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Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in algorithms, numerical analysis and global analysis. Education and career Smale was born in Flint, Michigan and entered the University of Michigan in 1948. Initially, he was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. Smale obtained his Bachelor of Science degree in 1952. Despite his grades, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flint, Michigan
Flint is the largest city in Genesee County, Michigan, United States, and its county seat. Located along the Flint River (Michigan), Flint River northwest of Detroit, it is a principal city within the Central Michigan, Mid Michigan region. Flint had a population of 81,252 at the 2020 United States census, 2020 census, making it the List of municipalities in Michigan, 12th-most populous city in Michigan. The Flint metropolitan area is located entirely within Genesee County and is the Michigan statistical areas, third-largest metro area in Michigan, with a population of 406,892 in 2020. The city was Incorporated town, incorporated in 1855. Flint was founded as a Administrative divisions of Michigan#Villages, village by fur trader Jacob Smith (fur trader), Jacob Smith in 1819 and became a major lumbering area on the historic Saginaw Trail during the 19th century. From the late 19th century to the mid-20th century, the city was a leading manufacturer of carriages and later Car, auto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nancy Kopell
Nancy Jane Kopell (born November 8, 1942, New York City) is an American mathematician and professor at Boston University. She is co-director of the Center for Computational Neuroscience and Neural Technology (CompNet). She organized and directs the Cognitive Rhythms Collaborative (CRC). Kopell received her B.A. from Cornell University in 1963 and her Ph.D. from Berkeley in 1967. She held visiting positions at the Centre National de la Recherche Scientifique in France (1970), MIT (1975, 1976–1977), and the California Institute of Technology (1976). The focus of her research is the field of applied biomathematics and includes use of mathematical models to analyze the physiological mechanisms of brain dynamics. The techniques Kopell uses include extensions of invariant manifold theory, averaging theory, and geometric methods for singularly perturbed equations. From the peak of her career in 1990, she has contributed to over 200 published research articles in the field of biomathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse–Smale System
In dynamical systems theory, an area of pure mathematics, a Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic set, hyperbolic periodic orbits and satisfying a transversality condition on the stable manifold, stable and unstable manifolds. Morse–Smale systems are structural stability, structurally stable and form one of the simplest and best studied classes of smooth dynamical systems. They are named after Marston Morse, the creator of the Morse theory, and Stephen Smale, who emphasized their importance for smooth dynamics and algebraic topology. Definition Consider a smooth and Vector field#Complete vector fields, complete vector field ''X'' defined on a compact differentiable manifold ''M'' with dimension ''n''. The flow defined by this vector field is a Morse-Smale system if # ''X'' has only a finite number of singular points (i.e. equilibrium points of the flow), and all of them ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smale's Problems
Smale's problems is a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list of Hilbert's problems that had been published at the beginning of the 20th century. Table of problems In later versions, Smale also listed three additional problems, "that don't seem important enough to merit a place on our main list, but it would still be nice to solve them:" # Mean value problem #Is the three-sphere a minimal set ( Gottschalk's conjecture)? #Is an Anosov diffeomorphism of a compact manifold topologically the same as the Lie group model of John Franks? See also * Millennium Prize Problems * Simon problems * Taniyama's problems * Hilbert's problems Hilbert's problems are 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smale Conjecture
The Smale conjecture, named after Stephen Smale, is the statement that the diffeomorphism group of the 3-sphere has the homotopy-type of its isometry group, the orthogonal group O(4). It was proved in 1983 by Allen Hatcher. Equivalent statements There are several equivalent statements of the Smale conjecture. One is that the component of the unknot in the space of smooth embeddings of the circle in 3-space has the homotopy-type of the round circles, equivalently, O(3). Interestingly, this statement is not equivalent to the generalized Smale Conjecture, in higher dimensions. Another equivalent statement is that the group of diffeomorphisms of the 3-ball which restrict to the identity on the boundary is contractible. Yet another equivalent statement is that the space of constant-curvature Riemann metrics on the 3-sphere is contractible. Higher dimensions The (false) statement that the inclusion O(n+1) \to \text(S^n) is a weak equivalence for all n is sometimes meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horseshoe Map
In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator. The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe. Most points eventually leave the square under the action of the map. They go to the side caps where they will, under iteration, converge to a fixed point in one of the caps. The points that remain in the square under repeated iteration form a fractal set and are part of the invariant set of the map. The squishing, stretching and folding of the horseshoe map are typical of chaotic systems, but not necessary or even sufficient. In the horseshoe map, the squeezing and stretching are uniform. They compensate each oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homoclinic Orbit
In the study of dynamical systems, a homoclinic orbit is a path through phase space which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium. It is a heteroclinic orbit–a path between any two equilibrium points–in which the endpoints are one and the same. Consider the continuous dynamical system described by the ordinary differential equation :\dot x=f(x) Suppose there is an equilibrium at x=x_0, then a solution \Phi(t) is a homoclinic orbit if :\Phi(t)\rightarrow x_0\quad \mathrm\quad t\rightarrow\pm\infty If the phase space has three or more dimensions, then it is important to consider the topology of the unstable manifold of the saddle point. The figures show two cases. First, when the stable manifold is topologically a cylinder, and secondly, when the unstable manifold is topologically a Möbius strip; in this case the homoclinic orbit is called ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-cobordism
In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : M \hookrightarrow W \quad\mbox\quad N \hookrightarrow W are homotopy equivalences. The ''h''-cobordism theorem gives sufficient conditions for an ''h''-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder ''M'' × , 1 Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds. The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture. Background Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Handle Decomposition
In mathematics, a handle decomposition of an ''m''-manifold ''M'' is a union \emptyset = M_ \subset M_0 \subset M_1 \subset M_2 \subset \dots \subset M_ \subset M_m = M where each M_i is obtained from M_ by the attaching of i-handles. A handle decomposition is to a manifold what a CW-decomposition is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds. Thus an ''i''-handle is the smooth analogue of an ''i''-cell. Handle decompositions of manifolds arise naturally via Morse theory. The modification of handle structures is closely linked to Cerf theory. Motivation Consider the standard CW-decomposition of the ''n''-sphere, with one zero cell and a single ''n''-cell. From the point of view of smooth manifolds, this is a degenerate decomposition of the sphere, as there is no natural way to see the smooth structure of S^n from the eyes of this decomposition— ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Poincaré Conjecture
In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold that is a homotopy sphere a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is :Every homotopy sphere (a closed ''n''-manifold which is homotopy equivalent to the ''n''-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic, PL-isomorphic, or diffeomorphic) to the standard ''n''-sphere. The name derives from the Poincaré conjecture, which was made for (topological or PL) manifolds of dimension 3, where being a homotopy sphere is equivalent to being simply connected and closed. The generalized Poincaré conjecture is known to be true or false in a number of instances, due to the work of many distinguished topologists, including the Fields medal awardees John Milnor, Steve Smale, Mic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Shub
Michael Ira Shub (born August 17, 1943) is an American mathematician who has done research into dynamical systems and the complexity of real number algorithms. Career 1967: Ph.D. and early career In 1967, Shub obtained his Ph.D. degree at the University of California, Berkeley with a thesis entitled ''Endomorphisms of Compact Differentiable Manifolds''. In his Ph.D. thesis, he introduced the notion of expanding maps, which gave the first examples of structurally stable strange attractors. His advisor was Stephen Smale. From 1967 to 1985, he worked at Brandeis University, the University of California, Santa Cruz and the Queens College at the City University of New York. In 1974, he proposed the Entropy Conjecture, an open problem in dynamical systems, which was proved by Yosef Yomdin for C^\infty mappings in 1987. 1985–2004: IBM research From 1985 to 2004, he joined IBM's Thomas J. Watson Research Center. In 1987, Shub published his book ''Global Stability of Dynamical Syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siavash Shahshahani
Siavash Mirshams Shahshahani (; born May 31, 1942) is an Iranian mathematician. He is a professor of mathematics and head of Mathematical Sciences Department at Sharif University of Technology. He headed up the IRNIC registry for the .ir ccTLD until his retirement from that position in late 2008. He has also served as a director of APTLD (the Asia Pacific Top Level Domain Association) between 2007 and his retirement from that position in February 2009. Education Shahshahani completed his PhD under the supervision of Stephen Smale at the University of California at Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a public land-grant research university in Berkeley, California, United States. Founded in 1868 and named after the Anglo-Irish philosopher George Berkele ... in 1969. He has since devoted a substantial part of his career to mathematical education. Books * * * External links Siavash ShahshahaniHomepage at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
