Functional programming, the use and evaluation of functions as a programming paradigm, has a long and rich history in programming languages. Lisp came about in the search for a convenient language for representing mathematical concepts in programs. It borrowed from the lambda calculus of the logician Alonzo Church. More recent languages have in turn embraced many aspects of Lisp – in addition to Lisp's offspring such as Scheme and Haskell, you will find elements of functional constructs in Java, Python, Ruby, and Perl. Mathematica itself has clear bloodlines to Lisp, including the ability to operate on data structures such as lists as single objects and in its representation of mathematical properties through rules. Being able to express ideas in science, mathematics, and engineering in a language that naturally mirrors those fields is made much easier by the integration of these tools.

Functions not only offer a familiar paradigm to those representing ideas in science, mathematics, and engineering, they provide a consistent and efficient mechanism for computation and programming. In Mathematica, unlike many other languages, functions are considered “first class” objects, meaning they can be used as arguments to other functions, they can be returned as values, and they can be part of many other kinds of data objects such as arrays. In addition, you can create and use functions at runtime, that is, when you evaluate an expression. This functional style of programming distinguishes Mathematica from traditional procedural languages like C and Fortran.

A very brief historical note

Smale's 7th problem is the computational version of an old problem dating back to Thomson [30] and Tammes [29], see Whyte's early review [32] for its history, namely, the sensible distribution of points in the two- dimensional sphere. In Whyte's paper different possible definitions of “well-distributed points in the sphere” are suggested:

1. 1. Points which maximise the product of their mutual distances (called elliptic Fekete points after [14]).

2. 2. Points which minimise the sum of the inverse of their mutual distances (Thomson's problem), and more generally which minimise some sum of potentials which depend on the mutual distances (like Riesz potentials).

3. 3. Points which maximise the least distance between any pair.

4. 4. Points which are the center of the optimal packing problem, that is, the problem of finding the smallest radius of a sphere such that one can place on its surface k non-overlapping circles of a given radius.

This beautiful problem is terribly challenging! A first shocking result by Leech [19] showed that even though the set of N particles on the sphere which are critical points for the problem in item (2) for every possible potential can be completely described, this description is not enough to solve the problem for any particular potential. Namely, solving problem (2) for some particular potential may be completely meaningless for solving problem (2) for another, different potential.

Abstract

We provide a notion of finite element system, that enables the construction of spaces of differential forms, which can be used for the numerical solution of variationally posed partial differential equations. Within this framework, we introduce a form of upwinding, with the aim of stabilizing methods for the purposes of computational fluid dynamics, in the vanishing viscosity regime.

Foreword

I am deeply honored to have received the first Stephen Smale prize from the Society for the Foundations of Computational Mathematics.

I want to thank the jury for deciding, in what I understand was a difficult weighing process, to tip the balance in my favor. The tiny margins that similarly enable the Gömböc to find its way to equilibrium, give me equal pleasure to contemplate. It's a beautiful prize trophy.

It is a great joy to receive a prize that celebrates the unity of mathematics. I hope it will draw attention to the satisfaction there can be, in combining theoretical musings with potent applications. Differential geometry, which infuses most of my work, is a good example of a subject that defies perceived boundaries, equally appealing to craftsmen of various trades.

As I was entering the subject, rumors that Smale could turn spheres inside out without pinching, were among the legends that gave it a sense of surprise and mystery. I also remember reading about Turing machines built on rings other than ℤ/2ℤ, which, together with parallelism and quantum computing, convinced me that the foundations of our subject were still in the making.

The Society for the Foundations of Computational Mathematics supports and promotes fundamental research in computational mathematics and its applications, interpreted in the broadest sense. It fosters interaction among mathematics, computer science and other areas of computational science through its conferences, workshops and publications. As part of this endeavour to promote research across a wide spectrum of subjects concerned with computation, the Society brings together leading researchers working in diverse fields. Major conferences of the Society have been held in Park City (1995), Rio de Janeiro (1997), Oxford (1999), Minneapolis (2002), Santander (2005), Hong Kong (2008), and Budapest (2011). The next conference is expected to be held in 2014. More information about FoCM is available at its website http://focmsociety. org.

The conference in Budapest on July 4 – 14, 2011, was attended by some 450 scientists. FoCM conferences follow a set pattern: mornings are devoted to plenary talks, while in the afternoon the conference divides into a number of workshops, each devoted to a different theme within the broad theme of foundations of computational mathematics. This structure allows for a very high standard of presentation, while affording endless opportunities for cross-fertilization and communication across subject boundaries. Workshops at the Budapest conference were held in the following nineteen fields:

1. – Approximation theory

2. – Asymptotic analysis and high oscillation

3. – Computational algebraic geometry

4. – Computational dynamics

5. – Computational harmonic analysis, image and signal processing

6. – Computational number theory

7. – Continuous optimization

8. – Flocking, swarming, and control of distributed systems

9. – Foundations of numerical PDEs

10. – Geometric integration and computational mechanics

11. – Information-based complexity

12. – Learning theory

13. – Multiresolution and adaptivity in numerical PDEs

14. – Numerical linear algebra

15. – Random matrix theory, computations & applications

16. – Real-number complexity

17. – Special functions and orthogonal polynomials

18. – Stochastic computation

19. – Symbolic analysis

Abstract

In this paper we survey parts of group theory, with emphasis on structures that are important in design and analysis of numerical algorithms and in software design. In particular, we provide an extensive introduction to Fourier analysis on locally compact abelian groups, and point towards applications of this theory in computational mathematics. Fourier analysis on non-commutative groups, with applications, is discussed more briefly. In the final part of the paper we provide an introduction to multivariate Chebyshev polynomials. These are constructed by a kaleidoscope of mirrors acting upon an abelian group, and have recently been applied in numerical Clenshaw-Curtis type numerical quadrature and in spectral element solution of partial differential equations, based on triangular and simplicial subdivisions of the domain.

Introduction

Group theory is the mathematical language of symmetry. As a mature branch of mathematics, with roots going almost two centuries back, it has evolved into a highly technical discipline. Many texts on group theory and representation theory are not readily accessible to applied mathematicians and computational scientists, and the relevance of group theoretical techniques in computational mathematics is not widely recognized.

Nevertheless, it is our conviction that knowledge of central parts of group theory and harmonic analysis on groups is invaluable also for computational scientists, both as a language to unify, analyze and generalize computational algorithms and also as an organizing principle of mathematical software construction.

Abstract

This article highlights a coherent series of algorithmic tools to compute and work with algebraic and differential invariants.

Introduction

Group actions are ubiquitous in mathematics and arise in diverse fields of science and engineering, including physics, mechanics, and computer vision. Invariants of these group actions typically arise to reduce a problem or to decide if two objects, geometric or abstract, are obtained from one another by the action of a group element. [8, 9, 10, 11, 13, 15, 17, 39, 40, 42, 43, 45, 46, 52, 59] are a few recent references of applications. Both algebraic and differential invariant theories have become in recent years the subject of computational mathematics [13, 14, 17, 40, 60]. Algebraic invariant theory studies polynomial or rational invariants of algebraic group actions [18, 22, 23, 54]. A typical example is the discriminant of a quadratic binary form as an invariant of an action of the special linear group. The differential invariants appearing in differential geometry are smooth functions on a jet bundle that are invariant under a prolonged action of a Lie group [4, 16, 34, 48, 53]. A typical example is the curvature of a plane curve, invariant under the action of the group of the isometries on the plane. Curvature is not a rational function, but an algebraic function. Concomitantly the classical Lie groups are linear algebraic groups.

This article reviews results of [14, 28, 29, 30, 31, 32] in order to show their coherence in addressing algorithmically an algebraic description of the differential invariants of a group action. In the first section we show how to compute the rational invariants of a group action and give concrete expressions to a set of algebraic invariants that are of fundamental importance in the differential context. The second section addresses the question of finite representation of differential invariants with invariant derivations, a set of generating differential invariants and the differential relationships among them. In the last section we describe the algebraic structure that better serves the representation of differential invariants.

Abstract

We investigate the dual role of convection on the large time behavior of the 3D incompressible Navier-Stokes equations. On the one hand, convection is responsible for generating small scales dynamically. On the other hand, convection may play a stabilizing role in potentially depleting nonlinear vortex stretching for certain flow geometry. Our study is centered around a 3D model that was recently proposed by Hou and Lei in [23] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. This model is derived by neglecting the convection term from the reformulated Navier-Stokes equations and shares many properties with the 3D incompressible Navier-Stokes equations. In this paper, we review some of the recent progress in studying the singularity formation of this 3D model and how convection may destroy the mechanism that leads to singularity formation in the 3D model.

Introduction

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data with finite energy is one of the seven Millennium problems posted by the Clay Mathematical Institute [16]. This problem is challenging because the vortex stretching nonlinearity is super-critical for the 3D Navier-Stokes equation. Conventional functional analysis based on energy type estimates fails to provide a definite answer to this problem. Global regularity results are obtained only under certain smallness assumptions on the initial data or the solution itself. Due to the incompressibility condition, the convection term seems to be neutrally stable if one tries to estimate the Lp (1 < p≤∞) norm of the vorticity field. As a result, the main effort has been to use the diffusion term to control the nonlinear vortex stretching term by diffusion without making use of the convection term explicitly.

Abstract

This article reviews some of the phenomena and theoretical results on the long-time energy behaviour of continuous and discretized oscillatory systems that can be explained by modulated Fourier expansions: longtime preservation of total and oscillatory energies in oscillatory Hamiltonian systems and their numerical discretizations, near-conservation of energy and angular momentum of symmetric multistep methods for celestial mechanics, metastable energy strata in nonlinear wave equations. We describe what modulated Fourier expansions are and what they are good for.

Introduction

As a new analytical tool developed in the past decade, modulated Fourier expansions have been found useful to explain various long-time phenomena in both continuous and discretized oscillatory Hamiltonian systems, ordinary differential equations as well as partial differential equations. In addition, modulated Fourier expansions have turned out useful as a numerical approximation method in oscillatory systems.

In this review paper we first show some long-time phenomena in oscillatory systems, then give theoretical results that explain these phenomena, and finally outline the basics of modulated Fourier expansions with which these results are proved.

Abstract

Sage is a large free open source software package aimed at all areas of mathematical computation. Hundreds of people have contributed to the project, which has steadily grown in popularity since 2005. This paper describes the motivation for starting Sage and the history of the project.

Introduction

The goal of the Sage project (http://www.sagemath.org) is to create a viable free open source alternative to Magma, Maple™, Mathematica®, and MATLAB®, which are the most popular non-free closed source mathematical software systems. Magma is (by far) the most advanced non-free system for structured abstract algebraic computation, Mathematica and Maple are popular and highly developed systems that shine at symbolic manipulation, and MATLAB is the most popular system for applied numerical mathematics. Together there are over 3,000 employees working at the companies that produce the four Ma's listed above, which take in over a hundred million dollars of revenue annually.

By a viable free alternative to the Ma's, we mean a system that will have the important mathematical features of each Ma, with comparable speed. It will have 2d and 3d graphics, an interactive graphical user interface, and documentation, including books, papers, school and college curriculum materials, etc. A single alternative to all of the Ma's is not necessarily a drop-in replacement for any of the Ma's; in particular, it need not run programs written in the custom languages of those systems. Thus an alternative may be philosophically different than the open source system Octave, which understands the MATLAB source language and attempts to implement the entire MATLAB library.

Introduction

With sensors becoming ubiquitous, there is an increasing interest in mining the data from these sensors as the data are being collected. This analysis of streaming data, or data streams, is presenting new challenges to analysis algorithms. The size of the data can be massive, especially when the sensors number in the thousands and the data are sampled at a high frequency. The data can be non-stationary, with statistics that vary over time. Real-time analysis is often required, either to avoid untoward incidents or to understand an interesting phenomenon better. These factors make the analysis of streaming data, whether from sensors or other sources, very data- and compute-intensive. One possible approach to making this analysis tractable is to identify the important data streams to focus on them. This chapter describes the different ways in which this can be done, given that what makes a stream important varies from problem to problem and can often change with time in a single problem. The following illustrate these techniques by applying them to data from a real problem and discuss the challenges faced in this emerging field of streaming data analysis.

This chapter is organized as follows: first, I define what is meant by streaming data and use examples from practical problems to discuss the challenges in the analysis of these data. Next, I describe the two main approaches used to handle the streaming nature of the data – the sliding window approach and the forgetting factor approach.