1,127 research outputs found
Arithmetic Dynamics
This survey paper is aimed to describe a relatively new branch of symbolic
dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic
expansions of reals and vectors that have a "dynamical" sense. This means
precisely that they (semi-) conjugate a given continuous (or
measure-preserving) dynamical system and a symbolic one. The classes of
dynamical systems and their codings considered in the paper involve: (1)
Beta-expansions, i.e., the radix expansions in non-integer bases; (2)
"Rotational" expansions which arise in the problem of encoding of irrational
rotations of the circle; (3) Toral expansions which naturally appear in
arithmetic symbolic codings of algebraic toral automorphisms (mostly
hyperbolic).
We study ergodic-theoretic and probabilistic properties of these expansions
and their applications. Besides, in some cases we create "redundant"
representations (those whose space of "digits" is a priori larger than
necessary) and study their combinatorics.Comment: 45 pages in Latex + 3 figures in ep
Sums of residues on algebraic surfaces and application to coding theory
In this paper, we study residues of differential 2-forms on a smooth
algebraic surface over an arbitrary field and give several statements about
sums of residues. Afterwards, using these results we construct
algebraic-geometric codes which are an extension to surfaces of the well-known
differential codes on curves. We also study some properties of these codes and
extend to them some known properties for codes on curves.Comment: 31 page
On the proximity of large primes
By a sphere-packing argument, we show that there are infinitely many pairs of
primes that are close to each other for some metrics on the integers. In
particular, for any numeration basis , we show that there are infinitely
many pairs of primes the base expansion of which differ in at most two
digits. Likewise, for any fixed integer there are infinitely many pairs of
primes, the first digits of which are the same. In another direction, we
show that, there is a constant depending on such that for infinitely
many integers there are at least primes which differ from
by at most one base digit
A p-Adic Model of DNA Sequence and Genetic Code
Using basic properties of p-adic numbers, we consider a simple new approach
to describe main aspects of DNA sequence and genetic code. Central role in our
investigation plays an ultrametric p-adic information space which basic
elements are nucleotides, codons and genes. We show that a 5-adic model is
appropriate for DNA sequence. This 5-adic model, combined with 2-adic distance,
is also suitable for genetic code and for a more advanced employment in
genomics. We find that genetic code degeneracy is related to the p-adic
distance between codons.Comment: 13 pages, 2 table
Genetic Code and Number Theory
Living organisms are the most complex, interesting and significant objects
regarding all substructures of the universe. Life science is regarded as a
science of the 21st century and one can expect great new discoveries in the
near futures. This article contains an introductory brief review of genetic
information, its coding and translation of genes to proteins through the
genetic code. Some theoretical approaches to the modelling of the genetic code
are presented. In particular, connection of the genetic code with number theory
is considered and the role of -adic numbers is underlined.Comment: 15 pages. To apper in "Modern Topics in Science", a book of invited
papers (Eds. R. Constantinescu, G. Djordjevic, Lj. Nesic
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