Author

# John Smillie

Other affiliations: University of Illinois at Chicago, University of Warwick

Bio: John Smillie is an academic researcher from Cornell University. The author has contributed to research in topics: Polynomial & Locus (mathematics). The author has an hindex of 27, co-authored 56 publications receiving 2827 citations. Previous affiliations of John Smillie include University of Illinois at Chicago & University of Warwick.

##### Papers published on a yearly basis

##### Papers

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TL;DR: On considere un systeme billard de 2 objets de masses m 1 and m 2, on montre que pour un ensemble dense de paires (m 1,m 2 ) ce systeme est ergodique.

Abstract: On considere un systeme billard de 2 objets de masses m 1 et m 2 . On montre que pour un ensemble dense de paires (m 1 ,m 2 ) ce systeme est ergodique

314 citations

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231 citations

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TL;DR: In this paper, the algebraic degree of the polynomial is defined as the maximum of the degrees of the coordinate functions of the holomorphic polynomials of the system.

Abstract: The simplest holomorphic dynamical systems which display interesting behavior are the polynomial maps of C The dynamical study of these maps began with Fatou and Julia in the s and is currently a very active area of research If we are interested in studying invertible holomorphic dynamical systems then the simplest examples with interesting behavior are probably the polynomial di eomorphisms ofC These are maps f C C such that the coordinate functions of f and f are holomorphic polynomials For polynomial maps of C the algebraic degree of the polynomial is a useful dynamical invariant In particular the only dynamically interesting maps are those with degree d greater than one For polynomial di eomorphisms we can de ne the algebraic degree to be the maximum of the degrees of the coordinate functions This is not however a conjugacy invariant Friedland and Milnor FM gave an alternative de nition of a positive integer deg f which is more natural from a dynamical point of view If deg f then deg f coincides with the minimal algebraic degree of a di eomorphism in the conjugacy class of f As in the case of polynomial maps of C the polynomial di eomorphisms f with deg f are rather uninteresting We will make the standing assumption that deg f For a polynomial map of C the point at in nity is an attractor Thus the recurrent dynamics can take place only on the set K consisting of bounded orbits A normal families argument shows that there is no expansion on the interior of K so chaotic dynamics can occur only on J K This set is called the Julia set and plays a major role in the study of polynomial maps For di eomorphisms of C each of the objectsK and J has three analogs Correspond ing to the set K in one dimension we have the sets K resp K consisting of the points whose orbits are bounded in forward resp backward time and the set K K K consisting of points with bounded total orbits Each of these sets is invariant and K is compact As is in the one dimensional case recurrence can occur only on the set K Corresponding to the set J in dimension one we have the sets J K and the set J J J Each of these sets is invariant and J is compact A normal families argu ment shows that there is no forward instability in the interior of K and no backward instability in the interior of K Thus chaotic dynamics that is recurrent dynamics with instability in both forward and backward time can occur only on the set J The techniques that Fatou and Julia used in one dimension are based on Montel s theory of normal families and do not readily generalize to higher dimensions A di erent tool appears in the work of Brolin Br who made use of the theory of the logarithmic

228 citations

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205 citations

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Open University

^{1}TL;DR: Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis and will be of interest to applied mathematicians, engineers, and computer scientists.

Abstract: Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

2,586 citations

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TL;DR: In this paper, the authors presented a proof of the classification of surface automorphisms from the point of view of Teichmüller theory, generalizing Teichmiiller's theorem by allowing the Riemann surface to vary as well as the map.

Abstract: This article was widely circulated as a preprint, about 12 years ago. At that time the Bulletin did not accept research announcements, and after a couple of attempts to publish it, I gave up, and the preprint did not find a home. I very soon saw that there were many ramifications of this theory, and I talked extensively about it in a number of places. One year I devoted my graduate course to this theory, and notes of Bill Floyd and Michael Handel from that course were circulated for a while. The participants in a seminar at Orsay in 1976-1977 went over this material, and wrote a volume [FLP] including some original material as well. Another good general reference, from a somewhat different point of view, is a set of notes of lectures by A. Casson, taken by S. Bleiler [CasBlei]. There are by now several alternative ways to develop the classification of diffeomorphisms of surfaces described here. At the time I originally discovered the classification of diffeomorphism of surfaces, I was unfamiliar with two bodies of mathematics which were quite relevant: first, Riemann surfaces, quasiconformal maps and Teichmiiller's theory; and second, Nielsen's theory of the dynamical behavior of surface at infinity, and his near-understanding of geodesic laminations. After hearing about the classification of surface automorphisms from the point of view of the space of measured foliations, Lipman Bers [Bersl] developed a proof of the classification of surface automorphisms from the point of view of Teichmüller theory, generalizing Teichmiiller's theorem by allowing the Riemann surface to vary as well as the map. Dennis Sullivan first told me of some neglected articles by Nielsen which might be relevant. This point of view has been discussed by R. Miller, J. Gilman, M. Handel and me. The analogous theory, of measured laminations and 2-dimensional train tracks in three dimensions, has been considerable development. This has been applied to reinterpret some of Haken's work, to classify incompressible surfaces in particular classes of 3-manifolds in papers by me, Hatcher, Floyd, Oertel and others in various combinations. Shalen, Morgan, Culler and others have developed the related theory of groups acting on trees, and its relation to measured laminations, to define and analyze compactifications of representation spaces of groups in SL(2, C) and SO(n, 1); this has many interesting applications, including the theory of incompressible surfaces in 3-manifolds.

1,290 citations

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01 Feb 1994

TL;DR: Details of Matrix Eigenvalue Methods, including Double Bracket Isospectral Flows, and Singular Value Decomposition are revealed.

Abstract: Contents: Matrix Eigenvalue Methods.- Double Bracket Isospectral Flows.- Singular Value Decomposition.- Linear Programming.- Approximation and Control.- Balanced Matrix Factorizations.- Invariant Theory and System Balancing.- Balancing via Gradient Flows.- Sensitivity Optimization.- Linear Algebra.- Dynamical Systems.- Global Analysis.

800 citations

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TL;DR: The activation dynamics of nets are considered from a rigorous mathematical point of view and an extension of the Cohen-Grossberg convergence theorem is proved for certain nets with nonsymmetric weight matrices.

601 citations