Asymptotic behavior of unstable ARMA processes with application to least squares estimates of their parameters

A time series x(t), t≥1, is said to be an unstable ARMA process if x(t) satisfies an unstableARMA model such asx(t)=a_1x(t-1) a_2x(t-2) … a_8x(t-s) w(t)where w(t) is a stationary ARMA process; and the characteristic polynomial A(z)=1-a_1z-a_2z~2-…-a_3z~3 has all roots on the unit circle. Asymptotic behavior of sum form 1 to n (x~2(t)) will be studied by showing somerates of divergence of sum form 1 to n (x~2(t)). This kind of properties Will be used for getting the rates of convergenceof least squares estimates of parameters a_1, a_2,…, a_?     

关于B.E.Rhoades的两个不动点定理的反例

B.E.Rhoades[1]中定理10和定理16是错误的。现举反例说明之。 (一)[1]中定理10为: 设f∈(73),且诸α_i满足: (1) r(t)·s(t)=a_1(t)+a_3(t)+a_5(t)/1-a_2(t)-a_3(t)·a_2(t)+a_4(t)=a_5(t)/1-a_1(t)-a_4(t)<1对任何t>0均成立。又设x_0∈X。则f~p或f~q有不动点。若再有:     

Periodic Solution of a Nonautonomous Diffusive Food Chain System of Three Species with Time Delays

Abstract By using the continuation theorem of coincidence degree theory,the existence of a positive periodicsolution for a nonautonomous diffusive food chain system of three species. dx_1(t)/dt=x_1(t)[r_1(t)-a_(11)(t)x_1(t)-a_(12)(t)x_2(t)]+D_1(t)[y(t)-x_1(t)], dx_2(t)/dt=x_2(t)[-r_2(t)+a_(21)(t)x_1(t-r_1)-a_(22)(t)x_2(t)-a_(23)(t)x_3(t)], dx_3(t)/dt=x_3(t)[-r_3(t)+a_(32)(t)x_2(t-r_2)-a_(33)(t)x_3(t)], dy(t)/dt=y(t)[r_4(t)-a_(44)(t)y(t)]+D_2(t)[x_1(t)-y(t)]+D_2(t)[x_1(t)-y(t)],is established,where,r_i(t),a_(ii)(t)(i=1,2,3,4),D_i(t)(i=1,2),a_(12)(t),a_(21)(t),a_(23)(t)and a_(32)(t) are all positiveperiodic continuous functions with period w>0,T_i(i=1,2)are positive constants.     

一类Kolmogorov捕食系统的极限环

研究一类Kolmogorov捕食系统dx/dt=x(a_0-a_1x+a_2x~(n-1)-a_3x~n+a_4xy~m),dy/dt=y(b_1x~n-b2),得到了存在唯一极限环和不存在极限环的充要条件,从而推广了前人相关的结果.     

学会用化归法思考高中代数问题

<正>化归法是通过数学知识和方法将不熟悉的问题转化为熟悉的问题的数学方法.在下面的内容中,将重点介绍化归法在高中代数中的应用.例1(1999年高考试题理科)若(2x+√3)⁴=a_0+a_1x+a_2x4=a_0+a_1x+a_2x²+a_3x2+a_3x³+a_4x3+a_4x⁴,那么(a_0+a_2+a_4)4,那么(a_0+a_2+a_4)²-(a_1+a_3)2-(a_1+a_3)²的值是().(A)1(B)-1(C)0(D)2思考方法上述问题如果你能找到"(a_0+a_2+a_4)2的值是().(A)1(B)-1(C)0(D)2思考方法上述问题如果你能找到"(a_0+a_2+a_4)²-(a_1+a_3)2-(a_1+a_3)²=(a_0+a_1+a_2+a_3+a_4)(a_0-a_1+a_2-a_3+a_4)"之间的联系,就说明你学会化归法的使用方法.     

求多项式根的新劈因子法

本文给出了劈出多项式F(z)的二次因子的程序{w_n(z)}:若则取 w_(n+1)(z)=z~2+(a_2c_1-a_1c_2)/(a_2b_1-a_1b_2)z+(a_2d_1-a_1d_2)/(a_2b_1-a_1b_2);文中指出此法与Bairstow法效果相当,并提供了一些例子.     

通项·回归·解出

先看具体问题。例1 数列{a_n}满足a_1=a_2=1,且a_(n+2)=a_n+a_(n+1) 求证:(a_1/2)+(a_2/2~2)+(a_3/2~3)...+a_n/2~n<2 证明设S_n=a_1/2+a_2/2~2+...+a_n/2~n 由a_n+2=a_n+a_(n+1)得 a_n=a_(n+2)-a_(n+1)。 Sn=(a_3-a_2)/2+(a_4-a_3)/2~2+...+     

Initial value problem for a class of nonlinear pseudo-hyperbolic equations

The existence of global weak solutions to the periodic boundary problem or the initial value problem for the nonlinear Pseudo-hyperbolic equation u_(tt)-[a_1+a_2(u_x)~(2m)]u_(xx)-a_3u_(xxt)=f(x,t,u,u_x) is proved by the method of the vanishing of the additional diffusion terms, Leray-Schauder's fixedpoint argument and Sobolev's estimates,where m≥1 is a natural number and a_i>0(i=1,2,3)are constants.     

等差数列的一个有趣的性质

我们知道,如果a_1,a_2,a_3成等差数列,由等差数列的定义,就有:a_2-a_1=a_3-a_1由此可得a_1-2a_2 a_3=0 (A) 如果a_1,a_2,a_3,a_4成等差数列,应用上面的结论,对这个数列的前三项有a_1-2a_2 a_3=0 (1)而对它的后三项又有a_2-2a_3 a_4=0 (2)将(1)减去(2),便得     

巧妙代换,简捷证明

浙江教育出版社出版的《高中数学精编综合训练》一书P.18有这样一道试题: 例已知1/2≤a_k≤1,(k∈N)。求证:a_1a_2a_3…a_n (1-a_1)(1-a_2)…(1-a_n)≥1/2~(n-1)。该不等式证明确有一定的难度,原书采用数学归纳法证明,其过程十分繁杂,并且技巧性很强。如果我们作一个巧妙的代换,问题则十分简单,迎刃而解。证明令a_k=1/2 b_k,有0≤b_k≤1/2。则 a_1a_2…a_n (1-a_1)(1-a_2)…(1-a_n)=(1/2 b_1)(1/2 b_2)…(1/2 b_n) (1/2-     

一类特殊分布纪录值之和的中心极限定理

摘要：设{X，Xn,n≥1)为独立同分布的服从某连续分布F的随机变量序列，X^(1)=X1，X^(2),X^(3),…为其纪录值序列．令ψ(u)=F^-1(1-e^-u)．其中F^-1是F的反函数．本文研究当ψ(u)=log^pu时Tn=∑k=1^nX^(k)=^dn∑k=1^nψ(Sn)的极限性质．解决了户为所有正整数时Tn的中心极限定理．     

ADDITIVE FAMILIES OF INVARIANTS OF FORMS OF DEGREE d

Abstract

Let d be a positive integer, and F be a field of characteristic zero. Suppose that for each positive integer n, I _n, is a GL _n,(F)- invariant of forms of degree d in x₁, …, x _n, over F. We call {I _n} an additive family of invariants if I _p+q (f ⊥ g) = I _p(f).I _q(g) whenever f; g are forms of degree d over F in x _l, …, x _p; …, x _q respectively, and where (f ⊥ g)(x _l, …, x _p+q) = f(x ₁, …, x _p,) + g (x _p+1, …, x _p+q). It is well-known that the family of discriminants of the quadratic forms is additive. We prove that in odd degree d each invariant in an additive family must be a constant. We also give an example in each even degree d of a nontrivial family of invariants of the forms of degree d. The proofs depend on the symbolic method for representing invariants of a form, which we review.     

COMPARISON OF PRODUCTS OF THE VARIETIES OF LIE ALGEBRAS

Abstract

In Fortes (2001), we introduced a notion of order for associative pairs and we obtained a Goldie-like characterization of left orders in a semiprime pair coinciding with its socle. In this paper, we take up again that notion of order to establish a Faith-Utumi theorem, which studies left orders in a prime pair coinciding with its socle.     

有限Abel群上子集等价类的计数

设G为一有限Abel群，|G|=v,D1,D2是G的两个子集，如果存在t∈Zv,(t,v)=1,s∈G使D1＝tD2 s，则称D1与D2是等价的。文中给出了G的k-子集等价类的计数公式，同时也给出了G的的所有子集等价类的计数公式。     

ESTIMATES OF N-FUNCTION AND m-FUNCTION OF MEROMORPHIC SOLUTIONS OF SYSTEMS OF COMPLEX DIFFERENCE EQUATIONS

We apply Nevanlinna theory of the value distribution of meromorphic functions to study the properties of Nevanlinna counting function and proximity function of meromorphic solutions of a type of systems of complex difference equations. Our results can give estimates on the proximity function and the counting function of solutions of systems of difference equations. This implies that solutions have a relatively large number of poles. It extend some result concerning difference equations to the systems of difference equations.     

一类pnm阶群的构造

本文研究了一类特殊的pnm阶有限群的构造.利用求解数论同余方程的方法和群的扩张理论,得到了具有m阶循环正规子群,其补子群为循环群的Pnm阶有限群的构造及相关的计数定理.     

小康判别函数的建立与分析

近几年，国内在研究小康水平的定量指标分析中，应用较多的有综合评分法、层次分析法和模糊识别法。这几种方法比单项指标更全面、灵活，但在权数的确定上都难以克服人为因素的影响。当指标之间存在两个或两个以上的高度相关时，对问题的研究总存在一定的局限性。为了科学地研究小康问题，本文基于判别分析，建立城市小康的判别系数，来综合评价出我国城市小康水平的定量标准。