Rigidity and gluing for Morse and Novikov complexes

Octav Cornea; Andrew Ranicki

Displaying similar documents to “Rigidity and gluing for Morse and Novikov complexes”

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

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We show that if $(M, ω)$ is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of $(M, ω)$ has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in $M \times M$ ...

Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory

Marek Izydorek, Krzysztof P. Rybakowski (2003)

Fundamenta Mathematicae

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Let Ω be a bounded domain in $ℝ^{N}$ with smooth boundary. Consider the following elliptic system: $- Δ u = \partial_{v} H (u, v, x)$ in Ω, $- Δ v = \partial_{u} H (u, v, x)$ in Ω, u = 0, v = 0 in ∂Ω. (ES) We assume that H is an even "-"-type Hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions. We show that if (0,0) is a hyperbolic solution of (ES), then (ES) has at least 2|μ| nontrivial solutions, where μ = μ(0,0) is the renormalized Morse index of (0,0). This proves a conjecture by Angenent and van der Vorst. ...

An upper bound of a generalized upper Hamiltonian number of a graph

Martin Dzúrik (2021)

Archivum Mathematicum

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In this article we study graphs with ordering of vertices, we define a generalization called a pseudoordering, and for a graph $H$ we define the $H$ -Hamiltonian number of a graph $G$ . We will show that this concept is a generalization of both the Hamiltonian number and the traceable number. We will prove equivalent characteristics of an isomorphism of graphs $G$ and $H$ using $H$ -Hamiltonian number of $G$ . Furthermore, we will show that for a fixed number of vertices, each path has a maximal upper...

Linking and the Morse complex

Michael Usher (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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For a Morse function $f$ on a compact oriented manifold $M$ , we show that $f$ has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in $M$ whose components have nontrivial linking number, such that the minimal value of $f$ on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of $f$ in terms of the Betti numbers of $M$ and the behavior of $f$ with respect...

Problems remaining NP-complete for sparse or dense graphs

Ingo Schiermeyer (1995)

Discussiones Mathematicae Graph Theory

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For each fixed pair α,c > 0 let INDEPENDENT SET ( $m \leq c n^{α}$ ) and INDEPENDENT SET ( $m \geq (ⁿ ₂) - c n^{α}$ ) be the problem INDEPENDENT SET restricted to graphs on n vertices with $m \leq c n^{α}$ or $m \geq (ⁿ ₂) - c n^{α}$ edges, respectively. Analogously, HAMILTONIAN CIRCUIT ( $m \leq n + c n^{α}$ ) and HAMILTONIAN PATH ( $m \leq n + c n^{α}$ ) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with $m \leq n + c n^{α}$ edges. For each ϵ > 0 let HAMILTONIAN CIRCUIT (m ≥ (1 - ϵ)(ⁿ₂)) and HAMILTONIAN PATH (m ≥ (1 - ϵ)(ⁿ₂)) be the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH...

The cubic Szegő equation

Patrick Gérard, Sandrine Grellier (2010)

Annales scientifiques de l'École Normale Supérieure

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We consider the following Hamiltonian equation on the $L^{2}$ Hardy space on the circle, $i \partial_{t} u = {Π (| u |}^{2} u),$ where $Π$ is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several...

On some completions of the space of hamiltonian maps

Vincent Humilière (2008)

Bulletin de la Société Mathématique de France

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In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of $ℝ^{2 n}$ endowed with the standard symplectic form $ω_{0} = d p \land d q$ . We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example,...

Perturbation results for a class of singular Hamiltonian systems

Antonio Ambrosetti, Ivar Ekeland (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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The existence of solutions with prescribed period $T$ for a class of Hamiltonian systems with a Keplerian singularity is discussed.

Hamiltonian-colored powers of strong digraphs

Garry Johns, Ryan Jones, Kyle Kolasinski, Ping Zhang (2012)

Discussiones Mathematicae Graph Theory

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For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^{k}$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^{k}$ if the directed distance $^{\to} d_{D} (u, v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^{k}$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^{k}$ is distance-colored if each arc (u, v) of $D^{k}$ is assigned the color i where $i =^{\to} d_{D} (u, v)$ . The digraph...

Periodic solutions for a class of non-autonomous Hamiltonian systems with $p (t)$ -Laplacian

Zhiyong Wang, Zhengya Qian (2024)

Mathematica Bohemica

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We investigate the existence of infinitely many periodic solutions for the $p (t)$ -Laplacian Hamiltonian systems. By virtue of several auxiliary functions, we obtain a series of new super- $p^{+}$ growth and asymptotic- $p^{+}$ growth conditions. Using the minimax methods in critical point theory, some multiplicity theorems are established, which unify and generalize some known results in the literature. Meanwhile, we also present an example to illustrate our main results are new even in the case $p (t) \equiv p = 2$ . ...

Fredholm theory and transversality for the parametrized and for the $S^{1}$ -invariant symplectic action

Frédéric Bourgeois, Alexandru Oancea (2010)

Journal of the European Mathematical Society

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We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the $L^{2}$ -gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic $S^{1}$ -invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define $S^{1}$ -equivariant...

Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan Zeng, Chun Lei Tang (2016)

Annales Polonici Mathematici

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We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $- [a + b (\int_{Ω} {| \nabla u | ² d x)}^{m} {] Δ u = f (x, u) + | u |}^{2 * - 2} u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

Perturbation results for a class of singular Hamiltonian systems

Antonio Ambrosetti, Ivar Ekeland (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The existence of solutions with prescribed period $T$ for a class of Hamiltonian systems with a Keplerian singularity is discussed.

Rabinowitz Floer homology and symplectic homology

Kai Cieliebak, Urs Frauenfelder, Alexandru Oancea (2010)

Annales scientifiques de l'École Normale Supérieure

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The first two authors have recently defined Rabinowitz Floer homology groups $R F H_{*} (M, W)$ associated to a separating exact embedding of a contact manifold $(M, ξ)$ into a symplectic manifold $(W, ω)$ . These depend only on the bounded component $V$ of $W ∖ M$ . We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$ , which in turn maps to Rabinowitz Floer homology $R F H_{*} (M, W)$ , which then maps to symplectic cohomology of $V$ . We compute $R F H_{*} (S T^{*} L, T^{*} L)$ , where $S T^{*} L$ is the unit cosphere bundle of a closed...

Generalized Conley-Zehnder index

Jean Gutt (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit $1$ as an eigenvalue. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W, \bar{Ω})$ , having chosen a given...

Analytic invariants for the $1 : - 1$ resonance

José Pedro Gaivão (2013)

Annales de l’institut Fourier

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Associated to analytic Hamiltonian vector fields in $ℂ^{4}$ having an equilibrium point satisfying a non semisimple $1 : - 1$ resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

Derivation of Hartree’s theory for mean-field Bose gases

Mathieu Lewin (2013)

Journées Équations aux dérivées partielles

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This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of $N$ bosons with an interaction of intensity $1 / N$ (mean-field regime). In the limit $N \to \infty$ , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation...