26,470 research outputs found
Effective Capacity and Randomness of Closed Sets
We investigate the connection between measure and capacity for the space of
nonempty closed subsets of {0,1}*. For any computable measure, a computable
capacity T may be defined by letting T(Q) be the measure of the family of
closed sets which have nonempty intersection with Q. We prove an effective
version of Choquet's capacity theorem by showing that every computable capacity
may be obtained from a computable measure in this way. We establish conditions
that characterize when the capacity of a random closed set equals zero or is
>0. We construct for certain measures an effectively closed set with positive
capacity and with Lebesgue measure zero
Algorithmic Randomness and Capacity of Closed Sets
We investigate the connection between measure, capacity and algorithmic
randomness for the space of closed sets. For any computable measure m, a
computable capacity T may be defined by letting T(Q) be the measure of the
family of closed sets K which have nonempty intersection with Q. We prove an
effective version of Choquet's capacity theorem by showing that every
computable capacity may be obtained from a computable measure in this way. We
establish conditions on the measure m that characterize when the capacity of an
m-random closed set equals zero. This includes new results in classical
probability theory as well as results for algorithmic randomness. For certain
computable measures, we construct effectively closed sets with positive
capacity and with Lebesgue measure zero. We show that for computable measures,
a real q is upper semi-computable if and only if there is an effectively closed
set with capacity q
The interplay of classes of algorithmically random objects
We study algorithmically random closed subsets of , algorithmically
random continuous functions from to , and algorithmically
random Borel probability measures on , especially the interplay
between these three classes of objects. Our main tools are preservation of
randomness and its converse, the no randomness ex nihilo principle, which say
together that given an almost-everywhere defined computable map between an
effectively compact probability space and an effective Polish space, a real is
Martin-L\"of random for the pushforward measure if and only if its preimage is
random with respect to the measure on the domain. These tools allow us to prove
new facts, some of which answer previously open questions, and reprove some
known results more simply.
Our main results are the following. First we answer an open question of
Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that
is a random closed set if and only if it is the
set of zeros of a random continuous function on . As a corollary we
obtain the result that the collection of random continuous functions on
is not closed under composition. Next, we construct a computable
measure on the space of measures on such that
is a random closed set if and only if
is the support of a -random measure. We also establish a
correspondence between random closed sets and the random measures studied by
Culver in previous work. Lastly, we study the ranges of random continuous
functions, showing that the Lebesgue measure of the range of a random
continuous function is always contained in
An Upper Bound on Multi-hop Transmission Capacity with Dynamic Routing Selection
This paper develops upper bounds on the end-to-end transmission capacity of
multi-hop wireless networks. Potential source-destination paths are dynamically
selected from a pool of randomly located relays, from which a closed-form lower
bound on the outage probability is derived in terms of the expected number of
potential paths. This is in turn used to provide an upper bound on the number
of successful transmissions that can occur per unit area, which is known as the
transmission capacity. The upper bound results from assuming independence among
the potential paths, and can be viewed as the maximum diversity case. A useful
aspect of the upper bound is its simple form for an arbitrary-sized network,
which allows insights into how the number of hops and other network parameters
affect spatial throughput in the non-asymptotic regime. The outage probability
analysis is then extended to account for retransmissions with a maximum number
of allowed attempts. In contrast to prevailing wisdom, we show that
predetermined routing (such as nearest-neighbor) is suboptimal, since more hops
are not useful once the network is interference-limited. Our results also make
clear that randomness in the location of relay sets and dynamically varying
channel states is helpful in obtaining higher aggregate throughput, and that
dynamic route selection should be used to exploit path diversity.Comment: 14 pages, 5 figures, accepted to IEEE Transactions on Information
Theory, 201
On the Catalyzing Effect of Randomness on the Per-Flow Throughput in Wireless Networks
This paper investigates the throughput capacity of a flow crossing a
multi-hop wireless network, whose geometry is characterized by general
randomness laws including Uniform, Poisson, Heavy-Tailed distributions for both
the nodes' densities and the number of hops. The key contribution is to
demonstrate \textit{how} the \textit{per-flow throughput} depends on the
distribution of 1) the number of nodes inside hops' interference sets, 2)
the number of hops , and 3) the degree of spatial correlations. The
randomness in both 's and is advantageous, i.e., it can yield larger
scalings (as large as ) than in non-random settings. An interesting
consequence is that the per-flow capacity can exhibit the opposite behavior to
the network capacity, which was shown to suffer from a logarithmic decrease in
the presence of randomness. In turn, spatial correlations along the end-to-end
path are detrimental by a logarithmic term
Fourier spectra of measures associated with algorithmically random Brownian motion
In this paper we study the behaviour at infinity of the Fourier transform of
Radon measures supported by the images of fractal sets under an algorithmically
random Brownian motion. We show that, under some computability conditions on
these sets, the Fourier transform of the associated measures have, relative to
the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity.
The argument relies heavily on a direct characterisation, due to Asarin and
Pokrovskii, of algorithmically random Brownian motion in terms of the prefix
free Kolmogorov complexity of finite binary sequences. The study also
necessitates a closer look at the potential theory over fractals from a
computable point of view.Comment: 24 page
Fundamentals of Inter-cell Overhead Signaling in Heterogeneous Cellular Networks
Heterogeneous base stations (e.g. picocells, microcells, femtocells and
distributed antennas) will become increasingly essential for cellular network
capacity and coverage. Up until now, little basic research has been done on the
fundamentals of managing so much infrastructure -- much of it unplanned --
together with the carefully planned macro-cellular network. Inter-cell
coordination is in principle an effective way of ensuring different
infrastructure components behave in a way that increases, rather than
decreases, the key quality of service (QoS) metrics. The success of such
coordination depends heavily on how the overhead is shared, and the rate and
delay of the overhead sharing. We develop a novel framework to quantify
overhead signaling for inter-cell coordination, which is usually ignored in
traditional 1-tier networks, and assumes even more importance in multi-tier
heterogeneous cellular networks (HCNs). We derive the overhead quality contour
for general K-tier HCNs -- the achievable set of overhead packet rate, size,
delay and outage probability -- in closed-form expressions or computable
integrals under general assumptions on overhead arrivals and different overhead
signaling methods (backhaul and/or wireless). The overhead quality contour is
further simplified for two widely used models of overhead arrivals: Poisson and
deterministic arrival process. This framework can be used in the design and
evaluation of any inter-cell coordination scheme. It also provides design
insights on backhaul and wireless overhead channels to handle specific overhead
signaling requirements.Comment: 21 pages, 9 figure
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