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On the Average Number of Unitary Factors of Finite Abelian Groups (Ⅱ)

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Acta Mathematica Sinica,English Series 2000年第4期16卷 549-554页

作者： Wenguang Zhai Department of Mathematics, shandong normal university. jinan 250014. P. B. china Department of Mathematics Shandong Normal University Jinan 250014 P. R. China

Let t（G） be the number of unitary factors of finite abelian group G. In this paper we prove T（x）=∑(G≤（）t（G） =main terms+O(x(（1+2k）/（3+4k）for any exponent pair （k, 1/2+2K）. which improves on the exponen... 详细信息

关键词： Unitary factor Finite Abelian group Exponential sum Divisor problem

A note on a result of Abercrombie

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Chinese Science Bulletin 1997年第14期42卷 1151-1154页

作者： ZHAI WenguangDepartment of Mathematics, shandong normal university, jinan 250014. china Department of Mathematics Shandong Normal University Jinan 250014 China

SUPPOSE θ∈[1,+ ∞) is any real number. The sequence B_θ={[nθ]|n∈N}is calledBeatty’s sequence determined by θ. Beatty sequences have been the subject of intensive inves-tigation in recent years on account of the... 详细信息

关键词： Bealty’s sequence divisor problem rational approximation.

Reciprocal sums of a class of additive functions in short intervals

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Chinese Science Bulletin 1997年第2期42卷 102-105页

作者： ZHAI WenguangDepartment of Mathematics, shandong normal university, jinan 250014. china Department of Mathematics Shandong Normal University Jinan China

USING the method of probabilistic number-theory, De Koninck, and De Koninck and Galambos studied the reciprocal sum of the additive function f（n） satisfying f（n）≥t0>0 （n≥2） and f（p）≡1 and obtained an asymptotic formula, where t0 is an absolute positive constant. Let B（x） denote the number of n≤x not satisfying the inequality loglogn-R（x）≤f（n）≤loglogn+R（x）, （1） where R（x） is a function tending to infinity. Then in refs. [1, 2], it is proved that if R（x）=o（loglogx） and B（x）=o（x/loglogx）,

关键词： additive functions Riemann zeta-function asymptotic formula.

On the error term in Weyl’s law for the Heisenberg manifolds (Ⅱ)

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Science china Mathematics 2009年第5期52卷 857-874页

作者： ZHAI WenGuang School of Mathematical Sciences, shandong normal university, jinan 250014. china 1. School of Mathematical Sciences Shandong Normal University Jinan 250014 China

In this paper we study the mean square of the error term in the Weyl's law of an irrational (2l + 1)-dimensional Heisenberg manifold. An asymptotic formula is established.

关键词： Heisenberg manifold Weyl’s law error term mean square

Extraction of Surface Roughness from the Central δ-peak of Light Scattering Using a Multi-wavelength Ar~+ Laser

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Chinese Journal of Lasers 2000年第3期9卷 22-27页

作者： LIU Deli QI Dongping CHENG Chuanfu (Department of Physics, shandong normal university, jinan 250014. china) 山东师范大学山东济南

In this paper, a new light scattering method for measuring t he roughness of random surfaces is proposed. The method is based on the relation between the specularly reflected intensity I s, i.e., the central δ-peak intensity and wave vector component k z. The linear fit of the logarithm graph of the normalized central δ-peak intensity I sr versus k-2 z, whose variation is induced by changing illuminating wavelength, gives the square of the surface roughness w. The roughnesses of 6 silicon backside samples were measured.

关键词： random surface roughness light scattering multi-wavelength

The Existence of Periodic Orbits for Nonlinear Impulsive Differential Systems

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Communications in Nonlinear Science and Numerical Simulation 1999年第1期4卷 50-53页

作者： Xilin FU Jiangang QI and Yansheng LIU (Department of Mathematics, shandong normal university, jinan 250014. china) Department of Mathematics Shandong Normal University Jinan 250014 China

In this paper, we gave the necessary and sufficient condition for the existence of the periodic solutions of nonlinear impulsive differential systems by using a special function and study the stable properties of suc... 详细信息

关键词： impulsive differential equation periodic orbit stable

Discussion of Population Modernization Theory and Actuality Analysis of Population Modernization in china

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Chinese Journal of Population,Resources and Environment 2004年第3期2卷 57-59页

作者： Wang Shu shandong normal university, jinan 250014. china shandong normal university jinan 250014 China

This paper discusses the basic concept and connotation of population modernization. The author briefly analyzes the actuality of population modernization in china, gives some advice and puts forward some measures.

关键词： population modernization theory summarization connotation

CHARACTERIZATIONS OF PROPER EFFICIENT SOLUTIONS IN TERMS OF SADDLE POINTS AND DUALITY

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Systems Science and Mathematical Sciences 1992年第1期5卷 9-18页

作者： LI Shizheng (shandong normal university,jinan 250014.china) 山东师范大学山东济南

In this paper we characterize the proper efficient solution of a mul-tiobjective programming problem(P)in terms of the saddle point criterion of anew problem(S)and establish a pair of dual problems(S_α)and(D_α)in order todecide the proper efficient solution by the strong *** also give methodof characterization in terms of the saddle point criterion of(P).Finally,we provethat every efficient solution of the matrix linear programming problem is a properefficient solution.

关键词： Efficient solution proper efficient solution saddle point criterion duality perturbation function matrix linear programming problem

PROPER EFFICIENT SOLUTIONS AND PROPER SADDLE POINTS

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Systems Science and Mathematical Sciences 1993年第3期6卷 245-251页

作者： LI Shizheng shandong normal university, jinan 250014. china

This paper offers a characterization method for the proper efficient solution ofthe multiobjective programming problem in terms of the proper saddle point and proves anecessary and sufficient condition for the equivalence between the proper efficient solutionand the associated proper saddle point by using the duality of a pair of problems.

关键词： Proper efficient solution proper saddle point dual problems strong duality