The simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem. A multidimensional generalization of the simple continued fraction expanding procedure is ...The simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem. A multidimensional generalization of the simple continued fraction expanding procedure is the Jacobi-Perron algorithm (JPA). This algorithm and展开更多
Given two positive constants α and β, we prove that the integral inequality ∫_0^1f^α＋β（x）dx≥∫_0^1∫^α（x）x^β dx holds for all non-negative valued continuous functions ∫ satisfying ∫_x^1f（t）dt≥∫_x^1t...Given two positive constants α and β, we prove that the integral inequality ∫_0^1f^α＋β（x）dx≥∫_0^1∫^α（x）x^β dx holds for all non-negative valued continuous functions ∫ satisfying ∫_x^1f（t）dt≥∫_x^1tdt for x∈[0,1] if and only if α＋β≥1.This solves an open problem proposed recently by Ngo, Thang, Dat, and Tuan.展开更多
In this note, we would like to point out that （i） of Corollary 1 given by Zhang et al. （cf Commun. Theor. Phys. 39 （2003） 381） is incorrect in general.
Problems, which are studied in the paper, concern to theoretical aspects of interpolation theory. As is known, interpolation is one of the methods for approximate representation or recovery of functions on the basis o...Problems, which are studied in the paper, concern to theoretical aspects of interpolation theory. As is known, interpolation is one of the methods for approximate representation or recovery of functions on the basis of their given values at points of a grid. Interpolating functions can be chosen by many various ways. In the paper the authors are interested in interpolating functions, for which the Laplace operator, applied to them, has a minimal norm. The authors interpolate infinite bounded sequences at the knots of the square grid in Euclidian space. The considered problem is formulated as an extremal one. The main result of the paper is the theorem, in which certain estimates for the uniform norm of the Laplace operator applied to smooth interpolating functions of two real variables are established for the class of all bounded （in the corresponding discrete norm） interpolated sequences. Also connections of the considered interpolation problem with other problems and with embeddings of the Sobolev classes into the space of continuous functions are discussed. In the final part of the main section of the paper, the authors formulate some open problems in this area and sketch possible approaches to the search of solutions. In order to prove the main results, the authors use methods of classical mathematical analysis and the theory of polynomial splines of one variable with equidistant knots.展开更多
A Bernstein type theorem and a converse theorem of best approximation by polynomials in Bergman spaces Hq^p(p>0,q>1) are proved.Some proofs and results in [1] are in proved.
Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 ...Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 author proves the following theorem: Theorem. Let r > 1,R > 1 be two real numbers and q L = 1 + 1 + anan rR, K = 1 L + L2 ? 4 . r?1 R?1 +1 2 (r?1)(R?1) Then (i) Cn < r, Cn < R imply Cn > K; ?2 ?1 (ii) Cn > r, Cn > R imply Cn < K. ?2 ?1 This theorem generalizes the main result in [1].展开更多
With the notion of independent identically distributed(IID) random variables under sublinear expectations introduced by Peng,we investigate moment bounds for IID sequences under sublinear expectations. We obtain a mom...With the notion of independent identically distributed(IID) random variables under sublinear expectations introduced by Peng,we investigate moment bounds for IID sequences under sublinear expectations. We obtain a moment inequality for a sequence of IID random variables under sublinear expectations. As an application of this inequality,we get the following result:For any continuous functionsatisfying the growth condition |(x) | C(1 + |x|p) for some C > 0,p 1 depending on ,the central limit theorem under sublinear expectations obtained by Peng still holds.展开更多
The sparse nonlinear programming （SNP） is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qu...The sparse nonlinear programming （SNP） is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Frechet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn- Tucker （KKT） conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.展开更多
Recently, double projection methods for solving variational inequalities havereceived much attention due to their fewer projection times at each iteration. In this paper, weunify these double projection methods within...Recently, double projection methods for solving variational inequalities havereceived much attention due to their fewer projection times at each iteration. In this paper, weunify these double projection methods within two unified frameworks, which contain the existingdouble projection methods as special cases. On the basis of this unification, theoretical andnumerical comparison between these double projection methods is presented.展开更多
基金This work is partly supported by NSFC(No. 60173016)the National 973 Project(No.1999035804)
文摘The simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem. A multidimensional generalization of the simple continued fraction expanding procedure is the Jacobi-Perron algorithm (JPA). This algorithm and
文摘Given two positive constants α and β, we prove that the integral inequality ∫_0^1f^α＋β（x）dx≥∫_0^1∫^α（x）x^β dx holds for all non-negative valued continuous functions ∫ satisfying ∫_x^1f（t）dt≥∫_x^1tdt for x∈[0,1] if and only if α＋β≥1.This solves an open problem proposed recently by Ngo, Thang, Dat, and Tuan.
文摘In this note, we would like to point out that （i） of Corollary 1 given by Zhang et al. （cf Commun. Theor. Phys. 39 （2003） 381） is incorrect in general.
文摘Problems, which are studied in the paper, concern to theoretical aspects of interpolation theory. As is known, interpolation is one of the methods for approximate representation or recovery of functions on the basis of their given values at points of a grid. Interpolating functions can be chosen by many various ways. In the paper the authors are interested in interpolating functions, for which the Laplace operator, applied to them, has a minimal norm. The authors interpolate infinite bounded sequences at the knots of the square grid in Euclidian space. The considered problem is formulated as an extremal one. The main result of the paper is the theorem, in which certain estimates for the uniform norm of the Laplace operator applied to smooth interpolating functions of two real variables are established for the class of all bounded （in the corresponding discrete norm） interpolated sequences. Also connections of the considered interpolation problem with other problems and with embeddings of the Sobolev classes into the space of continuous functions are discussed. In the final part of the main section of the paper, the authors formulate some open problems in this area and sketch possible approaches to the search of solutions. In order to prove the main results, the authors use methods of classical mathematical analysis and the theory of polynomial splines of one variable with equidistant knots.
基金This paper is a part of the author's series of letures at the Mathematical Institute of the Hungarian Academy of Sciences while visiting Hungary sent by the state Education Committee,the People's Republic of China.
文摘A Bernstein type theorem and a converse theorem of best approximation by polynomials in Bergman spaces Hq^p(p>0,q>1) are proved.Some proofs and results in [1] are in proved.
文摘Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 author proves the following theorem: Theorem. Let r > 1,R > 1 be two real numbers and q L = 1 + 1 + anan rR, K = 1 L + L2 ? 4 . r?1 R?1 +1 2 (r?1)(R?1) Then (i) Cn < r, Cn < R imply Cn > K; ?2 ?1 (ii) Cn > r, Cn > R imply Cn < K. ?2 ?1 This theorem generalizes the main result in [1].
基金supported in part by National Basic Research Program of China (973 Program) (Grant No. 2007CB814901)the Natural Science Foundation of Shandong Province (Grant No. ZR2009AL015)
文摘With the notion of independent identically distributed(IID) random variables under sublinear expectations introduced by Peng,we investigate moment bounds for IID sequences under sublinear expectations. We obtain a moment inequality for a sequence of IID random variables under sublinear expectations. As an application of this inequality,we get the following result:For any continuous functionsatisfying the growth condition |(x) | C(1 + |x|p) for some C > 0,p 1 depending on ,the central limit theorem under sublinear expectations obtained by Peng still holds.
基金supported by National Natural Science Foundation of China(Grant No.11431002)Shandong Province Natural Science Foundation(Grant No.ZR2016AM07)
文摘The sparse nonlinear programming （SNP） is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Frechet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn- Tucker （KKT） conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.
基金This work is supported by the Natural Science Foundation of China (Grant No. 10171055, 10231060).
文摘Recently, double projection methods for solving variational inequalities havereceived much attention due to their fewer projection times at each iteration. In this paper, weunify these double projection methods within two unified frameworks, which contain the existingdouble projection methods as special cases. On the basis of this unification, theoretical andnumerical comparison between these double projection methods is presented.
