The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artifcial boundary method, we frst derive the exact artifcial nonrefecting boundary conditions....The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artifcial boundary method, we frst derive the exact artifcial nonrefecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458-3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonrefecting boundary condition and leads us to reformulate the original infnite discrete system into an equivalent fnite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are fnally provided to demonstrate the efectiveness of our approach.展开更多
We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function s...We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2.The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.展开更多
In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-or...In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time stepsτ1=O(τ2)for the first level solution u1.This is,the first-order consistence of the first level solution u1 like BDF1(i.e.Euler scheme)as a starting point does not cause the loss of global temporal accuracy,and the ratios are updated to rk≤4.8645.展开更多
1 Introduction and Main Results In this paper,we study the dynamics of compressible viscous charged particles consisting of two-species particles(e.g.,ions and electrons)under the influence of the self-consistent elec...1 Introduction and Main Results In this paper,we study the dynamics of compressible viscous charged particles consisting of two-species particles(e.g.,ions and electrons)under the influence of the self-consistent electrostatic potential force in semiconductor device or plasma physics(cf.[1,6])which is modeled by the compressible Navier-Stokes-Poisson system(called NSP system in the sequel for simplicity).展开更多
This paper is concerned with the construction of global, large amplitude solutions to the Cauchy problem of the one-dimensional compressible Navier–Stokes system for a viscous radiative gas when the viscosity and hea...This paper is concerned with the construction of global, large amplitude solutions to the Cauchy problem of the one-dimensional compressible Navier–Stokes system for a viscous radiative gas when the viscosity and heat conductivity coefficients depend on both specific volume and absolute temperature. The data are assumed to be without vacuum,mass concentrations, or vanishing temperatures, and the same is shown to be hold for the global solution constructed. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and absolute temperature.展开更多
We proposed absorbing interface conditions for the simulation of linear wave propagation on non-uniform meshes.Based on the superposition principle of second-order linear wave equations,we decompose the interface cond...We proposed absorbing interface conditions for the simulation of linear wave propagation on non-uniform meshes.Based on the superposition principle of second-order linear wave equations,we decompose the interface condition problem into two subproblems around the interface:for the first one the conventional artificial absorbing boundary conditions is applied,while for the second one,the local analytic solutions can be derived.The proposed interface conditions permit a two-way transmission of low-frequency waves across mesh interfaces which can be supported by both coarse and fine meshes,and perform a one-way absorption of high-frequency waves which can only be supported by fine meshes when they travel from fine mesh regions to coarse ones.Numerical examples are presented to illustrate the efficiency of the proposed absorbing interface conditions.展开更多
In this paper,we present an improved analysis of the Physics Informed Neural Networks(PINNs)method for solving second-order elliptic equations.By assuming an intrinsic sparse structure in the underlying solution,we pr...In this paper,we present an improved analysis of the Physics Informed Neural Networks(PINNs)method for solving second-order elliptic equations.By assuming an intrinsic sparse structure in the underlying solution,we provide a convergence rate analysis that can overcome the curse of dimensionality(CoD).Specifically,using some approximation theory in Sobolev space together with the multivariate Faa di Bruno formula,we first derive the approximation error for composition functions with a small degree of freedom in each compositional layer.Furthermore,by integrating several results on the statistical error of neural networks,we obtain a refined convergence rate analysis for PINNs in solving elliptic equations with compositional solutions.We also demonstrate the benefits of the intrinsic sparse structure with two simple numerical examples.展开更多
In recent years,physical informed neural networks(PINNs)have been shown to be a powerful tool for solving PDEs empirically.However,numerical analysis of PINNs is still missing.In this paper,we prove the convergence ra...In recent years,physical informed neural networks(PINNs)have been shown to be a powerful tool for solving PDEs empirically.However,numerical analysis of PINNs is still missing.In this paper,we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition,by establishing the upper bounds on the number of training samples,depth and width of the deep neural networks to achieve desired accuracy.The error of PINNs is decomposed into approximation error and statistical error,where the approximation error is given in C2 norm with ReLU^(3)networks(deep network with activation function max{0,x^(3)})and the statistical error is estimated by Rademacher complexity.We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ReLU^(3)network,which is of immense independent interest.展开更多
We consider the computation of a nonlocal Helmholtz equation by using perfectly matched layer(PML).We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal opera...We consider the computation of a nonlocal Helmholtz equation by using perfectly matched layer(PML).We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form.After that,we give stability estimates of some weighted-average values of the nonlocal Helmholtz solution and prove that(i)the weighted-average value of the nonlocal PML solution decays exponentially in PML layers in one case;(ii)in the other case,the weighted-average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain.Particularly for a typical kernel functionγ_(1)(s)=1/2 e^(−|s|),we obtain the Green’s function of the nonlocal Helmholtz equation,and use the Green’s function to further prove that(i)the nonlocal PML solution decays exponentially in PML layers in one case;(ii)in the other case,the nonlocal Helmholtz solution itself decays exponentially outside some domain.Based on our theoretical analysis,the truncated nonlocal problems are discussed and an asymptotic compatibility scheme is also introduced to solve the resulting truncated problems.Finally,numerical examples are provided to verify the effectiveness and validation of our nonlocal PML strategy and theoretical findings.展开更多
In this paper, the author proves that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature by using the Omori-Yau maximum princ...In this paper, the author proves that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature by using the Omori-Yau maximum principle.展开更多
Global synchronizability of duplex networks induced by three different intra-layer rewiring mechanisms is explored in this paper.The rewiring mechanisms are named as model-preserving rewiring(MPR), simply direct rewir...Global synchronizability of duplex networks induced by three different intra-layer rewiring mechanisms is explored in this paper.The rewiring mechanisms are named as model-preserving rewiring(MPR), simply direct rewiring(SDR), and degree-preserving rewiring(DPR), respectively. It is found that high switching frequencies will certainly enhance global synchronizability for WSWS duplex networks(i.e., each layer is independently formed by the algorithm proposed by Watts and Strogatz for generating small-world networks), ER-ER duplex networks(i.e., each layer is independently generated by the algorithm proposed by Erd ¨os and Renyi) and BA-BA duplex networks(i.e., each layer is independently formed by the classical BA algorithm). Namely,the faster the intra-layer couplings are reconnected, the faster the duplex networks reach global synchronization. Furthermore,we find that by increasing the intra-or inter-coupling strengths, the WS-WS time-varying network’s global synchronizability is enhanced. Take the WS-WS time-varying network as an example, we find that SDR mechanism has greater impact on global synchronizability than MPR mechanism and DPR mechanism. The related dynamical networks can arrive at synchronization faster by SDR than by MPR or DPR. Thus, we only study the effects of SDR on ER-ER duplex networks and BA-BA duplex networks. In addition, we obtain the fact via numerical simulations that, switching intra-layer coupling topologies under SDR mechanism has the greatest impact on the BA-BA duplex network, followed by the ER-ER network, and has the weakest influence on the WS-WS duplex network in terms of improving the global synchronizability when all the intra-layer networks are sparse and have the same average degree. Finally, the global synchronizability of WS-WS and BA-BA time-varying networks is improved compared with static duplex networks, the reason being that the networks tend to be randomized under SDR according to analysis of the networks’ average clustering coefficients and degree distributions.展开更多
We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional compressible Navier-Stokes type system for a viscous,compressible,radiative and reactive gas,w...We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional compressible Navier-Stokes type system for a viscous,compressible,radiative and reactive gas,where the constitutive relations for the pressure p,the speci c internal energy e,the speci c volume v,the absolute temperature θ,and the specific entropy s are given by p=Rθv+aθ^(4)/3,e=C_(v)θ+avθ^(4),and s=C_(v)lnθ+4avθ^(3)/3+Rln v with R>0,C_(v)>0 and a>0 being the perfect gas constant,the speci c heat and the radiation constant,respectively.For such a specific gas motion,a somewhat surprising fact is that,generally speaking,the pressure p(v,s)is not a convex function of the specific volume v and the specific entropy s.Even so,we show in this paper that the rarefaction waves are time-asymptotically stable for large initial perturbation provided that the radiation constant a and the strength of the rarefaction waves are sufficiently small.The key point in our analysis is to deduce the positive lower and upper bounds on the specific volume and the absolute temperature,which are uniform with respect to the space and the time variables,but are independent of the radiation constant a.展开更多
We study 2D and 3D Prandtl equations of degenerate hyperbolic type,and establish without any structural assumption the Gevrey well-posedness with Gevrey index≤2.Compared with the classical parabolic Prandtl equations...We study 2D and 3D Prandtl equations of degenerate hyperbolic type,and establish without any structural assumption the Gevrey well-posedness with Gevrey index≤2.Compared with the classical parabolic Prandtl equations,the loss of the derivatives,caused by the hyperbolic feature coupled with the degeneracy,cannot be overcame by virtue of the classical cancellation mechanism that developed for the parabolic counterpart.Inspired by the abstract Cauchy-Kowalewski theorem and by virtue of the hyperbolic feature,we give in this text a straightforward proof,basing on an elementary L^(2)energy estimate.In particular our argument does not involve the cancellation mechanism used efficiently for the classical Prandtl equations.展开更多
In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic ...In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic property is preserved along the gradient flow.Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form.When β=1,we derive a monotonicity formula for the flow.As applications,we show that the l-tangent cone of the flow consists of the finite flat planes.展开更多
In this paper,we prove gradient estimates for the positive solutions of Lu=0 and Lu=?u/?t on conformal solitons,where L(·)=Δ(·)+<X,?(·)>.We also give some applications for them.
Boundary integral equations provide a powerful tool for the solution of scattering problems.However,often a singular kernel arises,in which case the standard quadratures will give rise to unavoidable deteriorations in...Boundary integral equations provide a powerful tool for the solution of scattering problems.However,often a singular kernel arises,in which case the standard quadratures will give rise to unavoidable deteriorations in numerical precision,thus special treatment is needed to handle the singular behavior.Especially,for inhomogeneous media,it is difficult if not impossible to find out an analytical expression for Green’s function.In this paper,an efficient fourth-order accurate Cartesian grid-based method is proposed for the two-dimensional Helmholtz scattering and transmission problems with inhomogeneous media.This method provides an alternative approach to indirect integral evaluation by solving equivalent interface problems on Cartesian grid with a modified fourth-order accurate compact finite difference scheme and a fast Fourier transform preconditioned conjugate gradient(FFT-PCG)solver.A remarkable point of this method is that there is no need to know analytical expressions for Green’s function.Numerical experiments are provided to demonstrate the advantage of the current approach,including its simplicity in implementation,its high accuracy and efficiency.展开更多
We study deviation inequalities for some quadratic Wiener functionals and moderate deviations for parameter estimators in a linear stochastic differential equation model.Firstly,we give some estimates for Laplace inte...We study deviation inequalities for some quadratic Wiener functionals and moderate deviations for parameter estimators in a linear stochastic differential equation model.Firstly,we give some estimates for Laplace integrals of the quadratic Wiener functionals by calculating the eigenvalues of the associated HilbertSchmidt operators.Then applying the estimates,we establish deviation inequalities for the quadratic functionals and moderate deviation principles for the parameter estimators.展开更多
Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical ...Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical analysis on deep Ritz method(DRM)[47]for second order elliptic equations with Neumann boundary conditions.We establish the first nonasymptotic convergence rate in H^(1)norm for DRM using deep networks with ReLU^(2)activation functions.In addition to providing a theoretical justification of DRM,our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples.Technically,we derive bound on the approximation error of deep ReLU^(2)network in C^(1)norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ReLU^(2)network,both of which are of independent interest.展开更多
In this paper,we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator.With the help of the subcritical approach from the variational method,we obtain the ...In this paper,we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator.With the help of the subcritical approach from the variational method,we obtain the non-existence,existence,and multiplicity results under some given assumptions.展开更多
基金Jiwei Zhang is partially supported by the National Natural Science Foundation of China under Grant No.11771035the NSAF U1530401+3 种基金the Natural Science Foundation of Hubei Province No.2019CFA007Xiangtan University 2018ICIP01Chunxiong Zheng is partially supported by Natural Science Foundation of Xinjiang Autonom ous Region under No.2019D01C026the National Natural Science Foundation of China under Grant Nos.11771248 and 91630205。
文摘The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artifcial boundary method, we frst derive the exact artifcial nonrefecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458-3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonrefecting boundary condition and leads us to reformulate the original infnite discrete system into an equivalent fnite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are fnally provided to demonstrate the efectiveness of our approach.
基金W.-X.Li's research was supported by NSF of China(11871054,11961160716,12131017)the Natural Science Foundation of Hubei Province(2019CFA007)T.Yang's research was supported by the General Research Fund of Hong Kong CityU(11304419).
文摘We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer.A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2.The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.
基金Natural Science Foundation of Hubei Province(2019CFA007)Supported by NSFC(11771035).
文摘In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time stepsτ1=O(τ2)for the first level solution u1.This is,the first-order consistence of the first level solution u1 like BDF1(i.e.Euler scheme)as a starting point does not cause the loss of global temporal accuracy,and the ratios are updated to rk≤4.8645.
基金Supported by the National Natural Science Foundation of China(12101472,11731008)。
文摘1 Introduction and Main Results In this paper,we study the dynamics of compressible viscous charged particles consisting of two-species particles(e.g.,ions and electrons)under the influence of the self-consistent electrostatic potential force in semiconductor device or plasma physics(cf.[1,6])which is modeled by the compressible Navier-Stokes-Poisson system(called NSP system in the sequel for simplicity).
基金supported by National Natural Science Foundation of China(11601398,11671309,11731008)
文摘This paper is concerned with the construction of global, large amplitude solutions to the Cauchy problem of the one-dimensional compressible Navier–Stokes system for a viscous radiative gas when the viscosity and heat conductivity coefficients depend on both specific volume and absolute temperature. The data are assumed to be without vacuum,mass concentrations, or vanishing temperatures, and the same is shown to be hold for the global solution constructed. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and absolute temperature.
基金supported by the National Key Research and Development Program of China(No.2020YFA0714200)by the National Nature Science Foundation of China(Nos.12125103,12071362)+1 种基金by the Natural Science Foundation of Hubei Province(Nos.2021AAA010,2019CFA007)by the Fundamental Research Funds for the Central Universities.The numerical calculations have been done at the Supercomputing Center of Wuhan University.
文摘We proposed absorbing interface conditions for the simulation of linear wave propagation on non-uniform meshes.Based on the superposition principle of second-order linear wave equations,we decompose the interface condition problem into two subproblems around the interface:for the first one the conventional artificial absorbing boundary conditions is applied,while for the second one,the local analytic solutions can be derived.The proposed interface conditions permit a two-way transmission of low-frequency waves across mesh interfaces which can be supported by both coarse and fine meshes,and perform a one-way absorption of high-frequency waves which can only be supported by fine meshes when they travel from fine mesh regions to coarse ones.Numerical examples are presented to illustrate the efficiency of the proposed absorbing interface conditions.
基金supported by the National Key Research and Development Program of China(No.2020YFA0714200)by the Natural Science Foundation of Hubei Province(Nos.2021AAA010 and 2019CFA007)+1 种基金the National Nature Science Foundation of China(Nos.12301558,12288101,12125103 and 12071362)by the Fundamental Research Funds for the Central Universities.
文摘In this paper,we present an improved analysis of the Physics Informed Neural Networks(PINNs)method for solving second-order elliptic equations.By assuming an intrinsic sparse structure in the underlying solution,we provide a convergence rate analysis that can overcome the curse of dimensionality(CoD).Specifically,using some approximation theory in Sobolev space together with the multivariate Faa di Bruno formula,we first derive the approximation error for composition functions with a small degree of freedom in each compositional layer.Furthermore,by integrating several results on the statistical error of neural networks,we obtain a refined convergence rate analysis for PINNs in solving elliptic equations with compositional solutions.We also demonstrate the benefits of the intrinsic sparse structure with two simple numerical examples.
基金supported by the National Key Research and Development Program of China(No.2020YFA0714200)the National Science Foundation of China(No.12125103,No.12071362,No.11971468,No.11871474,No.11871385)+1 种基金the Natural Science Foundation of Hubei Province(No.2021AAA010,No.2019CFA007)the Fundamental Research Funds for the Central Universities.
文摘In recent years,physical informed neural networks(PINNs)have been shown to be a powerful tool for solving PDEs empirically.However,numerical analysis of PINNs is still missing.In this paper,we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition,by establishing the upper bounds on the number of training samples,depth and width of the deep neural networks to achieve desired accuracy.The error of PINNs is decomposed into approximation error and statistical error,where the approximation error is given in C2 norm with ReLU^(3)networks(deep network with activation function max{0,x^(3)})and the statistical error is estimated by Rademacher complexity.We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ReLU^(3)network,which is of immense independent interest.
基金supported by the NSFC(Grants No.11771035,12071401,12171376,2020-JCJQ-ZD-029)the Natural Science Foundation of Hunan Province(Grant No.2019JJ50572)+1 种基金the Natural Science Foundation of Hubei Province(Grant No.2019CFA007)the Xiangtan University(Grant No.2018ICIP01)。
文摘We consider the computation of a nonlocal Helmholtz equation by using perfectly matched layer(PML).We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form.After that,we give stability estimates of some weighted-average values of the nonlocal Helmholtz solution and prove that(i)the weighted-average value of the nonlocal PML solution decays exponentially in PML layers in one case;(ii)in the other case,the weighted-average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain.Particularly for a typical kernel functionγ_(1)(s)=1/2 e^(−|s|),we obtain the Green’s function of the nonlocal Helmholtz equation,and use the Green’s function to further prove that(i)the nonlocal PML solution decays exponentially in PML layers in one case;(ii)in the other case,the nonlocal Helmholtz solution itself decays exponentially outside some domain.Based on our theoretical analysis,the truncated nonlocal problems are discussed and an asymptotic compatibility scheme is also introduced to solve the resulting truncated problems.Finally,numerical examples are provided to verify the effectiveness and validation of our nonlocal PML strategy and theoretical findings.
基金This work was supported by the National Natural Science Foundation of China(No.11771339)the Fundamental Research Funds for the Central Universities(No.2042019kf0198)the Youth Talent Training Program of Wuhan University。
文摘In this paper, the author proves that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature by using the Omori-Yau maximum principle.
基金supported by the National Key Research and Development Program of China(Grant No.2018AAA0101100)the National Natural Science Foundation of China(Grant No.61973241)the Natural Science Foundation of Hubei Province(Grant No.2019CFA007)。
文摘Global synchronizability of duplex networks induced by three different intra-layer rewiring mechanisms is explored in this paper.The rewiring mechanisms are named as model-preserving rewiring(MPR), simply direct rewiring(SDR), and degree-preserving rewiring(DPR), respectively. It is found that high switching frequencies will certainly enhance global synchronizability for WSWS duplex networks(i.e., each layer is independently formed by the algorithm proposed by Watts and Strogatz for generating small-world networks), ER-ER duplex networks(i.e., each layer is independently generated by the algorithm proposed by Erd ¨os and Renyi) and BA-BA duplex networks(i.e., each layer is independently formed by the classical BA algorithm). Namely,the faster the intra-layer couplings are reconnected, the faster the duplex networks reach global synchronization. Furthermore,we find that by increasing the intra-or inter-coupling strengths, the WS-WS time-varying network’s global synchronizability is enhanced. Take the WS-WS time-varying network as an example, we find that SDR mechanism has greater impact on global synchronizability than MPR mechanism and DPR mechanism. The related dynamical networks can arrive at synchronization faster by SDR than by MPR or DPR. Thus, we only study the effects of SDR on ER-ER duplex networks and BA-BA duplex networks. In addition, we obtain the fact via numerical simulations that, switching intra-layer coupling topologies under SDR mechanism has the greatest impact on the BA-BA duplex network, followed by the ER-ER network, and has the weakest influence on the WS-WS duplex network in terms of improving the global synchronizability when all the intra-layer networks are sparse and have the same average degree. Finally, the global synchronizability of WS-WS and BA-BA time-varying networks is improved compared with static duplex networks, the reason being that the networks tend to be randomized under SDR according to analysis of the networks’ average clustering coefficients and degree distributions.
基金supported by the Fundamental Research Funds for the Central Universities and National Natural Science Foundation of China(Grant Nos.11731008 and 11671309)supported by the Fundamental Research Funds for the Central Universities(Grant No.YJ201962)supported by National Postdoctoral Program for Innovative Talents of China(Grant No.BX20180054).
文摘We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional compressible Navier-Stokes type system for a viscous,compressible,radiative and reactive gas,where the constitutive relations for the pressure p,the speci c internal energy e,the speci c volume v,the absolute temperature θ,and the specific entropy s are given by p=Rθv+aθ^(4)/3,e=C_(v)θ+avθ^(4),and s=C_(v)lnθ+4avθ^(3)/3+Rln v with R>0,C_(v)>0 and a>0 being the perfect gas constant,the speci c heat and the radiation constant,respectively.For such a specific gas motion,a somewhat surprising fact is that,generally speaking,the pressure p(v,s)is not a convex function of the specific volume v and the specific entropy s.Even so,we show in this paper that the rarefaction waves are time-asymptotically stable for large initial perturbation provided that the radiation constant a and the strength of the rarefaction waves are sufficiently small.The key point in our analysis is to deduce the positive lower and upper bounds on the specific volume and the absolute temperature,which are uniform with respect to the space and the time variables,but are independent of the radiation constant a.
基金supported by the NSF of China(Grant Nos.11961160716,11871054,12131017)the Natural Science Foundation of Hubei Province(Grant No.2019CFA007).
文摘We study 2D and 3D Prandtl equations of degenerate hyperbolic type,and establish without any structural assumption the Gevrey well-posedness with Gevrey index≤2.Compared with the classical parabolic Prandtl equations,the loss of the derivatives,caused by the hyperbolic feature coupled with the degeneracy,cannot be overcame by virtue of the classical cancellation mechanism that developed for the parabolic counterpart.Inspired by the abstract Cauchy-Kowalewski theorem and by virtue of the hyperbolic feature,we give in this text a straightforward proof,basing on an elementary L^(2)energy estimate.In particular our argument does not involve the cancellation mechanism used efficiently for the classical Prandtl equations.
基金supported by theNationalNatural Science Foundation of China,Nos.11721101,12071352,12031017。
文摘In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic property is preserved along the gradient flow.Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form.When β=1,we derive a monotonicity formula for the flow.As applications,we show that the l-tangent cone of the flow consists of the finite flat planes.
基金supported by the National Natural Science Foundation of China(Grant No.12071352)。
文摘In this paper,we prove gradient estimates for the positive solutions of Lu=0 and Lu=?u/?t on conformal solitons,where L(·)=Δ(·)+<X,?(·)>.We also give some applications for them.
基金supported by the NSFC(Grant No.12001193),by the Scientific Research Fund of Hunan Provincial Education Department(Grant No.20B376)by the Key Projects of Hunan Provincial Department of Education(Grant No.22A033)+4 种基金by the Changsha Municipal Natural Science Foundation(Grant Nos.kq2014073,kq2208158).W.Ying is supported by the NSFC(Grant No.DMS-11771290)by the Science Challenge Project of China(Grant No.TZ2016002)by the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA25000400).J.Zhang was partially supported by the National Natural Science Foundation of China(Grant No.12171376)by the Fundamental Research Funds for the Central Universities(Grant No.2042021kf0050)by the Natural Science Foundation of Hubei Province(Grant No.2019CFA007).
文摘Boundary integral equations provide a powerful tool for the solution of scattering problems.However,often a singular kernel arises,in which case the standard quadratures will give rise to unavoidable deteriorations in numerical precision,thus special treatment is needed to handle the singular behavior.Especially,for inhomogeneous media,it is difficult if not impossible to find out an analytical expression for Green’s function.In this paper,an efficient fourth-order accurate Cartesian grid-based method is proposed for the two-dimensional Helmholtz scattering and transmission problems with inhomogeneous media.This method provides an alternative approach to indirect integral evaluation by solving equivalent interface problems on Cartesian grid with a modified fourth-order accurate compact finite difference scheme and a fast Fourier transform preconditioned conjugate gradient(FFT-PCG)solver.A remarkable point of this method is that there is no need to know analytical expressions for Green’s function.Numerical experiments are provided to demonstrate the advantage of the current approach,including its simplicity in implementation,its high accuracy and efficiency.
基金supported by NSFC (Grant Nos.11971358,11571259,11771339)Hubei Provincial Natural Science Foundation of China (No.2021CFB400)+1 种基金Fundamental Research Funds for the Central Universities (No.2042019kf0198)the Youth Talent Training Program of Wuhan University。
文摘In this article,we prove that a quasi-isometric map between rank one symmetric spaces is within bounded distance from an f-harmonic map.
基金National Natural Science Foundation of China(Grant Nos. 11171262,11571262 and 11101210)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20130141110076)+1 种基金the Fundamental Research Funds for the Central Universities(Grant No.NS2015074)China Postdoctoral Science Foundation(Grant Nos.2013M531341 and 2016T90450)
文摘We study deviation inequalities for some quadratic Wiener functionals and moderate deviations for parameter estimators in a linear stochastic differential equation model.Firstly,we give some estimates for Laplace integrals of the quadratic Wiener functionals by calculating the eigenvalues of the associated HilbertSchmidt operators.Then applying the estimates,we establish deviation inequalities for the quadratic functionals and moderate deviation principles for the parameter estimators.
基金supported by the National Key Research and Development Program of China(No.2020YFA0714200)the Science and Technology Major Project of Hubei Province under Grant 2021AAA010+2 种基金the National Science Foundation of China(Nos.12125103,12071362,11871474,11871385)the Natural Science Foundation of Hubei Province(No.2019CFA007)by the research fund of KLATASDSMOE.
文摘Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical analysis on deep Ritz method(DRM)[47]for second order elliptic equations with Neumann boundary conditions.We establish the first nonasymptotic convergence rate in H^(1)norm for DRM using deep networks with ReLU^(2)activation functions.In addition to providing a theoretical justification of DRM,our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples.Technically,we derive bound on the approximation error of deep ReLU^(2)network in C^(1)norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ReLU^(2)network,both of which are of independent interest.
基金National Natural Science Foundation of China(Grant Nos.11771342 and 11571259)the Natural Science Foundation of Hubei Province(Grant No.2019CFA007)。
文摘In this paper,we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator.With the help of the subcritical approach from the variational method,we obtain the non-existence,existence,and multiplicity results under some given assumptions.