10,074 research outputs found
Sectional curvature and Weitzenb\"ock formulae
We establish a new algebraic characterization of sectional curvature bounds
and using only curvature terms in the Weitzenb\"ock
formulae for symmetric -tensors. By introducing a symmetric analogue of the
Kulkarni-Nomizu product, we provide a simple formula for such curvature terms.
We also give an application of the Bochner technique to closed -manifolds
with indefinite intersection form and or , obtaining new
insights into the Hopf Conjecture, without any symmetry assumptions.Comment: LaTeX2e, 25 pages, final version. To appear in Indiana Univ. Math.
Strongly positive curvature
We begin a systematic study of a curvature condition (strongly positive
curvature) which lies strictly between positive curvature operator and positive
sectional curvature, and stems from the work of Thorpe in the 1970s. We prove
that this condition is preserved under Riemannian submersions and Cheeger
deformations, and that most compact homogeneous spaces with positive sectional
curvature satisfy it.Comment: LaTeX2e, 26 page
Station coordinates in the standard earth 3 system and radiation-pressure perturbations from ISAGEX camera data
Simultaneous and individual camera observations of GEOS 1, GEOS 2, Pageos, and Midas 4 obtained during the International Satellite Geodesy Experiment are utilized to determine station coordinates. The Smithsonian Astrophysical Observatory Standard Earth III system of coordinates is used to tie the geometrical network to a geocentric system and as a reference for calculating satellite orbits. A solution for coordinates combining geometrical and dynamical methods is obtained, and a comparison between the solutions and terrestrial data is made. The radiation-pressure and earth-albedo perturbations for Pageos are very large, and Pageos' orbits are used to evaluate the analytical treatment of these perturbations. Residual effects, which are probably of interest to aeronomists, remain in the Pageos orbits
Symbolic Sequences and Tsallis Entropy
We address this work to investigate symbolic sequences with long-range
correlations by using computational simulation. We analyze sequences with two,
three and four symbols that could be repeated times, with the probability
distribution . For these sequences, we verified that
the usual entropy increases more slowly when the symbols are correlated and the
Tsallis entropy exhibits, for a suitable choice of , a linear behavior. We
also study the chain as a random walk-like process and observe a nonusual
diffusive behavior depending on the values of the parameter .Comment: Published in the Brazilian Journal of Physic
Critical dynamics of the k-core pruning process
We present the theory of the k-core pruning process (progressive removal of
nodes with degree less than k) in uncorrelated random networks. We derive exact
equations describing this process and the evolution of the network structure,
and solve them numerically and, in the critical regime of the process,
analytically. We show that the pruning process exhibits three different
behaviors depending on whether the mean degree of the initial network is
above, equal to, or below the threshold _c corresponding to the emergence of
the giant k-core. We find that above the threshold the network relaxes
exponentially to the k-core. The system manifests the phenomenon known as
"critical slowing down", as the relaxation time diverges when tends to
_c. At the threshold, the dynamics become critical characterized by a
power-law relaxation (1/t^2). Below the threshold, a long-lasting transient
process (a "plateau" stage) occurs. This transient process ends with a collapse
in which the entire network disappears completely. The duration of the process
diverges when tends to _c. We show that the critical dynamics of the
pruning are determined by branching processes of spreading damage. Clusters of
nodes of degree exactly k are the evolving substrate for these branching
processes. Our theory completely describes this branching cascade of damage in
uncorrelated networks by providing the time dependent distribution function of
branching. These theoretical results are supported by our simulations of the
-core pruning in Erdos-Renyi graphs.Comment: 12 pages, 10 figure
Bootstrap Percolation on Complex Networks
We consider bootstrap percolation on uncorrelated complex networks. We obtain
the phase diagram for this process with respect to two parameters: , the
fraction of vertices initially activated, and , the fraction of undamaged
vertices in the graph. We observe two transitions: the giant active component
appears continuously at a first threshold. There may also be a second,
discontinuous, hybrid transition at a higher threshold. Avalanches of
activations increase in size as this second critical point is approached,
finally diverging at this threshold. We describe the existence of a special
critical point at which this second transition first appears. In networks with
degree distributions whose second moment diverges (but whose first moment does
not), we find a qualitatively different behavior. In this case the giant active
component appears for any and , and the discontinuous transition is
absent. This means that the giant active component is robust to damage, and
also is very easily activated. We also formulate a generalized bootstrap
process in which each vertex can have an arbitrary threshold.Comment: 9 pages, 3 figure
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