Complete permutation polynomials with the form (xpm−x+δ)s+axpm+bx over Fpn

In this paper, several classes of complete permutation polynomials with the form ${(x^{p^m}-x+\delta )}^s+a{x}^{p^m}+bx$ over ${\mathbb{F}}_{p^n}$ are proposed by the AGW criterion and determining the number of solutions of some equations. Our results also enrich constructions of known permutation polynomials.     

高次矩阵方程f（X）=0的一种解法

关于m次矩阵方程Xm+a1Xm-1+…+am-1X+amEn=0,其中En是n阶单位矩阵,a1,a2,…,am∈R,X∈Cn×n,本文利用矩阵的化零多项式,最小多项式的相关结论以及Jordan标准形分解,讨论了该方程的所有可能解.     

Stochastic analysis method for hopf's equation ut+ uux=0

It is proved that the stochastic charactcrtstics of Burgers’ equation u _t+uu_x=μu_xxconverge in probability to the character is tcs of Hopf ’ s cqua lion u_t+uu_x=0 as the viscosity μ →0. It follows naturally that the solution of Burgers’ equation converges to the solution of Hopf 's equation satisfying entropy condition. This is well known result due to E. Hopf in 1950. The method here is new. This paper suggests that this method may be useful for proving the validity of “vanishing viscosity method” in certain cases. Those problems are usually extremely difficult.     

On the number of distinct roots of a lacunary polynomial over finite fields

We obtain new upper bounds on the number of distinct roots of lacunary polynomials over finite fields. Our focus will be on polynomials for which there is a large gap between consecutive exponents in the monomial expansion.     

Permutation binomials of the form xr(xq−1 + a) over Fqe

We present several existence and nonexistence results for permutation binomials of the form $x^r(x^{q-1}+a)$, where $e\ge 2$ and $a\in {\mathbb{F}}_{q^e}^{⁎}$. As a consequence, we obtain a complete characterization of such permutation binomials over ${\mathbb{F}}_{q^2}$, ${\mathbb{F}}_{q^3}$, ${\mathbb{F}}_{q^4}$, ${\mathbb{F}}_{p^5}$, and ${\mathbb{F}}_{p^6}$, where p is an odd prime.     

The coefficients of primitive polynomials over finite fields

For , we prove that there always exists a primitive polynomial of degree over a finite field with the first and second coefficients prescribed in advance.

     

Solutions of the matrix equation (1):X m+A 1Xm−1+…+A m=0

SupposeX and the coefficientsA ₁, …,A _m aren×n matrices. LetB be anmn×mn matrix as in (7). LetJ be the Jordan canonical matrix ofB andB=PJP ⁻. LetE _i denote thei×i unit matrix.V is defined bydV/dt=JV andV(t=0)=E _mn. ThenZ=PV satisfiesdZ/dt=BZ.P ^* is a matrix which consists of the firstn rows ofP. The author proves: There is a solution of (1) ↔ there are anmn×n matrixC, ann×n matrixQ and ann×n function matrixN such thatP ^*VC=QN, where detQ≠0 andN is defined byN(t=0)=E _n anddN/dt=RN, in whichR is ann×n Jordan canonical matrix. There are three cases regarding the solutions of (1): No solution, finitek solutions, 1<k<C _n ^m , and infinite solutions which containj parameters, 1<-j<-mn ².     

On the Hermitian positive defnite solution of the nonlinear matrix equation X + A*X −1 A + B*X −1 B = I

In this paper, we study the matrix equation X + A*X ⁻¹ A + B*X ⁻¹ B = I, where A, B are square matrices, and obtain some conditions for the existence of the positive definite solution of this equation. Two iterative algorithms to find the positive definite solution are given. Some numerical results are reported to illustrate the effectiveness of the algorithms. This research supported by the National Natural Science Foundation of China 10571047 and Doctorate Foundation of the Ministry of Education of China 20060532014.     

On a Matrix Equation AX＋XB=C over a Skew Field

In this paper we study a matrix equation AX BX=C （Ⅰ）over an arbitrary skew field,and give a consistency criterion of （Ⅰ） and an explicit expression of general solutions of （Ⅰ）.A convenient,simple and practical method of solving （Ⅰ） is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.