High-Precision Computation and Mathematical Physics (2009)
David H. Bailey, Jonathan M. Borwein
At the present time, IEEE 64-bit floating-point arithmetic is sufficiently accurate for most sci-entific applications. However, for a rapidly growing body of important scientific computing...
The Life of Pi: From Archimedes to Eniac and Beyond (2009)
Abstract. The desire to understand π, the challenge, and originally the need, to calculate ever more accurate values of π, the ratio of the circumference of a circle to its diameter, has challenged...
In this note we extend the solution to a recent Monthly problem to analyze a broad class of Dirichlet series and illustrate the result in action in various ways. More precisely, in [6] the following...
Ten Computational Challenge Problems (2009)
Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock our efforts. It should be to us a guidepost on the mazy path to hidden truths,...
Lipschitz functions with maximal Clarke subdierentials are generic (2009)
Jonathan M. Borwein, Xianfu Wang
ABSTRACT. In a recent paper we have shown that most non-expansive Lipschitz functions (in the sense of Baire’s category) have a maximal Clarke subdifferential. In the present paper, we show that in...
DIGITALLY-ASSISTED DISCOVERY AND PROOF (2009)
I will argue that the mathematical community (appropriately defined) is facing a great challenge to re-evaluate the role of proof in light of the power of current computer systems, of modern...
Lp Norms and the Sinc Function (2009)
Borwein, David, Borwein, Jonathan M., Leonard, I. E.
The object of this note is to study the behavior and properties of the following function $$ I(p) = \sqrt{p}\,\dsize\int_{0}^{\infty}\,\bigg|\dfrac{\sin x}{x}\bigg|^{p}\,dx $$
Advances in the theory of box integrals (2009)
Bailey, David H., Borwein, Jonathan M., Crandall, Richard E.
Box integrals, being expectations such as $\langle |\vec r|^s \rangle$ or $\langle |\vec r - \vec q|^s \rangle$ with $\vec r, \vec q$ chosen randomly over the unit $n$-cube, have over the years been...
DIGITALLY-ASSISTED DISCOVERY AND PROOF (2009)
I will argue that the mathematical community (appropriately defined) is facing a great challenge to re-evaluate the role of proof in light of the power of current computer systems, of modern...
Experimental computation with oscillatory integrals (2009)
Bailey, David H., Borwein, Jonathan M.
A previous study by one of the present authors, together with D. Borwein and I. Leonard [7], studied the asymptotic behavior of the p-norm of the sinc function: sinc(x) = (sin x)/x and along the way...
SIAM REVIEW of of the Princeton Companion to Mathematics edited by Timothy Gowers with June Barrow-Green and Imre Leader, associate editors.
Exploratory Experimentation: Digitally-Assisted Discovery and Proof (2009)
I believe that the mathematical community (appropriately defined) is facing a great challenge to re-evaluate the role of proof in light of the power of current computer systems, of modern...
Exploratory Experimentation and Computation (2009)
I believe our research community is facing a great challenge to re-evaluate the role of proof in light of the growing power of current computer systems, of modern mathematical computing packages, and...
Exploratory Experimentation: Digitally-Assisted Discovery and Proof (2009)
Our community (appropriately dened) is facing a great challenge to reevaluate the role of proof in light of the growing power of current computer systems, of modern mathematical computing packages...
Future Challenges for Variational Analysis (2009)
I will also discuss open problems and current challenges for the subject Boris Mordukhovich has played a key role in the development of modern Variational Analysis (VA) and its Applications. Modern...
Frechet-Legendre functions and reflexive Banach spaces (2009)
Borwein, Jonathan M., Vanderwer, Jon
A 2001 article by Bauschke, Borwein and Combettes [2] showed how to extend naturally the classical definitions of essential smoothness and essential strict convexity from functions on Rn in a...
BACAK, MIROSLAV, BORWEIN, JONATHAN M., EBERHARD, ANDREW, MORDUKHOVICH, BORIS S.
In this paper we study infimal convolutions of extended-real-valued functions in Hilbert spaces paying a special attention to the rather broad and remarkable class of prox-regular functions. Such...
Jonathan M. Borwein, M. Bradley
Abstract. We give thirty-two diverse proofs of a small mathematical gem—the fundamental Euler sum identity ζ(2, 1) = ζ(3) = 8 ζ(2, 1). We also discuss various generalizations for multiple...
THE OXFORD USERS ’ GUIDE TO MATHEMATICS SIAM REVIEW (2008)
outlines, primers, problem books, review journals, synopses, tables, and users guides, have played a central role in Mathematics– from the Rhind or Ahmes papyruses (see [3]) of the Egyptians (C...
(1.1) SPECIAL VALUES OF MULTIPLE POLYLOGARITHMS (2008)
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, Petr Lisonek
Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike withitslovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which,...
The Life of Pi: From Archimedes to Eniac and Beyond (2008)
The desire to understand π, the challenge, and originally the need, to calculate ever more accurate values of π, the ratio of the circumference of a circle to its diameter, has challenged...
Jonathan M. Borwein, Mayer Revised, Srinivasa Ramanujan, G. N. Watson, On Reading Ramanujan’s
“I feel so strongly about the wrongness of reading a lecture that my language may seem immoderate. · · · The spoken word and the written word are quite different arts. · · · I feel that to...
MAXIMALITY OF SUMS OF TWO MAXIMAL MONOTONE OPERATORS IN GENERAL BANACH SPACE (2008)
Abstract. We combine methods from convex analysis, based on a function of Simon Fitzpatrick, with a fine recent idea due to Voisei, to prove maximality of the sum of two maximal monotone operators in...
BOOK REVIEW Techniques of Variational Analysis (2008)
The title of this book may appear as rather mysterious to some readers. In fact, in spite of the impact of the book [9] of Rockafellar and Wets, “variational analysis ” is not a well delimited...
Elliptic integral evaluations of Bessel moments (2008)
David H. Bailey, Jonathan M. Borwein, David Broadhurst, M. L. Glasser
We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we...
PROXIMALITY AND CHEBYSHEV SETS (2008)
Abstract. This paper is a companion to a lecture given at the Prague Spring School in Analysis in April 2006. It highlights four distinct variational methods of proving that a finite dimensional...
NON-SMOOTH ANALYSIS, OPTIMISATION THEORY AND BANACH SPACE THEORY (2008)
Abstract. The questions listed here do not necessarily represent the most significant problems from the areas of Non-smooth Analysis, Optimisation theory and Banach space theory, but rather, they...
The Coast-to-Coast Seminar and Remote Mathematical Collaboration (2008)
Jonathan M. Borwein, David Langstroth, Mason Macklem, Scott Wilson
(WestGrid) and Dalhousie (ACEnet) seminar series which is now two years old, and is gradually expanding to include other Canadian universities. More generally we discuss current and future uses of...
Elliptic integral evaluations of Bessel moments (2008)
Bailey, David H., Borwein, Jonathan M., Broadhurst, David, Glasser, M.L.
We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we...
Effective Error Bounds in Euler-Maclaurin-Based Quadrature Schemes (2008)
Bailey, David H., Borwein, Jonathan M.
We analyze the behavior of Euler-Maclaurin-based integrationschemes with the intention of deriving accurate and economicestimations of the error term.
Hypergeometric Forms for Ising-Class Integrals (2008)
Bailey, David H., Borwein, David, Borwein, Jonathan M., Crandall, Richard E.
We apply experimental-mathematical principles to analyze certain integrals relevant to the Ising theory of solid-state physics. We find representations of the these integrals in terms of Meijer...
The Coast-to-Coast Seminar and Remote Mathematical Collaboration (2008)
Jonathan M. Borwein, David Langstroth, Mason Macklem, Scott Wilson
(WestGrid) and Dalhousie (ACEnet) seminar series which is now two years old, and is gradually expanding to include other Canadian universities. More generally we discuss current and future uses of...
Uniform Bounds for the Incomplete Complementary Gamma Function (2008)
Jonathan M. Borwein, O-yeat Chan
Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet...
Jonathan M. Borwein, O-yeat Chan
Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet...
Ten Problems in Experimental Mathematics (2008)
David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, Eric W. Weisstein
Challenge ” of Nick Trefethen, beautifully described in [12] (seealso[13]). Indeed, these ten numeric challenge problems are also listed in [15, pp. 22–26], where they are followed by the ten...
Jonathan M. Borwein, Richard E. Crandall, Carl Pomerance, D. H. Bailey
Résumé. En combinant des concepts de théorie additive des nombres avec des résultats sur les développements binaires et les séries partielles, nous établissons de nouvelles bornes pour la...
Jonathan M. Borwein, M. Bradley
Abstract. We give twenty eight diverse proofs of the fundamental Euler sum identity ζ(2, 1) = ζ(3) = 8 ζ(2, 1). We also discuss various generalizations for multiple harmonic (Euler) sums and some...
SIAM J. Control Optim., to appear BREGMAN MONOTONE OPTIMIZATION ALGORITHMS ∗ (2008)
Heinz H. Bauschke, Jonathan M. Borwein, L. Combettes
Abstract. A broad class of optimization algorithms based on Bregman distances in Banach spaces is unified around the notion of Bregman monotonicity. A systematic investigation of this notion leads to...
Prepared for ISSAC'99: International Symposium on Symbolic and Algebraic Computation (2008)
1 ABSTRACT. My intention is to discuss experimental computation from a mathematician's perspective. Using, for the most part, various of the zeta functions and related polylogarithmic functions...
Elliptic integral evaluations of Bessel moments (2008)
Bailey, David H., Borwein, Jonathan M., Broadhurst, David, Glasser, M. L.
We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we...
Elliptic integral evaluations of Bessel moments (2008)
Bailey, David H., Borwein, Jonathan M., Broadhurst, David, Glasser, M.L.
We record what is known about the closed forms for various Bessel function moments arising in quantum field theory,condensed matter theory and other parts of mathematical physics. More generally, we...
Duality in Tails of Multiple Zeta Values (2008)
Borwein, Jonathan M., Chan, O-Yeat
Duality relations are deduced for tails of multiple zeta values using elementary methods. These formulas extend the classical duality formulas for multiple zeta values.
High-Precision Numerical Integration:Progress and Challenges (2008)
Borwein, Jonathan M., Bailey, David H.
One of the most fruitful advances in the field of experimental mathematics has been the development of practical methods for very high-precision numerical integration, a quest initiated by Keith...
Differentiability of conjugate functions and perturbed minimization principles (2008)
Borwein, Jonathan M., Vanderwerff, Jon
We survey the tight connection between differentiability of conjugate functions and perturbed optimization principles.
Stability of closedness of convex cones under linear mappings (2008)
Borwein, Jonathan M., Moors, Warren B.
In this paper we reconsider the question of when the continuous linear image of a closed convex cone is closed in Euclidean space. In particular, we show that although it is not true that the...
An explicit non-expansive function whose subdifferential is the entire dual ball (2008)
Borwein, Jonathan M., Sciffer, Scott
We provide an explicit recipe for constructing a function on an arbitrary real Banach space whose Clarke and approximate subdifferentials are identically equal to the dual unit ball.
High-Precision Computation and Mathematical Physics (2008)
Bailey, David H., Borwein, Jonathan M.
At the present time, IEEE 64-bit floating-point arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing...
Elliptic integral evaluations of Bessel moments and applications (2008)
David H. Bailey, Jonathan M. Borwein, David Broadhurst, M. L. Glasser
We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical...
Jonathan M. Borwein, Bruno Salvy
We provide a proof of a conjecture in [2] on the existence and form of linear recursions for moments of powers of the Bessel function K0. 1
Elliptic integral evaluations of Bessel moments and applications (2008)
David H. Bailey, Jonathan M. Borwein, David Broadhurst, M. L. Glasser
We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical...
Elliptic integral evaluations of Bessel moments and applications (2008)
David H Bailey, Jonathan M Borwein, David Broadhurst, M L Glasser
We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical...
Heinz H. Bauschke, Jonathan M. Borwein
The convex feasibility problem, that is, finding a point in the intersection of finitely many closed convex sets in Euclidean space, arises in various areas of mathematics and physical sciences. It...
Heinz H. Bauschke, Dominikus Noll, Anna Celler, Jonathan M. Borwein
In this paper we present two varients of the EM-algorithm for dynamic SPECT imaging: (a) a version based on compartmental modeling which fits a sum of exponentials, and (b) a more general approach...
Metric regularity, strong CHIP, and CHIP are distinct properties (2007)
Heinz Bauschke Jonathan, Jonathan M. Borwein, Paul Tseng
Metric regularity, the strong conical hull intersection property (strong CHIP), and the conical hull intersection property (CHIP) are properties of a collection of finitely many closed convex...
Metric regularity, strong CHIP, and CHIP are distinct properties (2007)
Heinz Bauschke, Jonathan M. Borwein, Paul Tseng
Metric regularity, the strong conical hull intersection property (strong CHIP), and the conical hull intersection property (CHIP) are properties of a collection of finitely many closed convex...
A Pamphlet on Pi serving as a Supplement for the Third Edition of (2007)
J. Lennart Berggren, Jonathan M. Borwein, Peter B. Borwein
Our aim in preparing this pamphlet is to bring the material in the collection of papers in the second edition of our Pi: A Source Book [9] up to date. Moreover, several delightful pieces came...
ABSTRACT i I speak primarily from the per-spective of an informed consumer, and only secondarily as a producer of, mathematical soft-ware (www.mathresources. com). The emergence of powerful...
Jonathan M. Borwein, Algo Seminar, Wright Records Elegant
forms for the generating functions of the divisor functions k (n) = P
Centre for Experimental & (2007)
Jonathan M. Borwein, Seaway Sectional Meeting
\If my teachers had begun by telling me that mathematics was pure play with presuppositions, and wholly in the air, I might have become a good mathematician. But they were overworked drudges, and I...
Centre for Experimental & (2007)
\If my teachers had begun by telling me that mathematics was pure play with presuppositions, and wholly in the air, I might have become a good mathematician. But they were overworked drudges, and I...
History and Computation (2007)
Jonathan Borwein Frsc, Jonathan M. Borwein, Piscine Molitor Patel, The Simpsons
The desire, and originally the need, to calculate ever more accurate values of #, the ratio of the circumference of a circle to its diameter, has challenged mathematicians for many centuries and,...
The Experimental Mathematician: (2007)
Computational Guide To, Jonathan M. Borwein
I speak primarily from the perspective of an informed consumer, and only secondarily as a producer of, mathematical software (www.mathresources.com).
Euler-Boole Summation Revisited (2007)
Borwein, Jonathan M., Calkin, Neil J., Manna, Dante
We study a connection between Euler-MacLaurin Summation and Boole Summation suggested in an AMM note from 1960, which explains them as two cases of a general approach to approximation. Herein we give...
A Proof of a Recursion for Bessel Moments (2007)
Borwein, Jonathan M., Salvy, Bruno
We provide a proof of a conjecture in (Bailey, Borwein, Borwein, Crandall 2007) on the existence and form of linear recursions for moments of powers of the Bessel function $K_0$.
Uniform Bounds for the Complementary Incomplete Gamma Function (2007)
Borwein, Jonathan M., Chan, O-Yeat
We prove upper and lower bounds for the complementary incomplete gamma function $\G(a,z)$ with complex parameters $a$ and $z$. Our bounds are refined within the circular hyperboloid of one sheet...
UNIFORMLY CONVEX FUNCTIONS ON BANACH SPACES (2007)
Borwein, Jonathan M., Guirao, Antonio J., Vanderwerff, Jon D.
We study the connection between uniformly convex functions f : X -> R bounded above by ||x||^p, and the existence of norms on X with moduli of convexity of power type. In particular, we show that...
Resolution of the Quinn-Rand-Strogatz constant of nonlinear physics (2007)
Bailey, David H., Borwein, Jonathan M., Crandall, Richard E.
Herein we develop connections between zeta functions and some recent ``mysterious" constants of nonlinear physics. In an important analysis of coupled Winfree oscillators, Quinn, Rand, and Strogatz...
Resolution of the Quinn-Rand-Strogatz constant of nonlinear physics (2007)
Bailey, David H., Borwein, Jonathan M., Crandall, Richard E.
Herein we develop connections between zeta functions and some recent ``mysterious" constants of nonlinear physics. In an important analysis of coupled Winfree oscillators, Quinn, Rand, and Strogatz...
A PROOF OF A RECURSION FOR BESSEL MOMENTS (2007)
Borwein, Jonathan M., Salvy, Bruno
We provide a proof of a conjecture in [2] on the existence and form of linear recursions for moments of powers of the Bessel function K0.
THE EVALUATION OF BESSEL FUNCTIONS VIA EXP-ARC INTEGRALS (2007)
Borwein, David, Borwein, Jonathan M., Chan, O-Yeat
A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of...
The evaluation of Bessel functions via exp-arc integrals (2007)
Borwein, David, Borwein, Jonathan M., Chan, O-Yeat
A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of...
Second Order Cones for Maximal Monotone Operators via Representative Functions (2007)
Eberhard, Andrew C., Borwein, Jonathan M.
It is shown that various first and second order derivatives of the Fitzpatrick and Penot representative functions for a maximal monotone operator T, in a reflexive Banach space, can be used to...
A General Sum Theorem for Maximality of Monotone Operators (2007)
Eberhard, Andrew C., Borwein, Jonathan M.
We establish maximality of the sum of two maximal monotone operators in general Banach space, assuming only the Rockafellar qualification assumption. This resolves, affirmatively, a thirty-five year...
A Variation on the Banach-Dieudonne Theorem with Application to Maximal Monotone Operators (2007)
Eberhard, Andrew C., Borwein, Jonathan M.
A convex set $C\subseteq X^{\ast}\times X$ is said to admit the Generalised Banach-Dieudonn\'{e} property (GBDP) if the weak$^{\ast}$-strong closure $\overline{C}^{w^{\ast}\times s}$ is characterised...
Integer Powers of Arcsin (2007)
Jonathan M. Borwein, Marc Chamberland
New simple nested-sum representations for powers of the arcsin function are given. This generalization of Ramanujan's work makes connections to finite binomial sums and polylogarithms.
Integer powers of Arcsin (2007)
Jonathan M. Borwein, Marc Chamberl
Abstract: New simple nested sum representations for powers of the arcsin function are given. This generalization of Ramanujan’s work makes connections to finite binomial sums and polylogarithms....
Effective Laguerre asymptotics (2007)
D. Borwein, Jonathan M. Borwein, Richard E. Cr
It is known that the generalized Laguerre polynomials can enjoy sub-exponential growth for large primary index. Specifically, for certain fixed parameter pairs (a, z) one has the large-n asymptotic L...
The evaluation of Bessel functions via exp-arc integrals (2007)
David Borwein, Jonathan M. Borwein, O-yeat Chan
Abstract. A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the...
Effective Laguerre asymptotics (2007)
D. Borwein, Jonathan M. Borwein, Richard E. Cr
It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic...
Integer Powers of Arcsin (2007)
Jonathan M. Borwein, Marc Chamberland
New simple nested-sum representations for powers of the arcsin function are given. This generalization of Ramanujan's work makes connections to finite binomial sums and polylogarithms.
Hypergeometric forms for Ising-class integrals (2006)
Borwein, Jonathan M., Crandall, Richard, Bailey, David H.
We apply experimental-mathematical principles to analyze integrals Cn. These are generalizations of a previous integral Cn := Cn,1 relevant to the Ising theory of solid-state physics [8]. We find...
Integrals of the Ising class (2006)
Borwein, Jonathan M., Crandall, Richard, Bailey, David H.
From an experimental-mathematical perspective we analyze “Isingclass” integrals. These are structurally related n-dimensional integrals we call Cn, Dn, En, where Dn is a magnetic susceptibility...
Maximality of Monotone Operators in General Banach Space (2006)
Borwein, Jonathan M., Eberhard, Andrew C.
We establish maximality of the sum of two maximal monotone operators in general Banach space, assuming only the Rockafellar qualification assumption. As a by-product of our analysis, we also study...
Integrals of the Ising Class (2006)
Bailey, David H., Borwein, Jonathan M., Crandall, Richard E.
From an experimental-mathematical perspective we analyze "Ising-class" integrals. Our experimental results involved extreme-precision, multidimensional quadrature on intricate integrands; thus,...
Bailey, David H., Borwein, Jonathan M., Crandall, Richard E.
By a "box integral" we mean here an expectation $\langle |\vec r - \vec q|^s \rangle$ where $\vec r$runs over the unit $n$-cube, with $\vec q$ and $s$ fixed, explicitly:\begin eqnarray* &&\int_01...
Asplund Decomposition of Monotone Operators (2006)
Borwein, Jonathan M., Wiersma, Herre
We establish representations of a monotone mapping as the sum of a maximal subdifferential mapping and a `remainder' monotone mapping, where the remainder is either skew linear, or `acyclic', in the...
Maximality of Sums of Two Maximal Monotone Operators in General Banach Space (2006)
We combine methods from convex analysis, based on a function of Simon Fitzpatrick, with a fine recent idea due to Voisei, to prove maximality of the sum of two maximal monotone operators in Banach...
Proximality and Chebyshev sets (2006)
This paper is a companion to a lecture given at the Prague Spring School in Analysis in April 2006. It highlights four distinct variational methods of proving that a finite dimensional Chebyshev set...
Borwein, Jonathan M., Bailey, David H., Crandall, Richard
By a ``box integral'' we mean here an expectation $\langle |\vec r - \vec q|^s \rangle$ where $\vec r$ runs over the unit $n$-cube, with $\vec q$ and $s$ fixed, explicitly: \begin{eqnarray*}...
Fitzpatrick functions and continuous linear monotone operators (2006)
Borwein, Jonathan M., Bauschke, Heinz H., Wang, Shawn Xianfu
The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily...
Symbolic Computation of Fenchel Conjugates (2006)
Borwein, Jonathan M., Hamilton, Chris
Convex optimization is a branch of mathematics dealing with nonlinear optimization problems with additional geometric structure. This area has been the focus of considerable recent research due to...
Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative (2006)
Bartz, Sedi, Bauschke, Heinz H., Borwein, Jonathan M., Reich, Simeon, Wang, Shawn Xianfu
Several deeper results on maximal monotone operators have recently found simpler proofs using Fitzpatrick functions. In this paper, we study a sequence of Fitzpatrick functions associated with a...
Hilbert’s Inequality and Witten’s Zeta-function (2006)
We explore a variety of pleasing connections between analysis, number theory and operator theory, while revisiting a number of beautiful inequalities originating with Hilbert, Hardy and others. We...
Quadratic Convergence of the Tanh-sinh Quadrature Rule (2006)
Borwein, Jonathan M., Ye, Lingyun
In practice, the doubly exponential `tanh-sinh' transformation for numerical integration has remarkable effectiveness---even for quite unruly integrands. Our intention in this note is to provide a...
Ten Problems in Experimental Mathematics (2006)
Borwein, Jonathan M., Kapoor, Vishaal, Bailey, David H.
This article was stimulated by the recent SIAM “100 Digit Challenge” of Nick Trefethen, beautifully described in [12] (see also [13]). Indeed, these ten numeric challenge problems are also listed...
Non-smooth Analysis, Optimisation Theory and Banach Space Theory (2006)
Borwein, Jonathan M., Moors, Warren B.
The questions listed here do not necessarily represent the most significant problems from the areas of Non-smooth Analysis, Optimisation theory and Banach space theory, but rather, they represent a...
Quadratic Convergence of the Tanh-sinh Quadrature Rule (2006)
Borwein, Jonathan M., Ye, Lingyun
In practice, the doubly exponential `tanh-sinh' transformation for numerical integration has remarkable effectiveness---even for quite unruly integrands. Our intention in this note is to provide a...
Maximality of sums of two maximal monotone operators (2006)
Abstract. We use methods from convex analysis convex, relying on an ingenious function of Simon Fitzpatrick, to prove maximality of the sum of two maximal monotone operators on reflexive Banach space...
Fitzpatrick functions and continuous linear monotone operators (2006)
Heinz H. Bauschke, Jonathan M. Borwein, Xianfu Wang
The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily...
Ten Problems in Experimental Mathematics (2006)
David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, Eric W. Weisstein
Challenge ” of Nick Trefethen, beautifully described in [12] (see also [13]). Indeed, these ten numeric challenge problems are also listed in [15, pp. 22–26], where they are followed by the ten...
Jonathan M. Borwein, Lingyun Ye
Abstract. In [5] and [2] the authors describe the remarkable effectiveness of the doubly exponential ‘tanh-sinh ’ transformation for numerical integration —even for quite unruly integrands. Our...
Experimental Determination of Apéry-Like Identities for ζ(2n + 2) (2006)
David H. Bailey, Jonathan M. Borwein, David M. Bradley
We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others....
Ten Problems in Experimental Mathematics (2006)
David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, Eric W. Weisstein
Challenge ” of Nick Trefethen, beautifully described in [12] (see also [13]). Indeed, these ten numeric challenge problems are also listed in [15, pp. 22–26], where they are followed by the ten...
Maximal monotonicity via convex analysis (2006)
Abstract. In his ‘23 ’ “Mathematische Probleme ” 1 lecture to the Paris International Congress in 1900, David Hilbert wrote “Besides it is an error to believe that rigor in the proof is the...
Future Prospects for Computer-Assisted Mathematics (2005)
Borwein, Jonathan M., Bailey, David H.
The recent rise of ``computer-assisted'' and ``experimental'' mathematics raises intriguing questions as to the future role of computation in mathematics. These results also draw into question the...
Future Prospects for Computer-Assisted Mathematics (2005)
Bailey, David H., Borwein, Jonathan M.
The recent rise of "computer-assisted" and "experimental" mathematics raises intriguing questions as to the future role of computation in mathematics. These results also draw into question the...
The Oxford User's Guide To Mathematics, SIAM Review (2005)
For thousands of years, dictionaries, encyclopedias, handbooks, manuals, outlines, primers, problem books, review journals, synopses, tables, and users guides, have played a central role in...
Thirty-two Goldbach Variations (2005)
Borwein, Jonathan M., Bradley, David
We give thirty-two diverse proofs of a small mathematical gem—the fundamental Euler sum identity ζ(2, 1) = ζ(3) = 8 ζ(\overline{2}, 1). We also discuss various generalizations for multiple...
Lipschitz functions with maximal Clarke subdifferentials are staunch (2005)
Borwein, Jonathan M., Wang, Shawn Xianfu
In a recent paper we have shown that most non-expansive Lipschitz functions (in the sense of Baire's category) have a maximal Clarke subdifferential. In the present paper, we show that in a separable...
Asplund Decompositions of Monotone Operators (2005)
This paper is an extended version of my talk given at the associated 2004 conference in Guadeloupe. Further related matter can be found in [4]. In a largely forgotten 1968–1970 paper, Edgar...
Dynamics of a continued fraction of Ramanujan with random coefficients (2005)
Borwein, Jonathan M., Luke, D. Russell
We study a generalization of a continued fraction of Ramanujan with random, complex-valued coefficients. A study of the continued fraction is equivalent to an analysis of the convergence of certain...
Effective Error Bounds in Euler-Maclaurin-Based Quadrature Schemes (2005)
Borwein, Jonathan M., Bailey, David H.
We analyze the behavior of Euler-Maclaurin-based integration schemes with the intention of deriving accurate and economic estimations of the error term.
Experimental determination of Apery-like identities for zeta(2n+2) (2005)
Bailey, David H., Borwein, Jonathan M., Bradley, David M.
We document the discovery of two generating functions for the Riemann zeta values zeta(2n+2), analogous to earlier work for zeta(2n+1) and zeta(4n+3). This continues work initiated by Koecher and...
Experimental Determination of Apery-Like Identities for Zeta(2n+2) (2005)
Bailey, David H., Borwein, Jonathan M., Bradley, David M.
We document the discovery of two generating functions for zeta(2n+2), analogous to earlier work for zeta(2n+1) and zeta(4n+3), initiated by Koecher and pursued further by Borwein, Bradley and others.
Empirically determined Apery-like formulae for zeta(4n+3) (2005)
Borwein, Jonathan M., Bradley, David M.
Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for zeta(4n+3) which generalizes Apery's series for zeta(3), and...
Searching symbolically for Apery-like formulae for values of the Riemann zeta function (2005)
Borwein, Jonathan M., Bradley, David M.
We discuss some aspects of the search for identities using computer algebra and symbolic methods. The focus is on so-called Apery-like formulae for special values of the Riemann Zeta function. Much...
Parametric Euler Sum Identities (2005)
Borwein, David, Borwein, Jonathan M., Bradley, David M.
We consider some parametrized classes of multiple sums first studied by Euler. Identities between meromorphic functions of one or more variables generate reduction formulae for these sums.
Highly Parallel, High-Precision Numerical Integration (2005)
Borwein, Jonathan M., Bailey, David H.
This paper describes a scheme for rapidly computing numerical values of definite integrals to very high accuracy, ranging from ordinary machine precision to hundreds or thousands of digits, even for...
Experimental Determination of Apery-Like Identities for ζ(2n + 2) (2005)
Bailey, David H., Borwein, Jonathan M., Bradley, David
We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others.
Highly Parallel, High-Precision Numerical Integration (2005)
Bailey, David H., Borwein, Jonathan M.
This paper describes a scheme for rapidly computing numerical values of definite integrals to very high accuracy, ranging from ordinary machine precision to hundreds or thousands of digits, even for...
Review of the SIAM Hundred Digit Challenge Prepared for the Mathematical Intelligencer (2005)
Lists, challenges and competitions have a long and primarily lustrous history in mathematics. Consider the Hilbert and the Millennium problems. This is the story of a recent highly successful...
Integer Powers of Arcsin (2005)
Borwein, Jonathan M., Chamberland, Marc
This note discovers, derives, and then studies, closed-form series expressions for integer powers of arcsin(x).
Boundedness, Differentiability and Extensions of Convex Functions (2005)
Borwein, Jonathan M., Montesinos, Vicente, Vanderwerff, Jon
We survey various boundedness and differentiability properties of convex functions, and how they are related to sequential convergence with respect to various topologies in the dual space. We build...
On Two Fundamental Identities For Euler Sums (2005)
Borwein, Jonathan M., Bradley, David
We give diverse proofs of the fundamental identities ζ(2,1) =ζ(3) = 8ζ(2,1). We also discuss various generalizations for multiple harmonic (Euler) sums and some connections, thereby illustrating...
Maximal Monotonicity via Convex Analysis (2005)
In his ‘23’ “Mathematische Probleme” lecture to the Paris International Congress in 1900, David Hilbert wrote “Besides it is an error to believe that rigor in the proof is the enemy of...
A Prototype for the Federated World Directory of Mathematicians (2005)
Borwein, Jonathan M., Macklem, Mason, Paek, Jaehyun
In 1998, the International Math Union asked its Committee on Electronic Information Communication (CEIC) to consider an electronic World Directory of Mathematicians to replace the traditional...
Thirty-two Goldbach Variations (2005)
Borwein, Jonathan M., Bradley, David M.
We give thirty-two diverse proofs of a small mathematical gem--the fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also discuss various generalizations for multiple harmonic...
Experimental Mathemataics: Examples, Methods and Implications (2005)
Bailey, David H., Borwein, Jonathan M.
Recent years have seen the flowering of 'experimental' mathematics,namely the utilization of modern computer technology as an active toolin mathematical research. This development is not limited to a...
Implications of Experimental Mathematics for the Philosophy of Mathematics (2005)
Somewhat unusually, I can exactly place the day at registration that I became a mathematician and I recall the reason why. I was about to deposit my punch cards in the ‘honours history bin’. I...
Van der Pol Expansions of L-Series (2005)
Borwein, Jonathan M., Borwein, David
We provide concise series representations for various L-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying L-series.
Dynamics of a continued fraction of Ramanujan with random coefficients (2005)
Jonathan M. Borwein, D. Russell Luke
We study a generalization of a continued fraction of Ramanujan with random, complex-valued coefficients. A study of the continued fraction is equivalent to an analysis of the convergence of certain...
Future Prospects for Computer-Assisted Mathematics (2005)
David H. Bailey, Jonathan M. Borwein
The recent rise of “computer-assisted ” and “experimental ” mathematics raises intriguing questions as to the future role of computation in mathematics. These results also draw into question...
I’ll be glad if I have succeeded in impressing the idea that it is not only pleasant to read at times the works of the old mathematical authors, but this may occasionally be of use for the actual...
David H. Bailey, Jonathan M. Borwein
We analyze the behavior of Euler-Maclaurin-based integration schemes with the intention of deriving accurate and economic estimations of the error term. 1
REVIEW OF THE SIAM HUNDRED DIGIT CHALLENGE PREPARED FOR THE MATHEMATICAL INTELLIGENCER (2005)
Abstract. Lists, challenges and competitions have a long and primarily lustrous history in mathematics. Consider the Hilbert and the Millennium problems. This is the story of a recent highly...
David H. Bailey, Jonathan M. Borwein
We analyze the behavior of Euler-Maclaurin-based integration schemes with the intention of deriving accurate and economic estimations of the error term. 1
Jonathan M. Borwein, David Bailey, Neil Calkin, D. Russell Luke, Victor Moll
The last twenty years have been witness to a fundamental shift in the way mathematics is practiced. With the continued advance of computing power and accessibility, the view that “real...
Mathematics and Applications PLEASE ADD FUNCTIONS Monotone Operators as Convex Objects (2005)
Jonathan Borwein, Jonathan M. Borwein
"The formulas move in advance of thought, while the intuition often lags behind; in the oft-quoted words of d'Alembert, "L'algebre est genereuse, elle donne souvent plus...
Experimental Mathematics: Examples, Methods and Implications (2005)
Jonathan M. Borwein, David H. Bailey, David H. Bailey, Lawrence Berkeley National
The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it. —Jacques Hadamard 1 If mathematics describes an objective...
Dynamics of a continued fraction of Ramanujan with random coefficients (2005)
Jonathan M. Borwein, D. Russell Luke
We study a generalization of a continued fraction of Ramanujan with random, complex-valued coefficients. A study of the continued fraction is equivalent to an analysis of the convergence of certain...
Dynamics of Random Continued Fractions (2004)
Borwein, Jonathan M., Luke, D. Russell
We study a generalization of a continued fraction of Ramanujan with random coefficients. A study of the continued fraction is equivalent to an analysis of the convergence of certain stochastic...
Dynamics of generalizations of the AGM continued fraction of Ramanujan. Part I: divergence. (2004)
Borwein, Jonathan M., Luke, D. Russell
We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the...
The (Digital) Life of Pi (2004)
Borwein, Jonathan M., Macklem, Mason
The desire to understand $\pi$, the challenge, and originally the need, to calculate ever more accurate values of $\pi$, the ratio of the circumference of a circle to its diameter, has challenged...
Parametric Euler Sum Identities (2004)
Borwein, Jonathan M., Borwein, David, Bradley, David
A somewhat unlikely-looking identity is \begin{equation} \label{eu-dual} \sum_{n=1}^{\infty}\frac1{n(n-x)}\sum_{m=1}^{n-1}\frac{1}{m-x} =\sum _{n=1}^{\infty}\frac {1}{n^2(n-x)}, \end{equation} valid...
Experimental Mathematics: Examples, Methods and Directions (2004)
Bailey, David, Borwein, Jonathan M.
Recent years have seen the flowering of ``experimental'' mathematics, namely the utilization of modern computer technology as an active tool in mathematical research. This development is not limited...
Ten Problems in Experimental Mathematics (2004)
Borwein, Jonathan M., Bailey, David, Kapoor, Vishaal
This article was stimulated by the recent SIAM ``100 Digit Challenge'' of Nick Trefethen, beautifully described in \cite{digits}. Indeed, these ten numeric challenge problems are also listed in...
Asplund Decompositions of Monotone Operators (2004)
This paper is an extended version of my talk given in Guadeloupe. Further details will appear in: J. Borwein and H. Wiersma, Asplund Decompositions of Monotone Operators (on Asplund Spaces), in...
Mathematics by Experiment: Plausible Reasoning in the 21st Century (2004)
This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. I shall generally explore experimental and heuristic mathematics and give...
Constructive Minimal Cuscos (2004)
Borwein, Jonathan M., Kortezov, Ivaylo
A (set-valued) map F : Z → X is called an usco, if it is upper semicontinuous and F(z) is a nonempty compact set for every z ∈ Z. An usco map is called a minimal usco if it is minimal with...
Knowledge and Community in Mathematics (2004)
Borwein, Jonathan M., Stanway, Terry
This paper examines the relationship between ideas about mathematical knowledge, the formation of mathematical research communities, and the technologies that mediate mathematical activity. We...
Ten Problems in Experimental Mathematics (2004)
Bailey, David H., Borwein, Jonathan M., Kapoor, Vishaal, Weisstein, Eric
This article was stimulated by the recent SIAM "100 Digit Challenge" of Nick Trefethen, beautifully described in a recent book. Indeed, these ten numeric challenge problems are also listed in a...
The (Digital) Life of Pi (2004)
Borwein, Jonathan M., Macklem, Mason S.
The desire to understand pi, the challenge, and originally the need, to calculate ever more accurate values of pi, the ratio of the circumference of a circle to its diameter, has challenged...
A class of Dirichlet series integrals (2004)
We extend a recent \emph{Monthly} problem to analyse a broad class of Dirichlet series, and illustrate the result in action.
The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof (2004)
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...
The Life of Pi: From Archimedes to ENIAC and Beyond (2004)
The desire to understand pi, the challenge, and originally the need, to calculate ever more accurate values of pi, the ratio of the circumference of a circle to its diameter, has challenged...
Dynamics of generalizations of the AGM continued fraction of Ramanujan. Part I: divergence. (2004)
Borwein, Jonathan M., Luke, D.R.
We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the...
Variational methods in convex analysis (2004)
Borwein, Jonathan M., Zhu, Qiji J.
We use variational methods to provide a concise development of a number of basic results in convex and functional analysis. This illuminates the parallels between convex analysis and smooth...
The evaluation of character Euler double sums (2004)
Borwein, Jonathan M., Boersma, J., Zucker, I. J.
Euler considered sums of the form $$ \sum_{n=1}^{m-1}\f{1}{n^t} $$ Here natural generalizations of these sums namely $$ [p,q]:=[p,q](s,t)=\sum_{m=1}^{\infty}...
Algebraic dynamics of certain Gamma function values (2004)
Borwein, Jonathan M., Karamanos, Kostas
We present significant numerical evidence, based on the entropy analysis by lumping of the binary expansion of certain values of the Gamma function, that some of these values correspond to...
On the dynamics of certain recurrence relations (2004)
Borwein, Jonathan M., Crandall, Richard, Borwein, David, Mayer, Raymond
In previous analyses \cite{borcra1, borcra2} the remarkable AGM continued fraction of Ramanujan---denoted ${\cal R}_1(a,b)$---was proven to converge for almost all complex parameter pairs $(a,b)$. It...
The evaluation of character Euler double sums (2004)
Borwein, Jonathan M., Zucker, I. J., Boersma, J.
Euler considered sums of the form $$ \sum_{m=1}^{\infty}\f{1}{m^s}\sum_{n=1}^{m-1}\f{1}{n^t} $$ Here natural generalisations of these sums, namely $$...
On the binary expansions of algebraic numbers (2004)
Bailey, David H., Borwein, Jonathan M., Crandall, Richard E., Pomerance, Carl
Jonathan M. Borwein, Semidefinite Programming
www.dal.cs.ca/ ~ jborwein www.cecm.sfu.ca/preprints.html
OUTLINE of PRESENTATION (2004)
Jonathan M. Borwein, A. Part, Philosophy Experimental Math
If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt
FROM TROTSKY to GÖDEL: a LIFE (2004)
Jonathan M. Borwein, Its Philosophy
If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics.
MY INTENTIONS IN THIS TALK • to discuss Experimental Mathodology ∗ (2004)
Jonathan M. Borwein, Its Philosophy, Claude Chevalley
If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics.
GENERALIZATIONS OF CONTINUED FRACTIONS 2 Difference Equations (2004)
Jonathan M. Borwein, D. Russell Luke
Dynamics of generalizations of the AGM
Ten Problems in Experimental Mathematics: (2004)
David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, Eric Weisstein
This article was stimulated by the recent SIAM “100 Digit Challenge ” of Nick Trefethen,
Dynamics of generalizations of of the AGM continued fraction of Ramanujan. Part I: divergence (2004)
Part I Divergence, Jonathan M. Borwein, Jonathan M. Borwein, D. Russell Luke, D. Russell Luke
We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the...
Mathematics by Experiment: Plausible Reasoning in the 21st (2004)
If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics.
On the Ramanujan AGM fraction. Part I: the Real-parameter Case (2003)
Borwein, Jonathan M., Crandall, Richard, Fee, Greg
The Ramanujan AGM continued fraction is a construct enjoying attractive algebraic properties such as a striking arithmetic-geometric mean (AGM) relation and elegant connections with elliptic-function...
On the Ramanujan AGM fraction. Part II: the Complex-parameter Case (2003)
Borwein, Jonathan M., Crandall, Richard
The Ramanujan continued fraction is interesting in many ways; e.g. for certaiun complex parameters (eta, a, b) one has an attractive AGM relation Reta(a,b) + Reta(b,a) = 2Reta((a+b)/2, sqrt{ab})....
Borwein, Jonathan M., Bailey, David H., Girgensohn, Roland
This document is an adapted selection of excerpts from two newly published books, Mathematics by Experiment: Plausible Reasoning in the 21st Century, and Experimentation in Mathematics: Computational...
On the binary expansions of algebraic numbers (2003)
Bailey, David H., Borwein, Jonathan M., Crandall, Richard E., Pomerance, Carl
Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic...
On the Ramanujan AGM fraction (2003)
Borwein, Jonathan M., Crandall, Richard, Fee, Greg
The Ramanujan AGM fraction is a construct $${\cal R}_\eta(a,b) =\,\frac{a}{\displaystyle \eta+\frac{b^2}{\displaystyle \eta +\frac{4a^2}{\displaystyle \eta+\frac{9b^2}{\displaystyle...
Borwein, Jonathan M., Borwein, Peter, Berggren, Lennart
Our aim in preparing this pamphlet is to bring the material in the collection of papers in the second edition of our {\em Pi: A Source Book} (Springer-Verlag, 2000) up to date. Moreover, several...
Cone monotone functions: differentiability and continuity (2003)
Borwein, Jonathan M., Wang, Shawn Xianfu
We provide a porosity notion approach to the differentiability and continuity of real valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by...
On the binary expansions of algebraic numbers (2003)
Borwein, Jonathan M., Bailey, David H., Crandall, Richard, Pomerance, Carl
Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic...
Constructible Convex Sets (2003)
Borwein, Jonathan M., Vanderwerff, Jon
We investigate when closed convex sets can be written as countable intersections of closed half-spaces in Banach spaces. It is reasonable to consider this class to comprise the constructible convex...
Continued Fractions of Tails of Hypergeometric Series (2003)
Borwein, Jonathan M., Choi, Stephen, Pigulla, Wilfried
The tails of the Taylor series for many standard functions such as $\arctan$ and $\log$ can be expressed as continued fractions in a variety ways. A surprising side effect is that some of these...
A one perturbation variational principle and applications (2003)
Borwein, Jonathan M., Fabian, Marian, Revalski, Julian
We study a variational principle in which there is one common perturbation function $\vv$ for every proper lower semicontinuous extended real-valued function $f$ defined on a metric space $X$....
Dirichlet series for squares of sums of squares: a summary (2003)
Borwein, Jonathan M., Choi, Stephen
This note is a summary version of the results that the authors presented at CNTA VII in Montreal and detailed proofs may be found in J.M. Borwein and K.K.S. Choi, `` On Dirichlet Series for Sums of...
Australia June July, Jonathan M. Borwein
In the rst of these two lectures I shall talk generally about experimental mathematics. In Part II, I shall present some more detailed and sophisticated examples.
Jonathan M. Borwein, Srinivasa Ramanujan, G. N. Watson, On Reading Ramanujan’s
work, describes: a thrill which is indistinguishable from the thrill I feel when I enter the Sagrestia Nuovo of the Capella Medici and see before me the austere beauty of the four statues...
Mathematics by Experiment, I & II: Plausible Reasoning in the 21st Century (2003)
Abstract. In the first of these two lectures I shall talk generally about experimental mathematics. In Part II, I shall present some more detailed and sophisticated examples. The emergence of...
Mathematics by Experiment, I & II: Plausible Reasoning in the 21st Century (2003)
Abstract. In the first of these two lectures I shall talk generally about experimental mathematics. In Part II, I shall present some more detailed and sophisticated examples. The emergence of...
Jonathan M. Borwein, Centre For Experimental, Ramanujan J, J. Convex Analysis, ...
www.cecm.sfu.ca/ ~ jborwein/talks.html www.cecm.sfu.ca/preprints.html
Mathematical Prizes • There is no Nobel Prize in Mathematics. (2003)
Jonathan M. Borwein, The Nobels, Other Prizes, A. John, Charles Fields
• But we have our equivalents — the Fields Medals, the Nevanlinna Prize, and this year the new Abel Prize. • Ishall describe a little of the History of the prizes, and say something about the...
Jonathan M. Borwein, Srinivasa Ramanujan, G. N. Watson, On Reading Ramanujan’s, D. Borwein, G. Fee, ...
work, describes: a thrill which is indistinguishable from the thrill I feel when I enter the Sagrestia Nuovo of the Capella Medici and see before me the austere beauty of the four statues...
Finding and Excluding b-ary Machin-Type BBP Formulae (2002)
Borwein, Jonathan M., Galway, William F., Borwein, David
Missing
Differentiability of cone-monotone functions on separable Banach space (2002)
Borwein, Jonathan M., Lewis, Adrian, Burke, James V.
Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gateaux differentiability of real valued functions on separable Banach spaces,...
On Dirichlet series for sums of squares (2002)
Borwein, Jonathan M., Choi, Stephen
In \cite{HW}, Hardy and Wright recorded elegant closed forms for the generating functions of the divisor functions $\sigma_k(n)$ and $\sigma_k(n)^2$:\[\sum_{n=1}^\infty \frac{\sigma_k(n)}{n^s}=\zeta...
Borwein, Jonathan M., Choksi, Rustum, Marechal, Pierre
This article revisits the maximum entropy algorithm in the context of recovering the probability distribution of an asset from the prices of finitely many associated European call options, via...
The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof (2002)
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...
On the nondifferentiability of cone-monotone functions in Banach spaces (2002)
Borwein, Jonathan M., Goebel, Rafal
In finite-dimensional spaces, cone-monotone functions -- a special case of which are the coordinate-wise nondecreasing functions -- possess several regularity properties like almost everywhere...
Bregman Monotone Optimization Algorithms (2002)
Borwein, Jonathan M., Combettes, Patrick L., Bauschke, Heinz H.
A broad class of optimization algorithms based on Bregman distances in Banach spaces is unified around the notion of Bregman monotonicity. A systematic investigation of this notion leads to a...
On the Bartle-Graves Theorem (2002)
Borwein, Jonathan M., Dontchev, Asen L.
The Bartle-Graves theorem extends the Banach open mapping principle to a family of linear and bounded mappings, thus showing that surjectivity of each member of the family is equivalent to the...
Evaluations of Binomial Series (2002)
Borwein, Jonathan M., Girgensohn, Roland
We give closed forms, recursion formulas and experimental results for the series $\sum n^k/{2n\choose n}$ and $\sum 2^{-n}n^k/{3n\choose n}$ with $k\in\Z$
ABSTRACT The International Mathematical Union's (www.mathunion.org) Committee on Electronic Information and Communication, CEIC, (www.ceic.math.ca) was established in 1998 and given a very broad...
Bregman Monotone Optimization Algorithms (2002)
Heinz H. Bauschke, Jonathan M. Borwein, Patrick L. Combettes
A broad class of optimization algorithms based on Bregman distances in Banach spaces is uni ed around the notion of Bregman monotonicity. A systematic investigation of this notion leads to a simpli...
ffl There is no Nobel Prize in Mathematics. ffl But we have our equivalents-- the Fields (2002)
Medals and the Nevanlinna Prize. ffl I shall describe a little of the History of the prizes, and say something about the most recent winners and their work-- with very few formulae! ffl A Tom Lehrer...
Multifunctional and functional analytic techniques in nonsmooth analysis (2001)
Borwein, Jonathan M., Zhu, Qiji J.
These lectures center on the structure of real--valued Lipschitz functions, and their generalized derivatives on Banach spaces. We pay some attention to the role of measure and category and will try...
Subdifferentials whose Graphs are not Norm x Weak* Closed (2001)
Borwein, Jonathan M., Fitzpatrick, Simon, Girgensohn, Roland
In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a \pcf on separable Hilbert space such that the graph of its...
Notions of Relative Interior in Banach Spaces (2001)
Borwein, Jonathan M., Goebel, Rafal
Extensions to a Banach space of the equivalent notions of relatively absorbing, non-support, and relative interior points of a convex set in $\reals^n$ are presented. The relations between these...
Aesthetics for the Working Mathematician (2001)
Most research mathematicians neither think deeply about nor are terribly concerned about either pedagogy or the philosophy of mathematics. Nonetheless, as I hope to indicate, aesthetic notions have...
Special values of multiple polylogarithms (2001)
Borwein, Jonathan M., Bradey, David M., Broadhurst, David J., Lisonek, Petr
Local Lipschitz Constants and Maximal Subdifferentials (2001)
Borwein, Jonathan M., Vanderwerff, Jon, Wang, Shawn Xianfu
It is shown that if $k(x)$ is upper semicontinuous and quasi lower semicontinuou s on a Banach space $X$, then $k(x) B_{X^*}$ is the Clarke subdifferential of some locally Lipschitz function on $X$....
Multi-variable sinc integrals and volumes of polyhedra (2001)
Borwein, David, Borwein, Jonathan M., Mares Jr., Bernard A.
We investigate multi-variable integrals of products of sinc functions and show how they may be interpreted as volumes of symmetric convex polyhedra. We then derive an explicit formula for computing...
Challenges in Mathematical Computing (2001)
Borwein, Jonathan M., Borwein, Peter
Almost all interesting mathematical algorithmic questions relate to NP-hard questions and such computation is prone to explode exponentially. More space, more speed and processors, and even say...
A $C^1$ function even on the sphere with no critical point in the ball (2001)
Borwein, Jonathan M., Kortezov, Ivaylo, Wiersma, Herre
In this article we construct a real-valued $C^1$-function on the closed ball in $\R^2$ that is even on the boundary of the ball, and has no critical points inside the ball. This provides a...
The range of the gradient of a Lipschitz C^1-smooth bump in infinite dimensions (2001)
Borwein, Jonathan M., Fabian, Marian, Loewen, Philip D.
If a Banach space has a Lipschitz $\Ci$-smooth bump function, then it admits another bump of the same quality whose gradients exactly fill the dual unit ball or other reasonably looking figures. This...
A Characterization of Quasiconvex Vector-Valued Functions (2001)
Borwein, Jonathan M., Benoist, Joel, Popovici, Nicolae
The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are $K$-quasiconvex with respect to a closed convex cone $K$ in a Banach space. Our main...
A Class of Exponential Inequalities (2001)
Borwein, Jonathan M., Girgensohn, Roland
We prove that for reals $x_i$ with $\sum x_i \ge 0$, the estimate $\sum x_i\,e^{x_i} \ge \frac{C_N}N \sum x_i^2$ holds, where $C_N = \max\{2,e\left(1-1/N\right)\}$. We also prove analogues for the...
Essential Smoothness, Essential Strict Convexity, and Legendre functions in Banach spaces (2001)
Heinz H. Bauschke, Jonathan M. Borwein, Patrick L. Combettes
The classical notions of essential smoothness, essential strict convexity, and Legendreness for convex functions are extended from Euclidean to Banach spaces. A pertinent duality theory is developed...
Prepared for Seventh Canadian Number Theory Association Conference (2001)
ABSTRACT. Hardy & Wright records elegant forms for the generating functions of the divisor functions oek(n) = Pdjn dk and oe2k (n):
Experimental Mathematics: and Exact Computation (2001)
URL: www.cecm.sfu.ca/ ~ jborwein/talks.html 1 ABSTRACT. Computation in Mathematics is fast becoming ubiquitous. My intention is to discuss "pure and applied experimental computation...
Differentiability of Cone-Monotone Functions on Separable Banach Space (2001)
Jonathan M. Borwein, James V. Burke, Adrian S. Lewis, S. Lewis
Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real valued functions on separable Banach spaces,...
Prepared for Fields Workshop on Issues and Potential for Collaborative Online Mathematics (2001)
ABSTRACT After describing my background and situating mathematics in the `big picture', I intend to make brief assessments, and assertions, about the present state of and future prospects for...
Beauty and the Mathematical Beast: (2001)
URL: www.cecm.sfu.ca/ ~ jborwein/talks.html
Summary by Ludovic Meunier (2001)
F. Chyzak (ed, Jonathan M. Borwein
0 This talk investigates integrals of the form � ∞ n� τn:= sinc(akx) dx k=0 and their multi-dimensional analogues. These integrals are related to volumes of polyhedra, which allows to derive...
Summary by Ludovic Meunier (2001)
F. Chyzak (ed, Jonathan M. Borwein
0 This talk investigates integrals of the form � ∞ n� τn:= sinc(akx)dx k=0 and their multi-dimensional analogues. These integrals are related to volumes of polyhedra, which allows to derive...
Special Values of Multiple Polylogarithms (2001)
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, Lison Ěk
Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which,...
The Range of the Gradient of a Continuously Differentiable Bump (2000)
Borwein, Jonathan M., Fabian, Marian, Kortezov, Ivaylo, Loewen, Philip D.
For a continuously differentiable function with bounded nonempty support,the range of the gradient cannot be locally contained in a hyperplane.However, iterative methods allow the construction of...
Antiproximinal norms in Banach spaces (2000)
Borwein, Jonathan M., Jimenez-sevilla, M., Moreno, J.P
We prove that every Banach space containing a complemented copy of $c_0$ has an antipro\-ximinal body for a suitable norm. If, in addition, the space is separable,there is a pair of antiproximinal...
Refined Convexity and Special Cases of the Blaschke-Santalo Inequality (2000)
Borwein, David, Borwein, Jonathan M., Fee, Greg, Girgensohn, Roland
We derive the $l_p$ version of the classical Blaschke-Santalo inequality for polar volumes as a consequence of more subtle convexity estimates for the volume of the $p-$ball in Euclidean space. We...
Some Generic Results on Non-attaining Functionals (2000)
Borwein, Jonathan M., Kortezov, Ivaylo
We prove that (a) in a reflexive space, for any linearly bounded but unbounded closed convex subset the non-support functionals are a dense $G_\delta$ subset of the polar set, and (b) any...
Joint and Separate Convexity of the Bregman Distance (2000)
Bauschke, Heinz H., Borwein, Jonathan M.
Algorithms involving Bregman projections for solving optimization problems have been receiving much attention lately. Several of these methods rely crucially on the joint convexity of the Bregman...
Convex Functions of Legendre Type in General Banach Spaces (2000)
Convex functions of Legendre type are constructed on arbitrary open convex sets in Banach spaces that satisfy appropriate rotundity and smoothness conditions. A simple direct proof of universal...
On the continuity of biconjugate convex functions (2000)
Borwein, Jonathan M., Vanderwerff, Jon
We show that a Banach space is a Grothendieck space if and only if every continuous convex function on $X$ has a continuous biconjugate function on $X^{**}$, thus also answering a question raised by...
Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces (2000)
Bauschke, Heinz H., Borwein, Jonathan M., Combettes, Patrick L.
The classical notions of essential smoothness, essential strict convexity, and Legendreness for convex functions are extended from Euclidean to Banach spaces. A pertinent duality theory is developed...
Experimental Mathematics: Recent Developments and Future Outlook,” pp (2000)
David H. Bailey, Jonathan M. Borwein, Centre For Experimental
While extensive usage of high-performance computing has been a staple of other scientific and engineering disciplines for some time, research mathematics is one discipline that has heretofore not yet...
Experimental Mathematics: and Exact Computation (2000)
Jonathan M. Borwein, David Broadhurst
. Computation in Mathematics is fast becoming ubiquitous. My intention is to discuss \pure and applied experimental computation " from a mathematician's perspective. I shall try to...
(Partially) Smooth Variational Analysis and Variations (2000)
Jonathan M. Borwein, E. Minimality
I shall describe some basic Banach space ideas that allow one to fruitfully apply notions from smooth analysis, even in spaces which do not admit smooth renorms (or "bumps"), and so to...
Generalized subdifferentials: a Baire categorical approach (2000)
Jonathan M. Borwein, Warren B. Moors, Xianfu Wang
. We describe Baire categorical arguments to construct pathological locally Lipschitz functions. The method traces back to Banach and Mazurkiewicz (1931) who independently used a categorical approach...
Experimental Mathematics: Recent Developments and Future Outlook (2000)
David H. Bailey, Jonathan M. Borwein
This article briey discusses the nature of mathematical experiment. It then presents a few instances primarily of our own recent computeraided mathematical discoveries, and sketches the outlook for...
Generalized subdifferentials: a Baire (2000)
Jonathan M. Borwein, Warren B. Moors, Xianfu Wang
Abstract. We describe Baire categorical arguments to construct pathological locally Lipschitz functions. The method traces back to Banach and Mazurkiewicz (1931) who independently used a categorical...
Special Values of Multiple Polylogarithms (1999)
Borwein, Jonathan M., Bradley, David M., Broadhurst, David J., Lisonek, Petr
Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within...
Computational strategies for the Riemann zeta function (1999)
Borwein, Jonathan M., Bradley, David M., Crandall, Richard
We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new....
Borwein, David, Borwein, Jonathan M., Marechal, Pierre
The optimization problems arising from an information theoretic formulation of the {\it Surprise Examination} (or {\it Unexpected Hanging}) Paradox are examined and solved. They provide a nice...
Compactly epi-Lipschitzian convex sets and functions in normed spaces (1999)
Borwein, Jonathan M., Lucet, Yves, Mordukhovich, Boris S.
We provide several characterizations of compact epi-Lipschitzness for closed convex sets in normed vector spaces. In particular, we show that a closed convex set is compactly epi-Lipschitzian if and...
Generalized subdifferentials: a Baire categorical approach (1999)
Borwein, Jonathan M., Wang, Shawn Xianfu, Moors, Warren B.
We use Baire categorical arguments to construct dramatically pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who...
Generalized subdifferentials: a Baire categorical approach (extended abstract) (1999)
Borwein, Jonathan M., Moors, Warren B., Wang, Shawn Xianfu
Dans ce compte rendu, nous decrivons comment utiliser des arguments de type cat\'egorie de Baire pour construire certaines fonctions localement convexes pathologiques. Les origines de cette approche...
Convex spectral functions of compact operators, Part 1 (1999)
Borwein, Jonathan M., Lewis, Adrian, Read, John, Zhu, Qiji J.
We consider functions on the space of compact self-adjoint Hilbert space operators. Specifically, we study those extended-real functions which depend only on the operators' spectral sequences....
Subdifferentiability of Typical Continuous Functions (1999)
Borwein, Jonathan M., Wang, Shawn Xianfu
We prove a ``typical" subdifferentiability principle and apply it to a variety of complete metric spaces of continuous functions on separable Banach spaces; so as to obtain existence of functions...
Central Binomial Sums, Multiple Clausen Values and Zeta Values (1999)
Borwein, Jonathan M., Broadhurst, David J., Kamnitzer, Joel
We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Ap\'ery sums). The study of on-alternating sums leads...
Numerical and Computational Mathematics (at the undergraduate level) (1999)
Borwein, Jonathan M., Stanway, Terry
We present an argument for including numerical and computational methods in the undergraduate mathematics curriculum. The argument is supported by presenting a number of specific examples; some...
The Impact of Technology on the Doing of Mathematics (1999)
Borwein, Jonathan M., Stanway, Terry
Technology has repeatedly promised to transform mathematics pedagogically. More recently it has made similar promises to the research community. That said, mathematics in 1999 looks a lot more like...
Some Remarkable Properties of Sinc and Related Integrals (1999)
Borwein, Jonathan M., Borwein, David
Using Fourier transform techniques, we establish inequalities for integrals of the form $$\int_0^\infty\prod_{k=0}^n \frac{\sin(a_kx)}{a_kx}\,dx.$$ We then give quite striking closed form evaluations...
Experimental Mathematics: Recent Developments and Future Outlook (1999)
Borwein, Jonathan M., Bailey, David H.
This article briefly discusses the nature of mathematical experiment. It then presents a few instances primarily of our own recent computer-aided mathematical discoveries, and sketches the outlook...
Convex spectral functions of compact operators, Part II (1999)
Borwein, Jonathan M., Lewis, Adrian, Zhu, Qiji J.
It was shown in Part I of this work that the Gateaux differentiability of a convex unitarily invariant function is characterized by that of a similar induced rearrangement invariant function on the...
Emerging tools for experimental mathematics (1999)
Jonathan M. Borwein, Robert M. Corless
If I can give an abstract proof of something, I'm reasonably happy. But if I can get a concrete, computational proof and actually produce numbers I'm much happier. I'm rather an addict...
Experimental Mathematics and Exact Computation (1999)
Jonathan M. Borwein, Years David Bailey, David Bradley, Peter Borwein
. My intention is to discuss experimental computation from a mathematician's perspective. Using, for the most part, various of the zeta functions and related polylogarithmic functions with which...
Subdifferentiability of Typical Continuous Functions (1999)
Jonathan M. Borwein, Xianfu Wang
We prove a "typical" subdifferentiability principle and apply it to a variety of complete metric spaces of continuous functions on separable Banach spaces; so as to obtain existence of...
Generalized subdifferentials: a Baire categorical approach (1999)
Jonathan M. Borwein, Warren B. Moors, Xianfu Wang
. We use Baire categorical arguments to construct dramatically pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who...
Generalized subdifferentials: a Baire categorical approach (Extended Abstract) (1999)
Jonathan M. Borwein, Xianfu Wang
) Jonathan M. Borwein 1 , Warren B. Moors 2 , Xianfu Wang April 15, 1999 R'esum'e. Dans ce compte rendu, nous decrivons comment utiliser des arguments de type cat'egorie de Baire pour...
Partially Smooth Variational Analysis (1999)
Jonathan M. Borwein, Variational Principles, D. Minimality, E. An
I will describe the basic ideas from Banach space theory that allow one to fruitfully apply notions from smooth analysis in spaces which do not admit smooth renorms (or "bumps") and so to...
An EM-algorithm for dynamic SPEC tomography (1999)
Heinz Bauschke, Dominikus Noll, Anna Celler, Jonathan M. Borwein
Introduction Single Photon Emission Computed Tomography (SPECT) is a nuclear medicine diagnosis technique which measures the three dimensional distribution of a radioactively labeled pharmaceutical...
An EM algorithm for dynamic SPECT (1999)
Heinz H. Bauschke, Dominikus Noll, Anna Celler, Jonathan M. Borwein
Abstract — In this paper we present two variants of the EM algorithm for dynamic SPECT imaging. A version based on compartmental modeling which fits a sum of exponentials and a more general...
Emerging Tools for Experimental Mathematics (1998)
Borwein, Jonathan M., Corless, Robert M.
Using mostly elementary examples, we discuss the use of some recent ane emerging tools for experimental mathematics. The tools discussed include so-called ``inverse symbolic computation" used to find...
A convex dual approach to the computation of NMR complex spectra (1998)
Borwein, Jonathan M., Marechal, Pierre, Naugler, D.
The particular entropy method proposed by Hoch {\it et al}~\cite{hoch-90} for the computation of NMR complex spectra allows an elegant application of the concepts of duality theory. Correspondingly,...
Conical open mapping theorems and regularity (1998)
Bauschke, Heinz H., Borwein, Jonathan M.
Suppose $T$ is a continuous linear operator between two Hilbert spaces $X$ and $Y$ and let $K$ be a closed convex nonempty cone in $X$. We investigate the possible existence of $\delta > 0$ such that...
Borwein, Jonathan M., Broadhurst, David J.
We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $$\frac{a}{b}\,{\rm vol}({\cal...
Lipschitz Functions with Maximal Clarke Subdifferentials Are Generic (1998)
Borwein, Jonathan M., Wang, Shawn Xianfu
We show that on a separable Banach space most Lipschitz functions have maximal Clarke subdifferential mappings. In particular, the generic nonexpansive function has the dual unit ball as its Clarke...
Combinatorial Aspects of Euler Sums (1998)
Borwein, Jonathan M., Bradley, David, Broadhurst, David J., Lisonek, Petr
We state a conjecture about certain cyclic sums of multiple zeta values (MZVs, also called Euler sums or Zagier sums) that simultaneoulsy generalizes several other (proved or conjectured) identities,...
Duality Inequalities and Sandwiched Functions (1998)
Borwein, Jonathan M., Fitzpatrick, Simon
We establish a mixed convex-Lipschitz mean value inequality from which recent results of Clarke and Ledyaev and of Lewis and Ralph follow naturally. We also provide various refinements and...
Metric regularity, strong CHIP, and CHIP are distinct properties (1998)
Bauschke, Heinz H., Borwein, Jonathan M., Tseng, Paul
Metric regularity, the strong conical hull intersection property (strong CHIP), and the conical hull intersection property (CHIP) are properties of a collection of finitely many closed convex...
Special Values of Multidimensional Polylogarithms (1998)
Borwein, Jonathan M., Bradley, David, Broadhurst, David J., Lisonek, Petr
Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within...
Separable determination of integrability and minimality of the Clarke subdifferential mapping (1998)
Borwein, Jonathan M., Moors, Warren B.
$D$-representability of Lipschitz functions defined on arbitrary Banach spaces reduces to the study of these properties on separable Banach spaces.
A Survey of Subdifferential Calculus with Applications (1998)
Borwein, Jonathan M., Zhu, Qiji J.
This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied...
On the Representations of $xy+xz+yx$ (1998)
Borwein, Jonathan M., Choi, Stephen
Recently, Crandall in \cite{Cr} used Andrews' identity for the cube of the Jacobian theta function $\theta_4$: \[\theta_4^3(q)=\left(\sum_{n \in...
Numerical Assessment of the Stability of Reconstruction Processes for Computed Tomography (1998)
Marechal, Pierre, Togane, D., Celler, Anna, Borwein, Jonathan M.
This paper deals with the assessment of the stability of reconstruction methods for computed tomography, including Filtered Back Projection and `entropy-like' methods. For each of these methods, the...
Rotund Norms, Clarke Subdifferentials and Extensions of Lipschitz Functions (1998)
Borwein, Jonathan M., Giles, John, Vanderwerff, Jon
We show that a certain condition regarding the separation of points by Lipschitz functions is useful in extending a given Lipschitz function from a subspace of a separable Banach space to the whole...
A generalization of Young's l p inequality (1998)
(communicated by J. Pečarić) Abstract. We show that, for positive real numbers with a> 1 + P N i αi, the function r a a Q Ni=1 x αi i has a convex conjugate of the same form and so, in...
Combinatorial aspects of multiple zeta values, Electron (1998)
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, Petr Lison Ek
3
Combinatorial Aspects of Multiple Zeta Values (1998)
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, Petr Lisonek
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...
Special Values of Multidimensional Polylogarithms (1998)
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, Petr Lisonek, Petr Lison Ek
. Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within...
Emerging Tools for Experimental Mathematics (1998)
Jonathan M. Borwein, Robert M. Corless
Using mostly elementary examples, we discuss the use of some recent and emerging tools for experimental mathematics. The tools discussed include so-called \inverse symbolic computation", lattice...
Combinatorial Aspects of Multiple Zeta Values (1998)
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, Petr Lisonek
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...
Combinatorial Aspects of Multiple Zeta Values (1998)
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, Petr Lison Ek
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...
Lipschitz Functions with Maximal Clarke Subdifferentials Are Generic (1998)
Jonathan M. Borwein, Xianfu Wang
. We show that on a separable Banach space most Lipschitz functions have maximal Clarke subdifferential mappings. In particular, the generic nonexpansive function has the dual unit ball as its Clarke...
A Survey of Subdifferential Calculus with Applications (1998)
Jonathan M. Borwein, Qiji J. Zhu, J. Zhu
This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied...
A nonconvex separation property in Banach spaces (1997)
Borwein, Jonathan M., Jofre, Alejandro
We establish, in infinite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension of Hahn-Banach separation theorem. We provide some consequences in...
Applications of Integer Relation Algorithms (1997)
Borwein, Jonathan M., Lisonek, Petr
Let $a$ be a~vector of real numbers. By an integer relation for $a$ we mean a~non-zero integer vector $c$ such that $ca^T=0$. We discuss the algorithms for finding such integer relations from the...
Bauschke, Heinz H., Borwein, Jonathan M., Li, Wu
The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental...
Separable determination of integrability and minimality of the Clarke subdifferential mapping (1997)
Borwein, Jonathan M., Moors, Warren B.
In this paper we show that the study of integrability and $D$-representability of Lipschitz functions defined on arbitrary Banach spaces reduces to the study of these properties on separable Banach...
A generalization of Young's $L^p$ inequality (1997)
We show that, for positive real numbers with $a \ge 1+ \sum_i^N \alpha_i$, the function $${r^a \over{a}} \over{ \prod_{i=1}^N x_i^{\alpha_i}}$$ has a convex conjugate of the same form and so, in...
On the Construction of H\"older and Proximal Subderivatives (1997)
Borwein, Jonathan M., Girgensohn, Roland, Wang, Shawn Xianfu
We construct Lipschitz functions such that for all $s>0$ they are $s$--H\"older, and so proximally, subdifferentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz...
Subgradient Representation of Multifunctions (1997)
Borwein, Jonathan M., Moors, Warren B., Shao, Yongheng
We provide necessary and sufficient conditions for a minimal upper semicontinuous multifunction defined on a separable Banach space to be the subdifferential mapping of a Lipschitz function.
An EM-algorithm for dynamic SPECT tomography (1997)
Bauschke, Heinz H., Noll, Dominik, Celler, Anna, Borwein, Jonathan M.
An EM-algorithm for dynamic SPECT tomography
Limiting convex examples for nonconvex subdifferential calculus (1997)
Borwein, Jonathan M., Zhu, Qiji J.
We show, largely using convex examples, that most of the core results for limiting subdifferential calculus fail without additional restrictions in infinite dimensional Banach spaces.
On equivalence of some basic principles in variational analysis (1997)
Borwein, Jonathan M., Mordukhovich, Boris S., Shao, Yongheng
The primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization. Considering a broad...
Berggren, John Lennart (ed.), Borwein, Jonathan M. (ed.), Borwein, Peter B. (ed.)
Colección de textos que representa un extenso recuento sobre la noción matemática de Pi, desde el nacimiento de esta ciencia hasta el presente.
Pi : A Sourcebook / Ed. de J.L. Berggren, J.M. Borwein, P.B. Borwein. (1997)
Berggren, John Lennart (ed.), Borwein, Jonathan M. (ed.), Borwein, Peter B. (ed.)
Colección de textos que representa un extenso recuento sobre la noción matemática de Pi , desde el nacimiento de esta ciencia hasta el presente.
On the Khintchine constant (1997)
David H. Bailey, Jonathan M. Borwein, Richard E. Cr
We present rapidly converging series for the Khintchine constant and for general “Khintchine means ” of continued fractions. We show that each of these constants can be cast in terms of an...
Applications of integer relation algorithms (1997)
Jonathan M. Borwein, Petr Lisonek
Let a be a vector of real numbers. By an integer relation for a we mean a non-zero integer vector c such that ca T =0. Wediscuss the algorithms for nding such integer relations from the user's...
David H. Bailey, Jonathan M. Borwein, Peter B. Borwein, Simon Plouffe
Ref: Mathematical Intelligencer, vol. 19, no. 1 (Jan. 1997), pg. 50–57 This article gives a brief history of the analysis and computation of the mathematical constant π = 3.14159..., including a...
Problems L, Jonathan M. Borwein, Jonathan M. Borwein, Jonathan M. Borwein, Qiji Zhu, Qiji Zhu
. We provide a refined sensitivity analysis for finite and infinite horizon control problems where in both cases the perturbation space is L 1 . Our underlying technique relies on a recent sequential...
Limiting Convex Examples for Nonconvex Subdifferential Calculus (1997)
Jonathan M. Borwein, Qiji J. Zhu
. We show, largely using convex examples, that most of the core results for limiting subdifferential calculus fail without additional restrictions in infinite dimensional Banach spaces. Key Words....
Applications of Integer Relation Algorithms (1997)
Jonathan M. Borwein, Petr Lisonek
Let a be a vector of real numbers. By an integer relation for a we mean a non-zero integer vector c such that ca T = 0. We discuss the algorithms for finding such integer relations from the...
Heinz H. Bauschke, Jonathan M. Borwein, Wu Li
The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental...
A Nonconvex Separation Property In Banach Spaces (1997)
Jonathan M. Borwein, Alejandro Jofré
We establish, in infinite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension of Hahn-Banach separation theorem. We provide some consequences in...
Separable determination of integrability and minimality of the Clarke subdifferential mapping (1997)
Jonathan M. Borwein, Jonathan M. Borwein, Warren B. Moors, Warren B. Moors
In this paper we show that the study of integrability and D-representability of Lipschitz functions defined on arbitrary Banach spaces reduces to the study of these properties on separable Banach...
On the Khintchine Constant (1997)
David Bailey, Jonathan M. Borwein, Richard E. Crandall
We present rapidly converging series for the Khintchine constant and for general "Khintchine means" of continued fractions. We show that each of these constants can be cast in terms of an...
Ju Ne, David H. Bailey, Jonathan M. Borwein, Peter B. Borwein, Simon Plouffe
This article gives a brief history of the analysis and computation of the mathematical constant ß = 3:14159 : : :, including a number of the formulas that have been used to compute ß through the...
Subgradient Representation of Multifunctions (1997)
Jonathan M. Borwein, Jonathan M. Borwein, Warren B. Moors, Warren B. Moors, Yongheng Shao, Yongheng Shao
We provide necessary and sufficient conditions for a minimal upper semicontinuous multifunction defined on a separable Banach space to be the subdifferential mapping of a Lipschitz function. Keywords...
Null sets and essentially smooth Lipschitz functions (1997)
Jonathan M. Borwein, Warren B. Moors
In this paper we extend the notion of a Lebesgue-null set to a notion which is valid in any completely metrizable Abelian topological group. We then use this definition to introduce and study the...
An EM-algorithm for dynamic SPECT tomography (1997)
Heinz H. Bauschke, Dominikus Noll, Anna Celler, Jonathan M. Borwein
Introduction Single Photon Emission Computed Tomography (SPECT) is a nuclear medicine diagnosis technique which measures the three dimensional distribution of a radioactively labeled pharmaceutical...
Intelligencer Vol, David H. Bailey, Jonathan M. Borwein, Peter B. Borwein, Simon Plouffe
This article gives a brief history of the analysis and computation of the mathematical constant ß = 3:14159 : : :, including a number of the formulas that have been used to compute ß through the...
On the Khintchine Constant (1997)
David H. Bailey, Jonathan M. Borwein, Richard E. Crandall
We present rapidly converging series for the Khintchine constant and for general "Khintchine means" of continued fractions. We show that each of these constants can be cast in terms of an...
A Generalization Of Young's l p Inequality (1997)
. We show that, for positive real numbers with a 1+ P N i ff i , the function r a a Q N i=1 x ff i i has a convex conjugate of the same form and so, in particular, obtain a clean proof that f is...
On the Khintchine constant (1997)
David H. Bailey, Jonathan M. Borwein, E. Crandall
Abstract. We present rapidly converging series for the Khintchine constant and for general “Khintchine means ” of continued fractions. We show that each of these constants can be cast in terms of...
Bauschke, Heinz H., Borwein, Jonathan M.
The concept of a monotone operator --- which covers both linear positive semi-definite operators and subdifferentials fo convex functions --- is fundamental in various branches of mathematics. Over...
Characterizations of Banach Spaces via Convex and other Locally Lipschitz Functions (1996)
Borwein, Jonathan M., Fabian, Marian, Vanderwerff, Jon
Various properties of Banach spaces, including the reflexivity and the Schur property of a space, are characterized in terms of properties of corresponding classes of locally Lipschitz functions on...
Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k (1996)
Borwein, Jonathan M., Bradley, David, Broadhurst, David J.
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...
Null sets and essentially smooth Lipschitz functions (1996)
Borwein, Jonathan M., Moors, Warren B.
In this paper we extend the notion of a Lebesgue-null set to a notion which is valid in any completely metrizable Abelian topological group. We then use this definition to introduce and study the...
Bailey, David H., Borwein, Jonathan M., Borwein, Peter, Plouffe, Simon
This article gives a brief history of the analysis and computation of the mathematical constant $\pi = 3.14159 \ldots$, including a number of the formulas that have been used to compute $\pi$ through...
What is Organic Mathematics? (1996)
Borwein, Jonathan M., Borwein, Peter, Corless, Robert M., Jorgenson, Loki, Sinclair, Nathalie
Mathematics is more important now than it has ever been. More mathematics is done both inside universities and outside, in industry, than ever before. Admittedly not all of it is called...
Approximations to Pi via the Dedekind eta function (1996)
Borwein, Jonathan M., Garvan, F.G.
Arguably the most efficient algorithm currently known for the extended precision calculation of $\pi$ is a quartic iteration due to J.M. and P.B. Borwein. In their paper, the Borwein's show how this...
Approximate Subgradients and Coderivatives (1996)
Borwein, David, Borwein, Jonathan M., Wang, Shawn Xianfu
We show that in two dimensions or higher the Mordukhovich--Ioffe approximate subgradient and Clarke subgradient may differ almost everywhere for real--valued Lipschitz functions. Uncountably many...
Arithmetic-Geometric Means Revisited (1996)
Borwein, Jonathan M., Lisonek, Petr, Macdonald, John A.
For any integer $N\ge 2$ we study a two-term recurrence (iteration) $AG_N$ consisting of two sequences which converge rapidly ($N$th-order) to a common limit. Our goal is to identify this limit....
A chain rule for essentially strictly differentiable Lipschitz functions (1996)
Borwein, Jonathan M., Moors, Warren B.
In this paper we introduce a new class of real-valued locally Lipschitz functions, (that are similar in nature and definition to Valadier's {\sl saine} functions) which we call {\sl arc-wise...
Empirically Determined Apery like Formulae for Zeta(4n+3) (1996)
Borwein, Jonathan M., Bradley, David
Some rapidly convergent formulae for special values of the Riemann Zeta function are given. We obtain a generating function formula for $\z(4n+3)$ which generalizes Ap\'ery's series for $\z(3)$, and...
Searching symbolically for Apery-like Formulae for values of the Riemann Zeta Function (1996)
Borwein, Jonathan M., Bradley, David
We discuss some aspects of the search for identities using computer algebra and symbolic methods. To keep the discussion as concrete as possible, we shall focus on so-called Ap\'ery-like formulae for...
Partially Smooth Variational Principles and Applications (1996)
Borwein, Jonathan M., Treiman, Jay S., Zhu, Qiji J.
We discuss a smooth variational principle for partially smooth viscosity subdifferentials and explore its applications in nonsmooth analysis.
Evaluation of Triple Euler Sums (1996)
Jonathan M. Borwein, Roland Girgensohn
. Let a, b, c be positive integers and define the so-called triple, double and single Euler sums by #(a, b, c):= # # x=1 x-1 # y=1 y-1 # z=1 1 x a y b z c ,#(a, b):= # # x=1 x-1 # y=1 1 x a y b and...
Proximal Analysis in Smooth Spaces (1996)
Alexander Ioffe, Jonathan M. Borwein, Jonathan M. Borwein, Er Ioffe
. We provide a highly--refined sequential description of the generalized gradients of Clarke and approximate G--subdifferential of a lower semicontinuous extended--real--valued function defined on a...
Qiji J. Zhu, Viscosity Solutions, Jonathan M. Borwein, Jonathan M. Borwein, Jonathan M. Borwein, Qiji Zhu
. In Gateaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Fr'echet subderivative: In this paper we study the viscosity subderivative (which...
On Projection Algorithms for Solving Convex Feasibility Problems (1996)
Heinz H. Bauschke, Jonathan M. Borwein
Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex...
Heinz H. Bauschke, Jonathan M. Borwein
The concept of a monotone operator --- which covers both linear positive semi-definite operators and subdifferentials fo convex functions --- is fundamental in various branches of mathematics. Over...
A chain rule for essentially strictly differentiable Lipschitz functions (1996)
Jonathan M. Borwein, Warren B. Moors
In this paper we introduce a new class of real-valued locally Lipschitz functions, (that are similar in nature and definition to Valadier's saine functions) which we call arc-wise essentially...
Lipschitz functions with minimal Clarke subdifferential mappings (1996)
B. M. Glover, D. Ralph, Jonathan M. Borwein, Warren B. Moors
In this paper we characterise, in terms of the upper Dini derivative, when the Clarke subdifferential mapping of a real-valued locally Lipschitz function is a minimal weak cusco. We then use this...
On Entropy Maximization via Convex Programming (1996)
Jonathan M. Borwein, Mark A. Limber
Maximum entropy methods have received much attention in recent years. Many of the published derivations of characterizations of solutions to these infinite dimensional concave programming problems...
Closed convex Haar null sets (1995)
Borwein, Jonathan M., Fitzpatrick, Simon
If $E$ is a separable super-reflexive Banach space then every closed convex subset of $E$ with empty interior is a Haar null set.
Lipschitz functions with minimal Clarke subdifferential mappings (1995)
Borwein, Jonathan M., Moors, Warren B.
In this paper we characterise, in terms of the upper Dini derivative, when the Clarke subdifferential mapping of a real-valued locally Lipschitz function is a minimal weak$^*$ cusco. We then use this...
Convex Functions on ``Sequentially Reflexive'' Banach Spaces (1995)
Borwein, Jonathan M., Vanderwerff, Jon
There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains...
Borwein, Jonathan M., Treiman, Jay S., Zhu, Qiji J.
We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions {\sl without} constraint qualification in terms of...
Continuous Linear Monotone Operators on Banach Spaces (1995)
Bauschke, Heinz H., Borwein, Jonathan M.
The concept of a monotone operator --- which covers both linear positive semi-definite operators and subdifferentials of convex functions --- has turned out to be very powerful in various branches of...
Lipschitz Functions with Prescribed Derivatives and Subderivatives (1995)
Borwein, Jonathan M., Moors, Warren B., Wang, Shawn Xianfu
In general it is difficult to construct Lipschitz functions which are not directly built up from either convex or distance functions. One impediment to such constructions is that outside of the real...
Making Sense of Experimental Mathematics (1995)
Borwein, Jonathan M., Borwein, Peter, Girgensohn, Roland, Parnes, Sheldon
Philosophers have frequently distinguished mathematics from the physical sciences. While the sciences were constrained to fit themselves via experimentation to the `real' world, mathematicians were...
Convergence of Madelung-Like Lattice Sums (1995)
Borwein, David, Borwein, Jonathan M., Pinner, Christopher
We make a general study of the convergence properties of lattice sums, involving potentials, of the form occuring in Mathematical Chemistry and Physics. Many specific examples are studied in detail....
Distinct Differentiable Functions May Share the Same Clarke Subdifferential at All Points (1995)
Borwein, Jonathan M., Wang, Shawn Xianfu
We construct, using Zahorski's Theorem, two everywhere differentiable real Lipschitz functions differing by more than a constant but sharing the same Clarke subdifferential and the same approximate...
A survey of results relating to Giuga's conjecture on primality (1995)
Borwein, Jonathan M., Wong, Erick
This article is an expanded version of the talk given by the first author at the $25$th Anniversary Conference of the Centre de R\'echerches Math\'ematiques. In ~1950, G.~Giuga conjectured that if an...
Legendre functions and the method of random Bregman projections (1995)
Bauschke, Heinz H., Borwein, Jonathan M.
The convex feasibility problem, that is, finding a point in the intersection of finitely many closed convex sets in Euclidean space, arises in various areas of mathematics and physical sciences. It...
A Survey on Renorming and Set Convergence (1995)
Borwein, Jonathan M., Vanderwerff, Jon
This article presents a survey of some recent results relating set convergence and renorming. We focus most of our attention on norms having various Kadec--Klee properties and their interplay with...
Epigraphical and Uniform Convergence of Convex Functions (1995)
Borwein, Jonathan M., Vanderwerff, Jon
We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlev\'e-Kuratowski). We also...
Essentially Strictly Differentiable Lipschitz Functions (1995)
Borwein, Jonathan M., Moors, Warren B.
In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on Banach spaces. For example, we examine the...
The Stability Analysis of Dynamic SPECT Systems (1995)
SPECT (Single Photon Emission Computed Tomography) techniques have been applied to a wide range of medical studies. The stability of a SPECT model depends strongly upon the data collected. We show...
On Projection Algorithms for Solving Convex Feasibility Problems (1995)
Bauschke, Heinz H., Borwein, Jonathan M.
Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex...
On the Khintchine Constant (1995)
Bailey, David H., Borwein, Jonathan M., Crandall, Richard
We present rapidly converging series for the Khintchine constant and for general ``Khintchine means'' of continued fractions. We show that each of these constants can be cast in terms of an efficient...
Evaluation of Triple Euler Sums (1995)
Borwein, Jonathan M., Girgensohn, Roland
Key words: Riemann zeta function, Euler sums, polylogarithms, harmonic numbers, quantum field theory, knot theory
Lipschitz Functions with Prescribed Derivatives and Subderivatives (1995)
Jonathan M. Borwein, Warren B. Moors, Wang Xianfu
. In general it is difficult to construct Lipschitz functions which are not directly built up from either convex or distance functions. One impediment to such constructions is that outside of the...
A Survey of Examples of Convex Functions and Classifications of Normed Spaces (1995)
Jonathan M. Borwein, Jonathan M. Borwein
This paper represents a slightly extended version of the eponymous talk given at the VII Colloque Franco-Allemand d'Optimisation. My aim is to illustrate the tight connection between the...
Evaluation of Triple Euler Sums (1995)
Jonathan M. Borwein, Roland Girgensohn
. Let a; b; c be positive integers and define the so-called triple, double and single Euler sums by i(a; b; c) := 1 X x=1 x\Gamma1 X y=1 y\Gamma1 X z=1 1 x a y b z c ; i(a; b) := 1 X x=1 x\Gamma1 X...
Evaluation Of Triple Euler Sums (1995)
Jonathan Borwein Cecm, Jonathan M. Borwein, Roland Girgensohn
. Let a; b; c be positive integers and define the so-called triple, double and single Euler sums by i(a; b; c) := 1 X x=1 x\Gamma1 X y=1 y\Gamma1 X z=1 1 x a y b z c ; i(a; b) := 1 X x=1 x\Gamma1 X...
ON AN INTRIGUING INTEGRAL AND SOME SERIES RELATED TO zeta(4) (1995)
Jonathan M. Borwein, Jonathan M. Borwein
. An intriguing log-cosine integral is fully analyzed and shown to have value a rational multiple of i(4), i being the Riemann zeta function. From this we deduce by means of generating functions and...
A survey of results relating to Giuga's conjecture on primality (1995)
J. M. Borwein, E. Wong, Jonathan M. Borwein, Erick Wong
. This article is an expanded version of the talk given by the first author at the 25th Anniversary Conference of the Centre de R'echerches Math'ematiques. In 1950, G. Giuga conjectured...
Epigraphical and Uniform Convergence of Convex Functions (1995)
Jonathan M. Borwein, Jon D. Vanderwerff
. We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlev'e-Kuratowski)....
Jim Zhu, Jonathan M. Borwein, Jonathan M. Borwein, Jay S. Treiman, Jay S. Treiman, Qiji J. Zhu, ...
. We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions without constraint qualification in terms of smooth...
A Survey on Renorming and Set Convergence (1995)
Jonathan M. Borwein, Jon D. Vanderwerff
. This article presents a survey of some recent results relating set convergence and renorming. We focus most of our attention on norms having various Kadec--Klee properties and their interplay with...
Essentially Strictly Differentiable Lipschitz Functions (1995)
. In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on Banach spaces. For example, we examine the...
Legendre Functions and the Method of Random Bregman Projections (1995)
Heinz H. Bauschke, Jonathan M. Borwein
The convex feasibility problem, that is, finding a point in the intersection of finitely many closed convex sets in Euclidean space, arises in various areas of mathematics and physical sciences. It...
Convergence of Madelung-Like Lattice Sums (1995)
Jonathan M. Borwein, Jonathan M. Borwein, Christopher Pinner
We make a general study of the convergence properties of lattice sums, involving potentials, of the form occuring in Mathematical Chemistry and Physics. Many specific examples are studied in detail....
Evaluation of Triple Euler Sums (1995)
Jonathan M. Borwein, Roland Girgensohn
This paper is concerned with the discussion of sums of the type i(a; b; c) :=
Continuous Linear Monotone Operators on Banach Spaces (1995)
Heinz H. Bauschke, Jonathan M. Borwein
The concept of a monotone operator--- which covers both linear positive semi-definite operators and subdifferentials of convex functions---has turned out to be very powerful in various branches of...
A Survey of Examples of Convex Functions and Classifications of Normed Space (1994)
This paper represents a slightly extended version of the eponymous talk given at the \sl VII Colloque Franco-Allemand d'Optimisation. My aim is to illustrate the tight connection between the...
On the method of cyclic projections for convex sets in Hilbert space (1994)
Bauschke, Heinz H., Borwein, Jonathan M., Lewis, Adrian
The method of cyclic projections is a powerful tool for solving convex feasibility problems in Hilbert space. Although in many applications, in particular in the field of image reconstruction...
Characterization of Clarke subgradients among one-dimensional multifunctions (1994)
Borwein, Jonathan M., Fitzpatrick, Simon
We introduce the notions of an \elsc and an \eusc function of a real variable. The Clarke subgradients of locally Lipschitz functions on an open interval $I$ are shown to be exactly those...
Uniform Convergence for Moment Problems with Fermi-Dirac Type Entropies (1994)
Borwein, Jonathan M., Huang, W.
We consider the best entropic estimation to a unknown density $\bar{x}$, given some of its algebraic or trigonometric moments. A uniform convergence theorem is established in this paper for such...
Maximum Entropy Spectral Analysis Using Derivative Information Part 2: Computational Results (1994)
Borwein, Jonathan M., Lewis, Adrian, Limber, Mark A., Noll, Dominik
Maximum entropy density estimation, a technique for reconstructing an unknown density function on the basis of certain measurements, has applications in various areas of applied physical sciences and...
Borwein, Jonathan M., Zhu, Qiji J.
We provide a refined sensitivity analysis for finite and infinite horizon control problems where in both cases the perturbation space is $L^1$. Our underlying technique relies on a recent sequential...
Borwein, Jonathan M., Zhu, Qiji J.
In Gateaux or Bornologically differentiable spaces there are two natural natural generalizations of the concept of a Fr\' echet subderivative: In this paper we study the viscosity subderivative...
Borwein, Jonathan M., Lewis, Adrian, Noll, Dominik
Maximum entropy spectral density estimation is a technique for reconstructing an unknown density function from some known measurements by maximizing a given measure of entropy of the estimate. Here...
Banach Spaces That Admit Support Sets (1994)
Borwein, Jonathan M., Vanderwerff, Jon
It is shown that the existence of a closed convex set all of whose points are properly supported in a Banach space is equivalent to the existence of a certain type of uncountable ordered one-sided...
Giuga's conjecture on primality (1994)
Borwein, David, Borwein, Jonathan M., Borwein, Peter, Girgensohn, Roland
G. Giuga conjectured that if an integer n satisfies \sum\limits_{k=1}^{n-1} k^{n-1} \equiv -1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old...
Functional Equations and Distribution Functions (1994)
Borwein, Jonathan M., Girgensohn, Roland
We consider the functional equation \hspace*{\fill} $\ds f(t) = {1 \over b} \sum_{\nu=0}^{b-1} f\left({t-\beta_\nu \over a}\right) \quad \mbox{for all } t \in \R,$ \hfill (F) where $0
Experimental evaluation of Euler sums (1994)
Bailey, David H., Borwein, Jonathan M., Girgensohn, Roland
Euler expressed certain sums of the form \sum_{k=1}^\infty \Bigl(1 + {1 \over 2^m} + \cdots + {1 \over k^m}\Bigr) (k + 1)^{-n}\hbox{,} where m and n are positive integers, in terms of the Riemann...
Experimental evaluation of Euler sums (1994)
David H. Bailey, Jonathan M. Borwein
In response to a letter from Goldbach, Euler considered sums of the form 1
Explicit evaluation of Euler sums (1994)
Jonathan M. Borwein, Jonathan M. Borwein, Roland Girgensohn
. In response to a letter from Goldbach, Euler considered sums of the form oe h (s; t) := 1 X n=1 (1 + 1 2 s + : : : + 1 (n \Gamma 1) s ) n \Gammat ; where s and t are positive integers. As Euler...
Experimental Mathematics Vol, David H. Bailey, Jonathan M. Borwein
In response to a letter from Goldbach, Euler considered sums of the form 1 X k=1 ` 1 + 1 2 m + \Delta \Delta \Delta + 1 k m ' (k + 1) \Gamman for positive integers m and n. Euler was able to...
Functional Equations And Distribution Functions (1994)
Jonathan M. Borwein, Roland Girgensohn
. We consider the functional equation f(t) = 1 b b\Gamma1 X =0 f i t \Gamma fi a j for all t 2 IR; (F) where 0 ! a ! 1, b 2 IN n f1g and \Gamma1 = fi 0 fi 1 : : : fi b\Gamma1 = 1 are given...
Experimental Evaluation of Euler Sums (1994)
David H. Bailey, Jonathan M. Borwein, Roland Girgensohn
this paper. It involves the compound application of the Euler--Maclaurin summation formula [Abramowitz and Stegun 1972, p. 806; Atkinson 1989, p. 289; Knuth 1973, p. 108], which 20 Experimental...
Experimental Evaluation of Euler Sums (1994)
David H. Bailey, Jonathan M. Borwein, Roland Girgensohn
In response to a letter from Goldbach, Euler considered sums of the form 1 X k=1 ` 1 + 1 2 m + \Delta \Delta \Delta + 1 k m ' (k + 1) \Gamman for positive integers m and n. Euler was able to...
Locally Lipschitz Functions and Bornological Derivatives (1993)
Borwein, Jonathan M., Fabian, Marian, Vanderwerff, Jon
We study the relationships between Gateaux, Weak Hadamard and Fr\'{e}chet differentiability and their bornologies for Lipschitz and for convex functions.
Weak* sequential compactness and bornological limit derivatives (1993)
Borwein, Jonathan M., Fitzpatrick, Simon
In this note we give a self-contained account of the relationship between the sequential and topological constructions of bornological limit derivatives for locally Lipschitzian real-valued functions...
Experimental Evaluation of Euler Sums (1993)
Bailey, David H., Borwein, Jonathan M., Girgensohn, Roland
In response to a letter from Goldbach, Euler considered sums of the form \begin{eqnarray*} \sum_{k=1}^\infty \left(1 + {1 \over 2^m} + \cdots + {1 \over k^m}\right) \, (k + 1)^{-n}. \end{eqnarray*}...
Explicit evaluation of Euler sums (1993)
Borwein, David, Borwein, Jonathan M., Girgensohn, Roland
In response to a letter from Goldbach, Euler considered sums of the form $$\sh(s,t):=\sum\limits_{n=1}^\infty (1+{1 \over 2^s}+\dots+{1 \over (n-1)^s}) \, n^{-t},$$ where $s$ and $t$ are positive...
Locally Lipschitz Functions and Bornological Derivatives (1993)
Borwein, Jonathan M., Fabian, Marian, Vanderwerff, J.
We study the relationships between Gateaux, weak Hadamard and Frechet differentiability and their bornologies for Lipschitz and for convex functions. In particular, Frechet and weak Hadamard...
Dual Kadec-Klee norms and the relationships between Wijsman, slice and Mosco convergence (1993)
Borwein, Jonathan M., Vanderwerff, J.
In this paper, we completely settle several of the open questions regarding the relationships between the three most fundamental forms of set convergence. In particular, it is shown that Wijsman and...
Proximal Analysis in Smooth Spaces (1993)
Borwein, Jonathan M., Ioffe, Alexander
We provide a highly--refined sequential description of the generalized gradients of Clarke and approximate $G$--subdifferential of a lower semicontinuous extended--real--valued function defined on a...
Further arguments for slice convergence in nonreflexive spaces (1993)
Borwein, Jonathan M., Vanderwerff, Jon
It is shown that no notion of set convergence at least as strong as Wijsman convergence but not as strong as slice convergence can be preserved in superspaces. We also show that such intermediate...
Examples of Convex Functions and Classifications of Normed Spaces (1993)
Borwein, Jonathan M., Fitzpatrick, Simon, Vanderwerff, Jon
We study various properties of convex functions and their connections to the structure of the spaces on which they are defined. In particular, it is shown boundedness properties of convex functions...
A Comparison of Entropies in the Underdetermined Moment Problem (1993)
Jonathan M. Borwein, Robert K. Goodrich, Mark A. Limber
The problem of function reconstruction from a small number of accurate moments is reviewed and studied numerically. Since this problem is underdetermined, a number of selections have been proposed to...
Two-Point Step Size Gradient Methods (1988)
BARZILAI, JONATHAN, BORWEIN, JONATHAN M.
We derive two-point step sizes for the steepest-descent method by approximating the secant equation. At the cost of storage of an extra iterate and gradient, these algorithms achieve better...
Special Values Of Multiple Polylogarithms
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, Petr Lisonek, Lison Ek
. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within...
Some Remarkable Properties Of Sinc And Related Integrals
Jonathan M. Borwein, Jonathan M. Borwein
. Using Fourier transform techniques, we establish inequalities for integrals of the form Z 1 0 n Y k=1 sin(a k x) x dx: We then give quite striking closed form evaluations of such integrals and nish...
Partially Smooth Variational Principles and Applications
Jonathan M. Borwein, Jay S. Treiman, Qiji J. Zhu
. We discuss a smooth variational principle for partially smooth viscosity subdifferentials and explore its applications in nonsmooth analysis. Keywords: Smooth variational principle, fuzzy sum...
Conical Open Mapping Theorems and Regularity
Heinz Bauschke, Jonathan M. Borwein
Suppose T is a continuous linear operator between two Hilbert spaces X and Y and let K be a closed convex nonempty cone in X. We investigate the possible existence of ffi ? 0 such that ffiB Y "...
On Equivalence Of Some Basic Principles In Variational Analysis
Jonathan M. Borwein, Boris S. Mordukhovich, Yongheng Shao
. The primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization. Considering a broad...