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First-order optimality condition of basis pursuit denoise problem

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Applied Mathematics and Mechanics(English Edition) 2014年第10期35卷 1345-1352页

作者：朱玮舒适成礼智 School of Mathematics and computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Department of Mathematics and computational Science College of ScienceNational University of Defense Technology

A new first-order optimality condition for the basis pursuit denoise （BPDN） problem is derived. This condition provides a new approach to choose the penalty param- eters adaptively for a fixed point iteration algorithm. Meanwhile, the result is extended to matrix completion which is a new field on the heel of the compressed sensing. The numerical experiments of sparse vector recovery and low-rank matrix completion show validity of the theoretic results.

关键词： basis pursuit denoise (BPDN) fixed point iteration first-order optimality,matrix completion

Nonlinear instability for nonhomogeneous incompressible viscous fluids

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Science China Mathematics 2013年第4期56卷 665-686页

作者： JIANG Fei JIANG Song NI GuoXi College of Mathematics and Computer Science Fuzhou University Laboratory of computational Physics Institute of Applied Physics and Computational Mathematics

We investigate the nonlinear instability of a smooth steady density profile solution to the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field,including a Rayleigh-Taylor steady-state solution with heavier density with increasing height(referred to the Rayleigh-Taylor instability).We first analyze the equations obtained from linearization around the steady density profile *** we construct solutions to the linearized problem that grow in time in the Sobolev space H k,thus leading to a global instability result for the linearized *** the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations,we can then demonstrate the instability of the nonlinear problem in some *** analysis shows that the third component of the velocity already induces the instability,which is different from the previous known results.

关键词： nonhomogeneous Navier-Stokes equations steady density profile Rayieigh-Taylor instability incompressible viscous flows

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

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Science China Mathematics 2007年第10期50卷 1401-1417页

作者： Chang-xing MIAO~(1+) Bao-quan YUAN~2 ~1 Institute of Applied Physics and computational Mathematics,P.O.Box 8009,Beijing 100088,China ~2 College of Mathematics and informatics,Henan Polytechnic University,Jiaozuo 454000,China 1. Institute of Applied Physics and computational Mathematics P.O. Box 8009 Beijing 100088 China 2. College of Mathematics and informatics Henan Polytechnic University Jiaozuo 454000 China

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey *** first define a class of weak Morrey type spaces M_(p,λ)~*(R^n)on the basis of Lorentz space L_(p,∞)=L_p~*(R^n) (in particular,M_(?)~*(R^n)=L_(p,∞),if p>1),and study some fundamental properties of them;Second, we prove that the heat operator U(t)= e^(tΔ)and Calderón-Zygmund singular integral operators are bounded linear operators on weak Morrey spaces,and establish the bilinear estimate in weak Morrey ***,by means of Kato's method and the contraction mapping principle,we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces M_(p,λ)~*(R^n)(1

关键词： Navier-Stokes equations, weak Morrey spaces, Lorentz spaces, Cauchy problem,time-global well-posedness, self-similar solutions

R-T instability model of magnetic fluid and its numerical simulations

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Journal of Central South University 2008年第S1期15卷 266-270页

作者：郑秋云李明军舒适 School of Mathematics and computational Science Xiangtan University

The Rayleigh-Taylor(R-T) instability of ferrofluid has been the subject of recent research,because of its implications on the stability of *** neglecting the viscosity and rotation of magnetic fluid,and assuming that the magnetic particles are irrotational and temperature insensitive,we obtain a simplified R-T instability model of magnetic *** the interface tracing,we use five-order weighted essentially non-oscillatory(WENO) scheme to spatial direction and three-order TVD R-K method to time direction on the uniform mesh,*** the direction of the external magnetic field is the same as that of gravity,the velocities of the interface will be *** if the direction of the external magnetic field is in opposition to the direction of gravity,the velocities of the interface will be *** the direction of the external magnetic field is perpendicular to the direction of gravity,the symmetry of the interface will be *** of the action which is produced by perpendicular external magnetic field,there are other bubbles at the boudaries which parallel the direction of *** we increase the magnetic susceptibility of the magnetic fluids,the effects of external magnetic fields will be more distinct for the interface tracing.

关键词： magnetic fluid Rayleigh-Taylor instability weighted essentially non-oscillatory scheme TVD R-K method

A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics

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Science China Mathematics 2020年第7期63卷 1235-1258页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The program in Applied and computational Mathematics Princeton UniversityNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are *** initialization schemes as well as general regimes for the network width and training data size are *** the overparametrized regime,it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the *** addition,it is proved that throughout the training process the functions represented by the neural network model are uniformly close to those of a kernel *** general values of the network width and training data size,sharp estimates of the generalization error are established for target functions in the appropriate reproducing kernel Hilbert space.

关键词： two-layer neural network random feature model Gram matrix generalization error

An ab initio study of niobium (n=2-11) clusters: structure, stability and magnetism

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Chinese Physics B 2009年第4期18卷 1491-1497页

作者：任凤竹王渊旭张光彪魏书克罗有华 Institute of computational Materials Science School of Physics and Electronics Henan University

The ground-state configurations of the Nbn （n = 2-11） clusters are studied through the first-principles calculations. It is found that niobium clusters （n = 2-11） tend to form compact structures with low symmetry. The clusters with 4, 8 and 10 atoms axe found to be magic and have relatively large highest occupied-lowest unoccupied molecular orbital （HOMO-LUMO） gaps. The Nbn clusters possess low magnetic moments, which exhibit an odd-even oscillational character. The analyses of calculated electronic density and population of the lowest-energy niobium clusters for n =2, 3, 5, 7, 9, 11 show that the total magnetic moments of Nbn originate mainly from a few Nb atoms with longer spacings between them in most cases, while they are located on two Nb atoms for n = 2, 3, 5. The total magnetic moments come mainly from the 4d local moments but with the exception of the Nb5 cluster.

关键词： Nbn cluster density functional theory structure stability and magnetism

Proof of Kochen-Specker Theorem： Conversion of Product Rule to Sum Rule

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Chinese Physics Letters 2009年第7期26卷 29-31页

作者： S.P. Toh Faculty of Applied Science Inti International University College Persiaran Perdana BBN Putra Nilai 71800 Nilai Negeri Sembilan Malaysia Laboratory of computational Science and informatics Institute for Mathematical Research Universiti Putra Malaysia 43400 UPM Serdang Selangor Malaysia

Valuation functions of observables in quantum mechanics are often expected to obey two constraints called the sum rule and product rule. However, the Kochen-Specker （KS） theorem shows that for a Hilbert space of quantum mechanics of dimension d ≥ 3, these constraints contradict individually with the assumption of value definiteness. The two rules are not irrelated and Peres [Found. Phys. 26 （1996）807] has conceived a method of converting the product rule into a sum rule for the case of two qubits. Here we apply this method to a proof provided by Mermin based on the product rule for a three-qubit system involving nine operators. We provide the conversion of this proof to one based on sum rule involving ten operators.

关键词： Mathematical physics computational physics Quantum information and quantum mechanics

Applications of probability and statistics in cancer genomics

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Quantitative Biology 2020年第2期8卷 95-108页

作者： Xiaotu Ma Sasi Arunachalam Yanling Liu Department of computational Biology and Cancer Biology ProgramComprehensive Cancer CenterSt Jude Children's Research HospitalMemphisTN 38105USA

Background:The past decade has witnessed a rapid progress in our understanding of the genetics of cancer and its *** and statistical modeling played a pivotal role in the discovery of general patterns from cancer genomics datasets and continue to be of central importance for personalized ***:In this review we introduce cancer genomics from a probabilistic and statistical *** start from(1)functional classification of genes into oncogenes and tumor suppressor genes,then(2)demonstrate the importance of comprehensive analysis of different mutation types for individual cancer genomes,followed by(3)tumor purity analysis,which in turn leads to(4)the concept of ploidy and clonality,that is next connected to(5)tumor evolution under treatment pressure,which yields insights into cancer drug *** also discuss future challenges including the non-coding genomic regions,integrative analysis of genomics and epigenomics,as well as early cancer ***:We believe probabilistic and statistical modeling will continue to play important roles for novel discoveries in the field o f cancer genomics and personalized medicine.

关键词： cancer genomics sequence analysis probability and statistics

Mathematical Study of the Dynamics of the Development of HIV

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Journal of Applied Mathematics and Physics 2016年第1期4卷 66-72页

作者： Gabriel Iyam Ogban Konstantin Andreyevich Lebedev Department of computational Mathematics and informatics Kuban State University Krasnodar Russia

Over the history of humankind, they is no disease that has received so much attention as the HIV infection and mathematical models have been applied successfully to the investigation of HIV dynamics. It is, however, of note that, few of these investigations are able to explain the observation that host cell counts reduce while viral load increases as the infection progress. Also, various clinical studies of HIV infection have suggested that high T-cell activation levels are positively correlated with rapid disease development in untreated patients. This activation might be a major reason for the depletion of cells observed in most cases of long-term untreated HIV infection. In this paper, we use a simple mathematical model without treatment to investigate the stability of the system and compare the results with that obtained numerically by the use of MATHCAD. Our model which is a system of differential equations describing the interaction of the HIV and the immune system is divided into three compartments: uninfected CD4T cells, infected CD4Tcells and the virus population. This third compartment includes an extra source of the virus since it is believed that the virus in the blood constitute less than 2% of the total population. We obtain a linearization of the original system, and using Routh-Hurwitz condition for the non-linear system, the critical points are unstable.

关键词： Model HIV CD4+T Cell Infected Cell Uninfected Cell Stability Linearization Equilibrium