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Ángel Arroyo

Publications and source records attributed to Ángel Arroyo.

15 recordsLinked to original sources
arXiv2026

A counterexample to the pointwise validity of an asymptotic mean value property for $p$-harmonic functions

We give a counterexample showing that an asymptotic mean value property for $p$-harmonic functions in $\mathbb R^d$ does not necessarily hold in the pointwise sense for $d\geq 3$ and $p>2$ near $2$. The proof uses a perturbation argument from $p=2$ and the Implicit Function Theorem to establish the counterexample. We also use a computer assisted proof to give an explicit range of values of $p$ for which the counterexample holds. Finally, we also establish a counterexample for the pointwise asymptotic variational mean value property.

math.AP↗
arXiv2024

Enhancing web traffic attacks identification through ensemble methods and feature selection

Websites, as essential digital assets, are highly vulnerable to cyberattacks because of their high traffic volume and the significant impact of breaches. This study aims to enhance the identification of web traffic attacks by leveraging machine learning techniques. A methodology was proposed to extract relevant features from HTTP traces using the CSIC2010 v2 dataset, which simulates e-commerce web traffic. Ensemble methods, such as Random Forest and Extreme Gradient Boosting, were employed and compared against baseline classifiers, including k-nearest Neighbor, LASSO, and Support Vector Machines. The results demonstrate that the ensemble methods outperform baseline classifiers by approximately 20% in predictive accuracy, achieving an Area Under the ROC Curve (AUC) of 0.989. Feature selection methods such as Information Gain, LASSO, and Random Forest further enhance the robustness of these models. This study highlights the efficacy of ensemble models in improving attack detection while minimizing performance variability, offering a practical framework for securing web traffic in diverse application contexts.

cs.CR↗
arXiv2024

Krylov-Safonov theory for Pucci-type extremal inequalities on random data clouds

We establish Krylov-Safonov type Hölder regularity theory for solutions to quite general discrete dynamic programming equations or equivalently discrete stochastic processes on random geometric graphs. Such graphs arise for example from data clouds in graph-based machine learning. The results actually hold to functions satisfying Pucci-type extremal inequalities, and thus we cover many examples including tug-of-war games on random geometric graphs. As an application we show that under suitable assumptions when the number of data points increases, the graph functions converge to a solution of a partial differential equation.

math.AP↗
arXiv2024

Existence of solutions for a system with general Hardy--Sobolev singular criticalities

In this paper we study a class of Hardy--Sobolev type systems defined in $\mathbb{R}^N$ and coupled by a singular critical Hardy--Sobolev term. The main novelty of this work is that the orders of the singularities are independent and contained in a wide range. By means of variational techniques, we will prove the existence of positive bound and ground states for such a system. In particular, we find solutions as minimizers or Mountain--Pass critical points of the energy functional on the underlying Nehari manifold.

math.AP↗
arXiv2022

Hölder estimate for a tug-of-war game with $1<p<2$ from Krylov-Safonov regularity theory

We propose a new version of the tug-of-war game and a corresponding dynamic programming principle related to the $p$-Laplacian with $1<p<2$. For this version, the asymptotic Hölder continuity of solutions can be directly derived from recent Krylov-Safonov type regularity results in the singular case. Moreover, existence of a measurable solution can be obtained without using boundary corrections. We also establish a comparison principle.

math.AP↗
arXiv2022

Hölder regularity for stochastic processes with bounded and measurable increments

We obtain an asymptotic Hölder estimate for expectations of a quite general class of discrete stochastic processes. Such expectations can also be described as solutions to a dynamic programming principle or as solutions to discretized PDEs. The result, which is also generalized to functions satisfying Pucci-type inequalities for discrete extremal operators, is a counterpart to the Krylov-Safonov regularity result in PDEs. However, the discrete step size $\varepsilon$ has some crucial effects compared to the PDE setting. The proof combines analytic and probabilistic arguments.

math.AP↗
arXiv2022

Local regularity estimates for general discrete dynamic programming equations

We obtain an analytic proof for asymptotic Hölder estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.

math.AP↗
arXiv2022

Inverse problems on low-dimensional manifolds

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite discretization of the measurements is available. Then, a Landweber-type reconstruction algorithm from a finite number of measurements is proposed, for which we prove global convergence, thanks to a new criterion for finding a suitable initial guess. These general results are then applied to several examples, including two classical nonlinear ill-posed inverse boundary value problems. The first is Calderón's inverse conductivity problem, for which we prove a Lipschitz stability estimate from a finite number of measurements for piece-wise constant conductivities with discontinuities on an unknown triangle. A similar stability result is then obtained for Gel'fand-Calderón's problem for the Schrödinger equation, in the case of piece-wise constant potentials with discontinuities on a finite number of non-intersecting balls.

math.FA↗
arXiv2020

Asymptotic Hölder Regularity for the Ellipsoid Process

We obtain an asymptotic Hölder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.

math.AP↗
arXiv2020

$p$-harmonic functions by way of intrinsic mean value properties

Let $Ω\subset\mathbb{R}^n$ be a bounded domain satisfying the uniform exterior cone condition. We establish existence and uniqueness of continuous solutions of the Dirichlet Problem associated to certain intrinsic nonlinear mean value properties in $Ω$. Furthermore we show that, when properly normalized, such functions converge to the $p$-harmonic solution of the Dirichlet problem in $Ω$, for $p\in[2,\infty)$. The proof of existence is constructive and the methods are entirely analytic, a fundamental tool being the construction of explicit, $p$-independent barrier functions in $Ω$.

math.AP↗
arXiv2018

On a class of singular measures satisfying a strong annular decay condition

A metric measure space $(X,d,μ)$ is said to satisfy the strong annular decay condition if there is a constant $C>0$ such that $$ μ\big(B(x,R)\setminus B(x,r)\big)\leq C\, \frac{R-r}{R}\, μ(B(x,R)) $$ for each $x\in X$ and all $0<r \leq R$. If $d_{\infty}$ is the distance induced by the $\infty$-norm in $\mathbb{R}^N$, we construct examples of singular measures $μ$ on $\mathbb{R}^N$ such that $(\mathbb{R}^N, d_{\infty},μ)$ satisfies the strong annular decay condition.

math.MG↗
arXiv2018

Tug-of-war games with varying probabilities and the normalized $p(x)$-Laplacian

We study a tug-of-war game with varying probabilities. In particular, we show that the value of the game is locally asymptotically Hölder continuous. We also show the existence and uniqueness of values of the game. As an application, we prove that the value function of the game converges to a solution of the normalized $p(x)$-Laplacian.

math.AP↗
arXiv2017

A priori Hölder and Lipschitz regularity for generalized $p$-harmonious functions in metric measure spaces

Let $(\mathbb{X} , d, μ)$ be a proper metric measure space and let $Ω\subset \mathbb{X}$ be a bounded domain. For each $x\in Ω$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial Ω) $ and let $B_x$ be the closed ball centered at $x$ with radius $\varrho (x)$. If $α\in \mathbb{R}$, consider the following operator in $C( \overlineΩ )$, $$ \mathcal{T}_αu(x)=\fracα{2}\left(\sup_{B_x } u+\inf_{B_x } u\right)+(1-α)\,\frac{1}{μ(B_x)}\int_{B_x}\hspace{-0.1cm} u\ dμ. $$ Under appropriate assumptions on $α$, $\mathbb{X}$, $μ$ and the radius function $\varrho$ we show that solutions $u\in C( \overlineΩ )$ of the functional equation $\mathcal{T}_αu = u$ satisfy a local Hölder or Lipschitz condition in $Ω$. The motivation comes from the so called $p$-harmonious functions in euclidean domains.

math.AP↗