Coincidences in numbers of graph vertices corresponding to regular planar hyperbolic mosaics

The aim of this paper is to determine the elements which are in two pairs of sequences linked to the regular mosaics $\{4,5\}$ and $\{p,q\}$ on the hyperbolic plane. The problem leads to the solution of diophantine equations of certain types.Comment: 10 pages, 2 figures, Annales Mathematicae et Informaticae 43 (2014

SMARANDACHE FUNCTION JOURNAL, 1

This journal is yearly published (in the Spring or Fall) in a 300-400 pages volume, and 800-1000 copies. SNJ is a referred journal: reviewed, indexed, cited, concerning any of Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, etc

Rational approximation and arithmetic progressions

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed

Logarithm laws and shrinking target properties

We survey some of the recent developments in the study of logarithm laws and shrinking target properties for various families of dynamical systems. We discuss connections to geometry, diophantine approximation, and probability theory.Comment: This is a survey paper written following the Conference on Measures and Dyanmics on groups and homogeneous spaces, at TIFR, Mumbai, in Dec. 2007. It is in honor of Prof. S.G. Dani's 60th Birthda

Metrical Diophantine approximation for quaternions

Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.Comment: 30 pages. Some minor improvement

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

Improved convergence estimates for the Schr\"oder-Siegel problem

We reconsider the Schr\"oder-Siegel problem of conjugating an analytic map in $\mathbb{C}$ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension $n>1$. Assuming a condition which is equivalent to Bruno's one on the eigenvalues $\lambda_1,\ldots,\lambda_n$ of the linear part we show that the convergence radius $\rho$ of the conjugating transformation satisfies $\ln \rho(\lambda )\geq -C\Gamma(\lambda)+C'$ with $\Gamma(\lambda)$ characterizing the eigenvalues $\lambda$, a constant $C'$ not depending on $\lambda$ and $C=1$. This improves the previous results for $n>1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for $n=1$.Comment: 21 page

Elliptic divisibility sequences and undecidable problems about rational points

Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0) where the \forall-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the \Sigma_5-theory of Q is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of Z in Q with quantifier complexity \forall \exists, involving only one universally quantified variable. This improves the complexity of defining Z in Q in two ways, and implies that the \Sigma_3-theory, and even the \Pi_2-theory, of Q is undecidable (recall that Hilbert's Tenth Problem for Q is the question whether the \Sigma_1-theory of Q is undecidable). In short, granting the conjecture, there is a one-parameter family of hypersurfaces over Q for which one cannot decide whether or not they all have a rational point. The conjecture is related to properties of elliptic divisibility sequences on an elliptic curve and its image under rational 2-descent, namely existence of primitive divisors in suitable residue classes, and we discuss how to prove weaker-in-density versions of the conjecture and present some heuristics.Comment: 39 pages, uses calrsfs. 3rd version: many small changes, change of titl