Nonlocal theories have emerged with powerful models and methods to analyze and predict complex phenomena. Different versions of nonlocal operators have been proposed, each with its advantages and challenges. In this talk I will give an introduction to main ideas in the nonlocality framework and present two sets of results for Helmholtz-Hodge type decompositions.
This talk focuses on using a multiwavelet representation of the discontinuous Galerkin (DG) approximation for trouble cell indication. The multiwavelet representation is related to the jumps in the (derivatives of) […]
Online social media networks have become extremely influential sources of news and information. Given the large audience and the ease of sharing content online, the content that spreads on online […]
In this talk, we will talk about the thin film model derived for liquid film resulting from a distributed source on a vertical wall and some distinct properties about the model. We will discuss the different behavior and properties of the model with and without considering surface tension. When the surface tension is neglected, a […]
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions for multi-dimensional hyperbolic systems of conservation laws. We present a numerical algorithm to approximate statistical solutions of conservation laws and show that under the assumption of 'weak statistical scaling', which is inspired by […]
In this talk, we will present a recent study of the Maxwell-Bloch equations that model the nonlinear interactions of light and matter, where the light is modeled classically by the Maxwell's equations with dispersions and the medium is modeled quantum-mechanically by the multilevel rate equations. We will show the connection between rate equations and the […]
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 […]
Abstract: In the presentation we will discuss our research program concerning the search for the most singular behaviors possible in viscous incompressible flows. These events are characterized by extremal growth of various quantities, such as the enstrophy, which control the regularity of the solution. They are therefore intimately related to the question of possible singularity […]
The immersed boundary method was first developed in the 1970s to model the motion of heart valves and has since been utilized to study many different biological systems. While the […]
We use a mathematical model to describe the delivery of a drug to a specific region of the brain. The drug is carried by liposomes that can release their cargo […]
Abstract: The identification of potential super-spreader nodes within a network is a critical part of the study and analysis of real-world networks. Motivated by a new interpretation of the “shortest […]
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