Searcharxiv⌕ Search

arXiv subjects

Daniel Restrepo

Publications and source records attributed to Daniel Restrepo.

14 recordsLinked to original sources
arXiv2026

Manipulation Testing in Boundary Discontinuity Designs

We propose the first manipulation test designed for boundary discontinuity designs (BDDs) with general boundary shapes. A BDD is a multidimensional extension of the regression discontinuity design (RDD) in which treatment assignment is determined by whether the multidimensional running variable crosses a lower-dimensional boundary set. The test avoids multivariate density estimation and builds on the observation that, in the absence of manipulation, observations near the boundary should be approximately evenly split between treatment and control within arbitrary groups defined by their projections onto the boundary. We test this implication using a collection of binomial balance tests on observations near the boundary, with groups formed by k-means clustering. We establish the asymptotic validity of the test under suitable regularity conditions. We also evaluate finite-sample performance through Monte Carlo simulations and illustrate the test in three empirical applications.

econ.EM↗
arXiv2026

Convergence of semilinear parabolic flows with general initial data

We analyze the long-time behavior of solutions to semilinear parabolic equations in Euclidean space that arise as gradient flows of an energy functional. We prove that, for general initial data (including data without compact support) the flow converges to a unique ground state. The argument relies on a sharp stability estimate for almost critical points of the energy, providing a flexible framework for establishing convergence of gradient flows associated with constrained minimization problems in R^n. As an application, we strengthen previous convergence results of Cortazar (1999) and Feireisl (1997).

math.AP↗
arXiv2025

On the asymptotic properties of solutions to one-phase free boundary problems

In this article we study the structure of solutions to the one-phase Bernoulli problem that are modeled either infinitesimally or at infinity by one-homogeneous solutions with an isolated singularity. In particular, we prove a uniqueness of blowups result under a natural symmetry condition on the one-homogeneous solution (\`a la Allard--Almgren) and we prove a rigidity result at infinity (\`a la Simon--Solomon) under additional constraints on the linearized operator around the one-homogeneous solution (which are satisfied by the only known examples of minimizing one-homogeneous solutions). We believe these are the first uniqueness of blow-up/blow-down results at singular points for non-minimizing solutions to the one-phase problem.

math.AP↗
arXiv2025

Teaching large language models to reason like expert diagnosticians

Differential diagnosis is an iterative process that integrates patient information with broader medical knowledge. Clinical case series such as the NEJM Clinicopathologic Conferences (CPCs), published continuously since 1923, feature expert physicians who demonstrate diagnostic reasoning to peers, and have been used for decades to evaluate AI. However, prior AI evaluations have largely focused on final diagnostic accuracy rather than nuanced clinical reasoning. Here, we introduce Dr. CaBot, an agentic AI system that emulates an expert diagnostician by generating written and narrated slide-based presentations from an initial case description alone. CaBot recently generated the first AI diagnosis published in the 100+ year history of the NEJM CPCs. In blinded evaluations, physicians misclassified the source of the differential (CaBot vs. physician-written) in 46/62 (74%) of trials and rated them favorably across quality dimensions. When tasked with solving cases for 72 patients with undiagnosed disease from the NIH Undiagnosed Diseases Network, CaBot identified the working diagnosis in 50/72 (69%) of cases from referral notes alone. To promote transparency and research, we also developed CPC-Bench, a physician-validated benchmark based on 7,102 CPCs and 47,648 questions across 10 tasks. We show that CaBot outperforms frontier models on CPC-Bench, and release both CaBot and CPC-Bench publicly to foster progress in clinical AI.

cs.AI↗
arXiv2025

On the Grad-Mercier equation and Semilinear Free Boundary Problems

In this paper, we establish regularity and uniqueness results for Grad-Mercier type equations that arise in the context of plasma physics. We show that solutions of this problem naturally develop a dead core, which corresponds to the set where the solutions become identically equal to their maximum. We prove uniqueness, sharp regularity, and non-degeneracy bounds for solutions under suitable assumptions on the reaction term. Of independent interest, our methods allow us to prove that the free boundaries of a broad class of semilinear equations have locally finite $H^{n-1}$ measure.

math.AP↗
arXiv2024

Superhuman performance of a large language model on the reasoning tasks of a physician

A seminal paper published by Ledley and Lusted in 1959 introduced complex clinical diagnostic reasoning cases as the gold standard for the evaluation of expert medical computing systems, a standard that has held ever since. Here, we report the results of a physician evaluation of a large language model (LLM) on challenging clinical cases against a baseline of hundreds of physicians. We conduct five experiments to measure clinical reasoning across differential diagnosis generation, display of diagnostic reasoning, triage differential diagnosis, probabilistic reasoning, and management reasoning, all adjudicated by physician experts with validated psychometrics. We then report a real-world study comparing human expert and AI second opinions in randomly-selected patients in the emergency room of a major tertiary academic medical center in Boston, MA. We compared LLMs and board-certified physicians at three predefined diagnostic touchpoints: triage in the emergency room, initial evaluation by a physician, and admission to the hospital or intensive care unit. In all experiments--both vignettes and emergency room second opinions--the LLM displayed superhuman diagnostic and reasoning abilities, as well as continued improvement from prior generations of AI clinical decision support. Our study suggests that LLMs have achieved superhuman performance on general medical diagnostic and management reasoning, fulfilling the vision put forth by Ledley and Lusted, and motivating the urgent need for prospective trials.

cs.AI↗
arXiv2024

Free boundary regularity for semilinear variational problems with a topological constraint

We study a class of semilinear free boundary problems in which admissible functions $u$ have a topological constraint, or spanning condition, on their 1-level set. This constraint forces $\{u=1\}$, which is the free boundary, to behave like a surface with some special types of singularities attached to a fixed boundary frame, in the spirit of the Plateau problem \cite{HP16}. Two such free boundary problems are the minimization of capacity among surfaces sharing a common boundary and an Allen-Cahn formulation of the Plateau problem. We establish the existence of minimizers and study their regularity properties, obtaining the optimal Lipschitz regularity of minimizers and analytic regularity for the free boundaries away from a codimension two singular set. The singularity models for these problems are given by conical critical points of the minimal capacity problem, which are closely related to spectral optimal partition and segregation problems.

math.AP↗
arXiv2024

$C^\infty$ regularity in semilinear free boundary problems

We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $\Delta u=u^{\gamma-1}$, with $\gamma\in(0,1)$. Our main results imply that, once free boundaries are $C^{1,\alpha}$, then they are $C^\infty$. In addition $u/d^{\frac{2}{2-\gamma}}$ and $u^{\frac{2-\gamma}{2}}$ are $C^\infty$ too. In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials $-\Delta v = \kappa v/d^2$ in $\Omega$, where $d$ is the distance to the boundary and $\kappa\leq\frac{1}{4}$. Interestingly, we need to include even the critical constant $\kappa=\frac{1}{4}$, which corresponds to $\gamma=\frac{2}{3}$.

math.AP↗
arXiv2023

A hierarchy of Plateau problems and the approximation of Plateau's laws via the Allen--Cahn equation

We introduce a diffused interface formulation of the Plateau problem, where the Allen--Cahn energy $\mathcal{AC}_\varepsilon$ is minimized under a volume constraint $v$ and a spanning condition on the level sets of the densities. We discuss two singular limits of these Allen--Cahn Plateau problems: when $\varepsilon\to 0^+$, we prove convergence to the Gauss' capillarity formulation of the Plateau problem with positive volume $v$; and when $\varepsilon\to 0^+$, $v\to 0^+$ and $\varepsilon/v\to 0^+$, we prove convergence to the classical Plateau problem (in the homotopic spanning formulation of Harrison and Pugh). As a corollary of our analysis we resolve the incompatibility between Plateau's laws and the Allen--Cahn equation implied by a regularity theorem of Tonegawa and Wickramasekera. In particular, we show that Plateau-type singularities can be approximated by energy minimizing solutions of the Allen--Cahn equation with a volume Lagrange multiplier and a transmission condition on a spanning free boundary.

math.AP↗
arXiv2023

Plateau borders in soap films and Gauss' capillarity theory

We provide, in the setting of Gauss' capillarity theory, a rigorous derivation of the equilibrium law for the three dimensional structures known as Plateau borders which arise in "wet" soap films and foams. A key step in our analysis is a complete measure-theoretic overhaul of the homotopic spanning condition introduced by Harrison and Pugh in the study of Plateau's laws for two-dimensional area minimizing surfaces ("dry" soap films). This new point of view allows us to obtain effective compactness theorems and energy representation formulae for the homotopic spanning relaxation of Gauss' capillarity theory which, in turn, lead to prove sharp regularity properties of energy minimizers. The equilibrium law for Plateau borders in wet foams is also addressed as a (simpler) variant of the theory for wet soap films.

math.AP↗
arXiv2022

Uniform stability in the Euclidean isoperimetric problem for the Allen--Cahn energy

We consider the isoperimetric problem defined on the whole $\mathbb{R}^n$ by the Allen--Cahn energy functional. For non-degenerate double well potentials, we prove sharp quantitative stability inequalities of quadratic type which are uniform in the length scale of the phase transitions. We also derive a rigidity theorem for critical points analogous to the classical Alexandrov's theorem for constant mean curvature boundaries.

math.AP↗
arXiv2018

Existence and Morse Index of least energy nodal solution of the $(p,2)$-laplacian

In this paper we study the quasilinear equation $- \ep^2 Δu-Δ_p u=f(u)$ in a smooth bounded domain $Ω$ with Dirichlet boundary condition. For $\ep \geq 0$, we review existence of a least energy nodal solution and then present information about the Morse Index of least nodal energy solutions this BVP. In particular we provide Morse Index information for the case $\ep =0$.

math.AP↗
arXiv2018

Multiplicity results and qualitative properties for semilinear elliptic problems with Neumann boundary conditions

In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of eigenvalues the derivative of the nonlinearity crosses to guarantee existence of at least five nontrivial solutions. The techniques we use are a combination of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la Castro-Lazer.

math.AP↗