3,316 research outputs found
Small overlap monoids II: automatic structures and normal forms
We show that any finite monoid or semigroup presentation satisfying the small
overlap condition C(4) has word problem which is a deterministic rational
relation. It follows that the set of lexicographically minimal words forms a
regular language of normal forms, and that these normal forms can be computed
in linear time. We also deduce that C(4) monoids and semigroups are rational
(in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in
the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy
analogues of Kleene's theorem, and admit decision algorithms for the rational
subset and finitely generated submonoid membership problems. We also prove some
automata-theoretic results which may be of independent interest.Comment: 17 page
Recognizing pro-R closures of regular languages
Given a regular language L, we effectively construct a unary semigroup that
recognizes the topological closure of L in the free unary semigroup relative to
the variety of unary semigroups generated by the pseudovariety R of all finite
R-trivial semigroups. In particular, we obtain a new effective solution of the
separation problem of regular languages by R-languages
Heavy-tailed targets and (ab)normal asymptotics in diffusive motion
We investigate temporal behavior of probability density functions (pdfs) of
paradigmatic jump-type and continuous processes that, under confining regimes,
share common heavy-tailed asymptotic (target) pdfs. Namely, we have shown that
under suitable confinement conditions, the ordinary Fokker-Planck equation may
generate non-Gaussian heavy-tailed pdfs (like e.g. Cauchy or more general
L\'evy stable distribution) in its long time asymptotics. For diffusion-type
processes, our main focus is on their transient regimes and specifically the
crossover features, when initially infinite number of the pdf moments drops
down to a few or none at all. The time-dependence of the variance (if in
existence), with , in principle may be
interpreted as a signature of sub-, normal or super-diffusive behavior under
confining conditions; the exponent is generically well defined in
substantial periods of time. However, there is no indication of any universal
time rate hierarchy, due to a proper choice of the driver and/or external
potential.Comment: Major revisio
Fragmentation arising from a distributional initial condition
A standard model for pure fragmentation is subjected to an initial condition of Dirac-delta type. Results for a corresponding abstract Cauchy problem are derived via the theory of equicontinuous semigroups of operators on locally convex spaces. An explicit solution is obtained for the case of a power-law kernel. Rigorous justification is thereby provided for results obtained more formally by Ziff and McGrady. Copyright © 2010 John Wiley & Sons, Ltd
- …