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Josep M. Brunat

Publications and source records attributed to Josep M. Brunat.

8 recordsLinked to original sources
arXiv2022

Periodicity of power Fibonacci sequences modulus a Fibonacci number

Let ${\mathcal F}=(F_i:i\ge 0)$ be the sequence of Fibonacci numbers, and $j$ and $e$ be non negative integers. We study the periodicity of the power Fibonacci sequences ${\mathcal F}^e(F_j)=(F_i^e\pmod{F_j}: i\ge 0)$. It is shown that for every $j,e\ge 1$ the sequence ${\mathcal F}^e(F_j)$ is periodic and its periodicity is computed. The result was previously known for ${\mathcal F}(F_j)$; that is, for $e=1$. For $e\in \{1, 2\}$, the values of the normalized residues $ρ_i\equiv F_i^e\pmod{F_j}$ with $0\le ρ_i<F_j-1$ are obtained.

math.NT↗
arXiv2021

A problem on concatenated integers

Motivated by a WhattsApp message, we find out the integers $x> y\ge 1$ such that $(x+1)/(y+1)=(x\circ(y+1))/(y\circ (x+1))$, where $\circ$ means the concatenation of the strings of two natural numbers (for instance $783\circ 56=78356$). The discussion involves the equation $x(x+1)=10y(y+1)$, a slight variation of Pell's equation related to the arithmetic of the Dedekind ring $\mathbb{Z}[\sqrt{10}]$. We obtain the infinite sequence $\mathcal{S}=\{(x_n,y_n)\}_{n\ge 1}$ of all the solutions of the equation $x(x+1)=10y(y+1)$, which tourn out to have limit $1/\sqrt{10}$. The solutions of the initial problem on concatenated integers form the infinite subsequence of $\mathcal{S}$ formed by the pairs $(x_n,y_n)$ such that $x_n$ has one more digit that $y_n$.

math.NT↗
arXiv2017

Small and Large Weights in Steinhaus Triangles

Let $\{0=w_0<w_1<w_2<\ldots<w_m\}$ be the set of weights of binary Steinhaus triangles of size $n$, and let $W_i$ be the set of sequences in $\mathbb{F}_2^n$ that generate triangles of weight $w_i$. We obtain the values of $w_i$ and the corresponding sets $W_i$ for $i\in\{1,2,3,m\}$, and partial results about $w_{m-1}$ and $W_{m-1}$.

math.CO↗
arXiv2012

Shaping Communities out of Triangles

Community detection has arisen as one of the most relevant topics in the field of graph data mining due to its importance in many fields such as biology, social networks or network traffic analysis. The metrics proposed to shape communities are generic and follow two approaches: maximizing the internal density of such communities or reducing the connectivity of the internal vertices with those outside the community. However, these metrics take the edges as a set and do not consider the internal layout of the edges in the community. We define a set of properties oriented to social networks that ensure that communities are cohesive, structured and well defined. Then, we propose the Weighted Community Clustering (WCC), which is a community metric based on triangles. We proof that analyzing communities by triangles gives communities that fulfill the listed set of properties, in contrast to previous metrics. Finally, we experimentally show that WCC correctly captures the concept of community in social networks using real and syntethic datasets, and compare statistically some of the most relevant community detection algorithms in the state of the art.

cs.SI↗
arXiv2006

A polynomial generalization of the power-compositions determinant

Let $C(n,p)$ be the set of $p$-compositions of an integer $n$, i.e., the set of $p$-tuples $\bmα=(α_1,...,α_p)$ of nonnegative integers such that $α_1+...+α_p=n$, and $\mathbf{x}=(x_1,...,x_p)$ a vector of indeterminates. For $\bmα$ and ${\bmβ}$ two $p$-compositions of $n$, define $(\mathbf{x}+\bmα)^{\bmβ} = (x_1+α_1)^{β_1}... x_p+α_p)^{β_p}$. In this paper we prove an explicit formula for the determinant $\det_{\bmα,{\bmβ}\in C(n,p)}((\mathbf{x}+\bmα)^{\bmβ})$. In the case $x_1=...=x_p$ the formula gives a proof of a conjecture by C.~Krattenthaler.

math.CO↗
arXiv2006

On polynomial digraphs

Let $Φ(x,y)$ be a bivariate polynomial with complex coefficients. The zeroes of $Φ(x,y)$ are given a combinatorial structure by considering them as arcs of a directed graph $G(Φ)$. This paper studies some relationship between the polynomial $Φ(x,y)$ and the structure of $G(Φ)$.

math.AC↗
arXiv2006

The characteristic ideal of a finite, connected, regular graph

Let $Φ(x,y)\in\mathbb{C}[x,y]$ be a symmetric polynomial of partial degree $d$. The graph $G(Φ)$ is defined by taking $\mathbb{C}$ as set of vertices and the points of $\mathbb{V}(Φ(x,y))$ as edges. We study the following problem: given a finite, connected, $d$-regular graph $H$, find the polynomials $Φ(x,y)$ such that $G(Φ)$ has some connected component isomorphic to $H$ and, in this case, if $G(Φ)$ has (almost) all components isomorphic to $H$. The problem is solved by associating to $H$ a characteristic ideal which offers a new perspective to the conjecture formulated in a previous paper, and allows to reduce its scope. In the second part, we determine the characteristic ideal for cycles of lengths $\le 5$ and for complete graphs of order $\le 6$. This results provide new evidence for the conjecture.

math.AC↗