Flow Invariance on Stratified Domains

This paper studies conditions for invariance of dynamical systems on stratified do- mains as originally introduced by Bressan and Hong. We establish Hamiltonian conditions for both weak and strong invariance of trajectories on systems with non-Lipschitz data. This is done via the identification of a new multifunction, the essential velocity multifunction. Properties of this multifunction are investigated and used to establish the relevant invariance criteria

On the Strong Invariance Property for Non-Lipschitz Dynamics

We provide a new sufficient condition for strong invariance for differential inclusions, under very general conditions on the dynamics, in terms of a Hamiltonian inequality. In lieu of the usual Lipschitzness assumption on the multifunction, we assume a feedback realization condition that can in particular be satisfied for measurable dynamics that are neither upper nor lower semicontinuous.Comment: 15 pages, 0 figures. For this revision, the authors added a remark about an alternative nonconstructive proof of the main resul

Acoustic Attenuation in High-TcT_c Superconductors

We analyze the acoustic attenuation rate in high-$T_c$ superconductors, and find that this method offers an additional way to examine the anisotropy of the superconducting order parameter in these materials. We argue that it should be possible to distinguish the electronic contribution to the acoustic attenuation, which has a strong temperature dependence near $T_c$, from the lattice contribution, which does not show a strong temperature dependence near $T_c$. We propose that this can be utilized to measure the anisotropy of the order parameter by measuring the attenuation rate near $T_c$ in different directions.Comment: 9 pages, latex, 2 postscript figures, in press Physica C, (uuencoded file consisting of paper and 2 figures, please contact J.C. Swihart ([email protected]) for a printed copy

Thurston's metric on Teichmüller space of semi-translation surfaces

We define two pseudo-metrics L_F and K_F on the Teichmüller space of semi-translation surfaces, which are the symmetric counterparts to the metrics defined by William Thurston. We prove some nice properties of L_F and K_F, most notably that they are complete pseudo-metrics. In the second part of the thesis we define their asymmetric analogues L_F^a and K_F^a and prove that their equality depends on two statements regarding 1-Lipschitz maps between polygons. We are able to prove the first statement, but the second one remains a conjecture: nonetheless, we explain why we believe it is true

Semiconcavity of the value function for a class of differential inclusions

We provide intrinsic sufficient conditions on a multifunction F and endpoint data phi so that the value function associated to the Mayer problem is semiconcave

Convexity and Duality in Hamilton-Jacobi Theory

Value functions propagated from initial or terminal costs and constraints by way of a differential or more broadly through a Lagrangian that may take on "alpha," are studied in the case where convexity persists in the state argument. Such value functions, themselves taking on "alpha," are shown to satisfy a subgradient form of the Hamilton-Jacobi equation which strongly supports properties of local Lipschitz continuity, semidifferentibility and Clarke regularity. An extended `method of characteristics' is developed which determines them from Hamiltonian dynamics underlying the given Lagrangian. Close relations with a dual value function are revealed

The Mayer and Minimum Time Problems with Stratified State Constraints *

International audienceThis paper studies optimal control problems with state constraints by imposing structural assumptions on the constraint domain coupled with a tangential restriction with the dynamics. These assumptions replace pointing or controllability assumptions that are common in the literature, and provide a framework under which feasible boundary trajectories can be analyzed directly. The value functions associated with the state constrained Mayer and minimal time problems are characterized as solutions to a pair of Hamilton-Jacobi inequalities with appropriate boundary conditions. The novel feature of these inequalities lies in the choice of the Hamiltonian

Jan Łukasiewicz, Sur la Logique des stoïciens

Jan Łukasiewicz a élaboré un programme de recherche portant sur l’histoire de la logique. À son avis, la logique mathématique contemporaine (modern) est en continuité directe avec la logique formelle du passé. En conséquence, une interprétation correcte et fidèle des idées anciennes exige une application des outils logiques contemporains. Autrement dit, de bonnes études historiques sur la logique doivent consister à examiner la logique d’autrefois à travers les lunettes logiques contemporaine..

Ultrasonic Attenuation in Clean d-Wave Superconductors

We calculate the low temperature longitudinal ultrasonic attenuation rate $\alpha_S$ in clean d-wave superconductors. We consider the contribution of previously ignored processes involving the excitation of a pair of quasi-holes or quasi-particles. These processes, which are forbidden by energy conservation in conventional s-wave superconductors, have a finite phase space in d-wave superconductors due to the presence of nodes in the gap which give rise to soft low-energy electronic excitations. We find the contribution to $\alpha_S$ from these processes to be proportional to $T$ in the regime $k_B T\ll Qv_{\Delta} \ll \Delta_0$,(ultra-low temperature regime) and to be proportional to 1/T in the region $Qv_F \ll k_BT \ll \Delta_0$, (low temperature regime) where ${\bf Q}$ is the ultrasound wave-vector and $\Delta_0$ is the maximum gap amplitude. We explicitly evaluate these terms, for parameters appropriate to the cuprates, for ${\bf Q}$ along the nodal and the antinodal directions and compare it with the contribution from processes considered earlier(I.Vekhter et al {\it Phys. Rev.}{\bf B59}, 7123(1999)). In the ultra-low temperature regime, the processes considered by us make a contribution which is smaller by about a factor of 10 for ${\bf Q}$ along the nodal direction, while along the antinodal direction it is larger by a factor of 100 or so. In the low temperature regime on the other hand the contribution made by these terms is small. However taken together with the original terms we describe a possible way to evaluate the parameter $v_F/v_\Delta$.Comment: 9 pages, RevTex, accepted for publication in Physica