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_?     

Interpolation with lagrange polynomials a simple proof of Markov inequality and some of its generalizations

The following theorem is provedTheorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying inthe interval[-1,1] and△'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}.If polynomial pP_n satisfies the inequalitythen for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality丨p~(k)(x)丨≤max{丨q~((k))(x)丨,丨1/k(x~2-1)q~(k+1)(x)+xq~((k))(x)丨}.This estimate leads to the Markov inequality for the higher order derivatives ofpolynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero.Some other results are established which gives evidence to the conjecture that under theconditions of Theorem 1 the inequality ‖p~((k))‖≤‖q~(k)‖holds.     

ASYMPTOTIC REPRESENTATION FOR REMAINDER OF QUASI-HERMITE-FEJER INTERPOLATION POLYNOMIAL

Let Q_(2n+1)(f,x)be the quasi-Hermite-Fejer interpolation polynomial of functionf(x)∈C_[-1,1]based on the zeros of the Chebyshev polynomial of the second kind U_n(x)=sin((n+l)arccosx)/sin(arc cosx). In this paper, the uniform asymptotic representation for thequantity| Q_(2n+l)(f, x) -f(x) |is given. A similar result for the Hermite-Fejer interpolationpolynomial based on the zeros of the Chebyshev polynomial of the first kind is alsoestablished.     

Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper,the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|~(2α)e~(-(x~4+tx~2)),x ∈ R,where α is a constant larger than -1/2 and t is any real number. They consider this problem in three separate cases:(i) c -2,(ii) c =-2,and(iii) c -2,where c := t N~(-1/2) is a constant,N = n + α and n is the degree of the polynomial. In the first two cases,the support of the associated equilibrium measure μ_t is a single interval,whereas in the third case the support of μ_t consists of two intervals. In each case,globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou(1993).     

ON CONVERGENCE OF PAL-TYPE INTERPOLATION POLYNOMIALS

Let {x_k~*}_(k=1)~(n-1) be the zeros of the (n-1) -th Legendre polynomial p_(n-1)(x) and {x_k}_(a=1)~n be the zeros of the polynomial w(x)= (1-x2~)p_(n-1)~1(x). By the theory of the Pal interpolation, for afunction f ∈ C_([-1,1])~1, there exists a unique polynomial Q_n(f, x) of degree 2n-1 satisfying conditions Q_n(f, x_k)=f(x_k), Q'_n(f, x_k~*)=f'(x_k~*), where k=1, 2, …, n and x_n~*=-1. The main result of this paper is that if f ∈ C_([-1,1])~r, thenf(x)-Q_n(f, x)=O(1)W(x)w(f~(r), 1/n)n~((1/2)-r), -1≤x≤1.Hence, if f ∈ C_[-1,1])~1, then Q_n(f, x) converges to the function f(x)uniformly on the interval [-1, 1].     

A GENERALIZATION OF SIMPSON''''S RULE

Let TA(f)=integral form n= to 1/2(P_~n(x) + P_b~n(x))dx and let TM(f)=integral form n= to P_((+b)/2)~(n+1)(x)dx, where P_c~n denotes theTaylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of theTrapezoidal rule and the midpoint rule, respectively. and are each exact for all polynomial of degree ≤n+1.We let L(f) = αTM(f) + (1-α)TA(f), where α =(2~(n+1)(n+1))/(2~(n+1)(n+1)+1), to obtain a numerical integrationrule L which is exact for all polynomials of degree≤n+3 (see Theorem l). The case n = 0 is just the classicolSimpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacingP_(+b)/2)~(n+1)(x) by the Hermite cabic interpolant at a and b. we obtain some known formulae by a different ap-proach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by takingpiecewise polynomials of odd degree, each piece being the Taylor polynomial off at a and b. respectively. Ofcourse all of our formulae can be compounded over subintervals of [a, b].     

On the Optimal Error Bounds for Derivatives of Cubic Interpolating Splines with Equidistant Nodes

Let ⊿n={i/n=x_i}be the uniform partition of the interval [0,1].Suppose s(x)is the Type Icubic subic spline interpolant of f(x),i.e.s(x)satisfies(i)s(x)∈C_2[0,1];(ii)s(x)is a polynomial of degree 3 in each subinterval[x_i,x_(i+1)];     

要什么,解什么

有这么一道函数方程:若f(x)-2f(1/x)=x,求f(x)。有这样一种解法: ∵ f(x)-2f(1/x)=x=(-3x~2/-3x)=(x~2 2-3-4x~2)/-3x=(x~2 2)/(-3x)-(2·(1 2x~2)/-3x)=(x~2 2)/(-3x)-(2·(1/x~2) 2/(-3·1/x)。∴ f(x)-(x~2 2)/(-3x)。我真不知道怎么想到要把x写成(-3x~2)/(-3x)?更不知道怎么想到要把-3x~2写成x~2 2 2-4x~2?我疑心的是编者事先求出f(x),再倒过来创造了这种解法的。我知道的是,要从f(x)-2f(1/x)=x中求出f(x)。于是要想办法消去f(1/x),因此还得     

On the Differential Polynomial of a Graph

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial B(G; x) :=∑?(G)k=-nB_k(G) x~(n+k), where B_k(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G;x) and its coefficients.In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.     

A polynomial algorithm for finding (g,f)-colorings orthogonal to stars in bipartite graphs

Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two nonnegative integer-valued functions defined on V(G) such that g(x)≤ f(x) for every vertex x of V(G). A (g. f)-coloring of G is a generalized edge-coloring in which each color appears at each vertex x at least g(x) and at most f(x) times. In this paper a polynomial algorithm to find a (g. f)-coloring of a bipartite graph with some constraints using the minimum number of colors is given. Furthermore, we show that the results in this paper are best possible.     

General ARMA Models

1.Introduction Consider the following model A(B)x(t)=w(t) t=1,2,…(1)where B denotes the Backwards shift operation,i.e.Bx(t)=x(t-1),B~2x(t)=x(t-2),and so on; A(B)is a polynomial of B, A(B)=1-a_1B-a_2B~2-…-a,B~r (2)with its roots laid on and at inside of the unit circle; and w(t) is a stationaryARMA(p,q)process,i. e. w(t) satisfies     

应用平均值换元巧解方程

本文通过举例说明平均值换元在解一类方程中的妙用。 例1 解方程 (x~2 2x-2)(x_2 4x 6)=3(x 4)~2 解 设t=1/a[(x~2 2x-2) (x~2 4x 6)]=x~2 3x 2,则原方程化为[(t-(x 4)]·[t (x-4)]-3(x 4)=0 t~2-4(x 4)~2=0,即[t 2(x 4)][t-2(x 4)]=0,     

On the Schinzel Identity of Cyclotomic Polynomial

For integer n>0, let n(x) denote the nth cyclotomic polynomial n(x)=tackrel{01 be an odd square-free number.Aurifeuille and Le Lasseur［1］ proved thatequationn(x)=An2(x)-(-1)n-12)nxBn2(x).equation     

On the Approximation of Continuous Functions by Pal-Type Interpolation Polynomials

Let us denote by -1=x_n相似文献   

Integral closure of a quartic extension

Let R be a Noetherian unique factorization domain such that 2 and 3 are units,and let A=R[α]be a quartic extension over R by adding a rootαof an irreducible quartic polynomial p(z)=z4+az2+bz+c over R.We will compute explicitly the integral closure of A in its fraction field,which is based on a proper factorization of the coefficients and the algebraic invariants of p(z).In fact,we get the factorization by resolving the singularities of a plane curve defined by z4+a(x)z2+b(x)z+c(x)=0.The integral closure is expressed as a syzygy module and the syzygy equations are given explicitly.We compute also the ramifications of the integral closure over R.     

从(x±1)(x~2±x+1)=x~3±1谈起

乘法公式中有 (x+1)(x~2-x+1)=x~3+1,(x-1)(x~2+x+1)=x~3-1。等式两边互换,就得到因式分解 x~3+1=(x+1)(x~2-x+1),x~3-1=(x-1)(x~2+x+1)。进而有 x~4+1=(x+1)(x~3-x~2+x-1),x~4-1=(x-1)(x~3+x~2+x+1)。推广这些公式,可以得到定理1 (1)对任意正整数n,有 x~n-1=(x-1)(x~(n-1)+x~(n-2)+…+x+1)     

关于有限区间上二次函数的最大(小)值

先看一道问题的解答: 问题:x、y是实数,且满足等式3x~2 2y~2=6x,求x~2 y~2的最大值. 解由3x~2 2y~2=6x,得y~2=-3/2x~2 3x,从而K=x~2 y~2=x~2-3/2x~2 3x=1/2x~2 3x.故由-1/2<0,可知当x=3/2×(-1/2)=3时,有(x~2 y~2)_(max)=4(-1/2)×0-3~2/4(-1/2)=9/2. 这是一道在约束条件下可化为求二次函数最大值的问题.上述解题过程显然是错误的,而这种错误不易被学生所觉察,常常出现在作业中.错误的根源在于没有考虑到“约束条件”,而乱用二次函数y=ax~2 bx c的极值公式来求在有限区间上该函数的     

Inequalities for trigonometric polynomials

Let tn(x) be any real trigonometric polynomial of degreen n such that , Here we are concerned with obtaining the best possible upper estimate ofwhere q>2. In addition, we shall obtain the estimate of in terms of and     

Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

Primitive Roots and Linearized Polynomials

Let f(x) = sum from t=0 to n α_ix~i∈GF(p)[x],we associate it with a ploynomial f~*(x)=sum from i=0 to n α_ix~(p~i),f(x) and f~*(x)are called p-associates of each other. f~*(x) is called a p-ploynomial,customary to speak of linearized polynomial. Let f(x)=x~m- 1/g(x), m = m_1~r, q = p~m, g(x)∈GF(p)[x],r be the order of g(x). Cohen and the author observed that if m_1≥2, there alwaysexsists a primitive roots ζ∈GF(q) suck that f~*(ζ) = f~*(c), here f~*(c)≠0. In fact