An attraction result and an index theorem for continuous flows on $ℝ^n × [0,∞)$

Klaudiusz Wójcik

Displaying similar documents to “An attraction result and an index theorem for continuous flows on $ℝ^{n} \times [0, \infty)$ ”

Monotonicity of first eigenvalues along the Yamabe flow

Liangdi Zhang (2021)

Czechoslovak Mathematical Journal

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We construct some nondecreasing quantities associated to the first eigenvalue of $- Δ_{φ} + c R$ $(c \geq \frac{1}{2} (n - 2) / (n - 1))$ along the Yamabe flow, where $Δ_{φ}$ is the Witten-Laplacian operator with a $C^{2}$ function $φ$ . We also prove a monotonic result on the first eigenvalue of $- Δ_{φ} + \frac{1}{4} (n / (n - 1)) R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $- Δ_{φ} + c R^{a}$ with $a \in (0, 1)$ along the Yamabe flow.

Three examples of brownian flows on $ℝ$

Yves Le Jan, Olivier Raimond (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We show that the only flow solving the stochastic differential equation (SDE) on $ℝ$ $d X_{t} = 1_{{X_{t} g t; 0}} W^{+} (d t) + 1_{{X_{t} l t; 0}} d W^{-} (d t),$ where $W^{+}$ and $W^{-}$ are two independent white noises, is a coalescing flow we will denote by $ϕ^{\pm}$ . The flow $ϕ^{\pm}$ is a Wiener solution of the SDE. Moreover, $K^{+} = 𝖤 [δ_{ϕ^{\pm}} | W^{+}]$ is the unique solution (it is also a Wiener solution) of the SDE $K_{s, t}^{+} f (x) = f (x) + \int_{s}^{t} K_{s, u} (1_{ℝ}^{+} f^{'}) (x) W^{+} (d u) + \frac{1}{2} \int_{s}^{t} K_{s, u} f ` ` (x) d u$ for $s l t; t$ , $x \in ℝ$ and $f$ a twice continuously differentiable function. A third flow $ϕ^{+}$ can be constructed out of the $n$ -point motions of $K^{+}$ . This flow is coalescing and its $n$ -point motion...

On Schrödinger maps from $T^{1}$ to $S^{2}$

Robert L. Jerrard, Didier Smets (2012)

Annales scientifiques de l'École Normale Supérieure

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We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^{1}$ to $S^{2} .$ This estimate yields some continuity properties of the flow map for the topology of $L^{2} (T^{1}, S^{2})$ , provided one takes its quotient by the continuous group action of $T^{1}$ given by translations. We also prove that without taking this quotient, for any $t > 0$ the flow map at time $t$ is discontinuous as a map from $𝒞^{\infty} (T^{1}, S^{2})$ , equipped with the weak topology of $H^{1 / 2},$ to the space of distributions ${(𝒞^{\infty} (T^{1}, ℝ^{3}))}^{*} .$ The argument relies...

Generalized gradient flow and singularities of the Riemannian distance function

Piermarco Cannarsa (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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Significant information about the topology of a bounded domain $Ω$ of a Riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial Ω}$ , from the boundary of $Ω$ . We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of $d_{\partial Ω}$ , as well as applications to homotopy equivalence.

Topological dynamics of unordered Ramsey structures

Moritz Müller, András Pongrácz (2015)

Fundamenta Mathematicae

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We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M (G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M (G_{)}$ . We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M (G_{)}$ admits...

Local-in-time existence for the non-resistive incompressible magneto-micropolar fluids

Peixin Zhang, Mingxuan Zhu (2022)

Applications of Mathematics

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We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_{0} \in H^{s - 1 + ε}$ , $w_{0} \in H^{s - 1}$ and $b_{0} \in H^{s}$ for $s > \frac{3}{2}$ and any $0 < ε < 1$ . The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$ .

Proximality in Pisot tiling spaces

Marcy Barge, Beverly Diamond (2007)

Fundamenta Mathematicae

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A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space $_{φ}$ has pure discrete spectrum, we describe the collection $_{φ}^{P}$ of pairs of proximal tilings in $_{φ}$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then $_{φ}$ and $_{ψ}$ are homeomorphic...

A second order unconditionally positive space-time residual distribution method for solving compressible flows on moving meshes

Dobeš, Jiří, Deconinck, Herman

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A space-time formulation for unsteady inviscid compressible flow computations in 2D moving geometries is presented. The governing equations in Arbitrary Lagrangian-Eulerian formulation (ALE) are discretized on two layers of space-time finite elements connecting levels $n$ , $n + 1 / 2$ and $n + 1$ . The solution is approximated with linear variation in space (P1 triangle) combined with linear variation in time. The space-time residual from the lower layer of elements is distributed to the nodes at level...

Measure-preserving homeomorphisms of noncompact manifolds and mass flow toward ends

Tatsuhiko Yagasaki (2007)

Fundamenta Mathematicae

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Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and $ℋ_{E} (M, ω)$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism $c^{ω} : ℋ_{E} (M, ω) \to (M, ω)$ , which measures for each $h \in ℋ_{E} (M, ω)$ the mass flow toward ends induced by h. We show that the...

Index of the Dirac operator in $R^{n}$

Jan A. Rempała (1985)

Annales Polonici Mathematici

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Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane Baladi, Daniel Smania (2012)

Annales scientifiques de l'École Normale Supérieure

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We consider $C^{2}$ families $t \mapsto f_{t}$ of $C^{4}$ unimodal maps $f_{t}$ whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure $μ_{t}$ of $f_{t}$ depends differentiably on $t$ , as a distribution of order $1$ . The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of $μ_{t}$ for a Benedicks-Carleson map $f_{t}$ , in terms of a single smooth function and the...

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

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.

Geometric rigidity of $\times m$ invariant measures

Michael Hochman (2012)

Journal of the European Mathematical Society

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Let $μ$ be a probability measure on $[0, 1]$ which is invariant and ergodic for $T_{a} (x) = a x 𝚖𝚘𝚍 1$ , and $0 < 𝚍𝚒𝚖 μ < 1$ . Let $f$ be a local diffeomorphism on some open set. We show that if $E \subseteq ℝ$ and $(f μ) {|_{E} \sim μ|}_{E}$ , then $f^{'} (x) \in \{\pm a^{r} : r \in ℚ\}$ at $μ$ -a.e. point $x \in f^{- 1} E$ . In particular, if $g$ is a piecewise-analytic map preserving $μ$ then there is an open $g$ -invariant set $U$ containing supp $μ$ such that $g |_{U}$ is piecewise-linear with slopes which are rational powers of $a$ . In a similar vein, for $μ$ as above, if $b$ is another integer and $a, b$ are not powers of a common integer, and if $ν$ is...

$C^{*}$ -points vs $P$ -points and $P^{♭}$ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space $X$ , the point $p \in X$ is called a $C^{*}$ -point if every real-valued continuous function on $C ∖ {p}$ can be extended continuously to $p$ . Every point in an extremally disconnected space is a $C^{*}$ -point. A classic example is the space $𝐖^{*} = ω_{1} + 1$ consisting of the countable ordinals together with $ω_{1}$ . The point $ω_{1}$ is known to be a $C^{*}$ -point as well as a $P$ -point. We supply a characterization of $C^{*}$ -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Displaying similar documents to “An attraction result and an index theorem for continuous flows on ℝ n × [ 0 , ∞ ) ”

Displaying similar documents to “An attraction result and an index theorem for continuous flows on $ℝ^{n} \times [0, \infty)$ ”