE ON EPSILON-SUBGRADIENTS AND MAXIMAL MONOTONICITY', PACIFIC JOURNAL OF MATHEMATICS, 103 307-314 (1982)
Citations Web of Science - 29
1982 BORWEIN JM, STYAN GPH, FOSTER LV, WOLKOWICZ H, 'SOME INEQUALITIES INVOLVING STATISTICAL EXPRESSIONS', SIAM REVIEW, 24 340-342 (1982)
DOI 10.1137/1024072
Citations Web of Science - 4
1982 BORWEIN JM, 'NECESSARY AND SUFFICIENT CONDITIONS FOR QUADRATIC MINIMALITY', NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, 5 127-140 (1982)
DOI 10.1080/01630568208816135
Citations Scopus - 17Web of Science - 17
1982 BORWEIN JM, WOLKOWICZ H, 'CHARACTERIZATIONS OF OPTIMALITY WITHOUT CONSTRAINT QUALIFICATION FOR THE ABSTRACT CONVEX PROGRAM', MATHEMATICAL PROGRAMMING STUDY, 19 77-100 (1982)
Citations Web of Science - 24
1982 Borwein JM, 'NOTE ON THE EXISTENCE OF SUBGRADIENTS.', Mathematical Programming, 24 225-228 (1982)
Citations Scopus - 3
1981 BORWEIN J, WOLKOWICZ H, 'REGULARIZING THE ABSTRACT CONVEX PROGRAM', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 83 495-530 (1981)
DOI 10.1016/0022-247X(81)90138-4
Citations Web of Science - 30
1981 BORWEIN JM, 'THE LIMITING LAGRANGIAN AS A CONSEQUENCE OF HELLY THEOREM', JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 33 497-513 (1981)
DOI 10.1007/BF00935755
Citations Scopus - 8Web of Science - 9
1981 BORWEIN JM, 'A LAGRANGE MULTIPLIER THEOREM AND A SANDWICH THEOREM FOR CONVEX RELATIONS', MATHEMATICA SCANDINAVICA, 48 189-204 (1981)
Citations Web of Science - 45
1981 BORWEIN JM, 'DIRECT THEOREMS IN SEMI-INFINITE CONVEX-PROGRAMMING', MATHEMATICAL PROGRAMMING, 21 301-318 (1981)
DOI 10.1007/BF01584251
Citations Scopus - 10Web of Science - 23
1981 BORWEIN 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)
Citations Web of Science - 11
1981 BORWEIN 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)
Citations Web of Science - 17
1980 GRIPENBERG G, WILANSKY A, JOYNER D, ROUSH F, WALSH B, BORWEIN J, 'RANGES IN BANACH-SPACES', AMERICAN MATHEMATICAL MONTHLY, 87 68-68 (1980)
1980 BORWEIN J, KEENER L, 'THE HAUSDORFF METRIC AND CEBYSEV CENTERS', JOURNAL OF APPROXIMATION THEORY, 28 366-376 (1980)
DOI 10.1016/0021-9045(80)90071-4
Citations Web of Science - 9
1980 BORWEIN JM, 'LEXICOGRAPHIC MULTIPLIERS', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 78 309-327 (1980)
DOI 10.1016/0022-247X(80)90228-0
Citations Web of Science - 2
1980 BORWEIN JM, 'A STRONG DUALITY THEOREM FOR THE MINIMUM OF A FAMILY OF CONVEX-PROGRAMS', JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 31 453-472 (1980)
DOI 10.1007/BF00934472
Citations Scopus - 4Web of Science - 7
1980 BORWEIN JM, 'A NOTE ON PERFECT DUALITY AND LIMITING LAGRANGEANS', MATHEMATICAL PROGRAMMING, 18 330-337 (1980)
DOI 10.1007/BF01588327
Citations Scopus - 8Web of Science - 15
1979 BORWEIN JM, 'WEAK LOCAL SUPPORTABILITY AND APPLICATIONS TO APPROXIMATION', PACIFIC JOURNAL OF MATHEMATICS, 82 323-338 (1979)
Citations Web of Science - 11
1979 BORWEIN JM, 'MULTIVALUED APPROACH TO THE FARKAS-LEMMA', MATHEMATICAL PROGRAMMING STUDY, 10 42-47 (1979)
Citations Web of Science - 2
1978 BORWEIN JM, 'STRONGLY EXPOSING FUNCTIONALS', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 69 46-48 (1978)
DOI 10.2307/2043185
Citations Web of Science - 2
1978 BORWEIN J, 'WEAK TANGENT CONES AND OPTIMIZATION IN A BANACH-SPACE', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 16 512-522 (1978)
DOI 10.1137/0316034
Citations Web of Science - 19
1977 BORWEIN J, EDELSTEIN M, OBRIEN R, 'SOME REMARKS ON VISIBILITY AND STARSHAPE', JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 15 342-344 (1977)
1977 BORWEIN J, 'MULTIVALUED CONVEXITY AND OPTIMIZATION - UNIFIED APPROACH TO INEQUALITY AND EQUALITY CONSTRAINTS', MATHEMATICAL PROGRAMMING, 13 183-199 (1977)
DOI 10.1007/BF01584336
Citations Web of Science - 40
1977 BORWEIN J, 'PROPER EFFICIENT POINTS FOR MAXIMIZATIONS WITH RESPECT TO CONES', SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 15 57-63 (1977)
DOI 10.1137/0315004
Citations Web of Science - 174
1976 BORWEIN J, EDELSTEIN M, OBRIEN R, 'VISIBILITY AND STARSHAPE', JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 14 313-318 (1976)
Citations Web of Science - 2
1976 BORWEIN JM, 'FRACTIONAL PROGRAMMING WITHOUT DIFFERENTIABILITY', MATHEMATICAL PROGRAMMING, 11 283-290 (1976)
DOI 10.1007/BF01580396
Citations Scopus - 6Web of Science - 10
Show 361 more journal articles

Conference (29 outputs)

Year Citation Altmetrics Link
2014 Borwein 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 ... [more]

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.

DOI 10.1007/978-3-662-44199-2_1
2013 Borwein 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]
2013 Borwein JM, Rose M, Bailey D, Crandall R, 'Expectations on fractal sets', Proceedings of the 49th ANZIAM Conference, Newcastle, NSW (2013) [E3]
2011 Henstridge 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]
2011 Borwein 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]
DOI 10.1145/1993886.1993899
Citations Scopus - 2Web of Science - 3
2010 Borwein 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]
Citations Scopus - 1
2010 Borwein 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]
DOI 10.1007/978-1-4419-0437-9_4
Citations Web of Science - 1
2010 Borwein 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]
DOI 10.1090/conm/514
Citations Web of Science - 1
2009 Borwein JM, 'Digitally-assisted discovery and proof', Proof and Proving in Mathematics Education: ICMI Study 19 Conference Proceedings, Taipei, Taiwan (2009) [E1]
2009 Borwein JM, Hamilton CH, 'Symbolic fenchel conjugation', Mathematical Programming, Univ Chile, Santiago, CHILE (2009) [E1]
DOI 10.1007/s10107-007-0134-4
Citations Scopus - 6Web of Science - 5
2008 Bailey DH, Borwein JM, 'Computer-assisted discovery and proof', TAPAS IN EXPERIMENTAL MATHEMATICS, New Orleans, LA (2008) [E1]
Citations Web of Science - 4
2006 Borwein JM, 'Aesthetics for the working mathematician', Mathematics and the Aesthetic: New Approaches to and Ancient Affinity, Queens Univ, Kingston, CANADA (2006)
Citations Web of Science - 1
2006 Borwein 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 rec... [more]

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.

Citations Scopus - 3
2005 Borwein JM, Karamanos K, 'Algebraic dynamics of certain gamma function values', GENERALIZED CONVEXITY, GENERALIZED MONOTONICITY AND APPLICATIONS, Hanoi, VIETNAM (2005)
Citations Web of Science - 6
2004 Borwein JM, Choi KKS, 'Dirichet series for squares of sums of squares: A summary', NUMBER THEORY, Univ Montreal, Ctr Rech Math, Montreal, CANADA (2004)
2003 Kellar 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]
2003 Borwein J, Stanway T, 'Managing digital mathematical discourse', MATHEMATICAL KNOWLEDGE MANAGEMENT, PROCEEDINGS, BERTINORO, ITALY (2003) [E1]
2002 Borwein JM, 'The experimental mathematician: A computational guide to the mathematical unknown', MATHEMATICAL SOFTWARE, PROCEEDINGS, Beijing, PEOPLES R CHINA (2002)
DOI 10.1142/9789812777171_0001
2000 Borwein JM, Lisonek P, 'Applications of integer relation algorithms', DISCRETE MATHEMATICS, UNIV VIENNA, VIENNA, AUSTRIA (2000)
DOI 10.1016/S0012-365X(99)00256-3
Citations Scopus - 6Web of Science - 5
1999 Borwein JM, Zhu QJ, 'Multifunctional and functional analytic techniques in nonsmooth analysis', NONLINEAR ANALYSIS, DIFFERENTIAL EQUATIONS AND CONTROL, MONTREAL, CANADA (1999)
Citations Web of Science - 7
1995 BAILEY 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)
1995 Borwein JM, 'A survey of examples of convex functions and classifications of normed spaces', RECENT DEVELOPMENTS IN OPTIMIZATION, DIJON, FRANCE (1995)
1995 LIMBER 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)
DOI 10.1109/23.467872
Citations Web of Science - 40
1994 BORWEIN 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)
1994 LIMBER 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)
1993 BORWEIN JM, LEWIS AS, 'A SURVEY OF CONVERGENCE RESULTS FOR MAXIMUM-ENTROPY METHODS', MAXIMUM ENTROPY AND BAYESIAN METHODS, CNRS, PARIS, FRANCE (1993)
Citations Web of Science - 5
1991 BORWEIN JM, 'MINIMAL CUSCOS AND SUBGRADIENTS OF LIPSCHITZ FUNCTIONS', FIXED POINT THEORY AND APPLICATIONS, CTR INT RECONTRES MATH, MARSEILLE, FRANCE (1991)
Citations Web of Science - 23
1991 BORWEIN JM, LEWIS AS, 'PRACTICAL CONDITIONS FOR FENCHEL DUALITY IN INFINITE DIMENSIONS', FIXED POINT THEORY AND APPLICATIONS, CTR INT RECONTRES MATH, MARSEILLE, FRANCE (1991)
Citations Web of Science - 2
1990 BORWEIN JM, BORWEIN PB, 'A REMARKABLE CUBIC MEAN ITERATION', COMPUTATIONAL METHODS AND FUNCTION THEORY, VALPARAISO, CHILE (1990)
Citations Web of Science - 6
Show 26 more conferences
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Grants and Funding

Summary

Number of grants 11
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 body University of Newcastle
Project Team Laureate Professor Jon Borwein, Professor Jeremy Avigad
Scheme International Research Visiting Fellowship
Role Lead
Funding Start 2015
Funding Finish 2015
GNo G1401314
Type Of Funding Internal
Category INTE
UON Y

20141 grants / $415,000

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

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Project Team Laureate Professor Jon Borwein, Conjoint Professor Richard Brent, Dr David Bailey
Scheme Discovery Projects
Role Lead
Funding Start 2014
Funding Finish 2014
GNo G1300158
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON Y

20123 grants / $657,664

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

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Project Team Laureate Professor Jon Borwein, Professor Andrew Eberhard, Professor Adrian Lewis, Associate Professor Brailey Sims, Professor Qiji Zhu
Scheme Discovery Projects
Role Lead
Funding Start 2012
Funding Finish 2012
GNo G1100082
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON Y

Project EVA: an environmentally responsible facility for interdisciplinary supercomputing applications$263,000

Funding body: University of Newcastle

Funding body University of Newcastle
Project Team Professor Pablo Moscato, Laureate Professor Jon Borwein, Conjoint Professor Keith Jones, Conjoint Professor Chris Levi, Professor Mark Parsons, Professor Michael Ostwald, Professor Hugh Craig, Conjoint Professor Peter Greer, Associate Professor Stephan Chalup, Associate Professor Regina Berretta
Scheme Equipment Grant
Role Investigator
Funding Start 2012
Funding Finish 2012
GNo G1100627
Type Of Funding Internal
Category INTE
UON Y

Faculty Visiting Fellowship 2012$4,664

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

Funding body University of Newcastle - Faculty of Science & IT
Project Team Laureate Professor Jon Borwein
Scheme Visiting Fellowship
Role Lead
Funding Start 2012
Funding Finish 2012
GNo G1401121
Type Of Funding Internal
Category INTE
UON Y

20112 grants / $85,000

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

Funding body: University of Newcastle

Funding body University of Newcastle
Project Team Professor Leo Radom, Associate Professor Frans Henskens, Laureate Professor Jon Borwein
Scheme Equipment Grant
Role Investigator
Funding Start 2011
Funding Finish 2011
GNo G1000459
Type Of Funding Internal
Category INTE
UON Y

DVCR Special Grant 2011$40,000

Funding body: University of Newcastle

Funding body University of Newcastle
Project Team Laureate Professor Jon Borwein
Scheme Special Project Grant
Role Lead
Funding Start 2011
Funding Finish 2011
GNo G1101012
Type Of Funding Internal
Category INTE
UON Y

20101 grants / $555,000

Computer Assisted Research Mathematics and its Applications$555,000

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Project Team Laureate Professor Jon Borwein
Scheme Discovery Projects
Role Lead
Funding Start 2010
Funding Finish 2010
GNo G0190001
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON Y

20093 grants / $533,961

Computer Assisted Research Mathematics and its Applications$267,361

Funding body: University of Newcastle

Funding body University of Newcastle
Project Team Laureate Professor Jon Borwein, Conjoint Professor Natashia Boland, Associate Professor Brailey Sims, Professor George Willis
Scheme Priority Research Centre
Role Lead
Funding Start 2009
Funding Finish 2009
GNo G0189866
Type Of Funding Internal
Category INTE
UON Y

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 body ARC (Australian Research Council)
Project Team Conjoint Professor Natashia Boland, Laureate Professor Jon Borwein, Professor Andrew Eberhard
Scheme Discovery Projects
Role Investigator
Funding Start 2009
Funding Finish 2009
GNo G0188718
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON Y

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

Funding body: University of Newcastle

Funding body University of Newcastle
Project Team Laureate Professor Jon Borwein
Scheme Special Project Grant
Role Lead
Funding Start 2009
Funding Finish 2009
GNo G0190019
Type Of Funding Internal
Category INTE
UON Y
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Research Supervision

Number of supervisions

Completed3
Current5

Total current UON EFTSL

PhD2.1

Current Supervision

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

Past Supervision

Year Level of Study Research Title / Program / Supervisor Type
2015 PhD Arithmetic Applications of Hankel Determinants
Mathematics, Faculty of Science and Information Technology, The University of Newcastle
Co-Supervisor
2014 Masters Tools for Teaching Computational Mathematics
Mathematics, Faculty of Science and Information Technology, The University of Newcastle
Principal Supervisor
2013 PhD Random Walks, Elliptic Integrals and Related Constants
Mathematics, Faculty of Science and Information Technology, The University of Newcastle
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

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

Office

Room V120
Building Mathematics
Location Callaghan
University Drive
Callaghan, NSW 2308
Australia
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