Estimation of parameters in ARUMA models

Let X(n) be a time series satisfying the following ARUMA(p, d, q) models:U (B) A (B)X (n)=C (B) W (n)where U(B)=1+u(1)B+…+u(d) B~d is a polynomial with all roots on the unit circle, A(B)=1+a(1)B+…+a(p)Bp is a polynomial with all roots outside the unit circle, C(B)=1+c(1) B+…+c(q)Bq is a polynomial which is relatively prime with the polynomial U(B)A(B), B is thebackshift operator such that BX(n)=X(n-1), and (W (n), F(n), n≥1) is a sequence of martingaledifferences satisfying the following conditions:lim E (W (n)~2|F(n-1))=σ~2 a.s.n→∞sup E |W(n)|γ<∞ for some γ>2.n≥1The purpose of this paper is to provide consistent estimates of the parameters p, d, q, u(j) (j=1,2,…,d), and a(k) (k=1, 2.…, p).     

On tensor spaces for rook monoid algebras

Let m,n∈N,and V be an m-dimensional vector space over a field F of characteristic 0.Let U=F⊕V and R_n be the rook monoid.In this paper,we construct a certain quasi-idempotent in the annihilator of U~(n) in FR_n,which comes from some one-dimensional two-sided ideal of rook monoid algebra.We show that the two-sided ideal generated by this element is indeed the whole annihilator of U~(n) in FR_n.     

Equivalent Separating Curves on Surfaces and Its Applications to Haken Spheres

1 Introduction Let A be a collection of n pairwise disjoint simple closed curves on an orientable closed surface F of genus n ≥2. We say that A is a complete system of F if the surface obtained by cutting F along A is a 2n-punctured sphere. Let A1, A2 be two non-empty subsets of A. We say that (A1,A2) is a partition of A if A1 ∩A2 = (?) and A1 ∪A2 = A. Let (A1, A2) be a partition of A on F, and C a simple closed curve on F. We say that C is separating with respect to (A1, A2) if it is disjoint from A and it cuts F into two pieces F1,F2 with A1(?) F1,A2 (?) F2.     

ON THE BOTTLENECK CAPACITY EXPANSION PROBLEMS ON NETWORKS

This article considers a class of bottleneck capacity expansion problems. Such problems aim to enhance bottleneck capacity to a certain level with minimum cost. Given a network G(V, A, C) consisting of a set of nodes V= {v1,v2,…,vn},a set of arcs A ■ {(vi, vj) | i=1,2,…, n;j=1,2,…,n} and a capacity vector C. The component cij of C is the capacity of arc (vi ,vj). Define the capacity of a subset A' of A as the minimum capacity of the arcs in A, the capacity of a family F of subsets of A is the maximum capacity of its members. There are two types of expanding models. In the arc-expanding model, the unit cost to increase the capacity of arc (vi, vj) is wij. In the node-expanding model, it is assumed that the capacities of all arcs (vi,vj) which start at the same node vi should be increased by the same amount and that the unit cost to make such expansion is wi. This article considers three kinds of bottleneck capacity expansion problems (path, spanning arborescence and maximum flow) in both expanding models. For each kind of expansion problems, this article discusses the characteristics of the problems and presents several results on the complexity of the problems.     

Equivalent Separating Curves on Closed Surface of Genus 2

Let {A,B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to (A, B) if it is disjoint from A∪B and it cuts F into two once-punctured tori X, Y with A(?)X, B(?)Y. Letγbe a simple closed curve on F which is disjoint from A∪B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hnγ(C):n∈Z} is the set containing all the simple closed curves on F which is separating with respect to (A,B), where hγis the Dehn twist alongγon F. This also shows how two simple closed curves on F which are separating with respect to (A,B) are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.     

Relationship Between the Stiffly Weighted Pseudoinverse and Multi-Level Constrained Rseudoinverse

It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, supw∈D‖（W^1/2A) W^1/2‖ = (miniσ (A^(i))^-1, in which (A^(i) is a submatrix of A formed with r = (rank(A)) rows of A, such that (A^(i) has full row rank r. In many practical applications this value is too large to be used. In this paper we consider the case that both A and W(∈D) are fixed with W severely stiff. We show that in this case the weighted pseudoinverse (W^1/2‖A) W^1/2‖ is close to a multilevel constrained weighted pseudoinverse therefore ‖(W^1/2A) W^1/‖2 is uniformly bounded.We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem.     

学英语（英文）

There are 6questions in total,presenting various different question types.While you attempt to resolve the problems,remember to be creative.During accomplishing these flexible mathematical exercises,you can inspire your mathematical thinking.1.What is the diameter of a circle whose area is A?(A)2A槡π(B)A槡π(C)A2π(D)Aπ(E)2槡Aπ2.If a,b,and c are positive numbers such that 3a=4b=5c,and if a+b+kc,what is the value of k?(A)1235(B)57(C)107(D)75(E)3512     

On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

Let F(x)=∑∞n=1 Tsi,s2,...,sk(n)x^n be the generating function for the number,Ts1bs2,...,sk(n) of spanning trees in the circulant graph Cn(s1,S2,...,Sk).We show that F(x)is a rational function with integer coefficients satisfying the property F(x)=F(l/x).A similar result is also true for the circulant graphs C2n(s1,S2,....,Sk,n)of odd valency.We illustrate the obtained results by a series of examples.     

A Note on First Order Differential Equation

In this note, a theorem and its three corollaries on solution of the first order ordinary differential equation are given. Theorem Suppose that b, F∈C,a∈C~1,b(y)≠0. If a(t) and b(t) satisfy the equality a′(t)b(t)=1, (1) then the first order differential equation y′=b(y)F(x,a(y)) (2) has a solution y=f(u) (3) where u=u(x) is a solution of the equatien     

Operator Matrix Forms of Positive Operators

If a 3-tuple (A:H1→H1,B:H2→H1,C:H2→H2）of operators on Hibert spaces is given,we proved that the operator ～↑A:=(↑A ↓B^*↑B ↓C) on H=H1 H2 is ≥0 is and only if A≥0,R(B)∪→R（A^1/2) and C≥B^* A^ b,where A^ is the generalized inverse of A.In general,A^ is a closed operator,but since R(B)∪→R(A^1/2,B^* A^ B is bounded yet.     

Rook Poset Equivalence of Ferrers Boards

A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of when two Ding Schubert varieties have identical Schubert cell structures. This also produces a complete classification of isomorphism types of lower intervals of 312-avoiding permutations in the Bruhat order.     

Cycles and perfect matchings

Fan Chung and Ron Graham (J. Combin. Theory Ser. B 65 (1995) 273–290) introduced the cover polynomial for a directed graph and showed that it was connected with classical rook theory. Dworkin (J. Combin. Theory Ser. B 71 (1997) 17–53) showed that the cover polynomial naturally factors for directed graphs associated with Ferrers boards. The authors (Adv. Appl. Math. 27 (2001) 438–481) developed a rook theory for shifted Ferrers boards where the analogue of a rook placement is replaced by a partial perfect matching of K_2n, the complete graph on 2n vertices. In this paper, we show that an analogue of Dworkin's result holds for shifted Ferrers boards in this setting. We also show how cycle-counting matching numbers are connected to cycle-counting “hit numbers” (which involve perfect matchings of K_2n).     

Rook placements and cellular decomposition of partition varieties

In this paper, we study the geometric implication of rook length polynomials introduced in the author's thesis. We introduce the idea of partition varieties. These are certain algebraic varieties which have CW-complex structures. We prove that the cell structure of a partition variety is in one-to-one correspondence with rook placements on a Ferrers board defined by a corresponding partition. This correspondence enables one to characterize the geometric attachment between a cell and the closure of another cell combinatorially. The main result of this paper is that the Poincaré polynomial of cohomology for a partition variety is given by the corresponding rook length polynomial.

This paper serves as a transition of our studies from combinatorial aspects to the geometric aspects. To make the transition accessible, we give three appendices on the known results on Grassmann manifolds and flag manifolds which are used frequently. One appendix is on a technical lemma on embeddings of manifolds.     

Further Investigations Involving Rook Polynomials With Only Real Zeros

We study the zeros of two families of polynomials related to rook theory and matchings in graphs. One of these families is based on the cover polynomial of a digraph introduced by Chung and Graham . Another involves a version of the ‘hit polynomial’ of rook theory, but which applies to weighted matchings in (non-bipartite) graphs. For both of these families we prove a result which is analogous to a theorem of the author, K. Ono, and D. G. Wagner, namely that for Ferrers boards the hit polynomial has only real zeros. We also show that for each of these families there is a general conjecture involving arrays of numbers satisfying inequalities which contains these theorems as special cases. We provide evidence for the truth of these conjectures by proving other special cases and discussing computational experiments.     

PARTIAL FERRERS MATRICES

ＰＡＲＴＩＡＬＦＥＲＲＥＲＳＭＡＴＲＩＣＥＳ￥ＲＩＣＨＡＲＤＡ；ＢＲＵＡＬＤＩ＆ＬＩＱＩＡＯ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＵｎｉｖｅｒｓｉｔｙｏｆＷｉｓｃｏｎｓｉｎ，Ｍａｄｉｓｏｎ，Ｗｉｓｃｏｎｓｉｎ５３７０６，Ｕ．Ｓ．Ａ．）（Ｄ...     

完全可控线性系统的谱分布

<正> 其中x(·)∈R_n,u(·)∈R_r,A,B分别为n×n,n×r阶复常数阵,R_n,R_r分别为n,r维酉空间,有如下熟知结果:任给n个复数λ_1,…,λ_n,恒存在r×n阶复数阵C,使阵的谱集σ(A+BC)={λ_1,…,λ_n}的充要条件为     

Fibonacci和Lucas序列的混合卷积公式

通过Fibonacci序列和Lucas序列的生成函数,利用导函数的性质,得到了Fibonacci序列和Lucas序列构成的混合卷积∑a1+a2+…+ak+b1+b2+…+b1+c1+c2+…+cm=na1Fa1+1…akFak+1.Fb1…Fb1.Lc1+1…Lcm+1的计算公式.     

格子点上随机场的回归系数的估计问题

<正> §1.引言U.Grenander 研究了随机叙列的回归系数的估计问题,最近 M.Rosenblatt 研究了随机向量叙列的回归系数的估计问题.我们这桌案里研究格子点上随机场的回归系数的估计问题.前二作者所采用的方法是一样的,但是对于随机场而言若采用同一方法则有     

Rook Theory and Hypergeometric Series

The number of ways of placingknon-attacking rooks on a Ferrers board is expressed as a hypergeometric series of a type originally studied by Karlsson (J. Math. Phys.12(1971), 270–271) and Minton (J. Math. Phys.11(1970), 1375–1376). Known transformation identities for series of this type translate into new theorems about rook polynomials.     

关于核完备空间的几个问题

<正> 1955年A.Grothendieck在建立核空间理论的同时,作为例子也具体给出了一类特殊的完备空间——gestufen空间具有核性的充要条件,十年之后,A.Pietsch和作者的一篇未发表的工作各自独立地得到了一般完备空间的核性条件,从此核完备空间的研究就展开了,如可参看[7—9]. 本文是继续探讨这方面的问题,共分四个部分:首先,为完整起见,在§1我们将重新叙述和证明完备空间为核的条件(我们原先的证明就与Pietsch不同);其次,在§2中我