arXiv subjects

Jacob Glidewell

Publications and source records attributed to Jacob Glidewell.

4 recordsLinked to original sources

arXiv2026

A Continuum of Small-cap Decouplings and Exponential Sums for the Moment Curve in $\mathbb{R}^4$

We use the high-low method and wavepacket pruning to prove new small-cap decoupling estimates for the moment curve in $\mathbb{R}^4$. As an application, we verify a conjecture of Demeter regarding the $L^{12}$ square-root cancellation of exponential sums associated with the moment curve in $\mathbb{R}^4$. This provides a continuum of square-root cancellation estimates that connects the Vinogradov MVT in $\mathbb{R}^3$ with a result of Bourgain, related to improving the best-known estimate for the Lindelöf hypothesis.

math.CA↗

arXiv2024

The Ross-Darboux-Stieltjes Integral

Motivated by the limitations of the traditional definitions of the Riemann-Stieltjes and Darboux-Stieltjes integrals, we introduce a generalized Darboux-Stieltjes integral that is equivalent to an earlier generalization by Ross \cite{Ross}. Our definition builds upon an approach to the Darboux-Stieltjes integral recently introduced by the first author and Convertito \cite{TSI}. We show that our definition agrees with all previous definitions, but that the class of integrable functions is much larger. We develop all the analogs of the classic results for the Riemann integral, and rectify the problems inherent in the definition of the Darboux-Stieltjes integral in \cite{TSI}. In particular, we show the Bounded Convergence Theorem holds for our definition and that it agrees with the Lebesgue-Stieltjes integral.

math.CA↗

arXiv2023

Additive Stability of Frames

Given a frame in a finite dimensional Hilbert space we construct additive perturbations which decrease the condition number of the frame. By iterating this perturbation, we introduce an algorithm that produces a tight frame in a finite number of steps. Additionally, we give sharp bounds on additive perturbations which preserve frames and we study the effect of appending and erasing vectors to a given tight frame. We also discuss under which conditions our finite-dimensional results are extendable to infinite-dimensional Hilbert spaces.

math.FA↗

arXiv2022

On the two-dimensional Jacobian conjecture: Magnus' formula revisited, II

This article is part of an ongoing investigation of the two-dimensional Jacobian conjecture. In the first paper of this series, we proved the generalized Magnus' formula. In this paper, inspired by cluster algebras, we introduce a sequence of new conjectures including the remainder vanishing conjecture. This makes the generalized Magnus' formula become a useful tool to show the two-dimensional Jacobian conjecture. In the forthcoming paper(s), we plan to prove the remainder vanishing conjecture.

math.AC↗