1,018 research outputs found
Shift-Symmetric Configurations in Two-Dimensional Cellular Automata: Irreversibility, Insolvability, and Enumeration
The search for symmetry as an unusual yet profoundly appealing phenomenon,
and the origin of regular, repeating configuration patterns have long been a
central focus of complexity science and physics. To better grasp and understand
symmetry of configurations in decentralized toroidal architectures, we employ
group-theoretic methods, which allow us to identify and enumerate these inputs,
and argue about irreversible system behaviors with undesired effects on many
computational problems. The concept of so-called configuration shift-symmetry
is applied to two-dimensional cellular automata as an ideal model of
computation. Regardless of the transition function, the results show the
universal insolvability of crucial distributed tasks, such as leader election,
pattern recognition, hashing, and encryption. By using compact enumeration
formulas and bounding the number of shift-symmetric configurations for a given
lattice size, we efficiently calculate the probability of a configuration being
shift-symmetric for a uniform or density-uniform distribution. Further, we
devise an algorithm detecting the presence of shift-symmetry in a
configuration.
Given the resource constraints, the enumeration and probability formulas can
directly help to lower the minimal expected error and provide recommendations
for system's size and initialization. Besides cellular automata, the
shift-symmetry analysis can be used to study the non-linear behavior in various
synchronous rule-based systems that include inference engines, Boolean
networks, neural networks, and systolic arrays.Comment: 22 pages, 9 figures, 2 appendice
A functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Levy-noise
We prove a functional non-central limit theorem for jump-diffusions with
periodic coefficients driven by strictly stable Levy-processes with stability
index bigger than one. The limit process turns out to be a strictly stable Levy
process with an averaged jump-measure. Unlike in the situation where the
diffusion is driven by Brownian motion, there is no drift related enhancement
of diffusivity.Comment: Accepted to Journal of Theoretical Probabilit
The Euler-Maruyama approximation for the absorption time of the CEV diffusion
A standard convergence analysis of the simulation schemes for the hitting
times of diffusions typically requires non-degeneracy of their coefficients on
the boundary, which excludes the possibility of absorption. In this paper we
consider the CEV diffusion from the mathematical finance and show how a weakly
consistent approximation for the absorption time can be constructed, using the
Euler-Maruyama scheme
Self-Averaging Scaling Limits of Two-Frequency Wigner Distribution for Random Paraxial Waves
Two-frequency Wigner distribution is introduced to capture the asymptotic
behavior of the space-frequency correlation of paraxial waves in the radiative
transfer limits. The scaling limits give rises to deterministic transport-like
equations. Depending on the ratio of the wavelength to the correlation length
the limiting equation is either a Boltzmann-like integral equation or a
Fokker-Planck-like differential equation in the phase space. The solutions to
these equations have a probabilistic representation which can be simulated by
Monte Carlo method. When the medium fluctuates more rapidly in the longitudinal
direction, the corresponding Fokker-Planck-like equation can be solved exactly.Comment: typos correcte
Rectification of thermal fluctuations in ideal gases
We calculate the systematic average speed of the adiabatic piston and a
thermal Brownian motor, introduced in [Van den Broeck, Kawai and Meurs,
\emph{Microscopic analysis of a thermal Brownian motor}, to appear in Phys.
Rev. Lett.], by an expansion of the Boltzmann equation and compare with the
exact numerical solution.Comment: 18 page
On the flow-level stability of data networks without congestion control: the case of linear networks and upstream trees
In this paper, flow models of networks without congestion control are
considered. Users generate data transfers according to some Poisson processes
and transmit corresponding packet at a fixed rate equal to their access rate
until the entire document is received at the destination; some erasure codes
are used to make the transmission robust to packet losses. We study the
stability of the stochastic process representing the number of active flows in
two particular cases: linear networks and upstream trees. For the case of
linear networks, we notably use fluid limits and an interesting phenomenon of
"time scale separation" occurs. Bounds on the stability region of linear
networks are given. For the case of upstream trees, underlying monotonic
properties are used. Finally, the asymptotic stability of those processes is
analyzed when the access rate of the users decreases to 0. An appropriate
scaling is introduced and used to prove that the stability region of those
networks is asymptotically maximized
Growth of uniform infinite causal triangulations
We introduce a growth process which samples sections of uniform infinite
causal triangulations by elementary moves in which a single triangle is added.
A relation to a random walk on the integer half line is shown. This relation is
used to estimate the geodesic distance of a given triangle to the rooted
boundary in terms of the time of the growth process and to determine from this
the fractal dimension. Furthermore, convergence of the boundary process to a
diffusion process is shown leading to an interesting duality relation between
the growth process and a corresponding branching process.Comment: 27 pages, 6 figures, small changes, as publishe
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