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## Über dieses Buch

The research of Jonathan Borwein has had a profound impact on optimization, functional analysis, operations research, mathematical programming, number theory, and experimental mathematics. Having authored more than a dozen books and more than 300 publications, Jonathan Borwein is one of the most productive Canadian mathematicians ever. His research spans pure, applied, and computational mathematics as well as high performance computing, and continues to have an enormous impact: MathSciNet lists more than 2500 citations by more than 1250 authors, and Borwein is one of the 250 most cited mathematicians of the period 1980-1999. He has served the Canadian Mathematics Community through his presidency (2000–02) as well as his 15 years of editing the CMS book series.

Jonathan Borwein’s vision and initiative have been crucial in initiating and developing several institutions that provide support for researchers with a wide range of scientific interests. A few notable examples include the Centre for Experimental and Constructive Mathematics and the IRMACS Centre at Simon Fraser University, the Dalhousie Distributed Research Institute at Dalhousie University, the Western Canada Research Grid, and the Centre for Computer Assisted Research Mathematics and its Applications, University of Newcastle.

The workshops that were held over the years in Dr. Borwein’s honor attracted high-caliber scientists from a wide range of mathematical fields. This present volume is an outgrowth of the workshop on ‘Computational and Analytical Mathematics’ held in May 2011 in celebration of Dr. Borwein’s 60th Birthday. The collection contains various state-of-the-art research manuscripts and surveys presenting contributions that have risen from the conference, and is an excellent opportunity to survey state-of-the-art research and discuss promising research directions and approaches.

## Inhaltsverzeichnis

### Chapter 1. Normal Numbers and Pseudorandom Generators

For an integer

b

≥ 2 a real number

α

is

b -normal

if, for all

m

> 0, every

m

-long string of digits in the base-

b

expansion of

α

appears, in the limit, with frequency

b

m

. Although almost all reals in [0, 1] are

b

-normal for every

b

, it has been rather difficult to exhibit explicit examples. No results whatsoever are known, one way or the other, for the class of “natural” mathematical constants, such as

$$\pi,\,e,\,\sqrt{2}$$

and log2. In this paper, we summarize some previous normality results for a certain class of explicit reals and then show that a specific member of this class, while provably 2-normal, is provably

not

6-normal. We then show that a practical and reasonably effective pseudorandom number generator can be defined based on the binary digits of this constant and conclude by sketching out some directions for further research.

David H. Bailey, Jonathan M. Borwein

### Chapter 2. New Demiclosedness Principles for (Firmly) Nonexpansive Operators

The demiclosedness principle is one of the key tools in nonlinear analysis and fixed point theory. In this note, this principle is extended and made more flexible by two mutually orthogonal affine subspaces. Versions for finitely many (firmly) nonexpansive operators are presented. As an application, a simple proof of the weak convergence of the Douglas-Rachford splitting algorithm is provided.

Heinz H. Bauschke

### Chapter 3. Champernowne’s Number, Strong Normality, and the X Chromosome

Champernowne’s number is the best-known example of a normal number, but its digits are far from random. The sequence of nucleotides in the human X chromosome appears nonrandom in a similar way. We give a new asymptotic test of pseudorandomness, based on the law of the iterated logarithm; we call this new criterion “strong normality.” We show that almost all numbers are strongly normal and that strong normality implies normality. However, Champernowne’s number is not strongly normal. We adapt a method of Sierpiński to construct an example of a strongly normal number.

Adrian Belshaw, Peter Borwein

### Chapter 4. Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems

We present optimality conditions for bilevel optimal control problems where the upper level is a scalar optimal control problem to be solved by a leader and the lower level is a multiobjective convex optimal control problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing amongst efficient optimal controls. We deal with the so-called optimistic case, when the followers are assumed to choose the best choice for the leader amongst their best responses, as well with the so-called pessimistic case, when the best response chosen by the followers can be the worst choice for the leader. This paper continues the research initiated in Bonnel (SIAM J. Control Optim.

50

(6), 3224–3241, 2012) where existence results for these problems have been obtained.

Henri Bonnel, Jacqueline Morgan

### Chapter 5. Monotone Operators Without Enlargements

Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the sum of two maximally monotone linear relations. We also present a new proof of the maximality of the sum of a maximally monotone linear relation and a normal cone operator when the domain of the linear relation intersects the interior of the domain of the normal cone.

Jonathan M. Borwein, Regina S. Burachik, Liangjin Yao

### Chapter 6. A Brøndsted–Rockafellar Theorem for Diagonal Subdifferential Operators

In this note we give a Brøndsted–Rockafellar Theorem for diagonal subdifferential operators in Banach spaces. To this end we apply an Ekeland-type variational principle for monotone bifunctions.

Radu Ioan Boţ, Ernö Robert Csetnek

### Chapter 7. A q-Analog of Euler’s Reduction Formula for the Double Zeta Function

The double zeta function is a function of two arguments defined by a double Dirichlet series and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed-form evaluation of the double zeta function in terms of values of the Riemann zeta function, in the case when the two arguments are positive integers with opposite parity. Here, we establish a

q

-analog of Euler’s evaluation. That is, we state and prove a 1-parameter generalization that reduces to Euler’s evaluation in the limit as the parameter

q

tends to 1.

David M. Bradley, Xia Zhou

### Chapter 8. Fast Computation of Bernoulli, Tangent and Secant Numbers

We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or Euler) numbers. In particular, we give asymptotically fast algorithms for computing the first

n

such numbers

O

(

n

2

(log

n

)

2 +

o

(1)

). We also give very short in-place algorithms for computing the first

n

Tangent or Secant numbers in

O

(

n

2

) integer operations. These algorithms are extremely simple and fast for moderate values of

n

. They are faster and use less space than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama and Tanigawa (for Bernoulli numbers).

Richard P. Brent, David Harvey

### Chapter 9. Monotone Operator Methods for Nash Equilibria in Non-potential Games

We observe that a significant class of Nash equilibrium problems in non-potential games can be associated with monotone inclusion problems. We propose splitting techniques to solve such problems and establish their convergence. Applications to generalized Nash equilibria, zero-sum games, and cyclic proximation problems are demonstrated.

Luis M. Briceño-Arias, Patrick L. Combettes

### Chapter 10. Compactness, Optimality, and Risk

This is a survey about one of the most important achievements in optimization in Banach space theory, namely, James’ weak compactness theorem, its relatives, and its applications. We present here a good number of topics related to James’ weak compactness theorem and try to keep the technicalities needed as simple as possible: Simons’ inequality is our preferred tool. Besides the expected applications to measures of weak noncompactness, compactness with respect to boundaries, size of sets of norm-attaining functionals, etc., we also exhibit other very recent developments in the area. In particular we deal with functions and their level sets to study a new Simons’ inequality on unbounded sets that appear as the epigraph of some fixed function

f

. Applications to variational problems for

f

and to risk measures associated with its Fenchel conjugate

f

are studied.

B. Cascales, J. Orihuela, M. Ruiz Galán

### Chapter 11. Logarithmic and Complex Constant Term Identities

In recent work on the representation theory of vertex algebras related to the Virasoro minimal models

M

(2,

p

), Adamović and Milas discovered logarithmic analogues of (special cases of) the famous Dyson and Morris constant term identities. In this paper we show how the identities of Adamović and Milas arise naturally by differentiating as-yet-conjectural complex analogues of the constant term identities of Dyson and Morris. We also discuss the existence of complex and logarithmic constant term identities for arbitrary root systems, and in particular prove such identities for the root system G

2

.

Tom Chappell, Alain Lascoux, S. Ole Warnaar, Wadim Zudilin

### Chapter 12. Preprocessing and Regularization for Degenerate Semidefinite Programs

This paper presents a backward stable preprocessing technique for (nearly) ill-posed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification (SCQ), the existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on

primal-dual interior-point, p-d i-p

, methods. These algorithms require the SCQ for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, well-posedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the SCQ fails or

nearly

fails. Our backward stable preprocessing technique is based on applying the Borwein–Wolkowicz facial reduction process to find a finite number,

k

, of

rank-revealing orthogonal rotations

of the problem. After an appropriate truncation, this results in a smaller, well-posed,

nearby

problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. The case

k

= 1 is of particular interest and is characterized by strict complementarity of an auxiliary problem.

Yuen-Lam Cheung, Simon Schurr, Henry Wolkowicz

### Chapter 13. The Largest Roots of the Mandelbrot Polynomials

This paper gives some details of the experimental discovery and partial proof of a simple asymptotic development for the largest magnitude roots of the Mandelbrot polynomials defined by

p

0

(

z

) = 1 and

$$p_{n+1}(z) = zp_{n}^{2}(z) + 1$$

.

Robert M. Corless, Piers W. Lawrence

### Chapter 14. On the Fractal Distribution of Brain Synapses

Herein we present mathematical ideas for assessing the fractal character of distributions of brain synapses. Remarkably, laboratory data are now available in the form of actual three-dimensional coordinates for millions of mouse-brain synapses (courtesy of Smithlab at Stanford Medical School). We analyze synapse datasets in regard to statistical moments and fractal measures. It is found that moments do not behave as if the distributions are uniformly random, and this observation can be quantified. Accordingly, we also find that the measured fractal dimension of each of two synapse datasets is 2.8 ± 0.05. Moreover, we are able to detect actual neural layers by generating what we call probagrams, paramegrams, and fractagrams—these are surfaces one of whose support axes is the

y

-depth (into the brain sample). Even the measured fractal dimension is evidently neural-layer dependent.

Richard Crandall

### Chapter 15. Visible Points in Convex Sets and Best Approximation

The concept of a

visible point

of a convex set relative to a given point is introduced. A number of basic properties of such visible point sets are developed. In particular, it is shown that this concept is useful in the study of best approximation, and it also seems to have potential value in the study of robotics.

Frank Deutsch, Hein Hundal, Ludmil Zikatanov

### Chapter 16. On Derivative Criteria for Metric Regularity

We give a simple self-contained proof of the equality which links directly the graphical derivative and coderivative criteria for metric regularity. Then we present a sharper form of the criterion for strong metric regularity involving the paratingent derivative.

Asen L. Dontchev, Hélène Frankowska

### Chapter 17. Five Classes of Monotone Linear Relations and Operators

The relationships among five classes of monotonicity, namely 3

-, 3-cyclic, strictly, para-, and maximal monotonicity, are explored for linear operators and linear relations in Hilbert space. Where classes overlap, examples are given; otherwise their relationships are noted for linear operators in

$${\mathbb{R}}^{2}$$

,

$${\mathbb{R}}^{3}$$

, and general Hilbert spaces. Along the way, some results for linear relations are obtained.

Mclean R. Edwards

### Chapter 18. Upper Semicontinuity of Duality and Preduality Mappings

In their paper studying Hausdorff weak upper semicontinuity of duality and preduality mappings on the dual of a Banach space, Godefroy and Indumathi related these by an interesting geometrical property. This property actually characterises Hausdorff upper semicontinuity of the preduality mapping. When the duality mapping is Hausdorff upper semicontinuous with weakly compact image, we investigate how this same property persists with natural embedding into higher duals.

J. R. Giles

### Chapter 19. Convexity and Variational Analysis

The paper focuses on the interplay between methods of general variational analysis, on the one hand, and convex analysis, on the other (with a special emphasis on the efficiency of the latter), in treating problems that come from the general variational analysis when applied in substantial or virtual presence of convexity. We discuss a number of relating problems: error bounds, metric regularity, linear openness, and stability as well as first-order necessary optimality conditions in semi-infinite programming.

A. D. Ioffe

### Chapter 20. Generic Existence of Solutions and Generic Well-Posedness of Optimization Problems

We exhibit a large class of topological spaces in which the generic attainability of the infimum by the bounded continuous perturbations of a lower semicontinuous function implies generic well-posedness of the perturbed optimization problems. The class consists of spaces which admit a winning strategy for one of the players in a certain topological game and contains, in particular, all metrizable spaces and all spaces that are homeomorphic to a Borel subset of a compact space.

P. S. Kenderov, J. P. Revalski

### Chapter 21. Legendre Functions Whose Gradients Map Convex Sets to Convex Sets

We show that a differentiable function on a Banach space of dimension at least two has an affine gradient provided the gradient is continuous and one-to-one and maps convex sets to convex sets.

Alexander Knecht, Jon Vanderwerff

### Chapter 22. On the Convergence of Iteration Processes for Semigroups of Nonlinear Mappings in Banach Spaces

Let

C

be a bounded, closed, convex subset of a uniformly convex Banach space

X

. We investigate the convergence of the generalized Krasnosel’skii-Mann and Ishikawa iteration processes to common fixed points of pointwise Lipschitzian semigroups of nonlinear mappings

T

t

:

C

C

. Each of

T

t

is assumed to be pointwise Lipschitzian, that is, there exists a family of functions

α

t

:

C

→ [0,

) such that

$$\|T_{t}(x) - T_{t}(y)\| \leq \alpha _{t}(x)\|x - y\|$$

for

x

,

y

C

.

W. M. Kozlowski, Brailey Sims

### Chapter 23. Techniques and Open Questions in Computational Convex Analysis

We present several techniques that take advantage of convexity and use optimal computational geometry algorithms to build fast (log-linear or linear) time algorithms in computational convex analysis. The techniques that have strong potential to be reused include: monotonicity of the argmax and injecting convexity to use that monotonicity, Lipschitzness of the argmin, exploiting various formulas in convex analysis, using a graph data structure to vectorize computation, and building a parametrization of the graph. We also point out the potential for parallelization. The techniques can be used as a check list on open problems to find an efficient algorithm. Finally, we list several currently open questions in computational convex analysis with links to computational geometry.

Yves Lucet

### Chapter 24. Existence and Approximation of Fixed Points of Right Bregman Nonexpansive Operators

We study the existence and approximation of fixed points of right Bregman nonexpansive operators in reflexive Banach space. We present, in particular, necessary and sufficient conditions for the existence of fixed points and an implicit scheme for approximating them.

Victoria Martín-Márquez, Simeon Reich, Shoham Sabach

### Chapter 25. Primal Lower Nice Functions and Their Moreau Envelopes

This paper studies two equivalent definitions of primal lower nice functions and some subdifferential characterizations of such functions. Various regularity properties of the associated Moreau envelopes and proximal mappings are also provided.

Marc Mazade, Lionel Thibault

### Chapter 26. Bundle Method for Non-Convex Minimization with Inexact Subgradients and Function Values

We discuss a bundle method to minimize locally Lipschitz functions which are both nonconvex and nonsmooth. We analyze situations where only inexact subgradients or function values are available. For suitable classes of such non-smooth functions we prove convergence of our algorithm to approximate critical points.

Dominikus Noll

### Chapter 27. Convergence of Linesearch and Trust-Region Methods Using the Kurdyka–Łojasiewicz Inequality

We discuss backtracking linesearch and trust-region descent algorithms for unconstrained optimization and prove convergence to a critical point if the objective is of class

C

1

and satisfies the Kurdyka–Łojasiewicz condition. For linesearch we investigate in which way an intelligent management memorizing the stepsize should be organized. For trust-regions we present a new curvature-based acceptance test which ensures convergence under rather weak assumptions.

Dominikus Noll, Aude Rondepierre

### Chapter 28. Strong Duality in Conic Linear Programming: Facial Reduction and Extended Duals

The facial reduction algorithm (FRA) of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program

P

$$\displaystyle{ \sup \,\{\,\langle c,x\rangle \,\vert \,Ax \leq _{K}b\,\} }$$

in the absence of any constraint qualification. The FRA solves a sequence of auxiliary optimization problems to obtain such a dual. Ramana’s dual is applicable when (P) is a semidefinite program (SDP) and is an explicit SDP itself. Ramana, Tunçel, and Wolkowicz showed that these approaches are closely related; in particular, they proved the correctness of Ramana’s dual using certificates from a facial reduction algorithm. Here we give a simple and self-contained exposition of facial reduction, of extended duals, and generalize Ramana’s dual:

We state a simple FRA and prove its correctness.

Building on this algorithm we construct a family of extended duals when

K

is a

nice

cone. This class of cones includes the semidefinite cone and other important cones.

Gábor Pataki

### Chapter 29. Towards a New Era in Subdifferential Analysis?

We give some new attention to the foundations of nonsmooth analysis. We endeavour to delineate the common features of usual subdifferentials. In particular, we stress calculus rules and properties linked with order. Our objective is to give the possibility of using subdifferentials without dealing with specific constructions.

Jean-Paul Penot

### Chapter 30. Modular Equations and Lattice Sums

We highlight modular equations due to Ramanujan and Somos and use them to prove new relations between lattice sums and hypergeometric functions. We also discuss progress towards solving Boyd’s Mahler measure conjectures. Finally, we conjecture a new formula for

L

(

E

, 2) of conductor 17 elliptic curves.

Mathew Rogers, Boonrod Yuttanan

### Chapter 31. An Epigraph-Based Approach to Sensitivity Analysis in Set-Valued Optimization

In this paper, we obtain estimates for the contingent and adjacent derivatives of the epigraph of the marginal multifunction in parametric set-valued optimization. These estimates generalize some sensitivity results from scalar-valued optimization and provide new information in the setting of multiobjective nonlinear programming.

Douglas E. Ward, Stephen E. Wright
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