In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve ...In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.展开更多
In various environmental studies, geoscience variables not only have the characteristics of time and space, but also are influenced by other variables. Multivariate spatiotemporal variables can improve the accuracy of...In various environmental studies, geoscience variables not only have the characteristics of time and space, but also are influenced by other variables. Multivariate spatiotemporal variables can improve the accuracy of spatiotemporal estimation. Taking the monthly mean ground observation data of the period 1960–2013 precipitation in the Xinjiang Uygur Autonomous Region, China, the spatiotemporal distribution from January to December in 2013 was respectively estimated by space-time Kriging and space-time CoKriging. Modeling spatiotemporal direct variograms and a cross variogram was a key step in space-time CoKriging. Taking the monthly mean air relative humidity of the same site at the same time as the covariates, the spatiotemporal direct variograms and the spatiotemporal cross variogram of the monthly mean precipitation for the period 1960–2013 were modeled. The experimental results show that the space-time CoKriging reduces the mean square error by 31.46% compared with the space-time ordinary Kriging. The correlation coefficient between the estimated values and the observed values of the space-time CoKriging is 5.07% higher than the one of the space-time ordinary Kriging. Therefore, a space-time CoKriging interpolation with air humidity as a covariate improves the interpolation accuracy.展开更多
For ill-posed bilevel programming problem,the optimistic solution is always the best decision for the upper level but it is not always the best choice for both levels if the authors consider the model's satisfacto...For ill-posed bilevel programming problem,the optimistic solution is always the best decision for the upper level but it is not always the best choice for both levels if the authors consider the model's satisfactory degree in application.To acquire a more satisfying solution than the optimistic one to realize the two levels' most profits,this paper considers both levels' satisfactory degree and constructs a minimization problem of the two objective functions by weighted summation.Then,using the duality gap of the lower level as the penalty function,the authors transfer these two levels problem to a single one and propose a corresponding algorithm.Finally,the authors give an example to show a more satisfying solution than the optimistic solution can be achieved by this algorithm.展开更多
Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N^2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q^(1/4+...Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N^2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q^(1/4+ε/2)N^(2/1)(log(6N))^(1/2)+N/(1/2)(loglog(6N)),where n is the multiplicative inverse of n such that nn ≡ 1(mod q),e(x)= exp(2πix),and τ(·)is the divisor function.展开更多
We continue our study on classification learning algorithms generated by Tikhonov regularization schemes associated with Gaussian kernels and general convex loss functions. Our main purpose of this paper is to improve...We continue our study on classification learning algorithms generated by Tikhonov regularization schemes associated with Gaussian kernels and general convex loss functions. Our main purpose of this paper is to improve error bounds by presenting a new comparison theorem associated with general convex loss functions and Tsybakov noise conditions. Some concrete examples are provided to illustrate the improved learning rates which demonstrate the effect of various loss functions for learning algorithms. In our analysis, the convexity of the loss functions plays a central role.展开更多
Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of o...Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.展开更多
基金supported in part by NSF of China N.10871131The Science and Technology Commission of Shanghai Municipality,Grant N.075105118+1 种基金Shanghai Leading Academic Discipline Project N.T0401Fund for E-institute of Shanghai Universities N.E03004.
文摘In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.
基金Project(17D02)supported by the Open Fund of State Laboratory of Information Engineering in Surveying,Mapping and Remote Sensing,Wuhan University,ChinaProject supported by the State Key Laboratory of Satellite Navigation System and Equipment Technology,China
文摘In various environmental studies, geoscience variables not only have the characteristics of time and space, but also are influenced by other variables. Multivariate spatiotemporal variables can improve the accuracy of spatiotemporal estimation. Taking the monthly mean ground observation data of the period 1960–2013 precipitation in the Xinjiang Uygur Autonomous Region, China, the spatiotemporal distribution from January to December in 2013 was respectively estimated by space-time Kriging and space-time CoKriging. Modeling spatiotemporal direct variograms and a cross variogram was a key step in space-time CoKriging. Taking the monthly mean air relative humidity of the same site at the same time as the covariates, the spatiotemporal direct variograms and the spatiotemporal cross variogram of the monthly mean precipitation for the period 1960–2013 were modeled. The experimental results show that the space-time CoKriging reduces the mean square error by 31.46% compared with the space-time ordinary Kriging. The correlation coefficient between the estimated values and the observed values of the space-time CoKriging is 5.07% higher than the one of the space-time ordinary Kriging. Therefore, a space-time CoKriging interpolation with air humidity as a covariate improves the interpolation accuracy.
基金supported by the National Science Foundation of China under Grant No.71171150the National Natural Science Foundation of ChinaTian Yuan Foundation under Grant No.11226226
文摘For ill-posed bilevel programming problem,the optimistic solution is always the best decision for the upper level but it is not always the best choice for both levels if the authors consider the model's satisfactory degree in application.To acquire a more satisfying solution than the optimistic one to realize the two levels' most profits,this paper considers both levels' satisfactory degree and constructs a minimization problem of the two objective functions by weighted summation.Then,using the duality gap of the lower level as the penalty function,the authors transfer these two levels problem to a single one and propose a corresponding algorithm.Finally,the authors give an example to show a more satisfying solution than the optimistic solution can be achieved by this algorithm.
基金supported by National Natural Science Foundation of China(Grant Nos.11201117 and 11126150),supported by National Natural Science Foundation of China(Grant Nos.11371344 and 11321101)National Key Basic Research Program of China(Grant No.2013CB834202)
文摘Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N^2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q^(1/4+ε/2)N^(2/1)(log(6N))^(1/2)+N/(1/2)(loglog(6N)),where n is the multiplicative inverse of n such that nn ≡ 1(mod q),e(x)= exp(2πix),and τ(·)is the divisor function.
文摘We continue our study on classification learning algorithms generated by Tikhonov regularization schemes associated with Gaussian kernels and general convex loss functions. Our main purpose of this paper is to improve error bounds by presenting a new comparison theorem associated with general convex loss functions and Tsybakov noise conditions. Some concrete examples are provided to illustrate the improved learning rates which demonstrate the effect of various loss functions for learning algorithms. In our analysis, the convexity of the loss functions plays a central role.
基金supported by National Research Foundation of Republic of Korea(Grant Nos.2011-0008976 and 2010-0011841)
文摘Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.
