We provide a refinement of the Poincaré inequality on the torus $\mathbb{T}^{d}$ : there exists a set $\mathcal{B} \subset \mathbb{T} ^{d}$ of directions such that for every $\alpha \in \mathcal{B}$ there is a $c_{\alpha } > 0$ with $$\begin{aligned} \|\nabla f\|_{L^{2}(\mathbb{T}^{d})}^{d-1} \| \langle \nabla f, \alpha \rangle \|_{L^{2}(\mathbb{T}^{d})} \geq c_{\alpha }\|f\| _{L^{2}(\mathbb{T}^{d})}^{d} \quad \mbox{for all}~f\in H^{1}\bigl( \mathbb{T}^{d}\bigr)~ \mbox{with mean 0.} \end{aligned}$$ The derivative $\langle \nabla f, \alpha \rangle $ does not detect any oscillation in directions orthogonal to $\alpha $ , however, for certain $\alpha $ the geodesic flow in direction $\alpha $ is sufficiently mixing to compensate for that defect. On the two-dimensional torus $\mathbb{T}^{2}$ the inequality holds for $\alpha = (1, \sqrt{2})$ but is not true for $\alpha = (1,e)$ . Similar results should hold at a great level of generality on very general domains.

As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to be shape densities (characteristic functions). The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: Any two shapes of equal volume can be approximately connected by an Euler spray---a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays. Each such Wasserstein geodesic is also the unique minimizer of a relaxed least-action principle for a two-fluid mixture theory corresponding to incompressible fluid mixed with vacuum.

No abstract uploaded!

In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $\bu = \n^\perp \Psi$, $\Psi(r,\th) = r^{\l} \psi(\th)$, for $\l >0$, we show that only trivial solutions exist in the range $0<\l<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $\l>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\l}) \cap \N$. The case $\l = 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity $1/3$.

A method for exploring the structure of populations of complex objects, such as images, is considered. The objects are summarized by feature vectors. The statistical backbone is Principal Component Analysis in the space of feature vectors. Visual insights come from representing the results in the original data space. In an ophthalmological example, endemic outliers motivate the development of a bounded influence approach to PCA.