Translating solutions to mean curvature flow with a forcing term in Minkowski space
Hongjie Ju Jian Lu Huaiyu Jian
We study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solutions to mean curvature flow with a forcing term in Minkowski space. As a result, a part of conjectures in [1] is proved.
keywords: elliptic equation asymptotic behavior. symmetric solution Spacelike hypersurface
An integrated cellular and sub-cellular model of cancer chemotherapy and therapies that target cell survival
Alexis B. Cook Daniel R. Ziazadeh Jianfeng Lu Trachette L. Jackson
Apoptosis resistance is a hallmark of human cancer, and tumor cells often become resistant due to defects in the programmed cell death machinery. Targeting key apoptosis regulators to overcome apoptotic resistance and promote rapid death of tumor cells is an exciting new strategy for cancer treatment, either alone or in combination with traditionally used anti-cancer drugs that target cell division. Here we present a multiscale modeling framework for investigating the synergism between traditional chemotherapy and targeted therapies aimed at critical regulators of apoptosis.
keywords: cisplatin. Multiscale model cancer therapy
Topological degree method for the rotationally symmetric $L_p$-Minkowski problem
Jian Lu Huaiyu Jian
Consider the existence of rotationally symmetric solutions to the $L_p$-Minkowski problem for $p=-n-1$. Recently a sufficient condition was obtained for the existence via the variational method and a blow-up analysis in [16]. In this paper we use a topological degree method to prove the same existence and show the result holds under a similar complementary sufficient condition. Moreover, by this degree method, we obtain the existence result in a perturbation case.
keywords: topological degree existence of solutions. Monge-Ampère equation Minkowski problem centro-affine Gauss curvature
Mathematical theory of solids: From quantum mechanics to continuum models
Weinan E Jianfeng Lu
keywords: density functional theory atomistic model Cauchy-Born rule multiscale modeling.
Multiplicative noise removal with a sparsity-aware optimization model
Jian Lu Lixin Shen Chen Xu Yuesheng Xu

Restoration of images contaminated by multiplicative noise (also known as speckle noise) is a key issue in coherent image processing. Notice that images under consideration are often highly compressible in certain suitably chosen transform domains. By exploring this intrinsic feature embedded in images, this paper introduces a variational restoration model for multiplicative noise reduction that consists of a term reflecting the observed image and multiplicative noise, a quadratic term measuring the closeness of the underlying image in a transform domain to a sparse vector, and a sparse regularizer for removing multiplicative noise. Being different from popular existing models which focus on pursuing convexity, the proposed sparsity-aware model may be nonconvex depending on the conditions of the parameters of the model for achieving the optimal denoising performance. An algorithm for finding a critical point of the objective function of the model is developed based on coupled fixed-point equations expressed in terms of the proximity operator of functions that appear in the objective function. Convergence analysis of the algorithm is provided. Experimental results are shown to demonstrate that the proposed iterative algorithm is sensitive to some initializations for obtaining the best restoration results. We observe that the proposed method with SAR-BM3D filtering images as initial estimates can remarkably outperform several state-of-art methods in terms of the quality of the restored images.

keywords: Multiplicative noise proximity operator nonconvex variational model image restoration

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