Let TtP2f(x) denote the solution to the linear Schrödinger equation at time t, with initial value function f, where P 2(ξ) = ∣ξ∣2. In 1980, Carleson asked for the minimal regularity of f that is required for the pointwise a.e. convergence of TtP2f(x) to f(x) as t → 0. This was recently resolved by… read more about this publication »

In volume transmission (or neuromodulation) neurons do not make one-to-one connections to other neurons, but instead simply release neurotransmitter into the extracellular space from numerous varicosities. Many well-known neurotransmitters including serotonin (5HT), dopamine (DA), histamine (HA),… read more about this publication »

Chemotaxis is a directed cell movement in response to external chemical stimuli. In this paper, we propose a simple model for the origin of chemotaxis - namely how a directed movement in response to an external chemical signal may occur based on purely reaction-diffusion equations reflecting inner… read more about this publication »

We introduce a family of stochastic models motivated by the study of nonequilibrium steady states of fluid equations. These models decompose the deterministic dynamics of interest into fundamental building blocks, i.e., minimal vector fields preserving some fundamental aspects of the original… read more about this publication »

In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space that implies pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose… read more about this publication »

Spectral Barron spaces have received considerable interest recently, as it is the natural function space for approximation theory of two-layer neural networks with a dimension-free convergence rate. In this paper, we study the regularity of solutions to the whole-space static Schrödinger equation… read more about this publication »

Chan, Durrett, and Lanchier introduced a multitype contact process with temporal heterogeneity involving two species competing for space on the d-dimensional integer lattice. Time is divided into two seasons. They proved that there is an open set of the parameters for which both species can coexist… read more about this publication »

The main mathematical result in this paper is that change of variables in the ordinary differential equation (ODE) for the competition of two infections in a Susceptible-Infected-Removed (SIR) model shows that the fraction of cases due to the new variant satisfies the logistic differential equation… read more about this publication »

A better understanding of various patterns in the coronavirus disease 2019 (COVID-19) spread in different parts of the world is crucial to its prevention and control. Motivated by the previously developed Global Epidemic and Mobility (GLEaM) model, this paper proposes a new stochastic dynamic model… read more about this publication »

We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time t, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity… read more about this publication »

The Susceptible-Infectious-Recovered (SIR) equations and their extensions comprise a commonly utilized set of models for understanding and predicting the course of an epidemic. In practice, it is of substantial interest to estimate the model parameters based on noisy observations early in the… read more about this publication »

The 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 »

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 »

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 »

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 »

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 »

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 »

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 »