Geometry and curvature of diffeomorphism groups with H1H^1 metric and mean hydrodynamics

Recently, Holm, Marsden, and Ratiu [1998] have derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation $\dot{V}(t) + \nabla_{U(t)} V(t) - \alpha^2 [\nabla U(t)]^t \cdot \triangle U(t) = -\text{grad} p(t)$ where $\text{div} U=0$, and $V = (1- \alpha^2 \triangle)U$. In this model, the momentum $V$ is transported by the velocity $U$, with the effect that nonlinear interaction between modes corresponding to length scales smaller than $\alpha$ is negligible. We generalize this equation to the setting of an $n$ dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincar\'{e} equation associated with the geodesic flow of the $H^1$ right invariant metric on ${\mathcal D}^s_\mu$, the group of volume preserving Hilbert diffeomorphisms of class $H^s$. We prove that the geodesic spray is continuously differentiable from $T{\mathcal D}_\mu^s(M)$ into $TT{\mathcal D}_\mu^s(M)$ so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold [1966]. To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariant $H^1$ metric on ${\mathcal D}^s_\mu$ is a bounded trilinear map in the $H^s$ topology, from which it follows that solutions to Jacobi's equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics.Comment: AMS-LaTeX, 22 pages, To appear in J. Func. Ana

Structural safety analysis of the aqueducts 'Coll de foix' and 'Capdevila' of the Canal of Aragon and Catalonia

The Canal of Aragon and Catalonia (CAC) is 134 km long and irrigates 105,000 ha (131 irrigation user communities) and it is owned by the River Ebro’s Water Agency. The aqueducts are located between km 67 and 71 of the canal and were designed by the civil engineer Félix de los Ríos Martín in 1907. The cross-section of both aqueducts, Coll de Foix and Capdevila, was extended within the framework of the project by Fernando Hué Herrero in 1962 in order to reach design flows of 26.1 m3/s and 25.7 m3/s, respectively. The structural performance of the aqueducts has been satisfactory; nevertheless, the hydraulic capacity has reduced over the years. As a result, the irrigation user communities have expressed the need to extend the cross-section of the aqueducts to meet the irrigation demands. Given the age of the structure and the different design considerations at the time, it is paramount to verify the structural reliability of the aqueducts in the new load configuration. Therefore, the objective of this contribution is to present the structural safety analysis conducted and to describe the new extended cross-section for both aqueducts (maintaining the original structural typology).Peer ReviewedPostprint (published version

Sequentiality vs. Concurrency in Games and Logic

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.Comment: 35 pages, appeared in Mathematical Structures in Computer Scienc

The unexpected resurgence of Weyl geometry in late 20-th century physics

Weyl's original scale geometry of 1918 ("purely infinitesimal geometry") was withdrawn by its author from physical theorizing in the early 1920s. It had a comeback in the last third of the 20th century in different contexts: scalar tensor theories of gravity, foundations of gravity, foundations of quantum mechanics, elementary particle physics, and cosmology. It seems that Weyl geometry continues to offer an open research potential for the foundations of physics even after the turn to the new millennium.Comment: Completely rewritten conference paper 'Beyond Einstein', Mainz Sep 2008. Preprint ELHC (Epistemology of the LHC) 2017-02, 92 pages, 1 figur