arXiv subjects

Steven Redolfi

Publications and source records attributed to Steven Redolfi.

4 recordsLinked to original sources

arXiv2026

A Review of the Receiver Operating Characteristic Curve and a Proof About the Area Beneath It

The Receiver Operating Characteristic (ROC) curve of a binary classifier has often been utilized to measure the performance of the classifier. The area beneath this curve is used in particular because of its quoted probabilistic interpretation as being equal to the probability that the classifier will rank a random positive observation above a random negative observation. This paper formalizes this claim, produces a bound on how far away from the truth it is if a hypothesis is not met, and gives a small literature review of the ROC curve.

cs.LG↗

arXiv2026

Self-adjoint extensions of symmetric relations associated with systems of ordinary differential equations with distributional coefficients

We study the extension theory for the two-dimensional first-order system $Ju' +qu = wf$ of differential equations on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are real distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we characterize the boundary conditions for solutions $u$ in the closure of the minimal relation, as well as describe the properties of quasi-boundary conditions which yield self-adjoint extensions. We then apply these ideas to a popular extension of non-negative minimal relations: the Krein-von Neumann extension. For more context on how the Krein-von Neumann is defined, an appendix shows a construction of the Friedrichs extension from which the Krein-von Neumann is traditionally defined.

math.SP↗

arXiv2023

On Fourier expansions for systems of ordinary differential equations with distributional coefficients

We study the spectral theory for the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we construct a generalized Weyl-Titchmarsh $m$-function with corresponding spectral measure $\tau$ and a generalized Fourier transform after imposing certain conditions on $J$, $q$, and $w$. Different conditions are motivated and studied in the later sections. A Fatou-type identity needed for our result is recorded in the appendix.

math.SP↗

arXiv2023

Green's functions for first-order systems of ordinary differential equations without the unique continuation property

This paper is a contribution to the spectral theory associated with the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with $Ju'+qu=wf$ and show the existence of Green's functions for self-adjoint relations even if unique continuation of solutions fails.

math.SP↗