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Part I


Some Remarks on Applied Mathematics

Encourage students to focus on fundamentals. In mathematics this means algebra, analysis and geometry, but it is also desirable for an applied mathematician to have a broad education in science and engineering, including the perspectives found in computer science. It is impossible to predict what kinds of mathematics will be especially useful for the problems yet to be encountered. Being prepared to read about the work of others and having a first hand knowledge of many good examples is about the best one can do. One of the very positive side effects associated with the increasing number of scientifically and mathematically literate people in the world is that there are now readable books and survey papers covering a vast array of scientific work. It is essential that students who aspire to be model builders and problem solvers should have the tools necessary to make use of this resource. This point is well illustrated by recent developments in mathematical physics involving the use of some of the very latest ideas in geometry to provide a language suitable for unifying field theories.
Roger Brockett

Mathematics is a Profession

It makes little sense to discuss future directions in applied mathematics without emphasizing similar opportunities and challenges in mathematics as a whole. A few decades ago, someone who should have known better wrote an article entitled “Applied mathematics is bad mathematics.” Nowadays, clearer thinking sometimes prevails. On one hand, it is clear that applied mathematicians have to be very good mathematicians. The reason for this is simple; applied mathematics tends to be interdisciplinary and one can’t do great interdisciplinary research without having core competency in a discipline in the first place. On the other hand, with the end of the Cold Wax and the evaporation of the longstanding rationale for national support of research, a grander debate between the relative merits of basic and applied research has subsumed any serious discussion about pure versus applied mathematics. Finally, the world is changing and mathematics, as one of the ways humans describe the world, will change as well. Henceforth, I won’t distinguish between pure and applied mathematics.
Christopher I. Byrnes

Comments on Applied Mathematics

Applied Mathematics is currently going in all directions, following many new applications as they arise. The advent of computers enables the numerical implementation of very abstract ideas and theories from core mathematics. For this reason the good applied mathematician must have a strong core mathematics background. He/she will find it extremely helpful to possess the skills and ability for abstraction. Such abstraction is often used to successfully solve applied problems, or to relate one set of problems to another.
Avner Friedman

Towards an Applied Mathematics for Computer Science

If you go up to a conventional engineer—someone who designs bridges or aeroplanes or concert halls or communication systems—and ask her what mathematical theorems she uses as part of the design process, you will, with probability 1, get a long list. For instance, a communications engineer might start with the sampling theorem of Shannon and Nyquist: to reconstruct a band limited signal from samples, it is necessary to sample at least at twice the highest frequency.
Jeremy Gunawardena

Infomercial for Applied Mathematics

According to Galileo, “Mathematics is the language in which God wrote the Universe”. As a living language, mathematics is still developing. Of course, any language develops to express new meanings. For example, English is no longer the same as it was in the time of Shakespeare or Chaucer. Now we have MTV, soundbites, factoids, psychobabble, technospeak and other types of jargon. Language usage helps promote language development. This is more than just a new word or a felicitous turn of phrase. Sometimes new language usage is the sign of an emerging new paradigm. Conversely, sometimes a new paradigm requires a whole new way of using language. Galileo’s analogy is apt — mathematics has many parallels with language. In mathematics there are parallels to linguists, translators, playwrights, poets, songwriters, storytellers, etc. (There are certainly parallels in mathematics to nonfiction and fiction!) Further, sometimes in mathematics, as in language, “the medium is the message”.
Darryl Holm

On Research in Mathematical Economics

I come to applied mathematics via economic theory, and it is difficult for me to obtain a global view, sufficiently removed from my current research interests, to single out the most promising directions for the future. It may be more productive to point the reader to the handbooks referenced below where the subject, as contemporaneously conceived, is collectively surveyed. I ought also point to a recent (1991) issue of The Economic Journal in which a distinguished subset of economists speculate on the shape of the discipline of economics in the next hundred years.
M. Ali Khan

Applied Mathematics in the Computer and Communications Industry

In the early ‘70’s, when I was a graduate student at Berkeley, I had the impression that mathematical problems of interest to industry were simply not of interest to mathematicians. I had read Hardy’s A Mathematician’s Apology and believed that pure mathematics ought not to be tainted by applications. But as my career progressed I found that I wanted to work on problems that bear more closely to science and engineering — partly in the hope that such work would be useful to people, but also that the connections would make the mathematics richer, more fun and more interesting. This was one of the reasons that I moved from a senior university position to IBM.
Brian Marcus

Trends in Applied Mathematics

Despite these uncertain times, talented young scientists can be encouraged to enter the complex field called applied mathematics. There has never been a greater need and use of mathematical ideas in science and technology than now. Indeed, one defining aspect of applied mathematics is to link mathematical ideas and techniques to the other sciences and to use them in that context in a productive way.
Jerrold E. Marsden

Applied Mathematics as an Interdisciplinary Subject

It is always a bit difficult to predict the future and it is particularly difficult to predict the future of applied mathematics because it is hard to even define what is meant by applied mathematics. It is even more difficult to attempt to lay out a future that will fit all departments and all of the practitioners of the art of applied mathematics. At any meeting of applied mathematicians or engineers we see many different aspects of applied mathematics and many ways of practicing the art. I will simply try to explain what we are doing at Texas Tech and present this approach as one possibility for the future of applied mathematics. I don’t believe that everything that we are doing at TTU will fit every other group in the country, but some of what we are doing could serve as a paradigm for any applied mathematics department in the United States.
Clyde F. Martin

Panel Discussion on Future Directions in Applied Mathematics

At the actual panel discussion, I was the moderator and as I am not an applied mathematician, I did little but introduce the panelists and keep the discussion going, the latter requiring very little effort on my part. I propose to use this forum to comment on an issue that was raised during the discussion and that all mathematics departments will face in the coming years.
Laurence R. Taylor

Part II


Feedback Stabilization of Relative Equilibria for Mechanical Systems with Symmetry

This paper is an outgrowth of the work of Bloch, Krishnaprasad, Marsden and Sánchez de Alvarez [1992], where a feedback control that stabilizes intermediate axis rigid body rotation using an internal rotor was found. Stabilization is determined by use of the energy-Casimir (Arnold) method. In the present paper we show that this feedback controlled system can be written as the Euler-Lagrange equations for a modified Lagrangian: a velocity shift associated with a change of connection turns the free (unforced) equations into the feedback controlled equations. We also show how stabilization of the inverted pendulum on a cart can be achieved in an analogous way. We provide a general systematic construction of such controlled Lagrangians.
The basic idea is to modify the kinetic energy of the free Lagrangian using a generalization of the Kaluza-Klein construction in such a way that the extra terms obtained in the Euler-Lagrange equations can be identified with control forces. The fact that the controlled system is Lagrangian by construction enables one to make use of energy techniques for a stability analysis. Once stabilization is achieved in a mechanical context, one can establish asymptotic stabilization by the further addition of dissipative controls. The methods here can be combined with symmetry breaking controls obtained by modifying the potential energy and also can be used for tracking.
Anthony M. Bloch, Jerrold E. Marsden, Gloria Sánchez de Alvarez

Oscillatory Descent for Function Minimization

Algorithms for minimizing a function based on continuous descent methods following the gradient relative to some riemannian metric suffer from the twin problems of converging to local, rather than global, minima and giving little indication about an approximate answer until the process has nearly converged. Simulated annealing addresses these problems through the introduction of stochastic terms, however the rate of convergence associated with the method can be unacceptably slow. In this paper we discuss a modification of simulated annealing which approaches a minimum through a damped oscillatory path. The characteristics of the path, including its tendency to be irregular, reflect the properties of the function being minimized. The oscillatory algorithm involves both a temperature and coupling parameters, giving it considerable flexibility.
Roger Brockett

On the Well-Posedness of the Rational Covariance Extension Problem

In this paper, we give a new proof of the solution of the rational covariance extension problem, an interpolation problem with historical roots in potential theory, and with recent application in speech synthesis, spectral estimation, stochastic systems theory, and systems identification. The heart of this problem is to parameterize, in useful systems theoretical terms, all rational, (strictly) positive real functions having a specified window of Laurent coefficients and a bounded degree. In the early 1980’s, Georgiou used degree theory to show, for any fixed “Laurent window”, that to each Schur polynomial there exists, in an intuitive systems-theoretic manner, a solution of the rational covariance extension problem. He also conjectured that this solution would be unique, so that the space of Schur polynomials would parameterize the solution set in a very useful form. In a recent paper, this problem was solved as a corollary to a theorem concerning the global geometry of rational, positive real functions. This corollary also asserts that the solutions are analytic functions of the Schur polynomials.
After giving an historical motivation and a survey of the rational covariance extension problem, we give a proof that the rational covariance extension problem is well-posed in the sense of Hadamard, i.e a proof of existence, uniqueness and continuity of solutions with respect to the problem data. While analytic dependence on the problem data is stronger than continuity, this proof is much more streamlined and also applies to a broader class of nonlinear problems. The paper concludes with a discussion of open problems.
Christopher I. Byrnes, Henry J. Landau, Anders Lindquist

Singular Limits in Fluid Mechanics

In this paper I would like to describe some of the mathematical problems encountered in the study of incompressible fluid turbulence. The equations of motion are the Navier-Stokes equations.
$$ \left( {{\partial _t} + u \cdot \nabla - v\Delta } \right)u + \nabla p = f. $$
Peter Constantin

Singularities and Defects in Patterns Far from Threshold

This is a report on recent work that examines the behaviour of a class of nonlinear partial differential equations which axe considered to provide a good qualitative model of significant aspects of pattern formation and defects in a diverse range of physical systems. This work was done in collaboration with C. Bowman, R. Indik, A. C. Newell at the University of Arizona and with T. Passot at the Observatoire de Nice. The details of the formal and numerical results mentioned in this introduction will appear in [15] and details of the analytical results mentioned in the last section will appear in [9].
N. M. Ercolani

Mathematical Modeling and Simulation for Applications of Fluid Flow in Porous Media

Mathematical models have been widely used to understand, predict, or optimize many complex physical processes. Here we address the need for developing models to understand the fate and transport of groundwater contaminants and to design in situ remediation strategies.
Three basic problem areas must be addressed in the modeling and simulation of the flow of groundwater contamination. One must first obtain effective model equations to describe the complex fluid/fluid and fluid/rock interactions that control the transport of contaminants in groundwater. This includes the problem of obtaining accurate reservoir descriptions at various length scales, modeling the effects of this heterogeneity of the porous medium, and developing effective parameters in the governing models that describe the effects of the heterogeneities in the reservoir simulators. Next, one must develop accurate discretization techniques that retain the important physical properties of the continuous models. Finally, one should develop efficient numerical solution algorithms that utilize the potential of the emerging computing architectures. We will discuss advances in these areas.
Richard E. Ewing

On Loeb Measure Spaces and their Significance for Non-Cooperative Game Theory

In this expository paper, Loeb measure spaces are constructed on the basis of sequences, and shown to satisfy many useful properties, including some regularity properties of correspondences involving distribution and integration. It is argued that Loeb measure spaces can be effectively and systematically used for the analysis of game-theoretic situations in which “strategic negligibility” and/or “diffuse-ness” of information are substantive and essential issues. Positive results are presented, and the failure of analogous results for identical models based on Lebesgue measure spaces is illustrated by several examples. It is also pointed out that the requirement of Lebesgue measurability, by going against the non-cooperative element in the situation being modelled, is partly responsible for this failure.
M. Ali Khan, Yeneng Sun

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov.
The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators.
Jerrold E. Marsden, Jeffrey M. Wendlandt
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