2,620 research outputs found
The Mermin fixed point
The most efficient known method for solving certain computational problems is
to construct an iterated map whose fixed points are by design the problem's
solution. Although the origins of this idea go back at least to Newton, the
clearest expression of its logical basis is an example due to Mermin. A
contemporary application in image recovery demonstrates the power of the
method.Comment: Contribution to Mermin Festschrift; 8 pages, 5 figure
Stochastic Models for the 3x+1 and 5x+1 Problems
This paper discusses stochastic models for predicting the long-time behavior
of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1
problem. The stochastic models are rigorously analyzable, and yield heuristic
predictions (conjectures) for the behavior of 3x+1 orbits and 5x+1 orbits.Comment: 68 pages, 9 figures, 4 table
Benford's Law, Values of L-functions and the 3x+1 Problem
We show the leading digits of a variety of systems satisfying certain
conditions follow Benford's Law. For each system proving this involves two main
ingredients. One is a structure theorem of the limiting distribution, specific
to the system. The other is a general technique of applying Poisson Summation
to the limiting distribution. We show the distribution of values of L-functions
near the central line and (in some sense) the iterates of the 3x+1 Problem are
Benford.Comment: 25 pages, 1 figure; replacement of earlier draft (corrected some
typos, added more exposition, added results for characteristic polynomials of
unitary matrices
Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps
We introduce a new universality class of one-dimensional unimodal dissipative
maps. The new family, from now on referred to as the ()-{\it
logarithmic map}, corresponds to a generalization of the -logistic map. The
Feigenbaum-like constants of these maps are determined. It has been recently
shown that the probability density of sums of iterates at the edge of chaos of
the -logistic map is numerically consistent with a -Gaussian, the
distribution which, under appropriate constraints, optimizes the nonadditive
entropy . We focus here on the presently generalized maps to check whether
they constitute a new universality class with regard to -Gaussian attractor
distributions. We also study the generalized -entropy production per unit
time on the new unimodal dissipative maps, both for strong and weak chaotic
cases. The -sensitivity indices are obtained as well. Our results are, like
those for the -logistic maps, numerically compatible with the
-generalization of a Pesin-like identity for ensemble averages.Comment: 17 pages, 10 figures. To appear in European Physical Journal
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