arXiv subjects

Alejandro Ortega

Publications and source records attributed to Alejandro Ortega.

At least 19 recordsLinked to original sources

arXiv2026

Eigendecomposition of the Hessian of the Robin function in orthogonally invariant domains

In this work we analyze the eigendecomposition of the Hessian matrix of the Robin function $\mathcal{R}(x)$ for the spectral fractional Laplacian in orthogonally invariant domains. We prove that, if $\Omega$ a smooth bounded convex domain invariant under the action of an orthogonal transformation $\mathcal{O}$ then, for $\overline{t}\in\{a\in\Omega:\mathcal{O}(a)=a\}$, the gradient vector $\nabla\mathcal{R}(\overline{t})$ is an eigenvector of the Jacobian $D\mathcal{O}$ associated to the eigenvalue $1$. Moreover, if $\Omega$ is a domain invariant under the reflection about a hyperplane $\pi_{v}=\{x\in\mathbb{R}^N:x\cdot v=0\}$, there exists $\eta>0$ such that $\mathbb{H}(\overline{t})v=\eta v$ where $\mathbb{H}$ denotes the Hessian matrix of $\mathcal{R}(x)$. Consequently, if $\Omega$ is invariant under the reflection about the hyperplanes $\pi_{v_i}$ for a linearly independent set $\{v_1,\ldots,v_N\}$, then the origin is a non degenerate critical point of $\mathcal{R}(x)$. A short proof of the Brezis-Peletier-like formulas for $\nabla \mathcal{R}$ is also provided which allows us to prove the former results for any $0<s<1$.

math.AP↗

arXiv2026

Asymptotically linear fractional problems with mixed boundary conditions

We derive the existence of solutions for an asymptotically linear equation driven by the spectral fractional Laplacian operator with mixed Dirichlet-Neumann boundary conditions. When the nonlinear term $f$ is odd and a suitable relation between the perturbation parameter, the limit of $f(\cdot,t)/t$ as $t\to 0$ and the eigenvalues occurs, we establish also a multiplicity result via the pseudo-index theory related to the genus.

math.AP↗

arXiv2025

The Loss of Control Playbook: Degrees, Dynamics, and Preparedness

This research report addresses the absence of an actionable definition for Loss of Control (LoC) in AI systems by developing a novel taxonomy and preparedness framework. Despite increasing policy and research attention, existing LoC definitions vary significantly in scope and timeline, hindering effective LoC assessment and mitigation. To address this issue, we draw from an extensive literature review and propose a graded LoC taxonomy, based on the metrics of severity and persistence, that distinguishes between Deviation, Bounded LoC, and Strict LoC. We model pathways toward a societal state of vulnerability in which sufficiently advanced AI systems have acquired or could acquire the means to cause Bounded or Strict LoC once a catalyst, either misalignment or pure malfunction, materializes. We argue that this state becomes increasingly likely over time, absent strategic intervention, and propose a strategy to avoid reaching a state of vulnerability. Rather than focusing solely on intervening on AI capabilities and propensities potentially relevant for LoC or on preventing potential catalysts, we introduce a complementary framework that emphasizes three extrinsic factors: Deployment context, Affordances, and Permissions (the DAP framework). Compared to work on intrinsic factors and catalysts, this framework has the unfair advantage of being actionable today. Finally, we put forward a plan to maintain preparedness and prevent the occurrence of LoC outcomes should a state of societal vulnerability be reached, focusing on governance measures (threat modeling, deployment policies, emergency response) and technical controls (pre-deployment testing, control measures, monitoring) that could maintain a condition of perennial suspension.

cs.CY↗

arXiv2025

Nonlocal critical problems with mixed boundary conditions and nearly resonant perturbations

We consider the following nonlocal critical problem with mixed Dirichlet-Neumann boundary conditions, \begin{equation} \left\{ \begin{array}{ll} (-\Delta)^su=\lambda u+|u|^{2_s^*-2}u &\text{in}\ \Omega,\\ \mkern+38.5mu u=0& \text{on}\ \Sigma_{\mathcal{D}},\\ \mkern+24mu \displaystyle \frac{\partial u}{\partial \nu}=0 &\text{on}\ \Sigma_{\mathcal{N}}, \end{array} \right. \end{equation} where $(-\Delta)^s$, $s\in (1/2,1)$, is the spectral fractional Laplacian operator, $\Omega\subset\mathbb{R}^N$, $N>2s$, is a smooth bounded domain, $2_s^*=\frac{2N}{N-2s}$ denotes the critical fractional Sobolev exponent, $\lambda>0$ is a real parameter, $\nu$ is the outwards normal to $\partial\Omega$, $\Sigma_{\mathcal{D}}$, $\Sigma_{\mathcal{N}}$ are smooth $(N-1)$--dimensional submanifolds of $\partial\Omega$ such that $\Sigma_{\mathcal{D}}\cup\Sigma_{\mathcal{N}}=\partial\Omega$, $\Sigma_{\mathcal{D}}\cap\Sigma_{\mathcal{N}}=\emptyset$ and $\Sigma_{\mathcal{D}}\cap\overline{\Sigma}_{\mathcal{N}}=\Gamma$ is a smooth $(N-2)$--dimensional submanifold of $\partial\Omega$. By employing a $\nabla$-theorem we prove the existence of multiple solutions when the parameter $\lambda$ is in a left neighborhood of a given eigenvalue of $(-\Delta)^s$.

math.AP↗

arXiv2025

The Singapore Consensus on Global AI Safety Research Priorities

Rapidly improving AI capabilities and autonomy hold significant promise of transformation, but are also driving vigorous debate on how to ensure that AI is safe, i.e., trustworthy, reliable, and secure. Building a trusted ecosystem is therefore essential -- it helps people embrace AI with confidence and gives maximal space for innovation while avoiding backlash. The "2025 Singapore Conference on AI (SCAI): International Scientific Exchange on AI Safety" aimed to support research in this space by bringing together AI scientists across geographies to identify and synthesise research priorities in AI safety. This resulting report builds on the International AI Safety Report chaired by Yoshua Bengio and backed by 33 governments. By adopting a defence-in-depth model, this report organises AI safety research domains into three types: challenges with creating trustworthy AI systems (Development), challenges with evaluating their risks (Assessment), and challenges with monitoring and intervening after deployment (Control).

cs.AI↗

arXiv2025

AI Behind Closed Doors: a Primer on The Governance of Internal Deployment

The most advanced future AI systems will first be deployed inside the frontier AI companies developing them. According to these companies and independent experts, AI systems may reach or even surpass human intelligence and capabilities by 2030. Internal deployment is, therefore, a key source of benefits and risks from frontier AI systems. Despite this, the governance of the internal deployment of highly advanced frontier AI systems appears absent. This report aims to address this absence by priming a conversation around the governance of internal deployment. It presents a conceptualization of internal deployment, learnings from other sectors, reviews of existing legal frameworks and their applicability, and illustrative examples of the type of scenarios we are most concerned about. Specifically, it discusses the risks correlated to the loss of control via the internal application of a misaligned AI system to the AI research and development pipeline, and unconstrained and undetected power concentration behind closed doors. The report culminates with a small number of targeted recommendations that provide a first blueprint for the governance of internal deployment.

cs.CY↗

arXiv2025

AI threats to national security can be countered through an incident regime

Recent progress in AI capabilities has heightened concerns that AI systems could pose a threat to national security, for example, by making it easier for malicious actors to perform cyberattacks on critical national infrastructure, or through loss of control of autonomous AI systems. In parallel, federal legislators in the US have proposed nascent 'AI incident regimes' to identify and counter similar threats. In this paper, we consolidate these two trends and present a timely proposal for a legally mandated post-deployment AI incident regime that aims to counter potential national security threats from AI systems. We start the paper by introducing the concept of 'security-critical' to describe sectors that pose extreme risks to national security, before arguing that 'security-critical' describes civilian nuclear power, aviation, life science dual-use research of concern, and frontier AI development. We then present in detail our AI incident regime proposal, justifying each component of the proposal by demonstrating its similarity to US domestic incident regimes in other 'security-critical' sectors. Finally, we sketch a hypothetical scenario where our proposed AI incident regime deals with an AI cyber incident. Our proposed AI incident regime is split into three phases. The first phase revolves around a novel operationalization of what counts as an 'AI incident' and we suggest that AI providers must create a 'national security case' before deploying a frontier AI system. The second and third phases spell out that AI providers should notify a government agency about incidents, and that the government agency should be involved in amending AI providers' security and safety procedures, in order to counter future threats to national security.

cs.CY↗

arXiv2025

Open Problems in Mechanistic Interpretability

Mechanistic interpretability aims to understand the computational mechanisms underlying neural networks' capabilities in order to accomplish concrete scientific and engineering goals. Progress in this field thus promises to provide greater assurance over AI system behavior and shed light on exciting scientific questions about the nature of intelligence. Despite recent progress toward these goals, there are many open problems in the field that require solutions before many scientific and practical benefits can be realized: Our methods require both conceptual and practical improvements to reveal deeper insights; we must figure out how best to apply our methods in pursuit of specific goals; and the field must grapple with socio-technical challenges that influence and are influenced by our work. This forward-facing review discusses the current frontier of mechanistic interpretability and the open problems that the field may benefit from prioritizing.

cs.LG↗

arXiv2024

Positive solutions for a weighted critical problem with mixed boundary conditions

We analyze the existence and multiplicity of positive solutions to a nonlocal elliptic problem involving the spectral fractional Laplace operator endowed with homogeneous mixed Dirichlet-Neumann boundary conditions and weighted critical nonlinearities. By means of variational methods and the Nehari manifold approach, we deduce the existence of multiple positive solutions under some assumptions on the behavior of the weight function around its maximum points. Such a behavior, formulated in terms of some rate growth, is explicitly determined and depends on the relation between the dimension, the order of the operator and the subcritical perturbation. In this way we extend and improve the results in "J.F. Liao, J. Liu, P. Zhang, C.L. Tang, Existence and multiplicity of positive solutions for a class of elliptic equations involving critical Sobolev exponents, RACSAM 110 (2016) 483--501", dealing with the Dirichlet problem for the classical Laplace operator, to the nonlocal setting involving mixed boundary conditions.

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↗

arXiv2024

New functional inequalities with applications to the arctan-fast diffusion equation

In this paper, we prove a couple of new nonlinear functional inequalities of Sobolev type akin to the logarithmic Sobolev inequality. In particular, one of the inequalities reads $$ \int_{\mathbb{S}^1}\arctan\left(\frac{\partial_x u}{u}\right)\partial_xu \,dx\geq \arctan\left(\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}\right)\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}. $$ Then, these inequalities are used in the study of the nonlinear \emph{arctan}-fast diffusion equation $$ \partial_t u-\partial_x\arctan\left(\frac{\partial_x u}{u}\right)=0. $$ For this highly nonlinear PDE we establish a number of well-posedness results and qualitative properties.

math.AP↗

arXiv2023

Fractional Schr\"odinger systems coupled by Hardy-Sobolev critical terms

In this work we analyze a class of nonlinear fractional elliptic systems involving Hardy--type potentials and coupled by critical Hardy-Sobolev--type nonlinearities in $\mathbb{R}^N$. Due to the lack of compactness at the critical exponent the variational approach requires a careful analysis of the Palais-Smale sequences. In order to overcome this loss of compactness, by means of a concentration--compactness argument the compactness of PS sequences is derived. This, combined with a energy characterization of the semi-trivial solutions, allow us to conclude the existence of positive ground and bound state solutions en terms of coupling parameter $\nu>0$ and the involved exponents $\alpha,\beta$.

math.AP↗

arXiv2023

Subcritical nonlocal problems with mixed boundary conditions

In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{ \begin{array}{lcl} (-\Delta)^su=\lambda u+f(x,u) & &\text{in } \Omega, \\[2pt] \mkern+39mu u=0& &\text{on } \Sigma_{\mathcal{D}}, \\[2pt] \mkern+26mu \displaystyle \frac{\partial u}{\partial \nu}=0& &\text{on } \Sigma_{\mathcal{N}}, \end{array} \right. $$ where $(-\Delta)^s$, $s\in (1/2,1)$, is the spectral fractional Laplacian operator, $\Omega\subset\mathbb{R}^N$, $N>2s$, is a smooth bounded domain, $\lambda>0$ is a real parameter, $\nu$ is the outward normal to $\partial\Omega$, $\Sigma_{\mathcal{D}}$, $\Sigma_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partial\Omega$ such that $\Sigma_{\mathcal{D}}\cup\Sigma_{\mathcal{N}}=\partial\Omega$, $\Sigma_{\mathcal{D}}\cap\Sigma_{\mathcal{N}}=\emptyset$ and $\Sigma_{\mathcal{D}}\cap\overline{\Sigma}_{\mathcal{N}}=\Gamma$ is a smooth $(N-2)$-dimensional submanifold of $\partial\Omega$.

math.AP↗

arXiv2023

Pervasiveness of the $p$-Laplace operator under localization of fractional $g$-Laplace operators

In this work we analyze the behavior of truncated functionals as \begin{equation*} \int_{\mathbb{R}^N}\int_{B(x,\delta)} G\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)\frac{dydx}{|x-y|^N}\qquad\text{for }\delta\to0^+. \end{equation*} Here the function $G$ is an Orlicz function that in addition is assumed to be a regularly varying function at $0$. A prototype of such function is given by $G(t)=t^p(1+|\log(t)|)$ with $p\geq2$. These kind of functionals arise naturally in {\it peridynamics}, where long-range interactions are neglected and only those exerted at distance smaller than $\delta>0$ are taken into account, i.e., the {\it horizon} $\delta>0$ represents the range of interactions or nonlocality.\\ This work is inspired by the celebrated result by Bourgain, Brezis and Mironescu, who analyzed the limit $s\to1^-$ with $G(t)=t^p$. In particular, we prove that, under appropriate conditions, \begin{equation*} \lim\limits_{\delta\to0^+}\frac{p(1-s)}{G(\delta^{1-s})}\int_{\mathbb{R}^N}\int_{B(x,\delta)}G\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)\frac{dydx}{|x-y|^N}=K_{N,p}\int_{\mathbb{R}^N}|\nabla u(x)|^p dx, \end{equation*} for $p=index(G)$ and an explicit constant $K_{N,p}>0$. Moreover, the converse is also true, if the above localization limit exist as $\delta\to0^+$, the Orlicz function $G$ is a regularly varying function with $index(G)=p$.

math.FA↗

arXiv2022

Nonlinear elliptic systems involving Hardy-Sobolev Criticalities

This paper is focused on the solvability of a family of nonlinear elliptic systems defined in $\mathbb{R}^N$. Such equations contain Hardy potentials and Hardy-Sobolev criticalities coupled by a possible critical Hardy-Sobolev term. That problem arises as a generalization of Gross-Pitaevskii and Bose-Einstein type systems. By means of variational techniques, we shall find ground and bound states in terms of the coupling parameter $\nu$ and the order of the different parameters and exponents. In particular, for a wide range of parameters we find solutions as minimizers or Mountain-Pass critical points of the energy functional on the underlying Nehari manifold.

math.AP↗

arXiv2022

Concave-Convex critical problems for the spectral fractional Laplacian with mixed boundary conditions

In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, \begin{equation*} \left \{ \begin{array}{l} (-Δ)^su=λu^q+u^{2_s^*-1},\ u>0\quad\text{in }Ω,\\[3pt] \mkern+51mu u=0\quad\text{on } Σ_{\mathcal{D}}\\ \mkern+36mu \displaystyle \frac{\partial u}{\partial ν}=0\quad\text{on } Σ_{\mathcal{N}} \end{array} \right. \end{equation*} where $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $\frac{1}{2} 0$, $ν$ is the outwards normal to $\partialΩ$, $Σ_{\mathcal{D}}$, $Σ_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partialΩ$ such that $Σ_{\mathcal{D}}\cupΣ_{\mathcal{N}}=\partialΩ$, $Σ_{\mathcal{D}}\capΣ_{\mathcal{N}}=\emptyset$, and $Σ_{\mathcal{D}}\cap\overlineΣ_{\mathcal{N}}=Γ$ is a smooth $(N-2)$-dimensional submanifold of $\partialΩ$.\newline In particular, we will prove that, for the sublinear case $0 0$. We will also prove that solutions are bounded.

math.AP↗

arXiv2021

Nonlinear Fractional Schrödinger Equations coupled by power-type nonlinearities

In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schrödinger equations, \begin{equation*} \left \{ \begin{array}{l} (-Δ)^s u_1+ λ_1 u_1= μ_1 |u_1|^{2p-2}u_1+β|u_2|^{p} |u_1|^{p-2}u_1 \quad\text{in }\mathbb{R}^N,\\[3pt] (-Δ)^s u_2 + λ_2 u_2= μ_2 |u_2|^{2p-2}u_2+β|u_1|^{p}|u_2|^{p-2}u_2 \quad\text{in }\mathbb{R}^N, \end{array} \right. \end{equation*} where $ u_1,\, u_2\in W^{s,2}(\mathbb{R}^N)$, with $ N=1,\, 2,\, 3$; $λ_j,\,μ_j>0$, $j=1,2$, $β\in \mathbb{R}$, $p\geq 2$ and $\displaystyle\frac{p-1}{2p}N 0$ for $j=1,\ldots ,m\ge 3$, the coupling parameters $β_{jk}=β_{kj}\in \mathbb{R}$ for $j,k=1,\ldots,m$, $j\neq k$. For this system we prove similar results as for $m=2$, depending on the values of the parameters $β_{jk}, p, λ_j,μ_j$, (for $j,k=1,\ldots,m$, $j\neq k$).

math.AP↗

arXiv2021

A Strong Maximum Principle for the fractional Laplace equation with mixed boundary condition

In this work we prove a strong maximum principle for fractional elliptic problems with mixed Dirichlet-Neumann boundary data which extends the one proved by J. Dávila to the fractional setting. In particular, we present a comparison result for two solutions of the fractional Laplace equation involving the spectral fractional Laplacian endowed with homogeneous mixed boundary condition. This result represents a non-local counterpart to a Hopf's Lemma for fractional elliptic problems with mixed boundary data.

math.AP↗