一个递推公式及其应用

本文介绍一个递推公式及其在解题中的广泛应用。1 递推公式设F(n)=a_1x_1~n+a_2x_2~n+…+a_kx_k~n(n≥0,n∈Z),构造以x_1,x_2,…,x_k为根的方程: x~k+m_1x~(k-1)+m_2x~(k-2)+…+m_k=0 我们称这个方程为F(n)的特征方程,则F(n)=a_1x_1~n+a_2x_2~n+…+a_kx_k~n(n≥k,x∈Z)满足下列递推公式:     

有限域上一类方程组解数的直接公式

本文给出有限域F=F_q(q=p~f,f≥1,p是一个奇素数)上一类方程组∑_(i=s_(r-1)+1~(s_r)∑_(j=1)~(m_i-m_(i-1))a_(m_(i-1)+j)x_1~(d_m(i-1)+j,1)…x_(n_i)~d_(m_(i-1)+j,n_i)=b_r,r=1,…,k当指数满足一定条件时,在F~(n_s_k)上解数的一个直接公式,这里d_(ij)＞0,a_i∈F~*,b_i∈F,0= s_0＜s_1＜…＜s_k,0=m_0＜m_1＜…＜m_(s_k),0=n_0＜n_1＜…＜n_(s_k), m_1≤n_1,…,m_(s_k)≤n_(s_k)．     

非线性复差分方程亚纯解的振荡性质

该文主要研究以下两类非线性复差分方程a_n(z)f(z+n)~(j_n)+…+a_1(z)f(z+1)~(j_1)+a_0(z)f(z)~(j_0)=b(z),a_n(z)f(q~nz)~(j_n)+…+a_1(z)f(qz)~(j_1)+a_0(z)f(z)~(j_0)=b(z),其中,a_i(z)(i=0,1,…,n)与b(z)为非零有理函数,j_i(i=0,1,…,n)为正整数,q为非零复常数.当上述方程的亚纯解的超级小于1并且极点较少时,对解的零点分布进行了估计.此外,当亚纯解具有无穷多个极点时,也对极点收敛指数给出下界.     

有限域上一类方程组的解数公式

设F_q为一个q元有限域,其中q=p~s(s≥1),p是一个奇素数.本文给出下列方程组在F_q上的解数公式:a_(k1)x_1~(d_(11)~((k)))...x_(n_1)~(d_(1n_1)~((k)))+...+a_(k,s_1)x_1~(d_(s_1,1)~((k)))...x_(n_1)~(d_(s_1,n_1)~((k)))+a_(k,s_1)+1x_1~(d_(s_1+1,1)~((k)))...x_(n_2)~(d_(s_1+1,n_2)~((k)))+...a_(k,s_2)x_1~(d_(s_2,1)~((k)))...x_(n_2)~(d_(s_2,1)~((k)))...x_(n_2)~(d_(s_2,n_2)~((k)))=b_k,k=1,...,m,其中0s_1s_2,0n_1n_2,a_(ki)∈F_q~*,b_k∈F_q,d_(ij)~(k)0(k=l,...,m,i=1,...,s_2,j=1,...,n_2).特别当ms_1≤n_1,ms_2≤n_2,d_(ij)~(k)满足一定条件时,得到了明确的解数公式.     

关于方程的几个问题

历年来在高等代数的教学中,总发現某些学生对方程有着模糊的概念。例如,按照現行教材,中学毕业生进入高等学校后第一次接触到方程概念的是克萊姆規則:n个未知量n个方程的綫性方程組 a_(11)x~1+a_(12)x_2+ …+a_(1n)x_n=b_1, a_(21)x_1+a_(22)x_2+…+a_(2n)x_n=b_2, a_(n1)+a_(n2)x_2+…+a_(nn)x_n=b_n (1)的系数行列式D=|aij≠0时,(1)有解且仅有一解,即x_i=Di/D,i=1,2,…,n。 証明分两步:第一步是假定(1)有解,得出xi=Di/D。第二步是用真x_i=Di/D代入(1),得出真的等式,因而x_i=Di/D的确是(1)的解。較多的同学感到第二步是多余的,沒有必要。另一个例子是在討論向量方程     

经典均值不等式的多重隔离

我们都知道下列经典均值不等式:设a_1,a_2,…,a_n是n个正数,n≥2,n∈N~*.则n/(1/(a_1)+1/(a_2)+…+1/(a_n))≤(a_1a_2…a_n)~(1/n)≤(a_1+a_2+…+a_n)/n≤((a_1~n+a_2~n+…+a_n~n)/n)~(1/n),等号当且仅当a_1=a_2=…=a_n取到.受文[1],[2]的启发,笔者给出下列经典均值不等式的多重隔离:     

一类三阶有理差分方程的奇点集和解的渐近性

研究三阶有理差分方程x_(n+1)=ax_(n-1)+x_(n-1)x_n/bx_(n-2)+cx_n,n=0,1,2,...的奇点集和解{x_n}_(n=-2)~∞的渐近性,其中a,b,c∈R,初始值x_(-2),x_(-1),x_0∈R.由a,b,c的取值的不同,而得到解的不同的渐近性.     

GF(p)上B_n的次数

行列式 B_n=∑±b_(i_1)~(m_1)b_(i_2)~(m_2)…b_(i_n)~(m_n)中各项含因子 b 的个数的最大值称为 B_n 的次数,其中,1≤t_k≤n,m_f≥0,b_(i_k)∈GF(p).当 p=2时,这是0-1矩阵的行列式,文[3]已有结果.本文在任意 p 的情形下给出 B_n 的次数 L(n)的公式:对任意正整数 r,当 n_r≤n≤n_(r+1)时,L(n)=r,其中,n_r=(r_0+1)(p~(q+1)-1)/(p-1)-(1+qp~(q+1),q=[r/(p-1)],r=q(p-1)+r_0。     

羅巴切夫斯基求實根近似值法

求方程各實根的近似值,往往先將各級分離而個別地進行,未有同時全部獲得者,有之,自俄羅斯伟大數學家羅巴切夫斯基創立方法始,茲依據於Я.C.貝吉克維奇著“近似計算”略述共法於下: 設已知一代數方程為a_0x~n+a_1x~(n-1)+a_2x~(n-2)+…+a_n-1~x+a_n=0 (1)其中n為自然數,a_0,a_1,a_2,…,a_n為整數,並設其僅有各不相等的實根而為 |x_1|>|x_2|>|x_3|>…>|x_n|。方程(1)也可寫作 (x-x_1)(x-x_2)(x-x_3)…(x-x_n)=0 (2) 現在讓我們來構成一新方程。以-x代原方程中之x,則必恒得a_0x~n-a_1x~(n-1)+a_2x~(n-2)-a_3~(n-3)+…+(-1)~na_n=O (3)其根為-x_1,-x_2,-x_3,…,-x_n,且由此得     

利用韦达定理证明某些三角恒等式

给定复数a_0,a_1,a_2,……a_n,则n次代数方程a_0x~n+a_1x~(n-1)+……+a_n=0 (a_0≠0)必存在n个根x_1,x_2,……x_n,韦达定理给出了这n个根与方程系数a_0,a_1,……a_n的关系如下:     

Study of the blow-up of solutions of systems of equations of hydrodynamic type

Mathematical Notes - We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from...     

Distribution of types of symmetry of tensors of high degree

The asymptotic distribution of tensors of degree N in symmetry types is studied in this paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 181–186, 1986.     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

Axiomatizability of reducts of algebras of relations

In this paper, we prove that any subreduct of the class of representable relation algebras whose similarity type includes intersection, relation composition and converse is a non-finitely axiomatizable quasivariety and that its equational theory is not finitely based. We show the same result for subreducts of the class of representable cylindric algebras of dimension at least three whose similarity types include intersection and cylindrifications. A similar result is proved for subreducts of the class of representable sequential algebras. Received October 7, 1998; accepted in final form September 10, 1999.     

Lengths of periods of continued fractions of square roots of integers

Empirical study of the period’s length T of the continued fractions of $\sqrt{Q}$ (for growing integers Q) shows several strange asymptotical results, for instance, $T\leq C\sqrt{Q}\ln{Q}$ . These results show important differences between the statistics of the elements of the continued fractions of random real numbers and of square roots of random integers.