The second Yamabe invariant

Let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 3$. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to $g$ and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation

Linking Linear Threshold Units with Quadratic Models of Motion Perception

Behavioral experiments on insects (Hassenstein and Reichardt 1956; Poggio and Reichardt 1976) as well as psychophysical evidence from human studies (Van Santen and Sperling 1985; Adelson and Bergen 1985; Watson and Ahumada 1985) support the notion that short-range motion perception is mediated by a system with a quadratic type of nonlinearity, as in correlation (Hassenstein and Reichardt 1956), multiplication (Torre and Poggio 1978), or squaring (Adelson and Bergen 1985). However, there is little physiological evidence for quadratic nonlinearities in directionally selective cells. For instance, the response of cortical simple cells to a moving sine grating is half-wave instead of full-wave rectified as it should be for a quadratic nonlinearity (Movshon ef al. 1978; Holub and Morton-Gibson 1981) and is linear for low contrast (Holub and Morton-Gibson 1981). Complex cells have full-wave rectified responses, but are also linear in contrast. Moreover, a detailed theoretical analysis of possible biophysical mechanisms underlying direction selectivity concludes that most do not have quadratic properties except under very limited conditions (Grzywacz and Koch 1987). Thus, it is presently mysterious how a system can show quadratic properties while its individual components do not. We briefly discuss here a simple population encoding scheme offering a possible solution to this problem

Optimal transportation between hypersurfaces bounding some strictly convex domains

Let $M,N$ be two smooth compact hypersurfaces of $\mathbb{R}^n$ which bound strictly convex domains equipped with two absolutely continuous measures $\mu$ and $\nu$ (with respect to the volume measures of $M$ and $N$). We consider the optimal transportation from $\mu$ to $\nu$ for the quadratic cost. Let $(\phi:m \to \mathbb{R},\psi:N \to \mathbb{R})$ be some functions which achieve the supremum in the Kantorovich formulation of the problem and which satisfy $$\psi (y) = \inf_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -\varphi(z)\Bigr); \varphi (x)=\inf_{z\in N} \Bigl( \frac{1}{2}|x-z|^2 -\psi(z)\Bigr).$$ Define for $y \in N$, $$\varphi^\Box(y) = \sup_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -\varphi(z)\Bigr).$$ In this short paper, we exhibit a relationship between the regularity of $\varphi^\Box$ and the existence of a solution to the Monge problem

The geometrical quantity in damped wave equations on a square

The energy in a square membrane $\Omega$ subject to constant viscous damping on a subset $\omega\subset \Omega$ decays exponentially in time as soon as $\omega$ satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate $\tau(\omega)$ of this decay satisfies $\tau(\omega)= 2 \min(-\mu(\omega), g(\omega))$ (see Lebeau [Math. Phys. Stud. 19 (1996) 73-109]). Here $\mu(\omega)$ denotes the spectral abscissa of the damped wave equation operator and $g(\omega)$ is a number called the geometrical quantity of $\omega$ and defined as follows. A ray in $\Omega$ is the trajectory generated by the free motion of a mass-point in $\Omega$ subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity $g(\omega)$ is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g(\omega)$ when $\omega$ is a finite union of squares