Representation of an Integer by a Quadratic Form through the Cornacchia Algorithm

Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2+y2=n). Starting from a quadratic form with two variables f(x,y)=ax2+bxy+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn/| D |) (possibly α itself) and the equation f(x,y)=n. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form n=ax2+bxy+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.

Two Classes of Quaternary Codes from Z_4-valued Quadratic Forms

Let R=GR (4,m)be a Galois ring with Teichmuller set T_m and Tr_m be the trace function fromRto Z_4.In this paper,two classes of quaternary codes C_1 = {c(a,b):a ∈R,b ∈ T_(m/2)},where c(a,b)=(Tr_m(ax)+Tr_(m/2)(2 bx^(2 m/2 +1)))_(x∈T_m),and C_2= {c(a,b):a ∈ R,b ∈ T_m}, where c(a,b)=(Tr_m(ax+2 bx^(2 k+1)))_(x∈T_m),and m/gcd(m,k)is even,are investigated,respectively.The Lee weight distributions,Hamming weight distributions and complete weight distributions of the codes are completely given.

Bell’s Ternary Quadratic Forms and Tunnel’s Congruent Number Criterion Revisited

Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.

Representations by Certain Sextenary Quadratic Forms Whose Coefficients Are 1, 2, 3 and 6

Here, we determine formulae, for the numbers of representations of a positive integer by certain sextenary quadratic forms whose coefficients are 1, 2, 3 and 6.

Generalized Theta-Functions with Characteristics and Cusp Form Corresponding to Quadratic Forms in Ten Variables

The modular properties of generalized theta-functions with characteristics are used to build cusp form corresponding to quadratic forms in ten variables.

On n-universal quadratic forms over dyadic local fields

Let n≥2 be an integer. We give necessary and sufficient conditions for an integral quadratic form over dyadic local fields to be n-universal by using invariants from Beli's theory of bases of norm generators.Also, we provide a minimal set for testing n-universal quadratic forms over dyadic local fields, as an analogue of Bhargava and Hanke's 290-theorem(or Conway and Schneeberger's 15-theorem) on universal quadratic forms with integer coefficients.

高层结构-混联Ⅱ型惯容阻尼耗能体系减震性分析

针对高层结构中设置混联Ⅱ型惯容阻尼器组成耗能体系的随机响应分析复杂的问题,提出了结构位移及阻尼器阻尼力响应的封闭解,并基于所获得的封闭解探究了实模态振型数对分析精度的影响和在原结构层间位移超限楼层设置阻尼器能有效减震的布置策略。首先,根据混联Ⅱ型惯容阻尼器力学构造图及设置方式,建立阻尼器阻尼力与结构节点水平位移的微分型本构关系进而重构耗能体系的地震动方程。其次,利用实模态解耦法获得高层结构简明等效的动力学参数,运用功率谱二次式分解法推导出结构节点相对于地面的绝对位移、竖向构件层间位移及阻尼器阻尼力等响应的功率谱精确二次式解进而推导出耗能体系基于双过滤白噪声激励下系列响应的0~2阶谱矩简明封闭解。最后,通过算例,验证了所提封闭解的正确性;研究了实模态振型数对系列响应谱矩分析精度的影响和阻尼器设置位置对结构减震效果的影响。研究表明:对于多自由度结构响应分析时,建议采用原结构自由振动分析时质量参与系数累计达100%时所对应的振型数,可获得稳定的精度和提高大型复杂结构设置混联Ⅱ型惯容阻尼器的响应的分析效率;通过在原结构层间位移超限楼层或连续多层层间位移超限的楼层中间隔一层设置参数合适的混联Ⅱ型惯容阻尼器可有效降低结构的地震动响应,所提惯容阻尼器设置策略具有较好的经济性。所提解析解和惯容系统在高层结构的设置策略可为混联Ⅱ型惯容阻尼器在实际工程应用中提供有益的参考价值。

素变量三元二次型除数函数的均值问题

在数论中,除数函数是最基础和最重要的算术函数.借助经典圆法,研究了素变量三元二次型除数函数的均值问题H(X):=∑1≤p_(1),p_(2),p_(3)≤X d(p^(2)_(1)+p^(2)_(2)+p^(2)_(3)),得到了均值估计的渐近公式,其中X是一充分大的正数,d(n)是除数函数,p1,p2,p3是素数.

基变换在几何问题上的应用

本文使用矩阵的工具将平面内准圆的结论推广到n维欧氏空间,并且论证了n维欧氏空间中顶点在原点的二次锥面有n条相互垂直的直母线的充要条件是该锥面所对应二次型的迹为零.

基于LQR的一级倒立摆控制器优化设计

沿用文献的数学模型及物理参数,以经验法自行设计LQR控制器加权矩阵Q、P,利用MATLAB完成反馈矩阵K的计算,通过逐项比对分析,对矩阵Q、P进行调整,使控制器性能更加优越。从Matlab模拟结果来看,与所选文献相比,该文的LQR控制器稳定性更好。

1∶2 INTERNAL RESONANCE OF COUPLED DYNAMIC SYSTEM WITH QUADRATIC AND CUBIC NONLINEARITIES

The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1 :2 internal resonance were derived by using the direct method of normal form. In the normal,forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the 4-dimension bifurcation equations were reduced to 3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4-dimension center manifolds.

MAXIMUM TEST FOR A SEQUENCE OF QUADRATIC FORM STATISTICS ABOUT SCORE TEST IN LOGISTIC REGRESSION MODEL

This article proposes the maximum test for a sequence of quadratic form statistics about score test in logistic regression model which can be applied to genetic and medicine fields.Theoretical properties about the maximum test are derived.Extensive simulation studies are conducted to testify powers robustness of the maximum test compared to other two existed test.We also apply the maximum test to a real dataset about multiple gene variables association analysis.

The quadratic-form identity for constructing Hamiltonian structures of the Guo hierarchy

The trace identity is extended to the quadratic-form identity. The Hamiltonian structures of the multi-component Guo hierarchy, integrable coupling of Guo hierarchy and （2＋l）-dimensional Guo hierarchy are obtained by the quadraticform identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.

FORMABILITY OF TERNARY INTERMEDIATE COMPOUNDS IN SOME MOLTEN SALT PHASE DIAGRAMS

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ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

Suppose that X is a right process which is associated with a semi-Dirichlet form （ε, D（ε）） on L2（E; m）. Let J be the jumping measure of （ε, D（ε）） satisfying J（E x E- d） 〈 ∞. Let u E D（ε）b ：= D（ε） N L（E; m）, we have the following Pukushima＇s decomposition u（Xt）-u（X0） --- Mut ＋ Nut. Define Pu f（x） = Ex[eNT f（Xt）]. Let Qu（f,g） = ε（f,g）＋ε（u, fg） for f, g E D（ε）b. In the first part, under some assumptions we show that （Qu, D（ε）b） is lower semi-bounded if and only if there exists a constant a0 〉 0 such that /Put/2 ≤eaot for every t 〉 0. If one of these assertions holds, then （Put〉0is strongly continuous on L2（E;m）. If X is equipped with a differential structure, then under some other assumptions, these conclusions remain valid without assuming J（E x E - d） 〈 ∞. Some examples are also given in this part. Let At be a local continuous additive functional with zero quadratic variation. In the second part, we get the representation of At and give two sufficient conditions for PAf（x） = Ex[eAtf（Xt）] to be strongly continuous.

CROSS-CORRELATIONS BETWEEN QUADRATIC FORM SEQUENCES AND GEOMETRIC SEQUENCES OVER FINITE FIELDS

For all types of quadratic forms,the cross-correlations between geometric sequences and the newly defined quadratic form sequences are determined to extend the results presented by Klapper in 1993 and 1997.The technique for computing cross-correlations is based on counting the number of solutions for a system of equations that consists of a quadratic form and a linear function.

Amorphous in a Zr-IQC-DQC pseudo-ternary alloy system

A pseudo-ternary alloy system was constructed by combining an icosahedralquasicrystal (IQC), a decagonal quasicrystal (DQC), and Zr into one alloy system. Differentproportions of Zr were added to pseudo-binary alloy IQC_(80)DQC_(20) (mass fraction in %);Structural evolution in these alloys was discussed. An amorphous alloy composition was found in thissystem and a melt-spinning amorphous alloy was produced in this composition. Through DSC analysis,the amorphous alloy exhibits high glass forming ability comparable to that of the InoueZr_(65)Al_(7.5)Cu_(17.5)Ni_(10) amorphous alloy.

组合体结构随机地震动响应的简明解法

在两相邻结构间设置耗能阻尼器能有效降低结构在地震作用下的响应,文中对相邻建筑结构设置Maxwell型阻尼器而形成组合体的结构地震动响应进行了研究。频域法中,结构响应功率谱表示为频率响应特征函数模值平方项(称之为频率响应特征值函数)与随机激励功率谱的乘积,文中提出了响应功率谱二次正交化法并获得了结构系列响应(绝对位移与层间位移)0~2阶谱矩及方差的简明封闭解。首先,综合复模态法与虚拟激励法提出了组合体结构频率响应特征值函数的二次正交式;其次,运用留数定律给出胡聿贤地震谱的二次正交式,由此求解出组合体结构响应功率谱密度函数的二次正交式;最后,得到了组合体结构系列响应0~2阶谱矩及方差的简明封闭解。通过算例验证了所提方法的正确性,并验证了相邻结构间设置连接阻尼器具有良好的减震性能。

单自由度混联Ⅱ型惯容系统随机地震动响应分析

提出了基于Clough-Penzien谱的混联Ⅱ型惯容结构地震动响应封闭解的计算方法,并基于所提方法研究了单自由度混联Ⅱ型惯容结构的减震性能及影响因素.首先,建立混联Ⅱ型惯容结构的运动方程,获得了结构位移与惯容出力等结构响应频域解的统一表达式.其次,基于有理式分解与留数定理分别获得了频率响应特征值函数与Clough-Penzien谱的二次正交式,进而获得了结构响应功率谱的二次正交式.最后,得到了结构与惯容随机地震动响应0~2阶谱矩的简明封闭解.利用所提方法和虚拟激励法分别对一算例进行分析,验证了所提方法的正确性;同时,利用所提方法分析了惯容各参数对结构抗震性能的影响.研究表明:该文方法获得的响应解为封闭解,在计算效率与计算准确性上都优于虚拟激励法.当μ_(ω)不变时,随着μ_(m)与μ_(ξ)增大,减震性能增强;当μ_(ω)=1时,对减震性能的影响最大.

混联Ⅰ型惯容系统基于李鸿晶谱的地震响应分析

针对设有混联Ⅰ型惯容系统的耗能结构基于李鸿晶谱的响应分析较为繁琐的问题,提出求解地震响应的简明封闭解法,并研究了惯容系统基本参数对减震性能的影响。首先,建立惯容系统和结构之间的微分型本构关系,并运用复模态法和虚拟激励法获得结构系列响应的统一解。其次,将结构系列响应的功率谱密度函数精确转化为复特征值及结构自振频率平方和表示的二次分解式,进而获得响应0阶~2阶谱矩的简明封闭解。最后,通过数值算例验证了所提方法的正确性及计算的高效性,对惯容系统参数的研究表明,当惯容系数和阻尼系数越大,结构减震效果越好,而刚度系数对结构的减震效果取决于惯容系统名义圆频率与结构自振圆频率的比值,当两者比值为0.9时减震效果最好。