A theorem on the transcendency and its applications

A theorem on the transcendency, for a limit of a sequence of algebraic numbers, is given and used to test the transcendencies of some classes of numbers. Project supported by the National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province.     

Exact traveling wave solutions and dynamical behavior for the (<Emphasis Type="Italic">n</Emphasis> + 1)-dimensional multiple sine-Gordon equation

Using the methods of dynamical systems for the (n + 1)-dimensional multiple sine-Gordon equation, the existences of uncountably infinite many periodic wave solutions and breaking bounded wave solutions are obtained. For the double sine-Gordon equation, the exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined. This work was supported by the National Natural Science Foundation of China (Grant No. 11671179) and the Natural Science Foundation of Yunnan Province (Grant No. 2005A0092M).     

Nonstandard decision methods for the solvability of real polynomial equations

For a multivariate polynomial equation with coefficients in a computable ordered field, two criteria of this equation having real solutions are given. Based on the criteria, decision methods for the existence of real zeros and the semidefiniteness of binaly polynomials are provided. With the aid of computers, these methods are used to solve several examples. The technique is to extend the original field involved in the question to a computable non-Archimedean ordered field containing infinitesimal elements. Project supported by the National Natural Science Foundation of China (Grant No. 19661002) and the Climbing Project.     

Genus distribution of ladder type and cross type graphs

In this paper a method is given to calculate the explicit expressions of embedding genus distribution for ladder type graphs and cross type graphs. As an example, we refind the genus distribution of the graph J _n which is the first class of graphs studied for genus distribution where its genus depends on n. This work was supported National Natural Science Foundation of China (Grant Nos. 10571013, 60433050) and the State Key Development Program of Basic Research of China (Grant No. 2004CB318004)     

Necessary optimality conditions for a class of nonsmooth vector optimization

The Kuhn-Tucker type necessary conditions of weak efficiency are given for the problem of minimizing a vector function whose each component is the sum of a differentiable function and a convex function, subject to a set of differentiable nonlinear inequalities on a convex subset C of ℝ ⁿ , under the conditions similar to the Abadie constraint qualification, or the Kuhn-Tucker constraint qualification, or the Arrow-Hurwicz-Uzawa constraint qualification. Supported by the National Natural Science Foundation of China (No. 70671064, No. 60673177), the Province Natural Science Foundation of Zhejiang (No.Y7080184) and the Education Department Foundation of Zhejiang Province (No. 20070306).     

L p -dual Quermassintegral sums

In this paper, we first introduce a concept of L _p -dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L _p -dual Quermassintegral sums. Moreover, by using Lutwak’s width-integral of index i, we establish the L _p -Brunn-Minkowski inequality for the polar mixed projection bodies. As applications, we prove some interrelated results. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10271071), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605065) and Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Consistency and asymptotic normality of profilekernel and backfitting estimators in semiparametric reproductive dispersion nonlinear models

Semiparametric reproductive dispersion nonlinear model (SRDNM) is an extension of nonlinear reproductive dispersion models and semiparametric nonlinear regression models, and includes semiparametric nonlinear model and semiparametric generalized linear model as its special cases. Based on the local kernel estimate of nonparametric component, profile-kernel and backfitting estimators of parameters of interest are proposed in SRDNM, and theoretical comparison of both estimators is also investigated in this paper. Under some regularity conditions, strong consistency and asymptotic normality of two estimators are proved. It is shown that the backfitting method produces a larger asymptotic variance than that for the profile-kernel method. A simulation study and a real example are used to illustrate the proposed methodologies. This work was supported by National Natural Science Foundation of China (Grant Nos. 10561008, 10761011), Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. Y200805073), PhD Special Scientific Research Foundation of Chinese University (Grant No. 20060673002) and Program for New Century Excellent Talents in University (Grant No. NCET-07-0737)     

Frame Wavelets with Compact Supports for L^2（R^n）

The construction of frame wavelets with compact supports is a meaningful problem in wavelet analysis. In particular, it is a hard work to construct the frame wavelets with explicit analytic forms. For a given n × n real expansive matrix A, the frame-sets with respect to A are a family of sets in R^n. Based on the frame-sets, a class of high-dimensional frame wavelets with analytic forms are constructed, which can be non-bandlimited, or even compactly supported. As an application, the construction is illustrated by several examples, in which some new frame wavelets with compact supports are constructed. Moreover, since the main result of this paper is about general dilation matrices, in the examples we present a family of frame wavelets associated with some non-integer dilation matrices that is meaningful in computational geometry.     

On Cycles Containing a Given Arc in Regular Multipartite Tournaments

In this paper we prove that if T is a regular n-partite tournament with n≥4, then each arc of T lies on a cycle whose vertices are from exactly κ partite sets for κ=4,5,…,n. Our result, in a sense, generalizes a theorem due to Alspach.     

On the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

A maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information

The paper is concerned with a stochastic optimal control problem in which the controlled system is described by a fully coupled nonlinear forward-backward stochastic differential equation driven by a Brownian motion. It is required that all admissible control processes are adapted to a given subfiltration of the filtration generated by the underlying Brownian motion. For this type of partial information control, one sufficient (a verification theorem) and one necessary conditions of optimality are proved. The control domain need to be convex and the forward diffusion coefficient of the system can contain the control variable. This work was partially supported by Basic Research Program of China (Grant No. 2007CB814904), National Natural Science Foundation of China (Grant No. 10325101) and Natural Science Foundation of Zhejiang Province (Grant No. Y605478, Y606667)     

Finite groups in which elements of the same order outside the center are conjugate

We prove that if G is a finite group in which the elements of the same order outside the center are conjugate,then either G is abelian or G(?)S_3.     

Circular neighbor-balanced designs universally optimal for total effects

In many experiments, the performance of a subject may be affected by some previous treatments applied to it apart from the current treatment. This motivates the studies of the residual effects of the treatments in a block design. This paper shows that a circular block design neighbor-balanced at distances up toγ≤k - 1, where k is the block size, is universally optimal for total effects under the linear models containing the neighbor effects at distances up toγamong the class of all circular binary block designs. Some combinatorial approaches to constructing these circular block designs neighbor-balanced at distances up to k - 1 are provided.     

Convergence of Newton‘’s Method and Uniqueness of the Solution of Equations in Banach SpacesⅡ

Some results on convergence of Newton‘s method in Banach spaces are established under the assumption that the derivative of the opderators satisfies the radius or center Lipschitz condition with a weak L average.     

Partial regularity of weak solutions for Landau-Lifshitz system with potential

In this note, we prove the partial regularity of stationary weak solutions for the Landau-Lifshitz system with general potential in four or three dimensional space. As well known, in general, since the constraint of the methods, in order to get the partial regularity of stationary weak solution of the Landau-Lifshitz system with potential, we need to add some very strongly conditions on the potential. The main difficulty caused by potential is how to find the equation satisfied by the scaling function, which breaks down the blow-up processing. We estimate directly Morrey’s energy to avoid the difficulties by blowing up. This work was supported by National Natural Science Foundation of China (Grant Nos. 10631020, 60850005) and the Natural Science Foundation of Zhejiang Province (Grant No. D7080080)     

On (2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript> + 1)-variable symmetric Boolean functions with submaximum algebraic immunity 2<Superscript><Emphasis Type="Italic">m</Emphasis>−1</Superscript>

All (2 ^m + 1)-variable symmetric Boolean functions with submaximal algebraic immunity 2 ^m−1 are described and constructed. The total number of such Boolean functions is 3² · 2^2m−3 +3 ^m−2 · 2⁴ − 2 for m ⩾ 2. This work was supported by the Major State Basic Research Development Program of China (Grant No. 2004CB3180004) and National Natural Science Foundation of China (Grant No. 60433050)     

The decomposition of Kv into K2×K5''''s

The decomposition of the complete graph Kv into Kr×Kc's, the products of Kr and Kc,is originated from the use of DNA library screening. In this paper, we consider the case where r=2 and c = 5, and show that such a decomposition exists if and only if v ≡ 1 (mod 25).     

Some results on 4<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> 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 2 ^n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2 ^n−(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 2 ^n−(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 4 ^m 2 ⁿ 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 ^m 2 ⁿ designs under the clear effects criterion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10571093, 10671099 and 10771123), the Research Foundation for Doctor Programme (Grant No. 20050055038) and the Natural Science Foundation of Shandong Province of China (Grant No. Q2007A05). Zhang’s research was also supported by the Visiting Scholar Program at Chern Institute of Mathematics.     

<Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript>(R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)-boundedness for maximal functions

In this paper, we get W ^1,p (R ⁿ )-boundedness for tangential maximal function and nontangential maximal function, which improves J.Kinnunen, P.Lindqvist and Tananka’s results. Supported by the key Academic Discipline of Zhejiang Province of China under Grant No.2005 and the Zhejiang Provincial Natural Science Foundation of China.     

On the distribution of imaginary parts of zeros of the Riemann zeta function,II

We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Montgomery’s pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. The first author is supported by National Science Foundation Grant DMS-0555367. The second author is partially supported by the National Science Foundation and the American Institute of Mathematics (AIM). The third author is supported by National Science Foundation Grant DMS-0456615.