17,598 research outputs found
On a Dynamical Approach to Some Prime Number Sequences
In this paper we show how the cross-disciplinary transfer of techniques from
Dynamical Systems Theory to Number Theory can be a fruitful avenue for
research. We illustrate this idea by exploring from a nonlinear and symbolic
dynamics viewpoint certain patterns emerging in some residue sequences
generated from the prime number sequence. We show that the sequence formed by
the residues of the primes modulo are maximally chaotic and, while lacking
forbidden patterns, display a non-trivial spectrum of Renyi entropies which
suggest that every block of size , while admissible, occurs with different
probability. This non-uniform distribution of blocks for contrasts
Dirichlet's theorem that guarantees equiprobability for . We then explore
in a similar fashion the sequence of prime gap residues. This sequence is again
chaotic (positivity of Kolmogorov-Sinai entropy), however chaos is weaker as we
find forbidden patterns for every block of size . We relate the onset of
these forbidden patterns with the divisibility properties of integers, and
estimate the densities of gap block residues via Hardy-Littlewood -tuple
conjecture. We use this estimation to argue that the amount of admissible
blocks is non-uniformly distributed, what supports the fact that the spectrum
of Renyi entropies is again non-trivial in this case. We complete our analysis
by applying the Chaos Game to these symbolic sequences, and comparing the IFS
attractors found for the experimental sequences with appropriate null models.Comment: 18 pages, 20 figure
Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences
The concept of symbolic sequences play important role in study of complex
systems. In the work we are interested in ultrametric structure of the set of
cyclic sequences naturally arising in theory of dynamical systems. Aimed at
construction of analytic and numerical methods for investigation of clusters we
introduce operator language on the space of symbolic sequences and propose an
approach based on wavelet analysis for study of the cluster hierarchy. The
analytic power of the approach is demonstrated by derivation of a formula for
counting of {\it two-fold de Bruijn sequences}, the extension of the notion of
de Bruijn sequences. Possible advantages of the developed description is also
discussed in context of applied
Replica Field Theory for Deterministic Models (II): A Non-Random Spin Glass with Glassy Behavior
We introduce and study a model which admits a complex landscape without
containing quenched disorder. Continuing our previous investigation we
introduce a disordered model which allows us to reconstruct all the main
features of the original phase diagram, including a low spin glass phase
and a complex dynamical behavior.Comment: 35 pages with uu figures, Roma 102
Using Dynamical Systems to Construct Infinitely Many Primes
Euclid's proof can be reworked to construct infinitely many primes, in many
different ways, using ideas from arithmetic dynamics.
After acceptance Soundararajan noted the beautiful and fast converging
formula: Comment: To appear in the American Mathematical Monthl
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