Hofer’s metrics and boundary depth

Michael Usher

Displaying similar documents to “Hofer’s metrics and boundary depth”

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...

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...

Special Lagrangian linear subspaces in product symplectic space

Małgorzata Mikosz (2004)

Banach Center Publications

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The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism $f : (ℝ^{2 n}, σ = \sum_{i = 1}^{n} d x_{i} \land d y_{i}) \to (ℝ^{2 n}, σ)$ to be a special Lagrangian linear subspace in $(ℝ^{2 n} \times ℝ^{2 n}, ω = π * ₂ σ - π * ₁ σ)$ . This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian $S Λ_{2 n} ≃ S U (2 n) / S O (2 n)$ is defined.

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...

Spectral radius and Hamiltonicity of graphs with large minimum degree

Vladimir Nikiforov (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a graph of order $n$ and $λ (G)$ the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in $G$ . One of the main results of the paper is the following theorem: Let $k \geq 2,$ $n \geq k^{3} + k + 4,$ and let $G$ be a graph of order $n$ , with minimum degree $δ (G) \geq k .$ If $λ (G) \geq n - k - 1,$ then $G$ has a Hamiltonian cycle, unless $G = K_{1} \lor (K_{n - k - 1} + K_{k})$ or $G = K_{k} \lor (K_{n - 2 k} + {\bar{K}}_{k}) .$

Rational symplectic field theory over $ℤ^{2}$ for exact Lagrangian cobordisms

Tobias Ekholm (2008)

Journal of the European Mathematical Society

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We construct a version of rational Symplectic Field Theory for pairs $(X, L)$ , where $X$ is an exact symplectic manifold, where $L \subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X, L)$ a $ℤ$ -graded chain complex of vector spaces over $ℤ_{2}$ , filtered with $k$ filtration levels. The corresponding $k$ -level spectral sequence is invariant under deformations of $(X, L)$ and has the following property: if $(X, L)$ is obtained by...

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,...

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...

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 $p (t) \equiv p = 2$ . ...

On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.

Aldo Bressan (1988)

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

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In [1] I and II various equivalence theorems are proved; e.g. an ODE $(\mathcal{E})\dot{z}=F(t,z,u,\dot{u})\,(\in\mathbb{R}^{m})$ with a scalar control $u=u(\cdot)$ is linear w.r.t. $\dot{u}$ iff $(\alpha)$ its solution $z(u,\cdot)$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta)$ $z(u,\cdot)$ satisfies certain conditions by which $1^{st}$ -order discontinuities of $u$ and $\dot{u}$ can be treated satisfactorily. In the case when, for $z=(q,p)$ equation $(\mathcal{E})$ is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type,...

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...

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia, Vadim Kaloshin (2015)

Journal of the European Mathematical Society

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We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s > 1$ . Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with $s$ -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c > 0$ such that for any $𝒦 ≫ 1$ we find a solution $u$ and a time $T$ such that ${∥ u (T) ∥}_{H^{s}} \geq 𝒦 {∥ u (0) ∥}_{H^{s}}$ . Moreover, the time $T$ satisfies the polynomial bound $0 < T < 𝒦^{C}$ .

Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville

Daniel Greb, Christian Lehn, Sönke Rollenske (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a compact hyperkähler manifold containing a complex torus $L$ as a Lagrangian subvariety. Beauville posed the question whether $X$ admits a Lagrangian fibration with fibre $L$ . We show that this is indeed the case if $X$ is not projective. If $X$ is projective we find an almost holomorphic Lagrangian fibration with fibre $L$ under additional assumptions on the pair $(X, L)$ , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic...

On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.

Aldo Bressan (1988)

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

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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...

Hamiltonicity of cubic Cayley graphs

Henry Glover, Dragan Marušič (2007)

Journal of the European Mathematical Society

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Following a problem posed by Lovász in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a $(2, s, 3)$ -presentation, that is, for groups $G = 〈 a, b ∣ a^{2} = 1, b^{s} = 1, {(a b)}^{3} = 1, \dots 〉$ generated by an involution $a$ and an element $b$ of order $s \geq 3$ such that their product $a b$ has order $3$ . More precisely, it is shown that the Cayley graph $X = Cay (G, {a, b, b^{- 1}})$ has a Hamilton cycle when $| G |$ (and thus $s$ ) is congruent to 2 modulo 4, and has a...