13,280 research outputs found
Middle divisors and -palindromic Dyck words
Given a real number , we say that is a -middle
divisor of if We
will prove that there are integers having an arbitrarily large number of
-middle divisors. Consider the word given by where is the set of
divisors of , and are the elements of the symmetric difference written in increasing order. We will prove that the language contains Dyck words having an
arbitrarily large number of centered tunnels. We will show a connection between
both results
On prime numbers of the form
Consider the set of integers for which there are infinitely
many primes such that is a power of . The aim of this paper is to
show a relationship between and the limits points of some set
rational numbers related to a sequence of polynomials introduced by
Kassel and Reutenauer [KasselReutenauer]
A relationship between the ideals of and the Fibonacci numbers
Let be the number of ideals of codimension of
, where is the
finite field with elements. Kassel and Reutenauer [KasselReutenauer2015A]
proved that is a polynomial in for any fixed value of .
For , this combinatorial interpretation of
is lost. Nevertheless, an unexpected connexion with Fibonacci numbers appears.
Let be the -th Fibonacci number (following the convention ,
). Define the series
We will prove that for each ,
where the integers are given by the following generating function
\prod_{m \geq 1} \left(1+F\left( t^m\right)\right) = 1 + \sum_{n \geq 1}
\lambda_n\,t^n. $
On a function introduced by Erd\"{o}s and Nicolas
Erd\"os and Nicolas [erdos1976methodes] introduced an arithmetical function
related to divisors of in short intervals . The aim of this note is to prove that is the largest
coefficient of polynomial introduced by Kassel and Reutenauer
[kassel2015counting]. We deduce that has a coefficient larger than
if and only if is the perimeter of a Pythagorean triangle. We improve a
result due to Vatne [vatne2017sequence] concerning the coefficients of
Factorization of Dyck words and the distribution of the divisors of an integer
In [CaballeroHooleyDelta], we associated a Dyck word to any pair consisting of an
integer and a real number . The goal of the present
paper is to show a relationship between the factorization of as the concatenation of irreducible Dyck words
and the distribution of the divisors of . In particular, we will provide a
characterization of -densely divisible numbers (these numbers were
introduced in [castryck2014new])
Relative positions of points on the real line and balanced parentheses
Consider a finite set of positive real numbers . For any real number
, a Dyck word denoted , was defined in [CaballeroWords2017] in
order to compute Hooley's -function and its generalization. The aim of
this paper is to prove that, given a real number , any Dyck word
can be expressed as for some
finite set of positive real numbers
Self-Similar Algebras with connections to Run-length Encoding and Rational Languages
A self-similar algebra is an associative
algebra with a morphism of algebras , where is the set of matrices with coefficients from
. We study the connection between self-similar algebras with
run-length encoding and rational languages. In particular, we provide a curious
relationship between the eigenvalues of a sequence of matrices related to a
specific self-similar algebra and the smooth words over a 2-letter alphabet. We
also consider the language of words in
where such that is a unit in . We
prove that is rational and provide an asymptotic formula for the number
of words of a given length in
Relativistic Quantum Scattering of High Energy Fermions in the Presence of Phase Transition
We study the high energy behaviour of fermions hitting a general wall caused
by a first-order phase transition. The wall profile is introduced through a
general analytic function. The reflection coefficient is computed in the high
energy limit and expressed in terms of the poles of the wall profile function.
It is shown that the leading singularity gives the high energy behaviour.Comment: 10 page
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