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Differential Equations, Dynamical Systems, and Linear Algebra
This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by theExpand
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Finding the Homology of Submanifolds with High Confidence from Random Samples
We show how to “learn” the homology of the submanifold with high confidence from random samples and provide learning-theoretic complexity bounds. Expand
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Differential Equations, Dynamical Systems, and an Introduction to Chaos
Hirsch, Devaney, and Smale's classic "Differential Equations, Dynamical Systems, and an Introduction to Chaos" has been used by professors as the primary text for undergraduate and graduate levelExpand
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Generalized Poincare's Conjecture in Dimensions Greater Than Four
Poincare has posed the problem as to whether every simply connected closed 3-manifold (triangulated) is homeomorphic to the 3-sphere, see [18] for example. This problem, still open, is usually calledExpand
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Topology and mechanics. I
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Morse inequalities for a dynamical system
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Best Choices for Regularization Parameters in Learning Theory: On the Bias—Variance Problem
We find a unique minimizer of E(γ ) which is found by an easy algorithm to yield the “best” regularization parameter γ = γ ∗. Expand
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The analogue of Theorem A for the topological case was proved by H. Kneser [2]. The problem in his case seems to be of a different nature from the differentiable case. J. Munkres [3] has proved thatExpand
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A generalized Morse theory
1. Abstract theory. Let M be a C-Riemannian manifold without boundary modeled on a separable Hubert space (see Lang [3]). For pÇzM we denote by ( , )p the inner product in the tangent space Mp and weExpand
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On Gradient Dynamical Systems
We consider in this paper a Co vector field X on a Co compact manifold Mn (&M, the boundary of M, may be empty or not) satisfying the following conditions: (1) At each singular point /8 of X, thereExpand
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