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José Manuel Rodríguez Caballero

Publications and source records attributed to José Manuel Rodríguez Caballero.

15 recordsLinked to original sources
arXiv2026

Complex Bounded Operators in Isabelle/HOL

We present a formalization of bounded operators on complex vector spaces in Isabelle/HOL. Our formalization contains material on complex vector spaces (normed spaces, Banach spaces, Hilbert spaces) that complements and goes beyond the developments of real vectors spaces in the Isabelle/HOL standard library. We define the type of bounded operators between complex vector spaces (cblinfun) and develop the theory of unitaries, projectors, extension of bounded linear functions (BLT theorem), adjoints, Loewner order, closed subspaces and more. For the finite-dimensional case, we provide code generation support by identifying finite-dimensional operators with matrices as formalized in the Jordan_Normal_Form AFP entry.

cs.LO↗
arXiv2023

On prime numbers of the form $2^n \pm k$

Consider the set $\mathcal{K}$ of integers $k$ for which there are infinitely many primes $p$ such that $p+k$ is a power of $2$. The aim of this paper is to show a relationship between $\mathcal{K}$ and the limits points of some set rational numbers related to a sequence of polynomials $C_n(q)$ introduced by Kassel and Reutenauer [KasselReutenauer].

math.NT↗
arXiv2023

A relationship between the ideals of $\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right]$ and the Fibonacci numbers

Let $C_n(q)$ be the number of ideals of codimension $n$ of $\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right]$, where $\mathbb{F}_q$ is the finite field with $q$ elements. Kassel and Reutenauer [KasselReutenauer2015A] proved that $C_n(q)$ is a polynomial in $q$ for any fixed value of $n \geq 1$. For $q = \frac{3+\sqrt{5}}{2}$, this combinatorial interpretation of $C_n(q)$ is lost. Nevertheless, an unexpected connexion with Fibonacci numbers appears. Let $f_n$ be the $n$-th Fibonacci number (following the convention $f_0 = 0$, $f_1 = 1$). Define the series $$ F(t) = \sum_{n \geq 1} f_{2n}\,t^n. $$ We will prove that for each $n \geq 1$, $$ C_n\left( \frac{3+\sqrt{5}}{2}\right) = λ_n \, \left(f_{2n} \frac{3+\sqrt{5}}{2} - f_{2n-2} \right) , $$ where the integers $λ_n \geq 0$ are given by the following generating function $$ \prod_{m \geq 1} \left(1+F\left( t^m\right)\right) = 1 + \sum_{n \geq 1} λ_n\,t^n. $$

math.NT↗
arXiv2023

Middle divisors and $σ$-palindromic Dyck words

Given a real number $λ> 1$, we say that $d|n$ is a $λ$-middle divisor of $n$ if $$ \sqrt{\frac{n}λ} < d \leq \sqrt{λn}. $$ We will prove that there are integers having an arbitrarily large number of $λ$-middle divisors. Consider the word $$ \langle\! \langle n \rangle\! \rangle_λ := w_1 w_2 ... w_k \in \{a,b\}^{\ast}, $$ given by $$ w_i := \left\{ \begin{array}{c l} a & \textrm{if } u_i \in D_n \backslash \left(λD_n\right), \\ b & \textrm{if } u_i \in \left(λD_n\right)\backslash D_n, \end{array} \right. $$ where $D_n$ is the set of divisors of $n$, $λD_n := \{λd: \quad d \in D_n\}$ and $u_1, u_2, ..., u_k$ are the elements of the symmetric difference $D_n \triangle λD_n$ written in increasing order. We will prove that the language $$ \mathcal{L}_λ := \left\{\langle\! \langle n \rangle\! \rangle_λ : \quad n \in \mathbb{Z}_{\geq 1} \right\} $$ contains Dyck words having an arbitrarily large number of centered tunnels. We will show a connection between both results.

math.NT↗
arXiv2023

Factorization of Dyck words and the distribution of the divisors of an integer

In [CaballeroHooleyDelta], we associated a Dyck word $\langle\! \langle n \rangle\! \rangle_λ$ to any pair $(n, λ)$ consisting of an integer $n \geq 1$ and a real number $λ> 1$. The goal of the present paper is to show a relationship between the factorization of $\langle\! \langle n \rangle\! \rangle_λ$ as the concatenation of irreducible Dyck words and the distribution of the divisors of $n$. In particular, we will provide a characterization of $λ$-densely divisible numbers (these numbers were introduced in [castryck2014new]).

math.NT↗
arXiv2023

Self-Similar Algebras with connections to Run-length Encoding and Rational Languages

A self-similar algebra $\left(\mathfrak{A}, ψ\right)$ is an associative algebra $\mathfrak{A}$ with a morphism of algebras $ψ: \mathfrak{A} \longrightarrow M_d \left( \mathfrak{A}\right)$, where $M_d \left( \mathfrak{A}\right)$ is the set of $d\times d$ matrices with coefficients from $\mathfrak{A}$. 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 $L(s)$ of words $u$ in $(Σ\times Σ)^*$ where $Σ=\{0,1\}$ such that $s\cdot u$ is a unit in $\mathfrak{A}$. We prove that $L(s)$ is rational and provide an asymptotic formula for the number of words of a given length in $L(s)$.

math.CO↗
arXiv2023

Relative positions of points on the real line and balanced parentheses

Consider a finite set of positive real numbers $S$. For any real number $λ> 1$, a Dyck word denoted $\langle\! \langle S \rangle\! \rangle_λ \in \{a,b\}^{\ast}$, was defined in [CaballeroWords2017] in order to compute Hooley's $Δ$-function and its generalization. The aim of this paper is to prove that, given a real number $λ> 1$, any Dyck word can be expressed as $\langle\! \langle S \rangle\! \rangle_λ$ for some finite set $S$ of positive real numbers.

math.CO↗
arXiv2023

Divisors on overlapped intervals and multiplicative functions

Consider the real numbers $$ \ell_{n,k} = \ln\left( \tfrac{3}{2}\,k+\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n} \right) $$ and the intervals $\mathcal{L}_{n,k} = \left]\ell_{n,k}-\ln 3,\ell_{n,k}\right]$. For all $n \geq 1$, define $$ \frac{L_n(q)}{q^{n-1}} = \sum_{d|n}\sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) \,q^k, $$ where $\boldsymbol{1}_{A}(x)$ is the characteristic function of the set $A$. Let $σ(n)$ be sum of divisors of $n$. We will prove that $\textbf{A002324}(n) = 4\,σ(n) - 3\,L_n(1)$ and $\textbf{A096936}(n) = L_n(-1)$, which are well-known multiplicative functions related to the number of representations of $n$ by a given quadratic form.

math.NT↗
arXiv2023

On a function introduced by Erdös and Nicolas

Erdös and Nicolas [erdos1976methodes] introduced an arithmetical function $F(n)$ related to divisors of $n$ in short intervals $\left] \frac{t}{2}, t\right]$. The aim of this note is to prove that $F(n)$ is the largest coefficient of polynomial $P_n(q)$ introduced by Kassel and Reutenauer [kassel2015counting]. We deduce that $P_n(q)$ has a coefficient larger than $1$ if and only if $2n$ is the perimeter of a Pythagorean triangle. We improve a result due to Vatne [vatne2017sequence] concerning the coefficients of $P_n(q)$.

math.NT↗
arXiv2022

A $q$-analog of Jacobi's two squares formula and its applications

We consider a $q$-analog $r_2(n, q)$ of the number of representations of an integer as a sum of two squares $r_2(n)$. This $q$-analog is generated by the expansion of a product that was studied by Kronecker and Jordan. We generalize Jacobi's two squares formula from $r_2(n)$ to $r_2(n, q)$. We characterize the signs in the coefficients of $r_2(n, q)$ using the prime factors of $n$. We use $r_2(n, q)$ to characterize the integers which are the length of the hypotenuse of a primitive Pythagorean triangle.

math.NT↗
arXiv2021

Incompatibility between 't Hooft's and Wolfram's models of quantum mechanics

Stephen Wolfram and Gerard 't Hooft developed classical models of quantum mechanics. We show that the descriptive complexity grows differently as a function of time in each model. Therefore, they cannot describe the same physical system. In addition, we propose an interpretation of the Wolfram model, which shares some characteristics with 't Hooft's model, but which involves a non-computable function.

quant-ph↗
arXiv2017

Elementary number-theoretical statements proved by Language Theory

We introduce a method to derive theorems from Elementary Number Theory by means of relationships among formal languages. Using $σ$-algebras, we define what a proof of a number-theoretical statement by Language Theory means. We prove that such a proof can be transformed into a traditional proof in $\textbf{ZFC}$. Finally, we show some examples of non-trivial number-theoretical theorems that can be proved by formal languages in a natural way. These number-theoretical results concern densely divisible numbers, semi-perimeters of Pythagorean triangles, middle divisors and partitions into consecutive parts.

math.LO↗