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| 1 | The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems显示文摘We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems. | Weinan E Bing Yu | 2018 | Communications in Mathematics and Statistics2018，6，1： | 22 |
| 2 | Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations显示文摘We study a new algorithm for solvingparabolic partial differential equations(PDEs)and backward stochastic differential equations(BSDEs)in high dimension,which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function,and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE.The policy function is then approximated by a neural network,as is done in deep reinforcement learning.Numerical results using TensorFlow illustrate the efficiency and accuracy of the studied algorithm for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen–Cahn equation,the Hamilton–Jacobi–Bellman equation,and a nonlinear pricing model for financial derivatives. | Weinan E Jiequn Han Arnulf Jentzen | 2017 | Communications in Mathematics and Statistics2017，5，4： | 20 |
| 3 | Donsker’s Invariance Principle Under the Sub-linear Expectation with an Application to Chung’s Law of the Iterated Logarithm显示文摘We prove a new Donsker’s invariance principle for independent and identically distributed random variables under the sub-linear expectation.As applications,the small deviations and Chung’s law of the iterated logarithm are obtained. | Li-Xin Zhang | 2015 | Communications in Mathematics and Statistics2015，3，2： | 17 |
| 4 | On Some Results in the Theory of Finite Partially Soluble Groups显示文摘This article provides an overview of some recent results and ideas relatedto the study of finite groups depending on the restrictions on some systems of theirsections.In particular,we discuss some properties of the lattice of all subgroups ofa finite group related with conditions of permutability and generalized subnormality for subgroups.The paper contains more than 30 open problems which were posed,atdifferent times,by some mathematicians working in the discussed direction. | Alexander N.Skiba | 2016 | Communications in Mathematics and Statistics2016，4，3： | 14 |
| 5 | Criteria for Discrete Spectrum of 1D Operators显示文摘For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators.Interestingly,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context.All of them are illustrated by examples.The main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper. | Mu-Fa Chen | 2014 | Communications in Mathematics and Statistics2014，2，3： | 10 |
| 6 | Finite p-Groups all of Whose Subgroups of Index p^(3) are Abelian显示文摘Suppose that G is a finite p-group.If all subgroups of index p^(t)of G are abelian and at least one subgroup of index p^(t−1)of G is not abelian,then G is called an A_(t)-group.We useA0-group to denote an abelian group.From the definition,we know every finite non-abelian p-group can be regarded as an A_(t)-group for some positive integer t.A_(1)-groups and A_(2)-groups have been classified.Classifying A_(3)-groups is an old problem.In this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to isomorphism.Moreover,we determine the Frattini subgroup,the derived subgroup and the center of every A_(3)-group,and give the number of A_(1)-subgroups and the triple(μ_(0),μ_(1),μ_(2))of every A_(3)-group,whereμi denotes the number of A_(i)-subgroups of index p of A_(3)-groups. | Qinhai Zhang Libo Zhao Miaomiao Li Yiqun Shen | 2015 | Communications in Mathematics and Statistics2015，3，1： | 9 |
| 7 | Isospectral Operators显示文摘For a large class of integral operators or second-order differential operators,their isospectral(or cospectral)operators are constructed explicitly in terms of htransform(duality).This provides us a simple way to extend the known knowledge on the spectrum(or the estimation of the principal eigenvalue)from a smaller class of operators to a much larger one.In particular,an open problem about the positivity of the principal eigenvalue for birth–death processes is solved in the paper. | Mu-Fa Chen Xu Zhang | 2014 | Communications in Mathematics and Statistics2014，2，1： | 7 |
| 8 | On the Linear Independence and Partition of Unity of Arbitrary Degree Analysis-Suitable T-splines显示文摘Analysis-suitable T-splines are a topological-restricted subset of T-splines,which are optimized to meet the needs both for design and analysis(Li and Scott ModelsMethods Appl Sci 24:1141-1164,2014;Li et al.Comput Aided Geom Design 29:63-76,2012;Scott et al.Comput Methods Appl Mech Eng 213-216,2012).The paper independently derives a class of bi-degree(d_(1),d_(2))T-splines for which no perpendicular T-junction extensions intersect,and provides a new proof for the linearly independence of the blending functions.We also prove that the sum of the basis functions is one for an analysis-suitable T-spline if the T-mesh is admissible based on a recursive relation. | Jingjing Zhang Xin Li | 2015 | Communications in Mathematics and Statistics2015，3，3： | 6 |
| 9 | Statistical inference for zero-and-one-inflated poisson models显示文摘In this paper, a zero-and-one-inflated Poisson (ZOIP) model is studied. The maximum likelihoodestimation and the Bayesian estimation of the model parameters are obtained based on dataaugmentation method. A simulation study based on proposed sampling algorithm is conductedto assess the performance of the proposed estimation for various sample sizes. Finally, two realdata-sets are analysed to illustrate the practicability of the proposed method. | Yincai Tang Wenchen Liu Ancha Xu | 2017 | Statistical Theory and Related Fields2017，1，2： | 5 |
| 10 | Finite Minimal Non-supersolvable Groups Decomposable into the Product of Two Normal Supersolvable Subgroups显示文摘In this paper,we determine the finite minimal non-supersolvable groups decomposable into the product of two normal supersolvable subgroups. | Wenbin Guo A.S.Kondrat’ev | 2015 | Communications in Mathematics and Statistics2015，3，2： | 5 |
| 11 | Dedicated to Professor Chi-Wang Shu on the occasion of his 60th birthday A Proposal on Machine Learning via Dynamical Systems显示文摘Wediscuss the idea of using continuous dynamicalsystemstomodel generalhigh-dimensional nonlinear functions used in machine learning.We also discuss theconnection with deep learning. | Weinan E | 2017 | Communications in Mathematics and Statistics2017，5，1： | 4 |
| 12 | Numerical Simulation of a Class of Nonlinear Wave Equations by Lattice Boltzmann Method显示文摘In this paper,we develop a lattice Boltzmann model for a class ofone-dimensional nonlinear wave equations,including the second-order hyperbolictelegraph equation,the nonlinear Klein-Gordon equation,the damped and undampedsine-Gordon equation and double sine-Gordon equation.By choosing properly theconservation condition between the macroscopic quantity u,and the distributionfunctions and applying the Chapman-Enskog expansion,the governing equation isrecovered correctly from the lattice Boltzmann equation.Moreover,the local equilib-rium distribution function is obtained.The results of numerical examples have beencompared with the analytical solutions to confirm the good accuracy and the applica-bility of our scheme. | Yali Duan Linghua Kong Min Guo | 2017 | Communications in Mathematics and Statistics2017，5，1： | 4 |
| 13 | On Boundary Factors and Traces of Subgroups of Finite Groups显示文摘A subgroup E of a finite group G is called hypercyclically embedded in G if every chief factor of G below E is cyclic.Let A be a subgroup of a group G.Then we call any chief factor H/AG of G a G-boundary factor of A.For any G-boundary factor H/AG of A,we call the subgroup(A∩H)/AG of G/AG a G-trace of A.On the basis of these notions,we give some new characterizations of hypercyclically embedded subgroups. | W.Guo Alexander N.Skiba X.Tang | 2014 | Communications in Mathematics and Statistics2014，2，3： | 4 |
| 14 | The Concavity of the Gaussian Curvature of the Convex Level Sets of p-Harmonic Functions with Respect to the Height显示文摘For the p-harmonic function with strictly convex level sets,we find an auxiliary function which comes from the combination of the norm of gradient of the p-harmonic function and the Gaussian curvature of the level sets of p-harmonic function.We prove that this curvature function is concave with respect to the height of the p-harmonic function.This auxiliary function is an affine function of the height when the p-harmonic function is the p-Green function on ball. | Xi-Nan Ma Wei Zhang | 2013 | Communications in Mathematics and Statistics2013，1，4： | 4 |
| 15 | Principal component analysis显示文摘 | HervéAbdi Lynne J.Williams | 2010 | Wiley Interdisciplinary Reviews Computational Statistics2010，，4： | 3 |
| 16 | A Note on Curvature Estimate of the Hermitian–Yang–Mills Flow Dedicated to celebrate the Sixtieth anniversary of USTC显示文摘In this paper,we study the curvature estimate of the Hermitian–Yang–Mills flow on holomorphic vector bundles.In one simple case,we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder–Narasimhan–Seshadri filtration of the holomorphic vector bundle. | Jiayu Li Chuanjing Zhang Xi Zhang | 2018 | Communications in Mathematics and Statistics2018，6，3： | 3 |
| 17 | Partial C^0-Estimate for Kähler–Einstein Metrics显示文摘In this short note,we give a proof of our partial C^0-estimate for Kähler-Einstein metrics.Our proof uses a compactness theorem of Cheeger-Colding-Tian and L^2-estimate for∂^--operator. | Gang Tian | 2013 | Communications in Mathematics and Statistics2013，1，2： | 3 |
| 18 | Extracting summary statistics to perform meta‐analyses of the published literature for survival endpoints显示文摘 | Mahesh K. B.Parmar ValterTorri LesleyStewart | 1999 | Statist Med1999，，24： | 3 |
| 19 | Evolution of the social network of scientific collaborations显示文摘 | A.L Barabási H Jeong Z Néda E Ravasz A Schubert T Vicsek | 2002 | Physica A: Statistical Mechanics and its Applications2002，，3： | 3 |
| 20 | Scale-space theory: a basic tool for analyzing structures at different scales显示文摘 | Tony Lindeberg | 1994 | Journal of Applied Statistics . 1994 (1-2)1994，，1： | 3 |
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