SU\G(𝔸)/K·U
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All published works (541)

Symmetry and the Monotonicity of Certain Riemann Sums

2020-01-01
David Borwein , Jonathan M. Borwein , Brailey Sims

Gamma and Factorial in the Monthly

2018-04-12
Jonathan M. Borwein , Robert M. Corless
Since its inception in 1894, the Monthly has printed 50 articles on the Γ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won … Since its inception in 1894, the Monthly has printed 50 articles on the Γ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the Γ function in future Monthly issues.11 It is a safe bet that there will be more proofs of Stirling's formula, for instance. We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of Γ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of n! and the first few terms of a new asymptotic formula for invΓ.
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Dynamics of the Douglas-Rachford Method for Ellipses and p-Spheres

2017-10-23
Jonathan M. Borwein , Scott B. Lindstrom , Brailey Sims +2 more
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Derivatives and fast evaluation of the Tornheim zeta function

2017-04-26
Jonathan M. Borwein , Karl Dilcher

Computation and Experimental Evaluation of Mordell–Tornheim–Witten Sum Derivatives

2017-03-17
David H. Bailey , Jonathan M. Borwein
In the previous work, the present authors and others have studied Mordell–Tornheim–Witten sums and their connections with multiple-zeta values. In this note, we describe the numerical computation of derivatives at … In the previous work, the present authors and others have studied Mordell–Tornheim–Witten sums and their connections with multiple-zeta values. In this note, we describe the numerical computation of derivatives at zero of a specialization originating in a preprint by Romik and the experimental evaluation of these numerical values in terms of well-known constants.

Gamma and Factorial in the Monthly

2017-03-15
Jonathan M. Borwein , Robert M. Corless
The Monthly has published roughly fifty papers on the $\Gamma$ function or Stirling's formula. We survey those papers (discussing only our favourites in any detail) and place them in the … The Monthly has published roughly fifty papers on the $\Gamma$ function or Stirling's formula. We survey those papers (discussing only our favourites in any detail) and place them in the context of the larger mathematical literature on $\Gamma$.
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Adventures with the OEIS

2017-01-01
Jonathan M. Borwein

Reflection Methods for Inverse Problems with Applications to Protein Conformation Determination

2017-01-01
Jonathan M. Borwein , Matthew K. Tam
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Generalized Continued Logarithms and Related Continued Fractions.

2017-01-01
Jonathan M. Borwein , Kevin G. Hare , Jason G. Lynch
We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary … We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base $b$. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchine's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchine constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize past continued logarithms.

Convergence Rate Analysis for Averaged Fixed Point Iterations in Common Fixed Point Problems

2017-01-01
Jonathan M. Borwein , Guoyin Li , Matthew K. Tam
In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded H\"older regularity … In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded H\"older regularity assumption which generalizes the well-known notion of bounded linear regularity. As an application of our results, we provide a convergence rate analysis for Krasnoselskii-Mann iterations, the cyclic projection algorithm, and the Douglas-Rachford feasibility algorithm along with some variants. In the important case in which the underlying sets are convex sets described by convex polynomials in a finite dimensional space, we show that the H\"older regularity properties are automatically satisfied, from which sublinear convergence follows.
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Ergodic behaviour of a Douglas-Rachford operator away from the origin

2017-01-01
Jonathan M. Borwein , Ohad Giladi
It is shown that away from the origin, the Douglas-Rachford operator with respect to a sphere and a convex set in a Hilbert space can be approximated by a another … It is shown that away from the origin, the Douglas-Rachford operator with respect to a sphere and a convex set in a Hilbert space can be approximated by a another operator which satisfies a weak ergodic theorem. Similar results for other projection and reflection operators are also discussed.
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Gamma and Factorial in the Monthly

2017-01-01
Jonathan M. Borwein , Robert M. Corless
The Monthly has published roughly fifty papers on the $\Gamma$ function or Stirling's formula. We survey those papers (discussing only our favourites in any detail) and place them in the … The Monthly has published roughly fifty papers on the $\Gamma$ function or Stirling's formula. We survey those papers (discussing only our favourites in any detail) and place them in the context of the larger mathematical literature on $\Gamma$.
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Continued Logarithms and Associated Continued Fractions

2016-09-16
Jonathan M. Borwein , Neil J. Calkin , Scott B. Lindstrom +1 more
We investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the continued … We investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the continued logarithm to base b. We show convergence for them using equivalent forms of their corresponding continued fractions. Through numerical experimentation, we discover that, for one such formulation, the exponent terms have finite arithmetic means for almost all real numbers. This set of means, which we call the logarithmic Khintchine numbers, has a pleasing relationship with the geometric means of the corresponding continued fraction terms. While the classical Khintchine's constant is believed not to be related to any naturally occurring number, we find surprisingly that the logarithmic Khintchine numbers are elementary.

Computer Discovery and Analysis of Large Poisson Polynomials

2016-08-24
David H. Bailey , Jonathan M. Borwein , Jason Kimberley +1 more
In two earlier studies of lattice sums arising from the Poisson equation of mathematical physics, it was established that the lattice sum 1/π · ∑m, n oddcos (mπx)cos (nπy)/(m2 + … In two earlier studies of lattice sums arising from the Poisson equation of mathematical physics, it was established that the lattice sum 1/π · ∑m, n oddcos (mπx)cos (nπy)/(m2 + n2) = log A, where A is an algebraic number, and explicit minimal polynomials associated with A were computed for a few specific rational arguments x and y. Based on these results, one of us (Kimberley) conjectured a number-theoretic formula for the degree of A in the case x = y = 1/s for some integer s. These earlier studies were hampered by the enormous cost and complexity of the requisite computations. In this study, we address the Poisson polynomial problem with significantly more capable computational tools. As a result of this improved capability, we have confirmed that Kimberley’s formula holds for all integers s up to 52 (except for s = 41, 43, 47, 49, 51, which are still too costly to test), and also for s = 60 and s = 64. As far as we are aware, these computations, which employed up to 64,000-digit precision, producing polynomials with degrees up to 512 and integer coefficients up to 10229, constitute the largest successful integer relation computations performed to date. By examining the computed results, we found connections to a sequence of polynomials defined in a 2010 paper by Savin and Quarfoot. These investigations subsequently led to a proof, given in the Appendix, of Kimberley’s formula and the fact that when s is even, the polynomial is palindromic (i.e., coefficient ak = am − k, where m is the degree).

Generalisations, Examples, and Counter-examples in Analysis and Optimisation

2016-08-05
Jonathan M. Borwein

Generalized Continued Logarithms and Related Continued Fractions

2016-06-22
Jonathan M. Borwein , Kevin G. Hare , Jason G. Lynch
We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary … We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base $b$. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchine's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchine constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize past continued logarithms.
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Convex analysis in groups and semigroups: a sampler

2016-04-30
Jonathan M. Borwein , Ohad Giladi
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Densities of short uniform random walks in higher dimensions

2016-01-13
Jonathan M. Borwein , Armin Straub , Christophe Vignat
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The computation of previously inaccessible digits of π2 and Catalan’s constant (2013)

2016-01-01
David H. Bailey , Jonathan M. Borwein , A. Mattingly +1 more

Pi day is upon us again and we still do not know if pi is normal (2014)

2016-01-01
David H. Bailey , Jonathan M. Borwein

I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly (2015)

2016-01-01
Jonathan M. Borwein

The Life of Modern Homo Habilis Mathematicus: Experimental Computation and Visual Theorems

2016-01-01
Jonathan M. Borwein

Discussions of Part I Chapters

2016-01-01
Luc Trouche , Jonathan M. Borwein , John Monaghan

Ramanujan and pi (1988)

2016-01-01
Jonathan M. Borwein , Peter Borwein

The Life of π (2014)

2016-01-01
Jonathan M. Borwein , Peter Borwein

Pi, Euler numbers, and asymptotic expansions (1989)

2016-01-01
Jonathan M. Borwein , Peter Borwein , Karl Dilcher

Ergodic behaviour of a Douglas-Rachford operator away from origin

2016-01-01
Jonathan M. Borwein , Ohad Giladi
Considering the Douglas-Rachford iteration scheme with respect to a line and a sphere $\R^d, d�\geq 2$, it is shown that by smoothing the operator in a neighbourhood of the origin, … Considering the Douglas-Rachford iteration scheme with respect to a line and a sphere $\R^d, d�\geq 2$, it is shown that by smoothing the operator in a neighbourhood of the origin, it can be approximated by another operator that satisfies a weak ergodic theorem. Other iteration schemes are also discussed.

Ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi (1989)

2016-01-01
Jonathan M. Borwein , Peter Borwein , David H. Bailey

The arithmetic-geometric mean and fast computation of elementary functions (1984)

2016-01-01
Jonathan M. Borwein , Peter Borwein

Generalized Continued Logarithms and Related Continued Fractions

2016-01-01
Jonathan M. Borwein , Kevin G. Hare , Jason G. Lynch
We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary … We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base $b$. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchine's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchine constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize past continued logarithms.
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Mathematics by experiment: Plausible reasoning in the 21st Century (2008)

2016-01-01
David H. Bailey , Jonathan M. Borwein

Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem

2015-11-04
Francisco J. Aragón Artacho , Jonathan M. Borwein , Matthew K. Tam
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Some remarks on convex analysis in topological groups

2015-10-15
Jonathan M. Borwein , Ohad Giladi
We discuss some key results from convex analysis in the setting of topological groups and monoids. These include separation theorems, Krein-Milman type theorems, and minimax theorems. We discuss some key results from convex analysis in the setting of topological groups and monoids. These include separation theorems, Krein-Milman type theorems, and minimax theorems.
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Convex analysis in groups and semigroups: a sampler

2015-10-15
Jonathan M. Borwein , Ohad Giladi
We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only … We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only on vector spaces. Some examples and counter-examples are also discussed.
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A CLOSED FORM FOR THE DENSITY FUNCTIONS OF RANDOM WALKS IN ODD DIMENSIONS

2015-10-02
Jonathan M. Borwein , Corwin Sinnamon
We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space. We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space.

Maximal Monotone Inclusions and Fitzpatrick Functions

2015-09-30
Jonathan M. Borwein , Joydeep Dutta

NEAREST POINTS AND DELTA CONVEX FUNCTIONS IN BANACH SPACES

2015-09-03
Jonathan M. Borwein , Ohad Giladi
Given a closed set $C$ in a Banach space $(X,\Vert \cdot \Vert )$ , a point $x\in X$ is said to have a nearest point in $C$ if there exists … Given a closed set $C$ in a Banach space $(X,\Vert \cdot \Vert )$ , a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_{C}(x)=\Vert x-z\Vert$ , where $d_{C}$ is the distance of $x$ from $C$ . We survey the problem of studying the size of the set of points in $X$ which have nearest points in $C$ . We then turn to the topic of delta convex functions and indicate how it is related to finding nearest points.
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Densities of short uniform random walks in higher dimensions

2015-08-19
Jonathan M. Borwein , Armin Straub , Christophe Vignat
We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of … We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of up to five steps. Somewhat to our surprise, we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the two-dimensional case.

Crandall’s computation of the incomplete Gamma function and the Hurwitz zeta function, with applications to Dirichlet L-series

2015-07-14
David H. Bailey , Jonathan M. Borwein
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Computation and structure of character polylogarithms with applications to character Mordell–Tornheim–Witten sums

2015-06-03
David H. Bailey , Jonathan M. Borwein
This paper extends tools developed to study character polylogarithms. These objects are used to compute Mordell–Tornheim–Witten character sums and to explore their connections with multiple-zeta values (MZVs) and with their … This paper extends tools developed to study character polylogarithms. These objects are used to compute Mordell–Tornheim–Witten character sums and to explore their connections with multiple-zeta values (MZVs) and with their character analogues.

A Variational Approach to Lagrange Multipliers

2015-05-26
Jonathan M. Borwein , Qiji Jim Zhu

High-Precision Arithmetic in Mathematical Physics

2015-05-12
David Bailey , Jonathan M. Borwein
For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for … For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture.
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Recent Progress on Monotone Operator Theory

2015-01-01
Jonathan M. Borwein , Liangjin Yao
In this paper, we survey recent progress on the theory of maximally monotone operators in general Banach space. We also extend various of the results and leave some open questions. In this paper, we survey recent progress on the theory of maximally monotone operators in general Banach space. We also extend various of the results and leave some open questions.
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Duality and Convex Programming

2015-01-01
Jonathan M. Borwein , D. Lüke

Some remarks on convex analysis in topological groups

2015-01-01
Jonathan M. Borwein , Ohad Giladi
We discuss some key results from convex analysis in the setting of topological groups and monoids. These include separation theorems, Krein-Milman type theorems, and minimax theorems. We discuss some key results from convex analysis in the setting of topological groups and monoids. These include separation theorems, Krein-Milman type theorems, and minimax theorems.
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Densities of short uniform random walks in higher dimensions

2015-01-01
Jonathan M. Borwein , Armin Straub , Christophe Vignat
We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of … We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of up to five steps. Somewhat to our surprise, we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the two-dimensional case.
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Convex analysis in groups and semigroups: a sampler

2015-01-01
Jonathan M. Borwein , Ohad Giladi
We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only … We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only on vector spaces. Some examples and counter-examples are also discussed.
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Standing Together for Reproducibility in Large-Scale Computing: Report on reproducibility@XSEDE

2014-12-17
Doug L. James , Nancy Wilkins‐Diehr , Victoria Stodden +7 more
This is the final report on reproducibility@xsede, a one-day workshop held in conjunction with XSEDE14, the annual conference of the Extreme Science and Engineering Discovery Environment (XSEDE). The workshop's discussion-oriented … This is the final report on reproducibility@xsede, a one-day workshop held in conjunction with XSEDE14, the annual conference of the Extreme Science and Engineering Discovery Environment (XSEDE). The workshop's discussion-oriented agenda focused on reproducibility in large-scale computational research. Two important themes capture the spirit of the workshop submissions and discussions: (1) organizational stakeholders, especially supercomputer centers, are in a unique position to promote, enable, and support reproducible research; and (2) individual researchers should conduct each experiment as though someone will replicate that experiment. Participants documented numerous issues, questions, technologies, practices, and potentially promising initiatives emerging from the discussion, but also highlighted four areas of particular interest to XSEDE: (1) documentation and training that promotes reproducible research; (2) system-level tools that provide build- and run-time information at the level of the individual job; (3) the need to model best practices in research collaborations involving XSEDE staff; and (4) continued work on gateways and related technologies. In addition, an intriguing question emerged from the day's interactions: would there be value in establishing an annual award for excellence in reproducible research?
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Computation and theory of Mordell–Tornheim–Witten sums II

2014-10-30
David H. Bailey , Jonathan M. Borwein

Douglas-Rachford feasibility methods for matrix completion problems

2014-10-03
Francisco J. Aragón Artacho , Jonathan M. Borwein , Matthew Kyle Tam
In this paper, we give general recommendations for successful application of the Douglas–Rachford reflection method to convex and nonconvex real matrix completion problems. These guidelines are demonstrated by various illustrative … In this paper, we give general recommendations for successful application of the Douglas–Rachford reflection method to convex and nonconvex real matrix completion problems. These guidelines are demonstrated by various illustrative examples. doi:10.1017/S1446181114000145
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Common Coauthors

Coauthor Papers Together
Peter Borwein 89
David H. Bailey 55
David Borwein 29
Lennart Berggren 29
Liangjin Yao 25
Adrian S. Lewis 22
Armin Straub 21
Jon D. Vanderwerff 20
Xianfu Wang 19
Heinz H. Bauschke 19
Richard E. Crandall 18
Richard Kane 18
David M. Bradley 18
J. G. Wan 13
Matthew K. Tam 12
Qiji Jim Zhu 12
Wadim Zudilin 11
Warren B. Moors 10
Roland Girgensohn 10
I. J. Zucker 9
Francisco J. Aragón Artacho 9
David Broadhurst 8
M. L. Glasser 8
Ohad Giladi 8
Alf van der Poorten 7
Jeffrey Shallit 7
Matthew P. Skerritt 7
Karl Dilcher 7
Brailey Sims 7
Marián Fabian 6
R. C. McPhedran 6
Petr Lisoněk 5
D. Lüke 5
H. M. Stròjwas 5
Henry Wolkowicz 5
Kwok-Kwong Stephen Choi 5
Simon Fitzpatrick 5
P. B. Borwein 5
Simon Fitzpatrick 4
Deming Zhuang 4
O-Yeat Chan 4
Robert M. Corless 4
Frank G. Garvan 4
Keith Devlin 3
Kevin G. Hare 3
A. Mattingly 3
D. J. Broadhurst 3
Guoyin Li 3
Boris S. Mordukhovich 3
Carl Pomerance 3

Commonly Cited References

Convex Functions, Monotone Operators and Differentiability

1989-01-01
R. R. Phelps
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Geometric Functional Analysis and its Applications

1975-01-01
Richard B. Holmes

Mathematics by experiment: plausible reasoning in the 21st century

2005-01-01
Jonathan M. Borwein
Paper 17: David H. Bailey and Jonathan M. Borwein, “Pi and its friends,” and “Normality: A stubborn question,” from Mathematics by Experiment: Plausible Reasoning in the 21st Century, A. K. … Paper 17: David H. Bailey and Jonathan M. Borwein, “Pi and its friends,” and “Normality: A stubborn question,” from Mathematics by Experiment: Plausible Reasoning in the 21st Century, A. K. Peters, Natick, MA, 2nd edition, 2008. Reproduced with permission of AK Peters.

Convex Analysis and Monotone Operator Theory in Hilbert Spaces

2017-01-01
Heinz H. Bauschke , Patrick L. Combettes

Convex Analysis and Nonlinear Optimization

2006-01-01
Jonathan M. Borwein , Adrian S. Lewis

An Introduction to Classical Real Analysis.

1983-04-01
Alberto Torchinsky , Karl Stromberg
* Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index * Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index

Parallel integer relation detection: Techniques and applications

2000-07-03
David H. Bailey , D.J. Broadhurst
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet x 1 comma x 2 comma midline-horizontal-ellipsis comma x Subscript n Baseline EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet x 1 comma x 2 comma midline-horizontal-ellipsis comma x Subscript n Baseline EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{x_1, x_2, \cdots , x_n\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a vector of real numbers. An <italic>integer relation algorithm</italic> is a computational scheme to find the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript k"> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">a_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, if they exist, such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a 1 x 1 plus a 2 x 2 plus midline-horizontal-ellipsis plus a Subscript n Baseline x Subscript n Baseline equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a_1 x_1 + a_2 x_2 + \cdots + a_n x_n= 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Single- and multi-level implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory.

Sequences and Series in Banach Spaces

1984-01-01
Joseph Diestel

Handbook of Mathematical Functions

1966-02-01
Donald A. McQuarrie
First Page First Page

Fast Multiple-Precision Evaluation of Elementary Functions

1976-04-01
Richard P. Brent
Let ƒ( x ) be one of the usual elementary functions (exp, log, artan, sin, cosh, etc.), and let M ( n ) be the number of single-precision operations required … Let ƒ( x ) be one of the usual elementary functions (exp, log, artan, sin, cosh, etc.), and let M ( n ) be the number of single-precision operations required to multiply n -bit integers. It is shown that ƒ( x ) can be evaluated, with relative error Ο (2 - n ), in Ο ( M ( n )log ( n )) operations as n → ∞, for any floating-point number x (with an n -bit fraction) in a suitable finite interval. From the Schönhage-Strassen bound on M ( n ), it follows that an n -bit approximation to ƒ( x ) may be evaluated in Ο ( n log 2 ( n ) log log( n )) operations. Special cases include the evaluation of constants such as π, e , and e π . The algorithms depend on the theory of elliptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations.
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From Hahn-Banach to Monotonicity

2008-01-01
Stephen Simons

Maximality of sums of two maximal monotone operators in general Banach space

2007-09-12
Jonathan M. Borwein
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 … 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 space under various natural domain and transversality conditions.

Geometry of Banach Spaces-Selected Topics

1975-01-01
Joseph Diestel
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Fifty years of maximal monotonicity

2010-02-13
Jonathan M. Borwein

Convex Analysis and Monotone Operator Theory in Hilbert Spaces

2011-01-01
Heinz H. Bauschke , Patrick L. Combettes

On the variational principle

1974-08-01
Ivar Ekeland

Continuity and Differentiability Properties of Convex Operators

1982-05-01
Jonathan M. Borwein
This paper provides some theorems on the generic differentiability of convex operators between topological vector spaces. These results are applied to give new conditions for subgradients of convex operators to … This paper provides some theorems on the generic differentiability of convex operators between topological vector spaces. These results are applied to give new conditions for subgradients of convex operators to be non-empty and well behaved. Limiting examples are given.

Analysis of PSLQ, an integer relation finding algorithm

1999-01-01
Helaman Ferguson , David A. Bailey , Steve Arno
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper K"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {K}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be either the real, complex, or quaternion … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper K"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {K}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be either the real, complex, or quaternion number system and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper O left-parenthesis double-struck upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {O}}({\mathbb {K}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the corresponding integers. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x equals left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x = (x_{1}, \dots , x_{n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a vector in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper K Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathbb {K}}^{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The vector <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has an integer relation if there exists a vector <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals left-parenthesis m 1 comma ellipsis comma m Subscript n Baseline right-parenthesis element-of double-struck upper O left-parenthesis double-struck upper K right-parenthesis Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">m = (m_{1}, \dots , m_{n}) \in {\mathbb {O}}({\mathbb {K}})^{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m not-equals 0"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m \ne 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m 1 x 1 plus m 2 x 2 plus ellipsis plus m Subscript n Baseline x Subscript n Baseline equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m_{1} x_{1} + m_{2} x_{2} + \ldots + m_{n} x_{n} = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper we define the parameterized integer relation construction algorithm PSLQ<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis tau right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\tau )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where the parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi>τ<!-- τ --></mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be freely chosen in a certain interval. Beginning with an arbitrary vector <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x equals left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis element-of double-struck upper K Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">x = (x_{1}, \dots , x_{n}) \in {\mathbb {K}}^{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, iterations of PSLQ<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis tau right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\tau )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will produce lower bounds on the norm of any possible relation for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Thus PSLQ<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis tau right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\tau )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be used to prove that there are no relations for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of norm less than a given size. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript x"> <mml:semantics> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">M_{x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the smallest norm of any relation for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For the real and complex case and each fixed parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi>τ<!-- τ --></mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a certain interval, we prove that PSLQ<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis tau right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\tau )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> constructs a relation in less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis n cubed plus n squared log upper M Subscript x Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>log</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(n^{3} + n^{2} \log M_{x})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> iterations.
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Ordered topological vector spaces

1967-01-01
Anthony L. Peressini

Minimax and Monotonicity

1998-01-01
Stephen Simons
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A Comparison of Three High-Precision Quadrature Schemes

2005-01-01
David H. Bailey , Karthik Jeyabalan , Xiaoye Sherry Li
Abstract The authors have implemented three numerical quadrature schemes, using the Arbitrary Precision (ARPREC) software package. The objective here is a quadrature facility that can efficiently evaluate to very high … Abstract The authors have implemented three numerical quadrature schemes, using the Arbitrary Precision (ARPREC) software package. The objective here is a quadrature facility that can efficiently evaluate to very high precision a large class of integrals typical of those encountered in experimental mathematics, relying on a minimum of a priori information regarding the function to be integrated. Such a facility is useful, for example, to permit the experimental identification of definite integrals based on their numerical values. The performance and accuracy of these three quadrature schemes are compared using a suite of 15 integrals, ranging from continuous, well-behaved functions on finite intervals to functions with infinite derivatives and blow-up singularities at endpoints, as well as several integrals on an infinite interval. In results using 412-digit arithmetic, we achieve at least 400-digit accuracy, using two of the programs, for all problems except one highly oscillatory function on an infinite interval. Similar results were obtained using 1,012-digit arithmetic. Keywords: Primary 65D30Numerical quadraturenumerical integrationarbitrary precision
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Convex functions: constructions, characterizations and counterexamples

2011-06-01
Jonathan M. Borwein , Jon D. Vanderwerff
Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from … Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.

On the maximality of sums of nonlinear monotone operators

1970-01-01
R. T. Rockafellar
R. T. ROCKAFELLAR(')= {*?+x% I xf e Tx(x), xt e T2(x)}.If Tx and F2 are maximal, it does not necessarily follow, however, that F», + T2 is maximal-some sort of … R. T. ROCKAFELLAR(')= {*?+x% I xf e Tx(x), xt e T2(x)}.If Tx and F2 are maximal, it does not necessarily follow, however, that F», + T2 is maximal-some sort of condition is needed, since for example the graph of Tx + T2 can even be empty (as happens when D(Tx) n D(T2)= 0).The problem of determining conditions under which Tx + T2 is maximal turns out to be of fundamental importance in the theory of monotone operators.Results in this direction have been proved by Lescarret [9] and Browder [5], [6], [7].The strongest result which is known at present is : Theorem (Browder [6],[7]).Let X be reflexive, and let Tx and T2 be monotone operators from X to X*. Suppose that Tx is maximal, D(T2) = X, T2 is single-valued
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1. Conjugate Duality and Optimization

1974-01-01
R. T. Rockafellar
1. The role of convexity and duality. In most situations involving optimization there is a great deal of mathematical structure to work with. However, in order to get to the … 1. The role of convexity and duality. In most situations involving optimization there is a great deal of mathematical structure to work with. However, in order to get to the fundamentals, it is convenient for us to begin by considering only the following abstract optimization problem: minimize ƒ(x) for x ∈ C, where C is a subset of a real linear space X, and ƒ:C → [−∞, +∞].

On Projection Algorithms for Solving Convex Feasibility Problems

1996-09-01
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 feasibility problems continue to … 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 feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated. Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given.
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A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions

1987-01-01
Jonathan M. Borwein , David Preiss
We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function … We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function on a space with a Gâteaux (weak Hadamard, Fréchet) smooth renorm is densely Gâteaux (weak Hadamard, Fréchet) differentiable. Our technique relies on a more powerful analogue of Ekeland’s variational principle in which the function is perturbed by a quadratic-like function. This "smooth" variational principle has very broad applicability in problems of nonsmooth analysis.
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Set-Valued Mappings and Enlargements of Monotone Operators

2007-11-14
Regina S. Burachik , Alfredo N. Iusem

Convex Functions, Monotone Operators and Differentiability

1989-01-01
R. R. Phelps
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On the maximal monotonicity of subdifferential mappings

1970-04-01
R. T. Rockafellar
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A Lagrange multiplier theorem and a sandwich theorem for convex relations.

1981-12-01
Jonathan M. Borwein
We formulate and prove various separation principles for convex relations taking values in an order complete vector space. These principles subsume the standard ones. We formulate and prove various separation principles for convex relations taking values in an order complete vector space. These principles subsume the standard ones.
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Regularity and Stability for Convex Multivalued Functions

1976-05-01
Stephen M. Robinson
Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points. These properties are applied to develop a perturbation theory for convex … Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points. These properties are applied to develop a perturbation theory for convex inequalities and to extend results on the continuity of convex functions.

Local boundedness of nonlinear, monotone operators.

1969-12-01
R. T. Rockafellar

Convex Analysis and Nonlinear Optimization

2000-01-01
Jonathan M. Borwein , Adrian S. Lewis

Central Binomial Sums, Multiple Clausen Values, and Zeta Values

2001-01-01
Jonathan M. Borwein , D.J. Broadhurst , Joel Kamnitzer
Abstract We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of nonalternating sums leads … Abstract We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of nonalternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experi mental resuIts involving polylogarithms in the golden ratio. Keywords: binomial sumsmultiple zeta valueslog-sine integralsClausen's functionmultiple Clausen valuespolylogarithmsApery sums
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Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization

2002-07-01
Heinz H. Bauschke , Patrick L. Combettes , D. Lüke
The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily … The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily on the pioneering work of Gerchberg, Saxton, and Fienup. Despite the widespread use of the algorithms proposed by these three researchers, current mathematical theory cannot explain their remarkable success. Nevertheless, great insight can be gained into the behavior, the shortcomings, and the performance of these algorithms from their possible counterparts in convex optimization theory. An important step in this direction was made two decades ago when the error reduction algorithm was identified as a nonconvex alternating projection algorithm. Our purpose is to formulate the phase retrieval problem with mathematical care and to establish new connections between well-established numerical phase retrieval schemes and classical convex optimization methods. Specifically, it is shown that Fienup's basic input-output algorithm corresponds to Dykstra's algorithm and that Fienup's hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. We provide a theoretical framework to better understand and, potentially, to improve existing phase recovery algorithms.
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Special values of multiple polylogarithms

2000-10-11
Jonathan M. Borwein , David M. Bradley , D.J. Broadhurst +1 more
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 the past … 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 the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.

Finding best approximation pairs relative to two closed convex sets in Hilbert spaces

2004-04-01
Heinz H. Bauschke , Patrick L. Combettes , D. Lüke

Convex analysis and nonlinear optimization : theory and examples

2000-01-01
Jonathan M. Borwein , Adrian S. Lewis

On the convergence of von Neumann's alternating projection algorithm for two sets

1993-01-01
Heinz H. Bauschke , Jonathan M. Borwein

Approximate subdifferentials and applications 3: the metric theory

1989-06-01
A. D. Ioffe

A Dual Approach to Multidimensional $L_p$ Spectral Estimation Problems

1988-07-01
Aharon Ben‐Tal , Jonathan M. Borwein , Marc Teboulle
A complete duality theory is presented for the multidimensional $L_p$ spectral estimation problem. The authors use a new constraint qualification (BWCQ) for infinite-dimensional convex programs with linear type constraints recently … A complete duality theory is presented for the multidimensional $L_p$ spectral estimation problem. The authors use a new constraint qualification (BWCQ) for infinite-dimensional convex programs with linear type constraints recently introduced in [Borwein and Wolkowicz, Math. Programming, 35 (1986), pp. 83–96]. This allows direct derivation of the explicit optimal solution of the problem as presented in [Goodrich and Steinhardt, SIAM J. Appl. Math., 46 (1986), pp. 417–426], and establishment of the existence of a simple and computationally tractable unconstrained Lagrangian dual problem. Moreover, the results illustrate that (BWCQ) is more appropriate to spectral estimation problems than the traditional Slater condition (which may only be applied after transformation of the problem into an $L_p$ space [Goodrich and Steinhardt, op. cit.] and which therefore yields only necessary conditions).
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Duality Relationships for Entropy-Like Minimization Problems

1991-03-01
Jonathan M. Borwein , Adrian S. Lewis
This paper considers the minimization of a convex integral functional over the positive cone of an $L_p $ space, subject to a finite number of linear equality constraints. Such problems … This paper considers the minimization of a convex integral functional over the positive cone of an $L_p $ space, subject to a finite number of linear equality constraints. Such problems arise in spectral estimation, where the bjective function is often entropy-like, and in constrained approximation. The Lagrangian dual problem is finite-dimensional and unconstrained. Under a quasi-interior constraint qualification, the primal and dual values are equal, with dual attainment. Examples show the primal value may not be attained. Conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum. These conditions are satisfied in many examples.
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Integrals of the Ising class

2006-09-19
David H. Bailey , Jonathan M. Borwein , Richard E. Crandall
From an experimental-mathematical perspective we analyzeIsing-class integrals. Our experimental results involvedextreme-precision, multidimensional quadrature on intricate integrands;thus, highly parallel computation was required. From an experimental-mathematical perspective we analyzeIsing-class integrals. Our experimental results involvedextreme-precision, multidimensional quadrature on intricate integrands;thus, highly parallel computation was required.
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Evaluations of $k$-fold Euler/Zagier sums: a compendium of results for arbitrary $k$

1996-10-31
Jonathan M. Borwein , David M. Bradley , D. J. Broadhurst
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 … Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble results for Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of arbitrary depth, including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations, we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking, we have sketched derivations of all results that have so far been proved.
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Minimal Convex Uscos and Monotone Operators on Small Sets

1991-06-01
Jonathan M. Borwein , Simon Fitzpatrick , Petar S. Kenderov
Abstract We generalize the generic single-valuedness and continuity of monotone operators defined on open subsets of Banach spaces of class (S) and Asplund spaces to monotone operators defined on convex … Abstract We generalize the generic single-valuedness and continuity of monotone operators defined on open subsets of Banach spaces of class (S) and Asplund spaces to monotone operators defined on convex subsets of such spaces which may even fail to have non-support points. This yields differentiability theorems for convex Lipschitzian functions on such sets. From a result about minimal convex uscos which are densely single-valued we obtain generic differentiability results for certain Lipschitzian realvalued functions.
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Essentially Smooth Lipschitz Functions

1997-10-01
Jonathan M. Borwein , Warren B. Moors

Handbook of Mathematical Functions

2018-02-06

Stability and regular points of inequality systems

1986-01-01
Jonathan M. Borwein

Convergence of convex sets and of solutions of variational inequalities

1969-10-01
Umberto Mosco

A Treatise on the Theory of Bessel Functions.

1923-09-01
Matthew Porter , G. N. Watson
1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions … 1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions of Bessel functions 8. Bessel functions of large order 9. Polynomials associated with Bessel functions 10. Functions associated with Bessel functions 11. Addition theorems 12. Definite integrals 13. Infinitive integrals 14. Multiple integrals 15. The zeros of Bessel functions 16. Neumann series and Lommel's functions of two variables 17. Kapteyn series 18. Series of Fourier-Bessel and Dini 19. Schlomlich series 20. The tabulation of Bessel functions Tables of Bessel functions Bibliography Indices.