P-Recursive Positivity and Numeric Analytic Continuation: An Application to the Uniqueness of Biomembranes

Date

Jun 15, 2026

Time

10:00 AM - 12:00 PM

Speaker

Stephen Melczer

Affiliation

University of Waterloo

Language

en

Main Topic

Biologie

Host

Math Groups

Description

Since the invention of the compound microscope in the early seventeenth century, scientists have marvelled over red blood cells and their surprising shape. An influential model of Canham predicts the shapes of blood cells and similar biomembranes come from a variational problem minimizing the “bending energy” of these surfaces. Because observed cells have the same shape in humans, it is natural to ask whether the model admits a unique solution. Yu and Chen reduced solution uniqueness for the genus one Canham problem (for a range of isoperimetric ratios) to proving positivity of a P-recursive sequence defined by an explicit linear recurrence relation with polynomial coefficients. In this talk we discuss a method of proving this positivity property, joint with Marc Mezzarobba, and its generalization, with Mezzarobba and Ruiwen Dong, to a wide range of P-recursive sequences. We combine rigorous numeric analytic continuation of D-finite functions with classic bounds from singularity analysis to derive an effective index where the asymptotic behaviour of the sequence, which is positive, dominates the sequence behaviour. Positivity of the finite number of remaining terms can then be checked computationally. Our work has been incorporated into the SageMath ore_algebra package, and can be used by researchers to automatically prove positivity for “generic” positive P-recursive sequences.

Last modified: Jun 9, 2026, 7:38:29 AM