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Topological data analysis and topological deep learning beyond persistent homology: a review. [PDF]
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Combinatorica, 2000
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Morse theory and Lyapunov stability on manifolds
Journal of Mathematical Sciences, 2011 International audienceThe aim of this article is to recall the main theorems of Morse theory and to infer some corollaries for the problem of Lyapunov stability on manifolds.
Moulay, Emmanuel, E. Moulay
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1991
This chapter is the heart of the book. Quite a few spectacular theorems will be proved. The main ones are the famous convexity theorem of Atiyah [7] and Guillemin—Sternberg [63] which asserts that the image of a compact connected symplectic manifold under the momentum mapping of a Hamiltonian torus action is a convex polyhedron (this is Theorem ...
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This chapter is the heart of the book. Quite a few spectacular theorems will be proved. The main ones are the famous convexity theorem of Atiyah [7] and Guillemin—Sternberg [63] which asserts that the image of a compact connected symplectic manifold under the momentum mapping of a Hamiltonian torus action is a convex polyhedron (this is Theorem ...
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Morse Theory, Discrete Morse Theory and Applications
Theoretical and Natural ScienceBy employing a specific class of smooth functions to study a space, Morse theory establishes deep connections between analysis and topology. It is a classical subject of pure mathematics, originally pioneered by Marston Morse in the 1920s. In this article, we use Morse theory to present a proof of an interesting result on the knots, known as the Fáry ...
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Denoising with discrete Morse theory
The Visual Computer, 2021Denoising noisy datasets is a crucial task in this data-driven world. In this paper, we develop a persistence-guided discrete Morse theoretic denoising framework. We use our method to denoise point-clouds and to extract surfaces from noisy volumes. In addition, we show that our method generally outperforms standard methods.
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COMBINATORIAL NOVIKOV–MORSE THEORY
International Journal of Mathematics, 2002In [7, 8, 9], we developed a combinatorial Morse theory which can be applied to any CW complex. In [25, 26] Novikov presented a generalization of classical Morse theory in which the Morse function is replaced by a closed 1-forms. In this paper we extend our combinatorial Morse theory to include a combinatorial analog of Novikov's theory.
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On Some Results in Morse Theory
Canadian Journal of Mathematics, 1975The /z-cobordism theorem in [8], the generalized Poincaré conjecture in higher dimensions in [20] and several other results in differential topology are proved by using the following theorems of Morse theory:(1) the elimination of critical points;(2) the existence of nondegenerate functions for which the descending and ascending bowls have normal ...
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manuscripta mathematica, 1991
It is shown that the assignment of a cell-complex to a Morse function on a compact smooth manifold can be achieved uniquely up to a contractible space of parameters and continuously in an appropriate sense with respect to these parameters among which being the Riemannian metrics on the manifold.
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It is shown that the assignment of a cell-complex to a Morse function on a compact smooth manifold can be achieved uniquely up to a contractible space of parameters and continuously in an appropriate sense with respect to these parameters among which being the Riemannian metrics on the manifold.
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