142,185 research outputs found
Dynamical entropy in Banach spaces
We introduce a version of Voiculescu-Brown approximation entropy for
isometric automorphisms of Banach spaces and develop within this framework the
connection between dynamics and the local theory of Banach spaces discovered by
Glasner and Weiss. Our fundamental result concerning this contractive
approximation entropy, or CA entropy, characterizes the occurrence of positive
values both geometrically and topologically. This leads to various
applications; for example, we obtain a geometric description of the topological
Pinsker factor and show that a C*-algebra is type I if and only if every
multiplier inner *-automorphism has zero CA entropy. We also examine the
behaviour of CA entropy under various product constructions and determine its
value in many examples, including isometric automorphisms of l_p spaces and
noncommutative tensor product shifts.Comment: 33 pages; unified approach to last three sections give
Beyond Geometry : Towards Fully Realistic Wireless Models
Signal-strength models of wireless communications capture the gradual fading
of signals and the additivity of interference. As such, they are closer to
reality than other models. However, nearly all theoretic work in the SINR model
depends on the assumption of smooth geometric decay, one that is true in free
space but is far off in actual environments. The challenge is to model
realistic environments, including walls, obstacles, reflections and anisotropic
antennas, without making the models algorithmically impractical or analytically
intractable.
We present a simple solution that allows the modeling of arbitrary static
situations by moving from geometry to arbitrary decay spaces. The complexity of
a setting is captured by a metricity parameter Z that indicates how far the
decay space is from satisfying the triangular inequality. All results that hold
in the SINR model in general metrics carry over to decay spaces, with the
resulting time complexity and approximation depending on Z in the same way that
the original results depends on the path loss term alpha. For distributed
algorithms, that to date have appeared to necessarily depend on the planarity,
we indicate how they can be adapted to arbitrary decay spaces.
Finally, we explore the dependence on Z in the approximability of core
problems. In particular, we observe that the capacity maximization problem has
exponential upper and lower bounds in terms of Z in general decay spaces. In
Euclidean metrics and related growth-bounded decay spaces, the performance
depends on the exact metricity definition, with a polynomial upper bound in
terms of Z, but an exponential lower bound in terms of a variant parameter phi.
On the plane, the upper bound result actually yields the first approximation of
a capacity-type SINR problem that is subexponential in alpha
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