arXiv subjects

James E. Pascoe

Publications and source records attributed to James E. Pascoe.

5 recordsLinked to original sources

arXiv2026

Positive-Allocation Companion Predictors for Nonlinear Dynamics and Their Finite-Difference Diagnostics

We introduce a positive-allocation companion construction for Koopman-inspired finite-dimensional prediction of nonlinear dynamical systems. The method determines recurrence coefficients by representing a target observable snapshot as a nonnegative, normalized combination of earlier training snapshots. These coefficients define a companion matrix whose spectral structure is induced by the allocation constraints at the construction stage. We prove that normalized positive allocation places the companion spectrum in the closed unit disk and, because the coefficients sum to one, includes $1$ as an eigenvalue. Additionally, we develop modal and non-modal diagnostics for the resulting model trajectory. When the companion matrix is diagonalizable, the modal representation shows that first and second finite differences act as spectral filters through factors of $λ_\ell-1$ and $(λ_\ell-1)^2$. We also derive $C$-based finite-difference bounds that avoid diagonalization and can be evaluated directly from the training data and companion matrix. Numerical experiments on the FitzHugh--Nagumo and susceptible--infectious--recovered (SIR) models illustrate the behavior of the construction in oscillatory and transient dissipative settings. The examples demonstrate both the interpretability of the companion recurrence and its limitations, particularly when pointwise trajectory agreement degrades while finite-difference and modal diagnostics remain informative.

math.DS↗

arXiv2026

Spectral constants for the quantum annulus

We find several new estimates for the spectral constants $K(\mathbb A_r)$ for which a closed annulus $\overline{\mathbb A}_r$ or closed polyannulus $\overline{\mathbb A}^n_r$ is a $K$-spectral set for operators in the quantum annulus $\mathbb Q \mathbb A_r$. We give two alternative proofs to an existing estimate of spectral constant. The first proof capitalizes a dilation theorem due to McCullough and Pascoe, while the second proof involves a certain variety in the Euclidean biball. For commuting and doubly commuting operators in $\mathbb Q \mathbb A_r$, we find upper and lower bounds for the smallest spectral constants.

math.FA↗

arXiv2023

Geometric Dilations and Operator Annuli

Fix 1<R. The dilation theory for the quantum annulus, consisting of those invertible Hilbert space operators T such that the norm of T and its inverse are both at most R is determined. The proof technique involves a geometric approach to dilation that applies to other well known dilation theorems. The dilation theory for the quantum annulus is compared, and contrasted, with the dilation theory for other canonical operator annuli.

math.FA↗

arXiv2017

A non-commutative Julia Inequality

We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in $\mathbb{C}^d$. We look at differentiability at a boundary point for functions that have a certain regularity there.

math.CV↗

arXiv2016

The Julia-Caratheodory theorem on the bidisk revisited

The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Carathéodory theorem on the bidisk states that if the Julia quotient is bounded along some sequence of nontangential approach to some point in the torus, the function must have directional derivatives in all directions pointing into the bidisk. The directional derivative, however, need not be a linear function of the direction in that case. In this note, we show that if the Julia quotient is uniformly bounded along every sequence of nontangential approach, the function must have a linear directional derivative. Additionally, we analyze a weaker condition, corresponding to being Lipschitz near the boundary, which implies the existence of a linear directional derivative for rational functions.

math.CV↗