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Latest Printed Issue
2008, Vol.51 Num.7
Online: 2008-07-29
Articles
 
Articles
LIU JianYa1, YE YangBo12
Correlation of zeros of automorphic L-functions
We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions: L(s,\pi_1)\cdots L(s,\pi_k)$, where $\pi_j$, $j=1,\ldots,k$, are automorphic cuspidal representations of ${\rm GL}_{m_j}(\Bbb Q_{\Bbb A})$. Here the sizes of the groups ${\rm GL}_{m_j}(\Bbb Q_{\Bbb A})$ are not necessarily the same. When these $L(s,\pi_j)$ are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When $L(s,\pi_j)$ are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when $m_1,\ldots,m_k\leq 4$, but are under Hypothesis H in other cases.
2008 Vol. 51 (7): 1147-1166 [Abstract] ( 75 ) [HTML 0KB] [PDF 303KB] ( 2 )
CUI GuiZhen, PENG WenJuan
On the structure of Fatou domains
Let U be a multiply-connected fixed attracting Fatou domain of a rational map f. We prove that there exist a rational map g and a completely invariant Fatou domain V of g such that (f,U) and (g,V) are holomorphically conjugate, and each non-trivial Julia component of g is a quasi-circle which bounds an eventually superattracting Fatou domain of g containing at most one postcritical point of g. Moreover, g is unique up to a holomorphic conjugation.
2008 Vol. 51 (7): 1167-1186 [Abstract] ( 81 ) [HTML 0KB] [PDF 330KB] ( 1 )
XU KeJian, LIU Min
On the torsion in K2 of a field
For a field F, let Gn(F)=$\{\{a, \Phi_{n}(a)\} \in K_{2}(F)\mid {a, \Phi_{n}(a)} \in F^{*}\},$ where Φn(x) is the n-th cyclotomic polynomial. At first, by using Faltings’ theorem on Mordell conjecture it is proved that if F is a number field and if n≠ 4, 8, 12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F), and then by using the results of Manin, Grauert, Samuel and Li on Mordell conjecture theorem for function fields, a similar result is established for function fields over an algebraically closed field.
2008 Vol. 51 (7): 1187-1195 [Abstract] ( 57 ) [HTML 0KB] [PDF 235KB] ( 1 )
FANG MingLiang1, Lawrence ZALCMAN2
On the value distribution of f+a(f’)n
Let f be a
transcendental meromorphic function, a a nonzero finite complex
number, and n≥ge 2 a positive integer. Then f+a(f’)n assumes
every complex value infinitely often. This answers a question of Ye
for n=2. A related normality criterion is also given.
2008 Vol. 51 (7): 1196-1202 [Abstract] ( 48 ) [HTML 0KB] [PDF 171KB] ( 1 )
ZHANG ChuanYi, YANG FengLin
Pseudo almost periodic solutions to parabolic boundary value inverse problems
We first define the pseudo almost periodic functions in a more general setting. Then we show the existence, uniqueness and stability of pseudo almost periodic solutions of parabolic inverse problems for a type of boundary value problems.
2008 Vol. 51 (7): 1203-1214 [Abstract] ( 59 ) [HTML 0KB] [PDF 206KB] ( 1 )
Veli B SHAKHMUROV
Carleman estimates and unique continuation property for the anisotropic differential-operator equations
The unique continuation theorems for the anisotropic partial differential-operator equations with variable coefficients in Banach-valued Lp-spaces are studied. To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations, the sufficient conditions are founded. By using these facts, the unique continuation properties are established. In the application part, the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.
2008 Vol. 51 (7): 1215-1231 [Abstract] ( 53 ) [HTML 0KB] [PDF 289KB] ( 2 )
YANG YiDu, CHEN Zhen
The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators
This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T. Under the condition that the approximate operator Th converges to T in norm, it is proven that the k-th eigenvalue of Th converges to the k-th eigenvalue of T. (We sorted the positive eigenvalues in decreasing order and negative eigenvalues in increasing order.) Then we apply this result to conforming elements, nonconforming elements and mixed elements of self-adjoint elliptic differential operators eigenvalue problems, and prove that the k-th approximate eigenvalue obtained by these methods converges to the k-th exact eigenvalue.
2008 Vol. 51 (7): 1232-1242 [Abstract] ( 70 ) [HTML 0KB] [PDF 208KB] ( 1 )
ZHENG XiYin1, YANG XiaoQi2
The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces
In general normed spaces, we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior. We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function. Moreover, we provide necessary and sufficient conditions about the existence of weak (sharp) Pareto solutions.
2008 Vol. 51 (7): 1243-1256 [Abstract] ( 54 ) [HTML 0KB] [PDF 235KB] ( 1 )
JIANG Tao

Large-deviation probabilities for maxima of sums of subexponential random variables with application to finite-time ruin probabilities
We establish an asymptotic relation for the large-deviation probabilities of the maxima of sums of subexponential random variables centered by multiples of order statistics of i.i.d. standard uniform random variables. This extends a corresponding result of Korshunov. As an application, we generalize a result of Tang, the uniform asymptotic estimate for the finite-time ruin probability, to the whole strongly subexponential class.
2008 Vol. 51 (7): 1257-1265 [Abstract] ( 52 ) [HTML 0KB] [PDF 170KB] ( 1 )
LI JunPing1 & CHEN AnYue23
Existence, uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection
We consider a modified Markov branching process incorporating with both stateindependent immigration and instantaneous resurrection. The existence criterion of the process is firstly considered. We prove that if the sum of the resurrection rates is finite, then there does not exist any process. An existence criterion is then established when the sum of the resurrection rates is infinite. Some equivalent criteria, possessing the advantage of being easily checked, are obtained for the latter case. The uniqueness criterion for such process is also investigated. We prove that although there exist infinitely many of them, there always exists a unique honest process for a given $q$-matrix. This unique honest process is then constructed. The ergodicity property of this honest process is analysed in detail. We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.
2008 Vol. 51 (7): 1266-1286 [Abstract] ( 65 ) [HTML 0KB] [PDF 269KB] ( 1 )
ZHANG SanGuo, LIAO Yuan
On some problems of weak consistency of quasi-maximum likelihood estimates in generalized linear models
In this paper, we explore some weakly consistent properties of quasi-maximum likelihood estimates (QMLE) concerning the quasi-likelihood equation $\sum_{i=1}^n X_i(y_i-\mu(X_i’\beta))=0$ for univariate generalized linear model ${\rm E}(\,y\,|X)= \mu(X’\beta)$. Given uncorrelated residuals $\{e_i=Y_i-\mu(X_i’\beta_0),1\leqslant i\leqslant n\}$ and other conditions, we prove that $\hat{\beta}_n-\beta_0=O_p(\underline{\lambda}_n^{-1/2})$ holds, where $\hat{\beta}_n$ is a root of the above equation, $\beta_0$ is the true value of parameter β and $\underline{\lambda}_n$ denotes the smallest eigenvalue of the matrix $S_n=\sum_{i=1}^nX_iX_i’$. We also show that the convergence rate above is sharp, provided independent non-asymptotically degenerate residual sequence and other conditions. More\/over, paralleling to the elegant result of Drygas (1976) for classical linear regression models, we point out that the necessary condition guaranteeing the weak consistency of QMLE is $\,S_n^{-1}\to 0$, as the sample size $n\to\infty$.
2008 Vol. 51 (7): 1287-1296 [Abstract] ( 55 ) [HTML 0KB] [PDF 220KB] ( 1 )
ZHAO ShengLi12, ZHANG RunChu2, LIU MinQian2
Some results on 4m2n designs with clear two-factor interaction components
Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and it has become an active research issue in recent years. Tang et al. derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi’s) in 2n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2n−(n−k) design for given k, which are only restricted for the symmetrical case. This paper proposes and studies the clear effects problem for the asymmetrical case. It improves the construction method of Tang et al. for 2n−(n−k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic’s) in 4m2n designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic’s in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct $4^m2^n$ designs under the clear effects criterion.
2008 Vol. 51 (7): 1297-1314 [Abstract] ( 60 ) [HTML 0KB] [PDF 297KB] ( 1 )
JIANG JianCheng12, LI JianTao1
Two-stage local M-estimation of additive models
This paper studies local M-estimation of the nonparametric components of additive models. A two-stage local M-estimation procedure is proposed for estimating the additive components and their derivatives. Under very mild conditions, the proposed estimators of each additive component and its derivative are jointly asymptotically normal and share the same asymptotic distributions as they would be if the other components were known. The established asymptotic results also hold for two particular local M-estimations: the local least squares and least absolute deviation estimations. However, for general two-stage local M-estimation with continuous and nonlinear Ψ-functions, its implementation is time-consuming. To reduce the computational burden, one-step approximations to the two-stage local M-estimators are developed. The one-step estimators are shown to achieve the same efficiency as the fully iterative two-stage local M-estimators, which makes the two-stage local M-estimation more feasible in practice. The proposed estimators inherit the advantages and at the same time overcome the disadvantages of the local least-squares based smoothers. In addition, the practical implementation of the proposed estimation is considered in details. Simulations demonstrate the merits of the two-stage local M-estimation, and a real example illustrates the performance of the methodology.
2008 Vol. 51 (7): 1315-1338 [Abstract] ( 102 ) [HTML 0KB] [PDF 376KB] ( 1 )
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