Department of MathematicsThe Fleming-Viot particle system consists of N identical particles diffusing in a domain U⊂Rd. Whenever a particle hits the boundary ∂U, that particle jumps onto another particle in the interior. It is known that this system provides a particle representation for both the Quasi-Stationary… read more about this publication »
We consider the variational problem of cross-entropy loss with n feature vectors on a unit hypersphere in Rd. We prove that when d≥n−1, the global minimum is given by the simplex equiangular tight frame, which justifies the neural collapse behavior. We also prove that, as n→∞ with fixed d, the… read more about this publication »
We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data [Formula: see text], [Formula: see text], we construct a divergence-free advecting velocity field [Formula: see text] (depending on [Formula: see text]) for which the… read more about this publication »
In this paper, we consider the dynamics of a 2D target-searching agent performing Brownian motion under the influence of fluid shear flow and chemical attraction. The analysis is motivated by numerous situations in biology where these effects are present, such as broadcast spawning of marine… read more about this publication »
What is the “right” topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from injective. We therefore propose that, in many cases, the collection of… read more about this publication »
Exponentially-localized Wannier functions (ELWFs) are an orthonormal basis of the Fermi projection of a material consisting of functions which decay exponentially fast away from their maxima. When the material is insulating and crystalline, conditions which guarantee existence of ELWFs in… read more about this publication »
The neurotransmitter dopamine (DA) is known to be influenced by the circadian timekeeping system in the mammalian brain. We have previously created a single-cell differential equations model to understand the mechanisms behind circadian rhythms of extracellular DA. In this paper, we investigate the… read more about this publication »
The 2-dimensional motion of a particle subject to Brownian motion and ambient shear flow transportation is considered. Numerical experiments are carried out to explore the relation between the shear strength, box size, and the particle’s expected first hitting time of a given target. The simulation… read more about this publication »
AbstractGiven a set P of n points and a constant k, we are interested in computing the persistent homology of the Čech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an open question of Sheehy (The persistent homology of… read more about this publication »
Background: The application of heart rate variability is problematic in patients with atrial fibrillation (AF). This study aims to explore the associations between all-cause mortality and the median hourly ambulatory heart rate range (ÃHRR24hr) compared with other parameters obtained from the… read more about this publication »
Braverman and Kazhdan (2000) introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands’… read more about this publication »
The oscillations observed in many time series, particularly in biomedicine, exhibit morphological variations over time. These morphological variations are caused by intrinsic or extrinsic changes to the state of the generating system, henceforth referred to as dynamics. To model these time series (… read more about this publication »
Recent work of Sottoriva, Graham, and collaborators have led to the controversial claim that exponentially growing tumors have a site frequency spectrum that follows the 1/f law consistent with neutral evolution. This conclusion has been criticized based on data quality issues, statistical… read more about this publication »
In this work, we analyze the global convergence property of coordinate gradient descent with random choice of coordinates and stepsizes for non-convex optimization problems. Under generic assumptions, we prove that the algorithm iterate will almost surely escape strict saddle points of the… read more about this publication »
We propose a novel numerical method for high dimensional Hamilton-Jacobi-Bellman (HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as optimal control problems, are tackled by the actor-critic framework inspired by reinforcement learning, based on neural network… read more about this publication »
Sucrose is among the main products of photosynthesis that are deemed necessary for plant growth and survival. It is produced in the mesophyll cells of leaves and translocated to different parts of the plant through the phloem. Progress in understanding this transport process remains fraught with… read more about this publication »
Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of d-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the… read more about this publication »
Multiple-spiral-wave solutions of the general cubic complex Ginzburg–Landau equation in bounded domains are considered. We investigate the effect of the boundaries on spiral motion under homogeneous Neumann boundary conditions, for small values of the twist parameter q. We derive explicit laws of… read more about this publication »
We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation… read more about this publication »
Abstract We study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different… read more about this publication »
Let F be a number field and let AF be its ring of adeles. Let B be a quaternion algebra over F and let ν: B → F be the reduced norm. Consider the reductive monoid M over F whose points in an F-algebra R are given by (Formula Presented). Motivated by an influential conjecture of Braverman and… read more about this publication »
Let V 1 ,V 2 ,V 3 be a triple of even dimensional vector spaces over a number field F equipped with nondegenerate quadratic forms Q 1 ,Q 2 ,Q 3 , respectively. Let Y⊂∏i=1V i be the closed subscheme consisting of (v 1 ,v 2 ,v 3 ) on which Q 1 (v 1 )=Q 2 (v 2 )=Q 3 (v 3 ). Motivated by conjectures of… read more about this publication »