The polynomial degree of the Grassmannian G(1,n,q) of lines in finite projective space PG(n, q)

Let G: = G(1,n,q) denote the Grassmannian of lines in PG(n,q), embedded as a point-set in PG(N, q) with For n = 2 or 3 the characteristic function of the complement of G is contained in the linear code generated by characteristic functions of complements of n-flats in PG(N, q). In this paper we prove this to be true for all cases (n, q) with q = 2 and we conjecture this to be true for all remaining cases (n, q). We show that the exact polynomial degree of is for δ: = δ(n, q) = 0 or 1, and that the possibility δ = 1 is ruled out if the above conjecture is true. The result deg( for the binary cases (n,2) can be used to construct quantum codes by intersecting G with subspaces of dimension at least      

Extensions of some polynomial inequalities to the polar derivative

Let p(z) be a polynomial of degree n and for any real or complex number α, let Dαp(z)=np(z)+(α−z)p^′(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros inside or outside a circle. Our results shall generalize and sharpen some well-known polynomial inequalities.     

Approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces

An attractive method for approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces is introduced. The main result is that the arbitrary given order derived vectors of a polynomial triangular surface converge uniformly to those of the approximated rational triangular Bézier surface as the elevated degree tends to infinity. The polynomial triangular surface is constructed as follows. Firstly, we elevate the degree of the approximated rational triangular Bézier surface, then a polynomial triangular Bézier surface is produced, which has the same order and new control points of the degree-elevated rational surface. The approximation method has theoretical significance and application value: it solves two shortcomings-fussy expression and uninsured convergence of the approximation-of Hybrid algorithms for rational polynomial curves and surfaces approximation.     

使用自正则度量的凸二次规划的原始对偶内点法的多项式复杂性

最近Peng等人使用新的搜索方向和自正则度量为求解线性规划问题提出了一个原始对偶内点法.本文将这个长步法延伸到凸二次规划.在线性规划情形时,原始空间和对偶空间中的尺度Newton方向是正交的,而在二次规划情形时这是不成立的.本文将处理这个问题并且证明多项式复杂性,并且得到复杂性的上界为O(n√log n log (n/ε)).     

The counting complexity of a simple scheduling problem

Let T be a set of tasks. Each task has a non-negative processing time and a deadline. The problem of determining whether or not there is a schedule of the tasks in T such that a single machine can finish processing each of them before its deadline is polynomially solvable. We prove that counting the number of schedules satisfying this condition is #P-complete.     

On closed invariant sets in local dynamics

We investigate the dynamical behaviour of a holomorphic map on an f-invariant subset C of U, where . We study two cases: when U is an open, connected and polynomially convex subset of Ck and C?U, closed in U, and when ∂U has a p.s.h. barrier at each of its points and C is not relatively compact in U. In the second part of the paper, we prove a Birkhoff's type theorem for holomorphic maps in several complex variables, i.e. given an injective holomorphic map f, defined in a neighborhood of , with U star-shaped and f(U) a Runge domain, we prove the existence of a unique, forward invariant, maximal, compact and connected subset of which touches ∂U.     

关于有向网络容量扩充问题

提出了有向网络最大容量的两种计算方法,将杨超等人（1998）的无向网络容量扩充问题,扩展到约束条件含固定费用的有向网络的扩充,并给出了强多项式算法。     

A computational comparison of the first nine members of a determinantal family of root-finding methods

For each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-finding iteration function, B_m^(k) defined as the ratio of two determinants that depend on the first m−k derivatives of the given function. This infinite family is derived in Kalantari (J. Comput. Appl. Math. 126 (2000) 287–318) and its order of convergence is analyzed in Kalantari (BIT 39 (1999) 96–109). In this paper we give a computational study of the first nine root-finding methods. These include Newton, secant, and Halley methods. Our computational results with polynomials of degree up to 30 reveal that for small degree polynomials B_m^(k−1) is more efficient than B_m^(k), but as the degree increases, B_m^(k) becomes more efficient than B_m^(k−1). The most efficient of the nine methods is B₄⁽⁴⁾, having theoretical order of convergence equal to 1.927. Newton's method which is often viewed as the method of choice is in fact the least efficient method.     

Power linear Keller maps of dimension three

In this paper it is proved that a power linear Keller map of dimension three over a field of characteristic zero is linearly triangularizable.