Second order tangent bundles of infinite dimensional manifolds

The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of the equivalent classes of curves on $M$ that agree up to their acceleration. It is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. Here we extend this result to $M$ modeled on an arbitrarily chosen Banach space and more generally to those Fr\'{e}chet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite-dimensional dynamical systems.Comment: 8 page

Wave Height Characteristics in the North Atlantic Ocean: A new approach based on Statistical and Geometrical techniques

Stochastic Environmental Research and Risk AssessmentThe article of record as published may be located at http://dx.doi.org/10.1007/s00477-011-0540-2The main characteristics of the significant wave height in an area of increased interest, the north Atlantic ocean, are studied based on satellite records and corresponding simulations obtained from the numerical wave prediction model WAM. The two data sets are analyzed by means of a variety of statistical measures mainly focusing on the distributions that they form. Moreover, new techniques for the estimation and minimization of the discrepancies between the observed and modeled values are proposed based on ideas and methodologies from a relatively new branch of mathematics, information geometry. The results obtained prove that the modeled values overestimate the corresponding observations through the whole study period. On the other hand, 2-parameter Weibull distributions fit well the data in the study. However, one cannot use the same probability density function for describing the whole study area since the corresponding scale and shape parameters deviate significantly for points belonging to different regions. This variation should be taken into account in optimization or assimilation procedures, which is possible by means of information geometry techniques

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra \mbox{{\cal A}} on the groupoid \Gamma = E \times G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of \mbox{{\cal A}}, is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein's equation. The algebra \mbox{{\cal A}}, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra \mathcal{M} of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which depends on the state \phi . The same state defines the noncommutative probability measure (in the sense of Voiculescu's free probability theory). Moreover, the state \phi satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra \mbox{{\cal A}}, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not ``feel'' singularities.Comment: 28 LaTex pages. Substantially enlarged version. Improved definition of generalized Einstein's field equation