36 research outputs found
Second order tangent bundles of infinite dimensional manifolds
The second order tangent bundle of a smooth manifold consists of
the equivalent classes of curves on that agree up to their acceleration. It
is known that in the case of a finite -dimensional manifold ,
becomes a vector bundle over if and only if is endowed with a linear
connection. Here we extend this result to 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