Paper proves minimizing movements match smooth droplet flow in 3D.
problem Consistency of minimizing movements with smooth mean curvature flow.
method Proved minimizing movements coincide with smooth droplet flow.
result Minimizing movements and smooth mean curvature flow are consistent in 3D.
Study on droplet flow on uneven surfaces, proving existence and properties.
problem Understanding droplet movement on irregular surfaces.
method Existence of smooth flow and 1/2-Hölder continuous minimizing movement solutions.
result Properties of minimizing movements including comparison principles and uniform boundedness.
New Heintze-Karcher inequality helps understand droplet shapes.
problem Characterize the shape of droplets inside smooth containers.
method Obtained a new form of the Heintze-Karcher inequality for mean convex hypersurfaces with boundary on curved substrates.
result New mathematical inequality aids in understanding droplet shapes.
Study phase transition in liquid crystal droplets using mathematical analysis.
problem Mathematical analysis of phase transition between isotropic and nematic states of liquid crystals.
method Rigorous mathematical analysis using the Ericksen model and Γ-convergence theory.
result Γ-limit provides geometric description and anchoring conditions for liquid crystal orientations.
Two oppositely charged droplets of (say) water in e.g. oil or air will tend to drift together under the influence of their charges. As they make contact, one might expect them to coalesce and form one large droplet, and this indeed happens when the charge difference is sufficiently small. However, Ristenpart et al disc…
Motivation: Single cell transcriptome sequencing (scRNA-Seq) has become a revolutionary tool to study cellular and molecular processes at single cell resolution. Among existing technologies, the recently developed droplet-based platform enables efficient parallel processing of thousands of single cells with direct coun…
Experimental fractal landscape dynamics observed in emulsions.
problem Understanding anomalous motions in soft glassy materials.
method Quantitative analysis of oil droplet trajectories in dense emulsions.
result Experimental fractal geometry matches computational model of soft glassy dynamics.
A theory of exceptional extreme events, characterized by their abnormal sizes compared with the rest of the distribution, is presented. Such outliers, called "dragon-kings", have been reported in the distribution of financial drawdowns, city-size distributions (e.g., Paris in France and London in the UK), in material f…
The study computes Bergman kernels and point process asymptotics on Kähler manifolds.
problem Computing asymptotics of Bergman kernels and point process distributions on Kähler manifolds.
method Equivariant and partial Bergman kernels, determinantal point processes, asymptotic analysis.
result The distribution of linear statistics converges to a centered normal variable with specific variances.
Deep learning detects cloud changes due to human aerosols.
problem Uncertainty in the effect of anthropogenic aerosols on cloud properties and Earth's energy balance.
method Deep convolutional neural networks to analyze cloud images.
result Identified and characterized specific cloud perturbations due to human aerosols.
Study on stability of 3D sessile drops, identifying degenerate kernel.
problem Linear stability of three-dimensional sessile drops with a free contact line.
method Derived constrained second variation, formulated Jacobi problem, combined geometric and Fourier analysis.
result Kernel of the constrained Jacobi operator is exactly the space of horizontal translations under pressure-volume nondegeneracy.
Motivation: With the development of droplet based systems, massive single cell transcriptome data has become available, which enables analysis of cellular and molecular processes at single cell resolution and is instrumental to understanding many biological processes. While state-of-the-art clustering methods have been…
This paper solves the normalizability crisis in sequential inference by introducing bounded information geometry.
problem Structural failure in standard sequential inference architectures when dealing with extreme outliers.
method Non-parametric field actions and bounded information geometry to truncate infinite tails of spatial distributions.
result Empirical benchmarks across three domains show robust estimation without infinite-tailed distributional assumptions.
MOAI evaluates indoor airflow's impact on COVID-19 transmission.
problem Understanding indoor airflow's role in COVID-19 transmission.
method Developed a privacy-preserving app and model to evaluate risk exposure.
result Quantified factors contributing to higher or lower contamination in settings.