The cubic Szegő equation

Patrick Gérard; Sandrine Grellier

Displaying similar documents to “The cubic Szegő equation”

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 $0 < T < 𝒦^{C}$ .

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...

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...

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...

Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure

Changxing Miao, Youbin Zhu (2006)

Applicationes Mathematicae

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We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧ $u_{t t} - Δ u + m ² u = - F ₁ (| u | ², | v | ²) u$ , ⎨ ⎩ $v_{t t} - Δ v + m ² v = - F ₂ (| u | ², | v | ²) v$ for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). Based on Morawetz-type identity, we prove that solutions to the above system decay to zero in local L²-norm, and local energy also decays to zero if the initial energy satisfies $E (u, v, ℝ ⁿ, 0) = 1 / 2 \int_{ℝ ⁿ} (| \nabla u (0) | ² + | u_{t} (0) | ² + m ² | u (0) | ² + | \nabla v (0) | ² + | v_{t} (0) | ² + m ² | v (0) | ² + F (| u (0) | ², | v (0) | ²)) d x < \infty$ , and F₁(|u|²,|v|²)|u|² + F₂(|u|²,|v|²)|v|² - F(|u|²,|v|²) ≥ aF(|u|²,|v|²) ≥ 0, a >...

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 $G = K_{k} \lor (K_{n - 2 k} + {\bar{K}}_{k}) .$

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...

Overview of the differential Galois integrability conditions for non-homogeneous potentials

Andrzej J. Maciejewski, Maria Przybylska (2011)

Banach Center Publications

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We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form $H = 1 / 2 \sum_{i = 1}^{n} p ²_{i} + V (q)$ , where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form $V = V_{k} + V_{K}$ , where $V_{k}$ and $V_{K}$ are homogeneous functions of integer degrees k and K > k, respectively. We present examples...

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 $M \times M$ ...

Copies of $l_{p}^{n}$ ’s uniformly in the spaces $Π_{2} (C [0, 1], X)$ and $Π_{1} (C [0, 1], X)$

Dumitru Popa (2017)

Czechoslovak Mathematical Journal

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We study the presence of copies of $l_{p}^{n}$ ’s uniformly in the spaces $Π_{2} (C [0, 1], X)$ and $Π_{1} (C [0, 1], X)$ . By using Dvoretzky’s theorem we deduce that if $X$ is an infinite-dimensional Banach space, then $Π_{2} (C [0, 1], X)$ contains $λ \sqrt{2}$ -uniformly copies of $l_{\infty}^{n}$ ’s and $Π_{1} (C [0, 1], X)$ contains $λ$ -uniformly copies of $l_{2}^{n}$ ’s for all $λ > 1$ . As an application, we show that if $X$ is an infinite-dimensional Banach space then the spaces $Π_{2} (C [0, 1], X)$ and $Π_{1} (C [0, 1], X)$ are distinct, extending the well-known result that the spaces $Π_{2} (C [0, 1], X)$ and $𝒩 (C [0, 1], X)$ are distinct.

Bounded evaluation operators from $H^{p}$ into $ℓ^{q}$

Martin Smith (2007)

Studia Mathematica

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Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by $T_{z, p} (f) = {(1 - | z ₙ | ²)}^{1 / p} f (z ₙ)$ . Necessary and sufficient conditions on zₙ are given such that $T_{z, p}$ maps the Hardy space $H^{p}$ boundedly into the sequence space $ℓ^{q}$ . A corresponding result for Bergman spaces is also stated.

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,...

On a family of cubic graphs containing the flower snarks

Jean-Luc Fouquet, Henri Thuillier, Jean-Marie Vanherpe (2010)

Discussiones Mathematicae Graph Theory

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We consider cubic graphs formed with k ≥ 2 disjoint claws $C_{i} K_{1, 3}$ (0 ≤ i ≤ k-1) such that for every integer i modulo k the three vertices of degree 1 of $C_{i}$ are joined to the three vertices of degree 1 of $C_{i - 1}$ and joined to the three vertices of degree 1 of $C_{i + 1}$ . Denote by $t_{i}$ the vertex of degree 3 of $C_{i}$ and by T the set $t ₁, t ₂, . . ., t_{k - 1}$ . In such a way we construct three distinct graphs, namely FS(1,k), FS(2,k) and FS(3,k). The graph FS(j,k) (j ∈ 1,2,3) is the graph where the set of vertices $⋃_{i = 0}^{i = k - 1} V (C_{i}) ∖ T$ induce j cycles (note...

Addendum to “On the $p^{λ}$ problem”

Stephan Baier (2005)

Acta Arithmetica

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Localization techniques in $L^{p}$ spaces

A. Pełczyński, H. Rosenthal (1975)

Studia Mathematica

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On a certain property of solution of the equation $u_{t} = u_{x x} + f (x, t, u)$

Z. Knap (1970)

Annales Polonici Mathematici

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The general methods of finding the sum $S_{n}$ for all kinds of series, II

K. Orlov (1981)

Matematički Vesnik

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Hardy spaces H¹ for Schrödinger operators with certain potentials

Jacek Dziubański, Jacek Zienkiewicz (2004)

Studia Mathematica

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Let ${K_{t}}_{t > 0}$ be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to $H ¹_{L}$ if $| | s u p_{t > 0} | K_{t} {f (x) | | |}_{L ¹ (d x)} < \infty$ . We state conditions on V and $K_{t}$ which allow us to give an atomic characterization of the space $H ¹_{L}$ .

Compactness in $C_{s} (T)$ and Applications

Richard Haydon (1972)

Publications du Département de mathématiques (Lyon)

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Геометрическая теория состояний, граница фон Неймана, двойственность $C^{*}$ -алгерб

А.М. Вершик (1972)

Zapiski naucnych seminarov Leningradskogo

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Some types of points in $N^{*}$

Gryzlov, A.

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