As an important type of polynomial approximation, approximation of functions by Bernstein operators is an important topic in approximation theory and computational theory. This paper gives global and pointwise estimat...As an important type of polynomial approximation, approximation of functions by Bernstein operators is an important topic in approximation theory and computational theory. This paper gives global and pointwise estimates for weighted approximation of functions with singularities by Bernstein operators. The main results are the Jackson＇s estimates of functions f∈ （Wwλ）2 andre Cw, which extends the result of （Della Vecchia et al., 2004）.展开更多
Let p and q be two distinct primes, epq（n） denotes the largest exponent of power pq which divides n. In this paper, we study the mean value properties of function epq（n）, and give some hybrid mean value formulas f...Let p and q be two distinct primes, epq（n） denotes the largest exponent of power pq which divides n. In this paper, we study the mean value properties of function epq（n）, and give some hybrid mean value formulas for epq（n） and Dirichlet divisor function d（n）. Key words： largest exponent; asymptotic formula; hybrid mean value; Dirichlet divisor function d（n）展开更多
In this paper, some necessary and sufficient optimality conditions are obtained for a fractional multiple objective programming involving semilocal E-convex and related functions. Also, some dual results are establish...In this paper, some necessary and sufficient optimality conditions are obtained for a fractional multiple objective programming involving semilocal E-convex and related functions. Also, some dual results are established under this kind of generalized convex functions. Our results generalize the ones obtained by Preda[J Math Anal Appl, 288（2003） 365-382].展开更多
We proved： Let F be a family of meromorphic functions in a domain D anda α ≠0, b ∈C. If f1（z） - α（f（z））^2 ≠ b, f≠ 0 and the poles of f（z） are of multiplicity ≥ 3 foreach f（z） ∈F, then F is normal in D.
It is one of the most interesting problems in number theory to compute someespecial series by using Zeta and Gamma functions, and results have been obtained for someespecial series. In this paper, we give an important...It is one of the most interesting problems in number theory to compute someespecial series by using Zeta and Gamma functions, and results have been obtained for someespecial series. In this paper, we give an important formula which is proved also by usingZeta and Gamma functions.展开更多
Let G = SU（n, 1）, K = S（U（n） × U（1））, and for l ∈Z, let {T;},l∈Z be a one- Dimensional K-type and let Et be the line bundle over G/K associated to Tl. In this work we obtain a central limit theorem for ...Let G = SU（n, 1）, K = S（U（n） × U（1））, and for l ∈Z, let {T;},l∈Z be a one- Dimensional K-type and let Et be the line bundle over G/K associated to Tl. In this work we obtain a central limit theorem for the space Et.展开更多
In this paper, we give the algebraic independence measures for the values ofMahler type functions in complex number field and p-adic number field, respectively.
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
By applying Lou＇s direct perturbation method to perturbed nonlinear Schroedinger equation and the critical nonlinear SchrSdinger equation with a small dispersion, their approximate analytical solutions including the ...By applying Lou＇s direct perturbation method to perturbed nonlinear Schroedinger equation and the critical nonlinear SchrSdinger equation with a small dispersion, their approximate analytical solutions including the zero-order and the first-order solutions are obtained. Based on these approximate solutions, the analytical forms of parameters of solitons are expressed and the effects of perturbations on solitons are briefly analyzed at the same time. In addition, the perturbed nonlinear Schroedinger equations is directly simulated by split-step Fourier method to check the validity of the direct perturbation method. It turns out that the analytical results given by the direct perturbation method are well supported by numerical calculations.展开更多
This paper is concerned with the determination of a unique scaling parameter in complex scaling analysis and with accurate calculation of dynamics resonances. In the preceding paper we have presented a theoretical ana...This paper is concerned with the determination of a unique scaling parameter in complex scaling analysis and with accurate calculation of dynamics resonances. In the preceding paper we have presented a theoretical analysis and provided a formalism for dynamical resonance calculations. In this paper we present accurate numerical results for two non-trivial dynamical processes, namely, models of diatomic molecular predissociation and of barrier potential scattering for resonances. The results presented in this paper confirm our theoretical analysis, remove a theoretical ambiguity on determination of the complex scaling parameter, and provide an improved understanding for dynamical resonance calculations in rigged Hilbert space.展开更多
We studied the normality criterion for families of meromorphic functions related to shared sets. Let F be a family of meromorphic functions on the unit disc △, a and b be distinct non-zero values, S={a,b}, and k be a...We studied the normality criterion for families of meromorphic functions related to shared sets. Let F be a family of meromorphic functions on the unit disc △, a and b be distinct non-zero values, S={a,b}, and k be a positive integer. If for every f∈ F, i） the zeros of f（z） have a multiplicity of at least k＋ 1, and ii） E^-f（k）（S） lohtain in E^-f（S）, then F is normal on .4. At the same time, the corresponding results of normal function are also proved.展开更多
In this paper, we consider an explicit iteration scheme with perturbed mapping for nonexpansive mappings in real q-uniformly smooth Banach spaces. Some weak and strong convergence theorems for this explicit iteration ...In this paper, we consider an explicit iteration scheme with perturbed mapping for nonexpansive mappings in real q-uniformly smooth Banach spaces. Some weak and strong convergence theorems for this explicit iteration scheme are established. In particular, necessary and sufficient conditions for strong convergence of this explicit iteration scheme are obtained. At last, some useful corollaries for strong convergence of this explicit iteration scheme are given.展开更多
The authors obtain some sufficient conditions for the stability of zero solutions to some types of the functional equation. (x)(t)+ p(t)-x(t)+q(t)x(t)+f (t, xt)=0 by transformations and the Liapunov's Second metho...The authors obtain some sufficient conditions for the stability of zero solutions to some types of the functional equation. (x)(t)+ p(t)-x(t)+q(t)x(t)+f (t, xt)=0 by transformations and the Liapunov's Second method. The obtained conclusions generalize some results of Stability of Equation (x)(t)+p(t)(x)(t)+q(t)x(t)=0 and Jack Hale in his paper of Theory of Functional Differential Equations.展开更多
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
Let G:Ω→Ω′be a closed unital map between commutative,unital quantales. G induces a functor G from the category of Ω-categories to that of Ω′-categories.This paper is concerned with some basic properties of G.Th...Let G:Ω→Ω′be a closed unital map between commutative,unital quantales. G induces a functor G from the category of Ω-categories to that of Ω′-categories.This paper is concerned with some basic properties of G.The main results are:(1) when Ω,Ω′are integral,G:Ω→Ω′and F:Ω′→Ωare closed unital maps,F is a left adjoint of G if and only if F is a left adjoint of G;(2) G is an equivalence of categories if and only if G is an isomorphism in the category of commutative unital quantales and closed unital maps; and (3) a sufficient condition is obtained for G to preserve completeness in the sense that GA is a complete Ω′-category whenever A is a complete Ω-category.展开更多
In this paper, we investigate the order of approximation by reproducing kernel spaces on （-1, 1） in weighted L^p spaces. We first restate the translation network from the view of reproducing kernel spaces and then c...In this paper, we investigate the order of approximation by reproducing kernel spaces on （-1, 1） in weighted L^p spaces. We first restate the translation network from the view of reproducing kernel spaces and then construct a sequence of approximating operators with the help of Jacobi orthogonal polynomials, with which we establish a kind of Jackson inequality to describe the error estimate. Finally, The results are used to discuss an approximation problem arising from learning theory.展开更多
文摘As an important type of polynomial approximation, approximation of functions by Bernstein operators is an important topic in approximation theory and computational theory. This paper gives global and pointwise estimates for weighted approximation of functions with singularities by Bernstein operators. The main results are the Jackson＇s estimates of functions f∈ （Wwλ）2 andre Cw, which extends the result of （Della Vecchia et al., 2004）.
文摘Let p and q be two distinct primes, epq（n） denotes the largest exponent of power pq which divides n. In this paper, we study the mean value properties of function epq（n）, and give some hybrid mean value formulas for epq（n） and Dirichlet divisor function d（n）. Key words： largest exponent; asymptotic formula; hybrid mean value; Dirichlet divisor function d（n）
基金Foundation item： Supported by Hunan Provincial Natural Science Foundation of China（05JJ40103） Supported by Soft Science Research Fund of Hunan Province（2006ZK3028） Supported by Scientific Research Fund of Hunan Provincial Education Department（105B0707, 08C470）
文摘In this paper, some necessary and sufficient optimality conditions are obtained for a fractional multiple objective programming involving semilocal E-convex and related functions. Also, some dual results are established under this kind of generalized convex functions. Our results generalize the ones obtained by Preda[J Math Anal Appl, 288（2003） 365-382].
文摘We proved： Let F be a family of meromorphic functions in a domain D anda α ≠0, b ∈C. If f1（z） - α（f（z））^2 ≠ b, f≠ 0 and the poles of f（z） are of multiplicity ≥ 3 foreach f（z） ∈F, then F is normal in D.
基金Supported by the Natural Science Foundation of China（10271093）
文摘It is one of the most interesting problems in number theory to compute someespecial series by using Zeta and Gamma functions, and results have been obtained for someespecial series. In this paper, we give an important formula which is proved also by usingZeta and Gamma functions.
基金the National Natural Science Foundation of China(70271069)
文摘Let G = SU（n, 1）, K = S（U（n） × U（1））, and for l ∈Z, let {T;},l∈Z be a one- Dimensional K-type and let Et be the line bundle over G/K associated to Tl. In this work we obtain a central limit theorem for the space Et.
基金Supported by the Natural Science Foundation of Henan University（05ZDZR001）
文摘In this paper, we give the algebraic independence measures for the values ofMahler type functions in complex number field and p-adic number field, respectively.
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.
基金The project supported by National Natural Science Foundation of China under Grant No. 10575087 and the Natural Science Foundation of Zhejiang Province of China under Grant No. 102053
文摘By applying Lou＇s direct perturbation method to perturbed nonlinear Schroedinger equation and the critical nonlinear SchrSdinger equation with a small dispersion, their approximate analytical solutions including the zero-order and the first-order solutions are obtained. Based on these approximate solutions, the analytical forms of parameters of solitons are expressed and the effects of perturbations on solitons are briefly analyzed at the same time. In addition, the perturbed nonlinear Schroedinger equations is directly simulated by split-step Fourier method to check the validity of the direct perturbation method. It turns out that the analytical results given by the direct perturbation method are well supported by numerical calculations.
文摘This paper is concerned with the determination of a unique scaling parameter in complex scaling analysis and with accurate calculation of dynamics resonances. In the preceding paper we have presented a theoretical analysis and provided a formalism for dynamical resonance calculations. In this paper we present accurate numerical results for two non-trivial dynamical processes, namely, models of diatomic molecular predissociation and of barrier potential scattering for resonances. The results presented in this paper confirm our theoretical analysis, remove a theoretical ambiguity on determination of the complex scaling parameter, and provide an improved understanding for dynamical resonance calculations in rigged Hilbert space.
文摘We studied the normality criterion for families of meromorphic functions related to shared sets. Let F be a family of meromorphic functions on the unit disc △, a and b be distinct non-zero values, S={a,b}, and k be a positive integer. If for every f∈ F, i） the zeros of f（z） have a multiplicity of at least k＋ 1, and ii） E^-f（k）（S） lohtain in E^-f（S）, then F is normal on .4. At the same time, the corresponding results of normal function are also proved.
文摘In this paper, we consider an explicit iteration scheme with perturbed mapping for nonexpansive mappings in real q-uniformly smooth Banach spaces. Some weak and strong convergence theorems for this explicit iteration scheme are established. In particular, necessary and sufficient conditions for strong convergence of this explicit iteration scheme are obtained. At last, some useful corollaries for strong convergence of this explicit iteration scheme are given.
文摘The authors obtain some sufficient conditions for the stability of zero solutions to some types of the functional equation. (x)(t)+ p(t)-x(t)+q(t)x(t)+f (t, xt)=0 by transformations and the Liapunov's Second method. The obtained conclusions generalize some results of Stability of Equation (x)(t)+p(t)(x)(t)+q(t)x(t)=0 and Jack Hale in his paper of Theory of Functional Differential Equations.
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the National Natural Science Foundation of China (No.10771147)the Program for New Century Excellent Talents in University (No.05-0779)
文摘Let G:Ω→Ω′be a closed unital map between commutative,unital quantales. G induces a functor G from the category of Ω-categories to that of Ω′-categories.This paper is concerned with some basic properties of G.The main results are:(1) when Ω,Ω′are integral,G:Ω→Ω′and F:Ω′→Ωare closed unital maps,F is a left adjoint of G if and only if F is a left adjoint of G;(2) G is an equivalence of categories if and only if G is an isomorphism in the category of commutative unital quantales and closed unital maps; and (3) a sufficient condition is obtained for G to preserve completeness in the sense that GA is a complete Ω′-category whenever A is a complete Ω-category.
基金The research is supported by the National Natural Science Foundation under Grant No. 10471130 and the Zhejiang Province Science Foundation under Grant No. Y604003. Acknowledgements The author thanks the referees for giving valuable comments on this paper which make him rewrite this paper in a better form.
文摘In this paper, we investigate the order of approximation by reproducing kernel spaces on （-1, 1） in weighted L^p spaces. We first restate the translation network from the view of reproducing kernel spaces and then construct a sequence of approximating operators with the help of Jacobi orthogonal polynomials, with which we establish a kind of Jackson inequality to describe the error estimate. Finally, The results are used to discuss an approximation problem arising from learning theory.
