In the present paper some multi-dimensional quadrature formulas of periodic functions are established by means of the number-theoretic method.Some results of Hua and Wang are generalized or improved.
Motivated by a random coverage problem from a real-life case study, this paper discusses and compares several number-theoretic methods including the ELP-net, NT-net, and others in statistical simulations. The methods ...Motivated by a random coverage problem from a real-life case study, this paper discusses and compares several number-theoretic methods including the ELP-net, NT-net, and others in statistical simulations. The methods and related conclusions studied in this paper can be extended into more general cases in statistical simulations.展开更多
In the present paper approximating functions for certain periodic functions in several variables with bounded mixed derivatives are constructed by making use of their values at the knots of some number-theoretic nets....In the present paper approximating functions for certain periodic functions in several variables with bounded mixed derivatives are constructed by making use of their values at the knots of some number-theoretic nets. The errors are estimated in the L_q-norm where l≤q≤∝.展开更多
To improve the numerical evaluation of integrals in Number-Theoretic Methods, we give a two-stage algorithm. The main idea is that we distribute the points according to the variations of the quadrature on the subdomai...To improve the numerical evaluation of integrals in Number-Theoretic Methods, we give a two-stage algorithm. The main idea is that we distribute the points according to the variations of the quadrature on the subdomains to reduce errors. The simulations results are also given.展开更多
Consider a design of experiments with mixtures:0≤ai【xi【bi≤1, 1≤i≤s,x1+…+xs=1,where ai, bi,1≤i≤s are given constants. A method is proposed to treat this model by the theory of uniform distribution in number th...Consider a design of experiments with mixtures:0≤ai【xi【bi≤1, 1≤i≤s,x1+…+xs=1,where ai, bi,1≤i≤s are given constants. A method is proposed to treat this model by the theory of uniform distribution in number theory.展开更多
An artificial neural network (ANN) is applied in the preliminary structural design of reticulated shells. Major efforts are made to enhance the generalization ability of networks through well-selected train- ing sam...An artificial neural network (ANN) is applied in the preliminary structural design of reticulated shells. Major efforts are made to enhance the generalization ability of networks through well-selected train- ing samples. Number-theoretic methods (NTMs) are adopted to generate samples with low discrepancy, i.e., uniformly scattered in the domain, where discrepancy is a quantitative measurement of the uniformity. The discrepancy of the NTM-based sample set is 1/6-1/7 that of samples with equal spacing. In a case study, networks trained by NTM-based samples are compared with those trained by equal-spaced samples in generalizing performance. The results show that both the computational precision and stability of the former ANNs are more satisfactory than those of the latter. It is concluded that the flexibility of ANNs in generalizing can be effectively increased by use of uniformly distributed training samples rather than simply piling data. More reliable uniformity should be obtained, however, through NTMs instead of equal-spaced samples.展开更多
We study the exponential sums involving l：burmr coeffcients ot Maass forms and exponential functions of the form e（anZ）, where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponent...We study the exponential sums involving l：burmr coeffcients ot Maass forms and exponential functions of the form e（anZ）, where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg（n）e（αnβ）, when β = 1/2 and ｜α｜ is close to 2√ q C Z＋, where Ag（n） is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 （Z）. The similar natures of the divisor function 7（n） and the representation function r（n） in the circle problem in nonlinear exponential sums of the above type are also studied.展开更多
THE range of usage of the quadrature formulae specially designed for the integrals of sometypes can often be extended by the variable transformation. For example, in a quadrature for-mula computing an integral with th...THE range of usage of the quadrature formulae specially designed for the integrals of sometypes can often be extended by the variable transformation. For example, in a quadrature for-mula computing an integral with the weight function g:展开更多
Quadrature formulas are considered for classes of smooth functions Wpr, Bpr,(?) with bounded mixed derivative or difference. For the classes of functions indicated above, the result that quadrature formulas constructe...Quadrature formulas are considered for classes of smooth functions Wpr, Bpr,(?) with bounded mixed derivative or difference. For the classes of functions indicated above, the result that quadrature formulas constructed with the help of number-theoretic methods are optimal (in the sense of order) is proved, and the optimal order of the error estimates is obtained.展开更多
This article investigates the test for linearity of a multivariate stochastic regression model.The use of nonparametric regression procedures for developing regression diagnostics has beenthe subject of several recent...This article investigates the test for linearity of a multivariate stochastic regression model.The use of nonparametric regression procedures for developing regression diagnostics has beenthe subject of several recent research efforts. However, when the dimension of the regressor islarge, some traditional nonparametric methods, such as kernel estimation, may be inefficient.We in this article suggest two test statistics based on projection pursuit technique and kernelmethod. The tests proposed are consistent against all fixed smooth alternatives to linearityand are asymptotically distribution-free for the distribution of the error. Furthermore, the testsare applied to an example of real-life data and some simulated data sets to demonstrate theavailability of the tests proposed.展开更多
In the present paper the approximate fuctions for certain multivariate periodic functions with bounded mired derivatives are constructed by making use of their values at the knots of number-theoretical nets.
基金This project is supported by the National Natural Science Foundation of China
文摘In the present paper some multi-dimensional quadrature formulas of periodic functions are established by means of the number-theoretic method.Some results of Hua and Wang are generalized or improved.
基金supported by the Award Research Foundation from Chinese Academy of Sciences
文摘Motivated by a random coverage problem from a real-life case study, this paper discusses and compares several number-theoretic methods including the ELP-net, NT-net, and others in statistical simulations. The methods and related conclusions studied in this paper can be extended into more general cases in statistical simulations.
基金This work is supported in part by the Alexander von Humboldt Foundationthe National Natural Science Foundation of China
文摘In the present paper approximating functions for certain periodic functions in several variables with bounded mixed derivatives are constructed by making use of their values at the knots of some number-theoretic nets. The errors are estimated in the L_q-norm where l≤q≤∝.
文摘To improve the numerical evaluation of integrals in Number-Theoretic Methods, we give a two-stage algorithm. The main idea is that we distribute the points according to the variations of the quadrature on the subdomains to reduce errors. The simulations results are also given.
基金Project partially supported by the National Natural Science Foundation of ChinaInstitute of Mathematics,Taipei.
文摘Consider a design of experiments with mixtures:0≤ai【xi【bi≤1, 1≤i≤s,x1+…+xs=1,where ai, bi,1≤i≤s are given constants. A method is proposed to treat this model by the theory of uniform distribution in number theory.
文摘An artificial neural network (ANN) is applied in the preliminary structural design of reticulated shells. Major efforts are made to enhance the generalization ability of networks through well-selected train- ing samples. Number-theoretic methods (NTMs) are adopted to generate samples with low discrepancy, i.e., uniformly scattered in the domain, where discrepancy is a quantitative measurement of the uniformity. The discrepancy of the NTM-based sample set is 1/6-1/7 that of samples with equal spacing. In a case study, networks trained by NTM-based samples are compared with those trained by equal-spaced samples in generalizing performance. The results show that both the computational precision and stability of the former ANNs are more satisfactory than those of the latter. It is concluded that the flexibility of ANNs in generalizing can be effectively increased by use of uniformly distributed training samples rather than simply piling data. More reliable uniformity should be obtained, however, through NTMs instead of equal-spaced samples.
基金Acknowledgements This work was partially supported by the National Natural Science Foundation of China （Grant Nos. 11101239, 10971119）, the Program for Changjiang Scholars and Innovative Research Team in University （IRT1264）, and the Independent Innovation Foundation of Shandong University （Grant No. 2012ZRYQ005）.
文摘We study the exponential sums involving l：burmr coeffcients ot Maass forms and exponential functions of the form e（anZ）, where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg（n）e（αnβ）, when β = 1/2 and ｜α｜ is close to 2√ q C Z＋, where Ag（n） is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 （Z）. The similar natures of the divisor function 7（n） and the representation function r（n） in the circle problem in nonlinear exponential sums of the above type are also studied.
文摘THE range of usage of the quadrature formulae specially designed for the integrals of sometypes can often be extended by the variable transformation. For example, in a quadrature for-mula computing an integral with the weight function g:
基金Project supported by the National Natural Science Foundation of China and the Doctoral Program Foundation of the State Education Commission of China.
文摘Quadrature formulas are considered for classes of smooth functions Wpr, Bpr,(?) with bounded mixed derivative or difference. For the classes of functions indicated above, the result that quadrature formulas constructed with the help of number-theoretic methods are optimal (in the sense of order) is proved, and the optimal order of the error estimates is obtained.
文摘This article investigates the test for linearity of a multivariate stochastic regression model.The use of nonparametric regression procedures for developing regression diagnostics has beenthe subject of several recent research efforts. However, when the dimension of the regressor islarge, some traditional nonparametric methods, such as kernel estimation, may be inefficient.We in this article suggest two test statistics based on projection pursuit technique and kernelmethod. The tests proposed are consistent against all fixed smooth alternatives to linearityand are asymptotically distribution-free for the distribution of the error. Furthermore, the testsare applied to an example of real-life data and some simulated data sets to demonstrate theavailability of the tests proposed.
文摘In the present paper the approximate fuctions for certain multivariate periodic functions with bounded mired derivatives are constructed by making use of their values at the knots of number-theoretical nets.
