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Incommensurate structures come from stacking the single layers of low-dimensional materials on top of one another with misalignment such as a twist in orientation. While these structures are of signif...
In this talk I will present one of our recent work in rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system. We establish the classical solution theory of the degenerate parabol...
In this talk, I will present our recent work on the convergence/generalization analysis for the popular optimizers in deep learning. (1) We establish the convergence for Adam under (L0,L1 ) smoothness...
We consider the semiclassical limit from the Hartree equation with Coulomb interaction potential to the Vlasov–Poisson equation. Using a new stability estimate for the difference of the square roots o...
Zero-One Composite Optimization (0/1-COP) is a prototype of nonsmooth, non- convex optimization problems and it has attracted much attention recently. Augmented Lagrangian Method (ALM) has stood out a...
We study the nonrelativistic limit of the cubic Klein-Gordon equations. We show the cubic Klein-Gordon equation converges to the cubic Schr\"odinger equation with a convergence rate of order $\epsilon...
In this talk, we consider a distributed interval optimization problem (DIOP) with uncertainties, whose global function is formed by local convex interval functions. In seeking Pareto solutions for dis...
In this talk, we present the pointwise convergence of one-point large deviations rate functions (LDRFs) of the spatial finite difference method and further the fully discrete method based on the tempo...
Deep neural networks, as a powerful system to represent high dimensional complex functions, play a key role in deep learning. Convergence of deep neural networks is a fundamental issue in building the...
The study of the spectrum of the Laplace operator has produced an extensive literature. (See [Cha] and the references therein.) Of special interest to recent applications has been the behavior of spec...
This paper studies Adaptive Finite Element Methods (AFEMs), based on piecewise liear elements and newest vertex bisection, for solving second order elliptic equations with piecewise constant coeʂ...
We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or q...
We design an adaptive finite element method (AFEM) for mixed boundary value problems associated with the differential operator A − ∇div in H(div,Ω). For A being a variab...
Dispersing billiards with cusps are deterministic dnamical systems with a mild degree of chaos, exhibiting “intermittent” behavior that alternates between regular and chaotic paterns.
The Hausdorff distance between a compact convex set K CRd and random sets K c lRd iS studied. Basic inequalities are derived for the case of K being a convex subset of K. If applied to special seq...

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