Banca de DEFESA: DESIANE MAIARA GOMES DOS SANTOS

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : DESIANE MAIARA GOMES DOS SANTOS
DATE: 23/04/2021
TIME: 14:30
LOCAL: Plataforma Google Meet
TITLE:

Scale-free behavior in hailstone sequences generated by the Collatz map

KEY WORDS:

avalanche, complex system, power law, Brownian motion, hailstone sequences

PAGES: 84
BIG AREA: Ciências Exatas e da Terra
AREA: Física
SUMMARY:

The Collatz conjecture, perhaps the most elementary unsolved problem in mathematics, claims that for all positive integers $n$, the map $n\mapsto n/2$ for even $n$ and $n \mapsto 3n+1$ for odd $n$ reaches 1 after a finite number of iterations. We examine the Collatz map's orbits, known as hailstone sequences, and ask whether or not they exhibit scale-invariant behavior, in analogy with certain processes observed in real physical systems. We develop an efficient way to generate orbits for extremely large $n$'s (e.g., higher than $n \approx 10^{3,000}$), allowing to statistically analyze very long sequences. We find strong evidence of a scale-free power law for the Collatz map. We analytically derive the scaling exponents, displaying excellent agreement with the numerical estimations. The scale-free sequences seen in the Collatz dynamics are consistent with geometric Brownian motion with drift, which is compatible with the validity of the Collatz conjecture. Our results lead to another conjecture (conceivably testable through direct, nonetheless very time consuming, numerical simulations): given an initial $n$, the average number of iterations needed to reach $1$ is proportional, to lowest order, to $log[n]$.

BANKING MEMBERS:
Interno - 1519258 - DORY HELIO AIRES DE LIMA ANSELMO
Externo à Instituição - ERNESTO CARNEIRO PESSOA RAPOSO - UFPE
Presidente - 1294916 - MADRAS VISWANATHAN GANDHI MOHAN
Interno - 1474380 - MARCIO ASSOLIN CORREA
Externo à Instituição - MARCOS GOMES ELEUTERIO DA LUZ - UFPR
Externo à Instituição - RICARDO VIANA - UFPR