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Complete solving of explicit evaluation of Gauss sums in the index 2 case

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Science china Mathematics 2010年第9期53卷 2525-2542页

作者： YANG Jing1,2 & XIA LingLi3,1Department of mathematical sciences,Tsinghua university,Beijing 100084,china 2Division of mathematical sciences,school of Physical and mathematical sciences,Nanyang Technological university,637371,Singapore 3Basic Courses Department,Beijing Union university,Beijing 100101,china Department of mathematical sciences Tsinghua University Beijing China Division of mathematical sciences School of Physical and Mathematical Sciences Nanyang Technological University Nanyang Singapore Basic Courses Department Beijing Union University Beijing China

Let p be a prime number,N be a positive integer such that gcd(N,p) = 1,q = pf where f is the multiplicative order of p modulo *** χ be a primitive multiplicative character of order N over finite field *** paper studies the problem of explicit evaluation of Gauss sums G(χ) in the "index 2 case"(i.e.[(Z/NZ):] = 2).Firstly,the classification of the Gauss sums in the index 2 case is ***,the explicit evaluation of Gauss sums G(χλ)(1 λ N-1) in the index 2 case with order N being general even integer(i.e.N = 2r·N0,where r,N0 are positive integers and N0 3 is odd) is ***,combining with the researches before,the problem of explicit evaluation of Gauss sums in the index 2 case is completely solved.

关键词： Gauss sum Stickelberger’s theorem Stickelberger congruence Davenport-Hasse lifting formula Davenport-Hasse product formula

The lifting scheme of 4-channel orthogonal wavelet transforms

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Progress in Natural Science:Materials International 2006年第1期16卷 100-104页

作者： PENG Lizhong and CHU Xiaoyong (LMAM, school of mathematical sciences, Peking university, Beijing 100871, china) LMAM School of Mathematical Sciences Peking University Beijing 100871 China

The 4-channel smooth wavelets with linear phase and orthogonality are designed from the 2-channel orthogonal wavelets with high transfer vanishing moments. Reversely, for simple lifting scheme of such 4-channel orthogonal wavelet transforms, a new 2-channel orthogonal wavelet associated with this 4-channel wavelet is constructed. The new 2-channel wavelet has at least the same number of vanishing moments as the associated 4-channel one. Finally, by combining the two such 2-channel wavelet systems, the lifting scheme of 4-channel orthogonal wavelet transform, which has simple structure and is easy to apply, is presented.

关键词： lifting scheme 4-channel orthogonal wavelet 2-channel orthogonal wavelet vanishing moment transfer vanishing moment linear phase.

THE REGULARITY OF THE FREE BOUNDARY FOR STRIKE RESET OPTION

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Acta Mathematica Scientia 2010年第5期30卷 1721-1729页

作者：杨舟 school of mathematical sciences South China Normal University

A strike reset option is an option that allows its holder to reset the strike price to the prevailing underlying asset price at a moment chosen by the holder. The pricing model of the option can be formulated as a parabolic variational inequality and the optimal reset strategy is the free boundary. The smoothness of the free boundary in some cases was showed in our article published in JDE. We would prove its smoothness in the other case in this paper by a generalized comparison principle for the variational inequality.

关键词： strike reset option free boundary variational inequality the comparison principle

The decomposition of K_v into K_2 × K_5's

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Science china Mathematics 2007年第10期50卷 1382-1388页

作者： Yang LI~1 Jian-xing YIN~1 Ru-cong ZHANG~2 Gen-nian GE~(2+) ~1 Department of Mathematics,suzhou university,suzhou 2.5006,china ~2 Department of Mathematics,Zhejiang university,Hangzhou 310027,china Department of Mathematics Suzhou University Suzhou China Department of Mathematics Zhejiang University Hangzhou China

The decomposition of the complete graph Kv into Kr×Kc's, the products of Kr and Kc,is originated from the use of DNA library screening. In this paper, we consider the case where r=2 and c = 5, and show that such a de... 详细信息

关键词： decomposition, the complete graph Kv, product graph Kr×Kc, DNA library screening

HIGH ACCURACY NUMERICAL METHOD OF THIN-FILM PROBLEMS IN MICROMAGNETICS

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Journal of Computational Mathematics 2003年第1期21卷 33-40页

作者： Zhongyi Huang (Deft, of mathematical sciences, Tsinghua university, Beijing, 100084, china) Dept. of mathematical sciences Tsinghua University 北京 100084

In this paper, a new high accuracy numerical method for the thin-film problems of micron and submicron size ferromagnetic elements is proposed. For the computation of stray field, we use the finite element method(FEM) by introducing a semi-discrete artificial boundary condition [1, 2]. In our numerical experiments about the domain patterns and their movement, we can see that the results are accordant to that of experiments and other numerical methods. Our method are very convenient to deal with arbitrary shape of thin films such as a polygon with high accuracy.

关键词： Thin-film Micromagnetics stray field Semi-discrete artificial boundary condition.

Romanoff theorem in a sparse set

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Science china Mathematics 2010年第9期53卷 2195-2202页

作者： CHEN Yong-Gao school of mathematical sciences,Nanjing Normal university,Nanjing 210046,china school of mathematical sciences Nanjing Normal University Nanjing China

Let A be any subset of positive integers,and P the set of all positive *** of our results are:(a) the number of positive integers which are less than x and can be represented as 2k + p(resp.p-2k) with k ∈ A and p ∈ P is more than 0.03A(log x/log 2)π(x) for all sufficiently large x;(b) the number of positive integers which are less than x and can be represented as 2q + p with p,q ∈ P is(1 + o(1))π(log x/log 2)π(x).Four related open problems and one conjecture are posed.

关键词： Romanoff theorem Polignac conjecture asymptotic density

Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance

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Science china Mathematics 2007年第8期50卷 1205-1216页

作者： Zai-hong WANG school of mathematical sciences,Capital Normal university,Beijing 100037,china school of mathematical sciences Capital Normal University Beijing China

In this paper, we deal with the existence of unbounded orbits of the mapping $$\left\{ \begin{gathered} \theta _1 = \theta + 2n\pi + \frac{1}{\rho }\mu (\theta ) + o(\rho ^{ - 1} ), \hfill \\ \rho _1 = \rho + c - \mu '(\theta ) + o(1), \rho \to \infty \hfill \\ \end{gathered} \right.$$ , where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c > 0 and μ(θ) ≠ 0, θ, ∈ [0, 2?], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″+f(x)x′+ax +?bx ?+?(x)=p(t) has unbounded solutions provided that a, b satisfy $1/\sqrt a + 1/\sqrt b = 2/n$ and ?(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.

关键词： Liénard equations unbounded solutions periodic solutions 34C11 34C25

The Geometric Structure of Strange Attractors in the Lozi Map

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Communications in Nonlinear Science and Numerical Simulation 1998年第2期3卷 67-71页

作者： Yongluo CAO and Zengrong LIU* (Department of Mathematics, suzhou university,suzhou 2.5006, china) *(Department of Mathematics, Shanghai university,Shanghai 200072, china) (E mail: ***) Department of Mathematics Suzhou University Suzhou 215006 China Department of Mathematics Shanghai University Shanghai 200072 China

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关键词： strange attractor transversal homoclinic point weak transversal homoclinic point

Asymptotic profile of solutions to the two-dimensional dissipative quasi-geostrophic equation

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Science china Mathematics 2010年第10期53卷 2733-2748页

作者： DONG BoQing & LIU QingQing school of mathematical sciences,Anhui university,Hefei 230039,china school of mathematical sciences Anhui University Hefei China

This paper is concerned with the asymptotic behavior of the two-dimensional dissipative quasigeostrophic *** on the spectral decomposition of the Laplacian operator and iterative techniques,we obtain improved L2 decay rates of weak solutions and derive more explicit upper bounds of higher order derivatives of *** also prove the asymptotic stability of the subcritical quasi-geostrophic equation under large initial and external perturbations.

关键词： quasi-geostrophic equation L2 decay asymptotic stability