Minimization of the First Nonzero Eigenvalue Problem for Two-Phase
Conductors with Neumann Boundary Conditions." SIAM Journal on Applied Mathematics, vol. 80, issue 4, 2020, pp. 1607-1628.
, Di, , and Chiu-Yen Kao. "
Abstract: We consider the problem of minimizing the first nonzero eigenvalue of an elliptic operator with Neumann boundary conditions with respect to the distribution of two conducting materials with a prescribed area
ratio in a given domain. In one dimension, we show monotone properties of the first nonzero eigenvalue with respect to various parameters and find the optimal distribution of two conducting materials on an interval under the assumption that
the region that has lower conductivity is simply connected. On a rectangular domain in two dimensions, we show that the strip configuration of two conducting materials can be a local minimizer. For general domains, we propose a
rearrangement algorithm to find the optimal distribution numerically. Many results on various domains are shown to demonstrate the efficiency and robustness of the algorithms. Topological changes of the optimal configurations are discussed
on circles, ellipses, annuli, and L-shaped domains.
Stokkermans, Thomas J., Jeremy C. Reitinger, George Tye, Chiu-Yen Kao, Sangeetha Ragupathy, Huachun A. Wang, and Carol B. Toris. “Accommodative Exercises to Lower Intraocular
Pressure.” Journal of Ophthalmology, vol. 2020, 2020, article 6613066.
Abstract: Purpose. This study investigated how a conscious change in ocular accommodation affects intraocular pressure (IOP) and ocular biometrics in healthy adult volunteers of different ages. Methods.
Thirty-five healthy volunteers without ocular disease or past ocular surgery, and with refractive error between −3.50 and +2.50 diopters, were stratified into 20, 40, and 60 year old (y.o.) age groups. Baseline measurements of central
cornea thickness, anterior chamber depth, anterior chamber angle, cornea diameter, pupil size, and ciliary muscle thickness were made by autorefraction and optical coherence tomography (OCT), while IOP was measured by pneumotonometry. Each
subject’s right eye focused on a target 40 cm away. Three different tests were performed in random order: (1) 10 minutes of nonaccommodation (gazing at the target through lenses that allowed clear vision without accommodating), (2) 10
minutes of accommodation (addition of a minus 3 diopter lens), and (3) 10 minutes of alternating between accommodation and nonaccommodation (1-minute intervals). IOP was measured immediately after each test. A 20-minute rest period was
provided between tests. Data from 31 subjects were included in the study. ANOVA and paired t-tests were used for statistical analyses. Results. Following alternating accommodation, IOP decreased by 0.7 mmHg in the right
eye when all age groups were combined ( = 0.029). Accommodation or nonaccommodation alone did not decrease IOP. Compared to the 20 y.o. group, the 60 y.o. group had a thicker ciliary muscle within 75 μm of the scleral spur, a
thinner ciliary muscle at 125–300 μm from the scleral spur, narrower anterior chamber angles, shallower anterior chambers, and smaller pupils during accommodation and nonaccommodation (’s < 0.01). Conclusion.
Alternating accommodation, but not constant accommodation, significantly decreased IOP. This effect was not lost with aging despite physical changes to the aging eye. A greater accommodative workload and/or longer test period may improve
External Grant: Kao, Chiu-Yen. “Research Pairs: Theoretical and Numerical Methods for Geometrical Optimization,” Centre International de Rencontres Mathematiques (CIRM),
Luminy, France, 2020.
Abstract: Geometrical optimization problems involve many areas of mathematics including differential geometry and geometric analysis, partial differential equations (PDE), numerical analysis, and scientific computation.
These optimization problems arise naturally in many different problems from science and technology, such as mechanical vibrations, photonic crystals, and population dynamics, to name just a few, but also in pure mathematics like in the
study of free boundary minimal surfaces, isometric embeddings, and extremal manifolds. The aim of this proposal is to bring together four mathematicians: (CYK) Chiu-Yen Kao, Claremont McKenna College, United States, (SAM) Seyyed Abbas
Mohammadi, Yasouj University, Iran, (BO) Braxton Osting, University of Utah, United States, and (EO) Edouard Oudet, Universite Grenoble Alpes, France, who share common research interests on geometric optimization, to work on the open
problems which are described below. The specifics of this research project requires specialists from different areas with different points of view to make significant progress. A research pairs context would give us the opportunity to focus
together on these questions and share our knowledge which seems to us mandatory to obtain new interesting results for the mathematical community. Our project will focus on the following two directions and the described goals: (1) Computing
free boundary minimal surfaces in the unit ball using extremal Steklov eigenfunctions: investigate numerically the existence of minimal surfaces with prescribed genius and a fixed number of boundary connected components. (2) A spectral
approach for computing Nash embeddings: provide an alternative constructive approach by spectral optimization to Gromov's convex integration theory to build C1 isometric embedding.