Gr?bner bases in difference-differential modules and difference-differential dimension polynomials

In this paper we extend the theory of Gr?bner bases to difference-differential modules and present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module equipped with the natural filtration. We present and verify algorithms for constructing these Gr?bner bases counterparts. To this aim we introduce the concept of “generalized term order” on ℕ ^m ×ℤ ⁿ and on difference-differential modules. Using Gr?bner bases on difference-differential modules we present a direct and algorithmic approach to computing the difference-differential dimension polynomials of a difference-differential module and of a system of linear partial difference-differential equations. This work was supported by the National Natural Science Foundation of China (Grant No. 60473019) and the KLMM (Grant No. 0705)     

Number of zeros of interval polynomials

In this paper, we develop a rigorous algorithm for counting the real interval zeros of polynomials with perturbed coefficients that lie within a given interval, without computing the roots of any polynomials. The result generalizes Sturm’s Theorem for counting the roots of univariate polynomials to univariate interval polynomials.     

微分多项式的唯一性和Lahiri的一个问题

讨论了亚纯函数的微分多项式分担一个 IM公共值的唯一性 ,回答了 LahiriL 提出的一个问题 .     

关于微分多项式的唯一性

研究了涉及亚纯函数及其导数具有三个公共值的唯一性问题,得到的结果改进了Lahiri I.,张庆彩和仪洪勋等的有关定理,并有例子表明所得结果是精确的.     

涉及微分多项式的亚纯函数分担值问题

This paper is devoted to studying the relationship between meromorphic functions f（z） and g（z） when their differential polynomials satisfy sharing condition weaker than sharing one value IM.     

Value-sharing and uniqueness problems for non-Archimedean differential polynomials in several variables

We consider differential polynomials of Fermat–Waring type, const-ructed using polynomials of Yi’s type for meromorphic functions in a non-Archimedean field. Similarly to the Hayman Conjecture, we prove that the considered differential polynomials assume all values. We established also a uniqueness theorem for these differential polynomials.     

Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds

The estimates of the radii of convergence balls of the Newton method and uniqueness balls of zeroes of vector fields on the Riemannian manifolds are given under the assumption that the covariant derivatives of the vector fields satisfy some kind of general Lipschitz conditions. Some classical results such as the Kantorovich's type theorem and the Smale's γ-theory are extended.     

Zeros of univariate interval polynomials

Polynomials with perturbed coefficients, which can be regarded as interval polynomials, are very common in the area of scientific computing due to floating point operations in a computer environment. In this paper, the zeros of interval polynomials are investigated. We show that, for a degree n interval polynomial, the number of interval zeros is at most n and the number of complex block zeros is exactly n if multiplicities are counted. The boundaries of complex block zeros on a complex plane are analyzed. Numeric algorithms to bound interval zeros and complex block zeros are presented.     

Non-existence and uniqueness results for boundary value problems for Yang-Mills connections

We show uniqueness results for the Dirichlet problem for Yang-Mills connections defined in -dimensional () star-shaped domains with flat boundary values. This result also shows the non-existence result for the Dirichlet problem in dimension 4, since in 4-dimension, there exist countably many connected components of connections with prescribed Dirichlet boundary value. We also show non-existence results for the Neumann problem. Examples of non-minimal Yang-Mills connections for the Dirichlet and the Neumann problems are also given.

     

Domination numbers and zeros of chromatic polynomials

In this paper, we shall prove that if the domination number of G is at most 2, then P(G,λ) is zero-free in the interval (1,β), where

     

Tutte uniqueness of line graphs

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result, we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.     

Extremal polynomials on arcs of the circle with zeros on these arcs

The paper gives a solution of an extremal problem of finding monic polynomial least deviating from zero on several arcs of the unit circle, under some restrictions on the location of zeros and additional conditions on mutual position of the arcs. The extremal polynomial is represented in the terms of density of harmonic measure. The work is done under the financial support of RFFI (Project 07-01-00167) and the President of RF Grant (Project NSh-2970.2008.1).     

关于Lucas多项式平方和的恒等式

利用广义Lucas多项式L n(x,y)的性质,通过构造组合和式T n(x,y;tx2),结合Bernoulli多项式的生成函数和Euler多项式的生成函数,采用分析学中的方法,得到两个有关L2n(x,y)的恒等式.并从这一结果出发,得到了两个推论,推广了相关文献的一些结果.     

Chromatic uniqueness of certain complete tripartite graphs

Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniqueness of certain complete t-partite graphs. Ars Comb., 92, 353–376 (2009)] show that K(p − k, p − i, p) for i = 0, 1 are chromatically unique if p ≥ k + 2 ≥ 4. In this paper, we show that if 2 ≤ i ≤ 4, the complete tripartite graph K(p − k, p − i, p) is chromatically unique for integers k ≥ i and p ≥ k ²/4 + i + 1.     

Tension polynomials of graphs

The tension polynomial F_G(k) of a graph G, evaluating the number of nowhere‐zero ?_k‐tensions in G, is the nontrivial divisor of the chromatic polynomial χ_G(k) of G, in that χ_G(k) = k^c(G)F_G(k), where c(G) denotes the number of components of G. We introduce the integral tension polynomial I_G(k), which evaluates the number of nowhere‐zero integral tensions in G with absolute values smaller than k. We show that 2^r(G)F_G(k)≥I_G(k)≥ (r(G)+1)F_G(k), where r(G)=|V(G)|?c(G), and, for every k>1, F_G(k+1)≥ F_G(k)˙k / (k?1) and I_G(k+1)≥I_G(k)˙k/(k?1). © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 137–146, 2002     

A survey on the study of real zeros of flow polynomials

For a bridgeless graph , its flow polynomial is defined to be the function , which counts the number of nonwhere-zero -flows on an orientation of whenever is a positive integer and is an additive Abelian group of order . It was introduced by Tutte in 1950, and the locations of zeros of this polynomial have been studied by many researchers. This paper gives a survey on the results and problems on the study of real zeros of flow polynomials.     

一类平面多项式系统极限环的存在唯一性

研究一类平面2n 1次多项式微分系统的极限环问题,利用Hopf分枝理论得到了该系统极限环存在性与稳定性的若干充分条件,利用Cherkas和Zheilevych的唯一性定理得到了极限环唯一性的若干充分条件.