5,881 research outputs found
Fractional Patlak-Keller-Segel equations for chemotactic superdiffusion
The long range movement of certain organisms in the presence of a
chemoattractant can be governed by long distance runs, according to an
approximate Levy distribution. This article clarifies the form of biologically
relevant model equations: We derive Patlak-Keller-Segel-like equations
involving nonlocal, fractional Laplacians from a microscopic model for cell
movement. Starting from a power-law distribution of run times, we derive a
kinetic equation in which the collision term takes into account the long range
behaviour of the individuals. A fractional chemotactic equation is obtained in
a biologically relevant regime. Apart from chemotaxis, our work has
implications for biological diffusion in numerous processes.Comment: 20 pages, 4 figures, to appear in SIAM Journal on Applied Mathematic
How to Couple from the Past Using a Read-Once Source of Randomness
We give a new method for generating perfectly random samples from the
stationary distribution of a Markov chain. The method is related to coupling
from the past (CFTP), but only runs the Markov chain forwards in time, and
never restarts it at previous times in the past. The method is also related to
an idea known as PASTA (Poisson arrivals see time averages) in the operations
research literature. Because the new algorithm can be run using a read-once
stream of randomness, we call it read-once CFTP. The memory and time
requirements of read-once CFTP are on par with the requirements of the usual
form of CFTP, and for a variety of applications the requirements may be
noticeably less. Some perfect sampling algorithms for point processes are based
on an extension of CFTP known as coupling into and from the past; for
completeness, we give a read-once version of coupling into and from the past,
but it remains unpractical. For these point process applications, we give an
alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure
Limits on Fundamental Limits to Computation
An indispensable part of our lives, computing has also become essential to
industries and governments. Steady improvements in computer hardware have been
supported by periodic doubling of transistor densities in integrated circuits
over the last fifty years. Such Moore scaling now requires increasingly heroic
efforts, stimulating research in alternative hardware and stirring controversy.
To help evaluate emerging technologies and enrich our understanding of
integrated-circuit scaling, we review fundamental limits to computation: in
manufacturing, energy, physical space, design and verification effort, and
algorithms. To outline what is achievable in principle and in practice, we
recall how some limits were circumvented, compare loose and tight limits. We
also point out that engineering difficulties encountered by emerging
technologies may indicate yet-unknown limits.Comment: 15 pages, 4 figures, 1 tabl
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