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Multiplicity results for Schrodinger type fractional p-Laplacian boundary value problems
Electronic Journal of Differential EquationsEmer Lopera +2 more
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Journal of Fixed Point Theory and Applications, 2011
Starting from the concept of Morse critical point, introduced in [A. Ioffe and E. Schwartzman, J. Math. Pures Appl. (9), 75 (1996), 125–153], we propose a purely topological approach to Morse theory.
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Starting from the concept of Morse critical point, introduced in [A. Ioffe and E. Schwartzman, J. Math. Pures Appl. (9), 75 (1996), 125–153], we propose a purely topological approach to Morse theory.
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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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2015
In this chapter we give an introduction to basic Morse theory. We define the notion of the Hessian of a smooth function at a critical point on a smooth manifold and show that if it is nondegenerate then there are local coordinates in which the function is equal to its second derivative. We also prove the Morse isotopy lemma which gives a criterion when
Larry Bates, Richard Cushman
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In this chapter we give an introduction to basic Morse theory. We define the notion of the Hessian of a smooth function at a critical point on a smooth manifold and show that if it is nondegenerate then there are local coordinates in which the function is equal to its second derivative. We also prove the Morse isotopy lemma which gives a criterion when
Larry Bates, Richard Cushman
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1988
The reader who is interested only in Morse theory for singular spaces or for nonproper Morse function may skip this chapter. We will consider the Morse theory of a composition $$ X\xrightarrow{R}Z\xrightarrow{f}\mathbb{R} $$ which will eventually be used (in Part II, Sects.
Robert MacPherson, Mark Goresky
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The reader who is interested only in Morse theory for singular spaces or for nonproper Morse function may skip this chapter. We will consider the Morse theory of a composition $$ X\xrightarrow{R}Z\xrightarrow{f}\mathbb{R} $$ which will eventually be used (in Part II, Sects.
Robert MacPherson, Mark Goresky
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Combinatorica, 2000
is a non-contractible subcomplex of a simplex S then M is evasive. In this paper we make this result quantitative, and show that the more non-contractible M is, the more evasive M is. Recall that M is evasive if for every decision tree algorithm A there is a face of S that requires that one examines all vertices of S (
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is a non-contractible subcomplex of a simplex S then M is evasive. In this paper we make this result quantitative, and show that the more non-contractible M is, the more evasive M is. Recall that M is evasive if for every decision tree algorithm A there is a face of S that requires that one examines all vertices of S (
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Morse theory (with constraints)
2001In Chapter 2 we developed Morse Theory for functions which are defined on the whole ℝ n . In this chapter we study Morse Theory for functions which need not to be defined on the whole ℝ n but merely on suitable subsets of it: C r -manifolds (see Section 2.1) or, more generally, “C r -Manifolds with Generalized Boundary”.
P. Jonker, Hubertus Th. Jongen, F. Twilt
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1988
Throughout this chapter we will fix an arrangement A = {A1A2,…,Am} of affine subspaces of ℝ n , let P denote the corresponding partially ordered set of flats which are the intersections of the affine spaces, let T denote the unique maximal element of P corresponding to ℝ n , and let K(P)denote the order complex of P.
Mark Goresky, Robert MacPherson
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Throughout this chapter we will fix an arrangement A = {A1A2,…,Am} of affine subspaces of ℝ n , let P denote the corresponding partially ordered set of flats which are the intersections of the affine spaces, let T denote the unique maximal element of P corresponding to ℝ n , and let K(P)denote the order complex of P.
Mark Goresky, Robert MacPherson
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