Artificial Intelligence • Computational Mathematics • Computational Number Theory • Discrete Dynamical Systems

Researcher with a background in software engineering, databases, artificial intelligence, digital image processing, scientific computing, and embedded systems.

Currículo Lattes • Google Scholar • ORCID • LinkedIn • CV

I am Renato Augusto Tavares, a computer scientist and researcher interested in the intersection of Artificial Intelligence, scientific computing, computational mathematics, and mathematical research.

My academic background includes a degree in Systems Analysis and Development, postgraduate specialization in Databases with emphasis on Big Data, an MBA in Software Engineering with Artificial Intelligence, and ongoing master's studies in Digital Image Processing.

My current research is divided into two complementary directions:

1. Artificial Intelligence and computational applications, especially Machine Learning, Deep Learning, Computer Vision, Digital Image Processing, precision agriculture, medical image analysis, autonomous systems, drones, and data-intensive scientific applications.
2. Mathematics and computational mathematics, with particular emphasis on computational number theory, discrete dynamical systems, 3-adic dynamics, the (3n+1) problem, asymptotic analysis, functional equations, Mellin/Laplace transforms, mathematical experimentation, and reproducible computer-assisted research.

A significant part of my current mathematical work is devoted to questions arising from the Collatz / (3n+1) dynamics and the program developed by Günther J. Wirsching for positive predecessor density.

I am especially interested in research where rigorous mathematics and high-performance computation reinforce each other: exact enumeration, computer-assisted proofs, interval arithmetic, symbolic and numerical verification, reproducible computational experiments, and AI-assisted exploration of mathematical structures.

• Computational Number Theory
• Discrete Dynamical Systems
• (3n+1) / Collatz Dynamics
• 3-adic Dynamics
• Mathematical Experimentation
• Computer-Assisted Mathematics
• Asymptotic Analysis
• Functional and Functional-Differential Equations
• Mellin and Laplace Transforms
• Fourier Analysis
• Probabilistic and Combinatorial Methods
• Scientific Computing
• Reproducible Computational Research

• Artificial Intelligence
• Machine Learning
• Deep Learning
• Computer Vision
• Digital Image Processing
• Explainable and Applied AI
• AI-assisted Scientific Discovery
• AI-assisted Mathematical Research

• Precision Agriculture
• Remote Sensing
• Drones and Autonomous Systems
• Medical Image Analysis
• Real-time Data Processing
• Large-scale Data Analysis
• Embedded and Low-power Systems

Wirsching's Conjecture 3, and the Periodic Correction to Berg–Krüppel's Density Asymptotic

This work studies one of the three conjectures appearing in Wirsching's program for positive predecessor density in the (3n+1) problem.

The research develops the asymptotic structure of the invariant density through its exact log-Laplace transform and identifies a log-periodic correction. The correction is expressed through a Fourier expansion with coefficients involving the Gamma and Riemann zeta functions.

Main themes include:

• invariant densities associated with (3n+1) predecessor dynamics;
• Mellin-transform analysis;
• saddle-point asymptotics;
• log-periodic phenomena;
• explicit Fourier coefficients involving (\Gamma) and (\zeta);
• rigorous non-constancy of the periodic correction;
• computer-assisted interval-arithmetic certificates;
• analysis of Wirsching's Conjecture 3 and of the stronger unrestricted asymptotic.

Research repository:
https://github.com/faculdade/wirsching-conjecture3-proof

Archived computational materials:
https://doi.org/10.5281/zenodo.21854549

Wirsching's Weak Covering Conjecture: An Extended Computation, a Conditional Bound, and the Structure of What Resists It

This project studies the Weak Covering Conjecture arising from the (3)-adic predecessor-set framework for the (3n+1) map.

The work combines exact computation and theoretical analysis of the covering budget (j^*(l)), including:

• exact computation of covering thresholds;
• large-scale bitset-based dynamic programming;
• analysis of the excess (e(l)=j^*(l)-l\log_4 3);
• unconditional lower bounds;
• mean-payoff-game relaxations;
• Fourier-analytic obstructions;
• structure of uncovered residue classes;
• doubling-chain phenomena;
• congruence restrictions on last holdouts;
• reproducible computational verification.

The project also provides source code, data, certificates, and archived computational artifacts intended to make the numerical claims independently reproducible.

Research repository:
https://github.com/faculdade/weak-covering-conjecture

Archived computational materials:
https://doi.org/10.5281/zenodo.21877643

I am interested in problems for which computation is not merely used to generate examples, but as part of a disciplined mathematical workflow.

My preferred research methodology combines:

• rigorous mathematical formulation;
• explicit separation between theorems, conditional results, empirical observations, and open questions;
• independent implementations whenever feasible;
• exact arithmetic or certified numerical computation when required;
• public source code and datasets;
• archived versions of research software and computational evidence;
• reproducible experiments;
• systematic falsification of candidate hypotheses before promoting them to conjectures.

For difficult mathematical problems, I also investigate the use of multi-agent artificial intelligence systems as tools for literature analysis, conjecture generation, symbolic manipulation, computational experimentation, proof auditing, and research orchestration.

My broader research background remains strongly connected to AI and image processing.

Current and previous interests include:

• supervised and unsupervised Machine Learning;
• Deep Neural Networks;
• Computer Vision;
• Digital Image Processing;
• feature extraction and image classification;
• object detection and segmentation;
• medical image analysis;
• precision-agriculture imaging;
• remote sensing;
• drone-based acquisition systems;
• multimodal and real-time data analysis.

I am particularly interested in the integration of AI, mathematical modeling, and high-performance computational methods for scientific and engineering applications.

• Master's studies in Digital Image Processing — in progress
• MBA in Software Engineering with Artificial Intelligence
• Postgraduate specialization in Databases with emphasis on Big Data
• Degree in Systems Analysis and Development

Detailed institutional and chronological information is available in my Currículo Lattes.

Python • C • C++ • Java • JavaScript • PHP • Shell

Python • LaTeX • Git • Linux • Arbitrary-precision arithmetic • Interval arithmetic • Numerical computing

PostgreSQL • MySQL • MariaDB • MongoDB • SQLite • Redis

Software Architecture • APIs • Distributed Systems • Web Applications • Automation • Data Pipelines

Atmel / AVR • Arduino • LoRa • Embedded Systems • Low-power Electronics • PCB Design • Altium Designer

I aim to make computational research independently verifiable whenever possible.

For mathematical and computational projects, I prioritize:

• version-controlled source code;
• deterministic scripts;
• explicit execution instructions;
• stored computational outputs when recomputation is expensive;
• independent implementations for cross-validation;
• persistent archival through DOI-based repositories;
• numerical certificates when floating-point computation contributes to a mathematical claim.

Research repositories are organized so that computational claims can be traced back to the scripts and data that produced them.

GitHub is used primarily as a platform for:

• research source code;
• mathematical experiments;
• reproducibility packages;
• scientific software;
• AI and computer-vision projects;
• data-processing tools;
• embedded-system projects.

Selected repositories are pinned on this profile according to their current research relevance.

I am open to genuine scientific collaboration in areas where my background can contribute substantially to the research question, particularly:

• Artificial Intelligence
• Computer Vision
• Digital Image Processing
• Scientific Computing
• Computational Mathematics
• Computational Number Theory
• Mathematical Experimentation
• Reproducible Computational Research
• Data-intensive scientific applications

Authorship in collaborative research should reflect a substantive intellectual or scientific contribution to the work.

• ORCID: https://orcid.org/0009-0002-0196-3311
• Google Scholar: https://scholar.google.com/citations?user=5L2RbhMAAAAJ&hl=en
• Currículo Lattes: https://lattes.cnpq.br/9521612341950206
• LinkedIn: https://www.linkedin.com/in/java
• GitHub: https://github.com/rat

For academic communication, institutional contact information can also be found in my current manuscripts and academic records.

Wirsching's Conjecture 3, and the Periodic Correction to Berg–Krüppel's Density Asymptotic

• 📄 Preprint: Figshare
• 💾 Code and reproducibility materials: GitHub
• 📦 Archived computational materials: Zenodo

Wirsching's Weak Covering Conjecture: An Extended Computation, a Conditional Bound, and the Structure of What Resists It

• 📄 Preprint: Figshare
• 💾 Code and reproducibility materials: GitHub
• 📦 Archived computational materials: Zenodo

_{Artificial Intelligence • Mathematics • Scientific Computing • Reproducible Research}