The discriminant and oscillation lengths for contact and Legendrian isotopies

Vincent Colin; Sheila Sandon

Displaying similar documents to “The discriminant and oscillation lengths for contact and Legendrian isotopies”

On the tangency of sets in generalized metric spaces for certain functions of the class $F_{ρ}^{*}$

T. Konik (1991)

Matematički Vesnik

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Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds

Neil Seshadri (2009)

Bulletin de la Société Mathématique de France

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To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$ , we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M \times (- 1, 0)$ . We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M \times (- 1, 0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice...

Metric unconditionality and Fourier analysis

Stefan Neuwirth (1998)

Studia Mathematica

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We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_{E}^{p} ()$ and $C_{E} ()$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_{E} ()$ , p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_{E}^{p} ()$ ...

Wasserstein metric and subordination

Philippe Clément, Wolfgang Desch (2008)

Studia Mathematica

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Let $(X, d_{X})$ , $(Ω, d_{Ω})$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_{p}$ the p-Wasserstein metric with some p ∈ [1,∞), and by $_{p} (X)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f : Ω \to_{p} (X)$ , $ω \mapsto f_{ω}$ , be a Borel map such that f is a contraction from $(Ω, d_{Ω})$ into $(_{p} (X), W_{p})$ . Let ν₁,ν₂ be probability measures on Ω with $W_{p} (ν ₁, ν ₂)$ finite. On X we consider the subordinated measures $μ_{i} = \int_{Ω} f_{ω} d ν_{i} (ω)$ . Then $W_{p} (μ ₁, μ ₂) \leq W_{p} (ν ₁, ν ₂)$ . As an application we show that the solution measures $ϱ_{α} (t)$ ...

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X, d, μ)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B (x, r)$ converge to $f (x)$ when $r$ converges to $0$ . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

General position properties in fiberwise geometric topology

Taras Banakh, Vesko Valov

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General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish $L C^{n - 1}$ -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...

Extending generalized Whitney maps

Ivan Lončar (2017)

Archivum Mathematicum

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For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^{X}$ can be extended to a Whitney map for $2^{X}$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.

Intrinsic pseudo-volume forms for logarithmic pairs

Thomas Dedieu (2010)

Bulletin de la Société Mathématique de France

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We study an adaptation to the logarithmic case of the Kobayashi-Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic $K$ -correspondences. We define an intrinsic logarithmic pseudo-volume form $Φ_{X, D}$ for every pair $(X, D)$ consisting of a complex manifold $X$ and a normal crossing Weil divisor $D$ on $X$ , the positive part of which is reduced. We then prove that $Φ_{X, D}$ is generically non-degenerate when $X$ is projective...

Adjacent dyadic systems and the $L^{p}$ -boundedness of shift operators in metric spaces revisited

Olli Tapiola (2016)

Colloquium Mathematicae

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With the help of recent adjacent dyadic constructions by Hytönen and the author, we give an alternative proof of results of Lechner, Müller and Passenbrunner about the $L^{p}$ -boundedness of shift operators acting on functions $f \in L^{p} (X; E)$ where 1 < p < ∞, X is a metric space and E is a UMD space.

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

Jan Kurek, Włodzimierz Mikulski (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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If $(M, g)$ is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism $T M \tilde{=} T^{*} M$ given by $v \to g (v, -)$ between the tangent $T M$ and the cotangent $T^{*} M$ bundles of $M$ . In the present note, we generalize this isomorphism to the one $T^{(r)} M \tilde{=} T^{r *} M$ between the $r$ -th order vector tangent $T^{(r)} M = {(J^{r} {(M, R)}_{0})}^{*}$ and the $r$ -th order cotangent $T^{r *} M = J^{r} {(M, R)}_{0}$ bundles of $M$ . Next, we describe all base preserving vector bundle maps $C_{M} (g) : T^{(r)} M \to T^{r *} M$ depending on a Riemannian metric $g$ in terms of natural (in $g$ ) tensor fields on $M$ .

Local density of diffeomorphisms with large centralizers

Christian Bonatti, Sylvain Crovisier, Gioia M. Vago, Amie Wilkinson (2008)

Annales scientifiques de l'École Normale Supérieure

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Given any compact manifold $M$ , we construct a non-empty open subset $𝒪$ of the space ${Diff}^{1} (M)$ of $C^{1}$ -diffeomorphisms and a dense subset $𝒟 \subset 𝒪$ such that the centralizer of every diffeomorphism in $𝒟$ is uncountable, hence non-trivial.

Shells of monotone curves

Josef Mikeš, Karl Strambach (2015)

Czechoslovak Mathematical Journal

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We determine in $ℝ^{n}$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla$ for which any image of $C$ under a compact-free group $Ω$ of affinities containing the translation group is a geodesic with respect to $\nabla$ . Special attention is paid to the case that $Ω$ contains many dilatations or that $C$ is a curve in $ℝ^{3}$ . If $C$ is a curve in $ℝ^{3}$ and $Ω$ is the translation group then we calculate not only the components of the curvature and the Weyl...

Neutral set differential equations

Umber Abbas, Vasile Lupulescu, Donald O&#039;Regan, Awais Younus (2015)

Czechoslovak Mathematical Journal

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The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type $\{\begin{matrix} D_{H} X (t) = F (t, X_{t}, D_{H} X_{t}), \\ {X |}_{[- r, 0]} = Ψ, \end{matrix}$ where $F : [0, b] \times 𝒞_{0} \times 𝔏_{0}^{1} \to K_{c} (E)$ is a given function, $K_{c} (E)$ is the family of all nonempty compact and convex subsets of a separable Banach space $E$ , $𝒞_{0}$ denotes the space of all continuous set-valued functions $X$ from $[- r, 0]$ into $K_{c} (E)$ , $𝔏_{0}^{1}$ is the space of all integrally bounded set-valued functions $X : [- r, 0] \to K_{c} (E)$ , $Ψ \in 𝒞_{0}$ and $D_{H}$ is the Hukuhara derivative. The continuous dependence of solutions on initial...

$𝒞^{k}$ -regularity for the $\bar{\partial}$ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let $D$ be a $𝒞^{d}$ $q$ -convex intersection, $d \geq 2$ , $0 \leq q \leq n - 1$ , in a complex manifold $X$ of complex dimension $n$ , $n \geq 2$ , and let $E$ be a holomorphic vector bundle of rank $N$ over $X$ . In this paper, $𝒞^{k}$ -estimates, $k = 2, 3, \dots, \infty$ , for solutions to the $\bar{\partial}$ -equation with small loss of smoothness are obtained for $E$ -valued $(0, s)$ -forms on $D$ when $n - q \leq s \leq n$ . In addition, we solve the $\bar{\partial}$ -equation with a support condition in $𝒞^{k}$ -spaces. More precisely, we prove that for a $\bar{\partial}$ -closed form $f$ in $𝒞_{0, q}^{k} (X ∖ D, E)$ , $1 \leq q \leq n - 2$ , $n \geq 3$ , with compact support and for $ε$ with $0 < ε < 1$ there...

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let $ℱ_{h}^{i} (k, n)$ be the $i$ -th ordered configuration space of all distinct points $H_{1}, ..., H_{h}$ in the Grassmannian $G r (k, n)$ of $k$ -dimensional subspaces of $ℂ^{n}$ , whose sum is a subspace of dimension $i$ . We prove that $ℱ_{h}^{i} (k, n)$ is (when non empty) a complex submanifold of $G r {(k, n)}^{h}$ of dimension $i (n - i) + h k (i - k)$ and its fundamental group is trivial if $i = m i n (n, h k)$ , $h k \neq n$ and $n > 2$ and equal to the braid group of the sphere $ℂ$ $P^{1}$ if $n = 2$ . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k = n - 1$ .

Operations between sets in geometry

Richard J. Gardner, Daniel Hug, Wolfgang Weil (2013)

Journal of the European Mathematical Society

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An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$ -dimensional Euclidean space $ℝ^{n}$ . It is proved that if $n \geq 2$ , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $G L (n)$ covariant, and associative if and only if it is $L_{p}$ addition for some $1 \leq p \leq \infty$ . It is also demonstrated...