Mathematics

I always thought that actions matter more than words. The ultimate expression of this idea is the stock market and, more recently, prediction markets (such as Polymarket). For example, you can bet on a wide variety of topics that stretch from who might win an election to how many tweets Elon Musk will post in a single day. Over the past few years, public trust in institutions and journalism has eroded [1], and this distrust has fueled the rise of new financial instruments like prediction markets. This data manipulation became clear during the last US election, where a big gap emerged between conventional polling data and the probabilities implied by market activity [2]. ...

This post is an extract of my Master’s Thesis in Mathematics, it can be found in this GitHub repository for a complete and autocomplete document: repo. In our continuous pursuit of enlightenment, knowledge, and understanding of the world around us, we turn to the tool of mathematics. For thousands of years, mathematicians have sought to translate our sensory experiences into a language that captures their essence in the most abstract, universal, and general form. ...

This work is part of project done for a class in the MSc Applied Mathematics in the Autonomous University of Madrid. You can find the complete work here. In this manuscript, we explore the application of dimensionality reduction algorithms to real-world datasets within the context of functional data analysis. We establish several theoretical results related to Principal Component Analysis (PCA) for functional data and introduce a novel variation, Fourier PCA, inspired by Fourier theory. Additionally, we extend Kernel PCA to the functional data setting by proposing new kernels, adapted from well-known finite-dimensional counterparts, and provide theoretical foundations for their use. Finally, we evaluate and compare the performance of these methods. All code associated with this study is available in a GitHub repository. ...

We have recently published a paper with GMV at Elsevier and I wanted to share it here also. Link to Article: https://www.sciencedirect.com/science/article/pii/S2352467724002248 Abstract This paper presents an exploration of the application of control theory, particularly utilizing a gradient-based algorithm, to automate and optimize the operation of photovoltaic panels and refrigeration systems in warehouse environments. The study emphasizes achieving coordination between energy generation and consumption, specifically harnessing surplus solar energy for efficient refrigeration. The complex interplay between fluctuating solar irradiance, thermal dynamics of the warehouse, and refrigeration needs underscores the significance of control theory in designing algorithms to dynamically adjust PV panel output and refrigeration system operation. The paper discusses foundational control theory principles, proposes a tailored framework for warehouse operations, and highlights the potential for sustainable energy practices. This paper explores the use of data-driven approaches based on NeuralODEs vs classical ones using physics equations. ...

What do traffic congestion, supermarket lines and fluid dynamics have in common? While we are driving, we are used to think of cars as single individuals/entities. Although, every individual has its own driving tendencies and peculiarities, at a higher-scale, we behave within certain constraints and collective behavior. This quantities can be interpreted in many cases as a homogeneous dense fluid of cars. In this post I will focus on modelling traffic flow using fluid dynamics principles. ...

Several non-linear problems relevant in practical applications can be expressed as a fixed point equation. In many cases, it is crucial to investigate how the model’s behavior changes with variations in a parameter, denoted as $\lambda$. In practical applications, $\lambda$ represents a physical or empirical magnitude of interest. Bifurcation Theory is a subfield in Nonlinear Functional Analysis that tries to study the general behavior of the equations that can be written as $\mathfrak{F}(\lambda, u)=0$ where $\lambda$ is the bifurcation parameter. ...