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B. Barrios

Publications and source records attributed to B. Barrios.

12 recordsLinked to original sources
arXiv2025

Minimum Sample Size Calculation for Multivariable Regression of Continuous Outcomes in Chemometrics for Astrobiology and Planetary Science

Over the last few decades, prediction models have become a fundamental tool in statistics, chemometrics, and related fields. However, to ensure that such models have high value, the inferences that they generate must be reliable. In this regard, the internal validity of a prediction model might be threatened if it is not calibrated with a sufficiently large sample size, as problems such as overfitting may occur. Such situations would be highly problematic in many fields, including space science, as the resulting inferences from prediction models often inform scientific inquiry about planetary bodies such as Mars. Therefore, to better inform the development of prediction models, we applied a theory-based guidance from the biomedical domain for establishing what the minimum sample size is under a range of conditions for continuous outcomes. This study aims to disseminate existing research criteria in biomedical research to a broader audience, specifically focusing on their potential applicability and utility within the field of chemometrics. As such, the paper emphasizes the importance of interdisciplinarity, bridging the gap between the medical domain and chemometrics. Lastly, we provide several examples of work in the context of space science. This work will be the foundation for more evidence-based model development and ensure rigorous predictive modelling in the search for life and possible habitable environments.

physics.space-ph↗
arXiv2023

Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities

In this paper we consider the heat semigroup $\{W_t\}_{t>0}$ defined by the combinatorial Laplacian and two subordinated families of $\{W_t\}_{t>0}$ on homogeneous trees $X$. We characterize the weights $u$ on $X$ for which the pointwise convergence to initial data of the above families holds for every $f\in L^{p}(X,μ,u)$ with $1\le p<\infty$, where $μ$ represents the counting measure in $X$ . We prove that this convergence property in $X$ is equivalent to the fact that the maximal operator on $t\in (0,R)$, for some $R>0$, defined by the semigroup is bounded from $L^{p}(X,μ,u)$ into $L^{p}(X,μ,v)$ for some weight $v$ on $X$.

math.AP↗
arXiv2020

Linear non-degeneracy and uniqueness of the bubble solution for the critical fractional Hénon equation in $\mathbb{R}^N$

We study the equation \begin{equation*}\label{P0} (-Δ)^s u = |x|^α u^{\frac{N+2s+2α}{N-2s}}\mbox{ in }\mathbb{R}^N,\tag{P} \end{equation*} where $(-Δ)^s$ is the fractional Laplacian operator with $0 < s < 1$, $α>-2s$ and $N>2s$. We prove the linear non-degeneracy of positive radially symmetric solutions of the equation (\ref{P0}) and, as a consequence, a uniqueness result of those solutions with Morse index equal to one. In particular, the ground state solution is unique. Our non-degeneracy result extends in the radial setting some known theorems done by Dávila, Del Pino and Sire (see \cite[Theorem 1.1]{Davila-DelPino-Sire}), and Gladiali, Grossi and Neves (see \cite[Theorem 1.3]{Gladiali-Grossi-Neves}).

math.AP↗
arXiv2019

The sharp exponent in the study of the nonlocal Hénon equation in $\mathbb{R}^{n}$. A Liouville theorem and an existence result

We will consider the nonlocal Hénon equation $$(-Δ)^s u= |x|^α u^{p},\quad \mathbb{R}^{N},$$ where $(-Δ)^s$ is the fractional Laplacian operator with $0 1$ and $N>2s$. We prove a nonexistence result for positive solutions in the optimal range of the nonlinearity, that is, when $$1<p<p^*_{α, s}:=\frac{N+2α+2s}{N-2s}.$$ Moreover, we prove that a bubble solution, that is a fast decay positive radially symmetric solutions, exists when $p=p_{α, s}^{*}$.

math.AP↗
arXiv2018

Periodic solutions for the one-dimensional fractional Laplacian

In this paper we are concerned with the construction of periodic solutions of the nonlocal problem $(-Δ)^s u= f(u)$ in $\mathbb{R}$, where $(-Δ)^s$ stands for the $s$-Laplacian, $s\in (0,1)$. We introduce a suitable framework which allows to reduce the search for such solutions to the resolution of a boundary value problem in a suitable Hilbert space, thereby making it possible to reach for the usual tools of nonlinear analysis, like bifurcation theory or variational methods. We obtain some existence theorems which are lately enlightened with the analysis of some examples.

math.AP↗
arXiv2017

Symmetry results in the half space for a semi-linear fractional Laplace equation through a one-dimensional analysis

In this paper we analyze the semi-linear fractional Laplace equation $$(-Δ)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\mathbb{R}^N_+=\{x=(x',x_N)\in \mathbb{R}^N:\ x_N>0\}$ stands for the half-space and $f$ is a locally Lipschitz nonlinearity. We completely characterize one-dimensional bounded solutions of this problem, and we prove among other things that if $u$ is a bounded solution with $ρ:=\sup_{\mathbb{R}^N}u$ verifying $f(ρ)=0$, then $u$ is necessarily one-dimensional.

math.AP↗
arXiv2017

A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions

In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-Δ)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-Δ)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the functions $h$ and $f$ are nondecreasing. The main feature is that the function $h$ changes sign in $\mathbb{R}$, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.

math.AP↗
arXiv2016

Monotonicity of solutions for some nonlocal elliptic problems in half-spaces

In this paper we consider classical solutions $u$ of the semilinear fractional problem $(-Δ)^s u = f(u)$ in $\mathbb{R}^N_+$ with $u=0$ in $\mathbb{R}^N \setminus \mathbb{R}^N_+$, where $(-Δ)^s$, $0 0\}$ is the half-space and $f\in C^1$ is a given function. With no additional restriction on the function $f$, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in $\mathbb{R}^N_+$ and verify $$ \frac{\partial u}{\partial x_N}>0 \quad \hbox{in } \mathbb{R}^N_+. $$ This is in contrast with previously known results for the local case $s=1$, where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when $f(0)<0$.

math.AP↗
arXiv2015

A priori bounds and existence of solutions for some nonlocal elliptic problems

In this paper we show existence of solutions for some elliptic problems with nonlocal diffusion by means of nonvariational tools. Our proof is based on the use of topological degree, which requires a priori bounds for the solutions. We obtain the a priori bounds by adapting the classical scaling method of Gidas and Spruck. We also deal with problems involving gradient terms.

math.AP↗
arXiv2013

A critical fractional equation with concave-convex power nonlinearities

In this work we study the following fractional critical problem $$ (P_λ)=\left\{\begin{array}{ll} (-Δ)^s u=λu^{q} + u^{2^*_{s}-1}, \quad u{>}0 & \mbox{in} Ω\\ u=0 & \mbox{in} \RR^n\setminus Ω\,, \end{array}\right. $$ where $Ω\subset \mathbb{R}^n$ is a regular bounded domain, $λ>0$, $0 2s$. Here $(-Δ)^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by $$ -(-Δ)^s u(x)={\rm P. V.} \int_{\RR^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \RR^n. $$ Our main results show the existence and multiplicity of solutions to problem $(P_λ)$ for different values of $λ$. The dependency on this parameter changes according to whether we consider the concave power case ($0<q<1$) or the convex power case ($1<q<2^*_s-1$). These two cases will be treated separately.

math.AP↗
arXiv2011

On some Critical Problems for the Fractional Laplacian Operator

We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: (-Δ)^{α/2}u=λu^q+u^{\frac{N+α}{N-α}}, \quad u>0 &\quad in Ω, u=0&\quad on \partialΩ,$$ where $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $N\ge1$, $λ>0$, $0 0$, at least one if $λ=Λ$, no solution if $λ>Λ$. For $q=1$ we show existence of at least one solution for $0<λ<λ_1$ and nonexistence for $λ\geλ_1$. When $q>1$ the existence is shown for every $λ>0$. Also we prove that the solutions are bounded and regular.

math.AP↗