UNFOLDING OF MULTIPARAMETER EQUIVARIANT BIFURCATION PROBLEMS WITH TWO GROUPS OF STATE VARIABLES UNDER LEFT-RIGHT EQUIVALENT GROUP

Based on the left_right equivalent relation of smooth map_germs in singularity theory, the unfoldings of multiparameter equivariant bifurcation problems with respect to left_right equivalence are discussed. The state variables of such an equivariant bifurcation problem were divided into two groups, in which the first can vary independently, while the others depend on the first in the varying process. By applying related methods and techniques in the unfolding theory of smooth map_germs, the necessary and sufficient condition for an unfolding of a multiparameter equivariant bifurcation problem with two groups of state variables to be versal is obtained.

The Exponent Set of Central Symmetric Primitive Matrices

This paper first establishes a distance inequality of the associated dia-graph of a central symmetric primitive matrix, then characters the exponent set of central symmetric primitive matrices, and proves that the exponent set of central symmetric primitive matrices of order n is {1, 2,…, n - 1}. There is no gap in it.

Global optimization design method for maximizing the capacity of V-belt drive

An optimization model is constructed to formulate the maximization problem on the capacity of V-belt drive. The concavity,the monotonicity and the global optimality condition are studied for the objective function,and it is proved that the feasible region of the model is bounded,closed and convex under some design conditions. Then,a solution method,called an optimal segment algorithm,is developed to find the global maximizer of the model. Under four different design conditions,solution methods are presented respectively. Some real case studies are employed to demonstrate that the model and the algorithm in this paper are promising.

A Representation of sl(2;C) on Clifford Algebra of Even Dimension

Under the foundation of Hermitean Clifford setting, we define the fundamental operators for complex Clifford algebra valued fimctions, obtain some properties of these operators, and discuss a representation of sl（2; C ） on Clifford algebra of even dimension.

The k-point Exponent Set of Primitive Digraphs with Girth 2

Let D = （V, E） be a primitive digraph. The vertex exponent of D at a vertex v∈ V, denoted by expD（v）, is the least integer p such that there is a v →u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exPD（V1） ≤ exPD（V2） …≤ exPD（Vn）. Then exPD（Vk） is called the k- point exponent of D and is denoted by exPD （k）, 1≤ k ≤ n. In this paper we define e（n, k） ：= max{expD （k） ｜ D ∈ PD（n, 2）} and E（n, k） ：= {exPD（k）｜ D ∈ PD（n, 2）}, where PD（n, 2） is the set of all primitive digraphs of order n with girth 2. We completely determine e（n, k） and E（n, k） for all n, k with n ≥ 3 and 1 ≤ k ≤ n.

TESTING SERIAL CORRELATION IN SEMIPARAMETRIC VARYING COEFFICIENT PARTIALLY LINEAR ERRORS-IN-VARIABLES MODEL

The authors propose a V_(N,p) test statistic for testing finite-order serial correlation in asemiparametric varying coefficient partially linear errors-in-variables model.The test statistic is shownto have asymptotic normal distribution under the null hypothesis of no serial correlation.Some MonteCarlo experiments are conducted to examine the finite sample performance of the proposed V_(N,p) teststatistic.Simulation results confirm that the proposed test performs satisfactorily in estimated sizeand power.

L_(p,q)-norm estimates associated with Burkholder’s inequalities

We prove Burkholder's inequalities in the frame of Lorentz spaces Lp,q(Ω), 1 < p < ∞, 1 < q < ∞. As application, we obtain the Lp,q-norm estimates on Rosenthal's inequalities. These estimates generalize the classical Rosenthal's inequalities.

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear EV Models

This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form Y = X^Tβ ＋Z^Tα（T） ＋ε,ξ = X ＋ η with the identifying condition E[（ε,η^T）^T] =0, Cov[（ε,η^T）^T] = σ^2Ip＋1. The estimators of interested regression parameters /3 , and the model error variance σ2, as well as the nonparametric components α（T）, are constructed. Under some regular conditions, we show that the estimators of the unknown vector β and the unknown parameter σ2 are strongly consistent and asymptotically normal and that the estimator of α（T） achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the VN,p test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.

Polyharmonic Boundary Value Problem in a Sector

In this article, a polyharmonic Neumann function in a sector with angle π n （n N） is studied by convolution. Especially, the outward normal derivatives at three corner points are defined properly. We give the recursive expressions for the polyharmonic Neumann function, obtaining the solution and the condition of solvability for the related polyharmonic Neumann problem.