This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of func...This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of functions which converges to the exact solution of problem. Our emphasis will be on an auxiliary parameter which directly affects on the rate of convergence. Comparison of PIM and the Variational Iteration Method (VIM) is given to show the preference of PIM over VIM. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.展开更多
We construct a modified Bernoulli iteration method for solving the quadratic matrix equation AX^2 ＋ BX ＋ C = 0, where A, B and C are square matrices. This method is motivated from the Gauss-Seidel iteration for solv...We construct a modified Bernoulli iteration method for solving the quadratic matrix equation AX^2 ＋ BX ＋ C = 0, where A, B and C are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the ShermanMorrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.展开更多
Let G(k) denote the smallest number s such that every sufficiently large natural number is the sum of, at most, s k-th powers of natural numbers. In this paper, we give new bounds for G(12), G(13) and G(19).
In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due...In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due to C. Bauer.展开更多
In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that wi...In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that with at most O(N2/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.展开更多
Let G (k) denote the smallest s such that every sufficiently large natural number is the sum of at most s fcth powers of natural numbers. It is proved that G (16)<111. This improves the result of T. D. Wooley's.
In this paper,we apply Ma’s variation of parameters method(VPM)for solving Fisher’s equations.The suggested algorithm proved to be very efficient and finds the solution without any discretization,linearization,pertu...In this paper,we apply Ma’s variation of parameters method(VPM)for solving Fisher’s equations.The suggested algorithm proved to be very efficient and finds the solution without any discretization,linearization,perturbation or restrictive assumptions.Numerical results reveal the complete reliability of the proposed VPM.展开更多
We consider a mathematical model which describes the static frictional contact between a piezoelectric body and a conductive foundation.A non linear electro-elastic constitutive law is used to model the piezoelectric ...We consider a mathematical model which describes the static frictional contact between a piezoelectric body and a conductive foundation.A non linear electro-elastic constitutive law is used to model the piezoelectric material.The unilateral contact is modelled using the Signorini condition,nonlocal Coulomb friction law with slip dependent friction coefficient and a regularized electrical conductivity condition.Existence and uniqueness of a weak solution is established.A finite elements approximation of the problem is presented,a priori error estimates of the solutions are derived and a convergent successive iteration technique is proposed.展开更多
It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous re...It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous results in this direction.展开更多
It is proved that every large integer N≡5（mod24）can be written as N=p<sub>1</sub><sup>2</sup>+…+p<sub>5</sub><sup>2</sup> with each prime p<sub>j</sub> s...It is proved that every large integer N≡5（mod24）can be written as N=p<sub>1</sub><sup>2</sup>+…+p<sub>5</sub><sup>2</sup> with each prime p<sub>j</sub> satisfying |p<sub>J</sub>-（N/5|）<sup>1/2</sup>≤N<sup>11/23</sup>.This gives a short interval version of Hua’s theorem on the quadratic Waring-Goldbach problem展开更多
In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent...In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12+p22+p32+p42+p52, with |pj-(N/5)^(1/2)|≤U = N1/2-1/20+ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.展开更多
文摘This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of functions which converges to the exact solution of problem. Our emphasis will be on an auxiliary parameter which directly affects on the rate of convergence. Comparison of PIM and the Variational Iteration Method (VIM) is given to show the preference of PIM over VIM. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.
基金Supported by The Special Funds For Major State Basic Research Projects （No. G1999032803） The China NNSF 0utstanding Young Scientist Foundation （No. 10525102）+1 种基金 The National Natural Science Foundation （No. 10471146） The National Basic Research Program （No. 2005CB321702）, P.R. China.
文摘We construct a modified Bernoulli iteration method for solving the quadratic matrix equation AX^2 ＋ BX ＋ C = 0, where A, B and C are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the ShermanMorrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.
文摘Let G(k) denote the smallest number s such that every sufficiently large natural number is the sum of, at most, s k-th powers of natural numbers. In this paper, we give new bounds for G(12), G(13) and G(19).
基金Supported by Post-Doctoral Fellowship of The University of Hong KongThe National Natural Science Foundation(Grant No.10571107)Supported by a grant from the Research Grant Council of Hong Kong(Project No.HKU7028/03P)
文摘In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due to C. Bauer.
基金The author is supported by Post-Doctoral Fellowsbip of The University of Hong Kong.
文摘In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that with at most O(N2/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.
基金Project supported by the National Natural Science Foundation of China.
文摘Let G (k) denote the smallest s such that every sufficiently large natural number is the sum of at most s fcth powers of natural numbers. It is proved that G (16)<111. This improves the result of T. D. Wooley's.
文摘In this paper,we apply Ma’s variation of parameters method(VPM)for solving Fisher’s equations.The suggested algorithm proved to be very efficient and finds the solution without any discretization,linearization,perturbation or restrictive assumptions.Numerical results reveal the complete reliability of the proposed VPM.
文摘We consider a mathematical model which describes the static frictional contact between a piezoelectric body and a conductive foundation.A non linear electro-elastic constitutive law is used to model the piezoelectric material.The unilateral contact is modelled using the Signorini condition,nonlocal Coulomb friction law with slip dependent friction coefficient and a regularized electrical conductivity condition.Existence and uniqueness of a weak solution is established.A finite elements approximation of the problem is presented,a priori error estimates of the solutions are derived and a convergent successive iteration technique is proposed.
基金Supported by The National Science Foundation(Grants #10125101 and #10131010)by a Ministry of Education Major Grant Program in Sciences and Technology
文摘It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous results in this direction.
基金Supported by MCSEC and the National Natural Science Foundation (Grant No. 19701019) Supported by MCSFC and the National Natural Science Foundation
文摘It is proved that every large integer N≡5（mod24）can be written as N=p<sub>1</sub><sup>2</sup>+…+p<sub>5</sub><sup>2</sup> with each prime p<sub>j</sub> satisfying |p<sub>J</sub>-（N/5|）<sup>1/2</sup>≤N<sup>11/23</sup>.This gives a short interval version of Hua’s theorem on the quadratic Waring-Goldbach problem
基金supported by the National Natural Science Foundation of China(Grant Nos.10125101&10531060)a Major Grant Program in Science and Technology by the Ministry of EducationTianyuan Mathematics Foundation(Grant No.10526028).
文摘In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12+p22+p32+p42+p52, with |pj-(N/5)^(1/2)|≤U = N1/2-1/20+ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.
