有限域上一类方程组解数的直接公式

本文给出有限域F=F_q(q=p~f,f≥1,p是一个奇素数)上一类方程组∑_(i=s_(r-1)+1~(s_r)∑_(j=1)~(m_i-m_(i-1))a_(m_(i-1)+j)x_1~(d_m(i-1)+j,1)…x_(n_i)~d_(m_(i-1)+j,n_i)=b_r,r=1,…,k当指数满足一定条件时,在F~(n_s_k)上解数的一个直接公式,这里d_(ij)＞0,a_i∈F~*,b_i∈F,0= s_0＜s_1＜…＜s_k,0=m_0＜m_1＜…＜m_(s_k),0=n_0＜n_1＜…＜n_(s_k), m_1≤n_1,…,m_(s_k)≤n_(s_k)．     

利用“投宿”模型解一类排列组合题

在排列组合教学中,构造模型解排列组合应用题,常常能使较复杂的问题明朗化,有利于学生的学习。现将“投宿”模型及其应用介绍如下: “投宿”模型:(1)若m个人到n家旅馆投宿,则有n~m种投宿方法;(2)若以上m个人中有m_0(m_0相似文献   

六类Oberwolfach问题OP(4~a,s~b)的解

完全图K_n(n为奇数)或K_n-I(n为偶数,I为K_n的1-因子)是否有2-因子分解称Oberwolfach问题.每个2-因子恰包含α_i个长为m_i的圈(i=1,2,…,t)的Oberwolfach问题记为OP(m_1~(α_1),m_2~(α_2),…,m_t~(α_t)).证明了对任意的a≥0,b=2,3和s=3,5,6,且(a,s,b)≠(0,3,2),都存在OP(4~a,s~b)的解.     

半线性高阶椭圆型方程非平凡解的存在性与不存在性

设(?)为 R~n 中的带光滑边界(?)的有界域。考察边值问题Δ~2u-aΔu bu=f(x,u),x∈(?),(1.1)或u=((?)u)/((?)v)=0,x∈(?) (1.2)u=Δu=0,x∈(?),(1.3)其中 a 和 b 为非负实数,((?)u)/((?)v)为沿(?)外法线的方向导数。当b=0,f(x,u)=cu~k,k为奇数且 c≤0时,文献[1]曾证明,方程(1.1)满足边值条件Δu|(?)=0的解满足极值原理;文献[2]则在对非线性项,f(x,u)加以某些限制的情况下,证明问题(1.1),(1.2)或(1.1),(1.3)存在非平凡解。本文的目的在于对上述问题作进一步讨论。在§2中,我们讨论了非平凡解存在性问题,代替[2]中的增长性条件     

局部(F_4)条件和两指标鞅a.s.收敛性

<正> §1.引言和记号 设(Ω,,P)为完备的概率空间.N+为非负整数集,N_+~2={Z=(m,n):m,n∈N+}.N_+~2依通常顺序构成定向集,在N_+~2上定义运算“∨”和“∧”如下:设Z_1,Z_2∈N_+~2,Z_1=(m_1,n_1),Z_2=(m_2,n_2),则     

椭球等高分布族下的非中心Cochran定理

本文在椭球等高分布假定下,讨论了二次型X′AX(A为对称阵)的非中心Cochran定理。主要结果如下: 若X～EC_n(μ,L_n;g),g(x)>0为x的连续函数,且X有有限的2n阶矩。A_i,i=1,2,…,m为n×n对称阵。A=∑A_i,λ_1,…,λ_k互不相同且非零。考虑下面的条件: (a) X′A_iX■sum from j=1 to k λ_jy_(ij),(y_(i1),…(y_(ik))′～Gχ~2(n_(i1),…,n_(ik);δ_(i1)~2,…,δ_(ik)~2;g)j=1,…,m。 (b) (X′A_1X,…,X′A_mX)■(sum from j=1 to k λ_jz_j…,sum from j=(m-1)k 1 to mk λ_(j-(m-1)k)z_j)(z_1…,z_(mk))′～Gχ~2(n_(11),n_(1k),n_(21)…,n_(mk);δ_(11)~2,…δ_(1k)~2,δ_(21)~2,…,δ_(mk)~2;g) (c) X′AX(?)sum from j=1 to k λ_jy_j,(y_1,…,y_k)′～Gχ(n_1,…,n_k;δ_1~2,…,δ_k~2;g) (d) r(A)=∑r(A_i)=∑∑r(A_iE_j),A=∑λ_jE_j,E_j~2=E_j,E_jE_(j′)=0,j≠j′=1,…,k, (e) k个等式n_j=∑n_(ij)中至少有k-1个成立。则 (Ⅰ) (a),(b)■(c),(d),(e), (Ⅱ) (a),(c),(e)■(b),(d), (Ⅲ) (b),(c)■(a),(d),(c), (Ⅳ) (c),(d)■(a),(b),(c)。     

有限域上一类方程组的解数公式

设F_q为一个q元有限域,其中q=p~s(s≥1),p是一个奇素数.本文给出下列方程组在F_q上的解数公式:a_(k1)x_1~(d_(11)~((k)))...x_(n_1)~(d_(1n_1)~((k)))+...+a_(k,s_1)x_1~(d_(s_1,1)~((k)))...x_(n_1)~(d_(s_1,n_1)~((k)))+a_(k,s_1)+1x_1~(d_(s_1+1,1)~((k)))...x_(n_2)~(d_(s_1+1,n_2)~((k)))+...a_(k,s_2)x_1~(d_(s_2,1)~((k)))...x_(n_2)~(d_(s_2,1)~((k)))...x_(n_2)~(d_(s_2,n_2)~((k)))=b_k,k=1,...,m,其中0s_1s_2,0n_1n_2,a_(ki)∈F_q~*,b_k∈F_q,d_(ij)~(k)0(k=l,...,m,i=1,...,s_2,j=1,...,n_2).特别当ms_1≤n_1,ms_2≤n_2,d_(ij)~(k)满足一定条件时,得到了明确的解数公式.     

负相协随机变量非随机和的差的精确大偏差

In this paper,we study precise large deviation for the non-random difference sum from j=1 to n_1(t) X_(1j)-sum from j=1 to n_2(t) X_(2j),where sum from j=1 to n_1(t) X_(1j) is the non-random sum of {X_(1j),j≥1} which is a sequence of negatively associated random variables with common distribution F_1(x),and sum from j=1 to n_2(t) X_(2j) is the non-random sum of {X_(2j),j≥1} which is a sequence of independent and identically distributed random variables,n_1(t) and n_2(t) are two positive integer functions.Under some other mild conditions,we establish the following uniformly asymptotic relation lim t→∞ sup x≥r(n_1(t))~(p+1)|(P(∑~(n_1(t)_(j=1)X_(1j)-∑~(n_2(t)_(j=1)X_(2j)-(μ_1n_1(t)-μ_2n_2(t)x))/(n_1(t)F_1(x))-1|=0.     

一类微分方程积分因子求法

—阶微分方程p(x,y)dx Q(x,y)dy=0,当它不是全微分方程但可化为形式x~(α_1)y~(β_1)(m_1ydx n_1xdy) x~(α_2)Y~(β_2)(m_2ydx n_2xdy)=0(1)(其中α_1,β_1,m_i,n_i,i=1,2,均为常数)时,若用观察法不易找到其积分因子.并且一般即方程也不存在仅与x或仅与y有关的积分因子.下面介绍这类方程(即方程(1))求积分因子的一个方法.     

关于矩阵张量积的谱

<正> 设 A=(a_(jk))_(m_1×m_2),B=(b_(jk))_(n_1×n_2),则 m_1n_1×m_2n_2矩阵(?)称为 A 与 B 的张量积(也称直积或 Kronecker 积).矩阵的张量积是多重线性代数的重要工具之一,在群表示论中也有重要应用.本文的主要目的是在作者过去工作的基础上给出矩阵张量积的一些谱性质.     

A microbial continuous culture system with diffusion and diversified growth

A reaction-diffusion model is presented to describe the microbial continuous culture with diversified growth. The existence of nonnegative solutions and attractors for the system is obtained, the stability of steady states and the steady state bifurcation are studied under three growth conditions. In the case of no growth inhibition or only product inhibition, the system admits one positive constant steady state which is stable; in the case of growth inhibition only by substrate, the system can have two positive constant steady states, explicit conditions of the stability and the steady state bifurcation are also determined. In addition, numerical simulations are given to exhibit the theoretical results.     

Positive steady states for a prey-predator model with some nonlinear diffusion terms

This paper discusses a prey-predator system with strongly coupled nonlinear diffusion terms. We give a sufficient condition for the existence of positive steady state solutions. Our proof is based on the bifurcation theory. Some a priori estimates for steady state solutions will play an important role in the proof.     

Bifurcation analysis in a diffusive Logistic population model with two delayed density-dependent feedback terms

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.     

Bifurcation from two equilibria of steady state solutions for non-reversible amplitude equations

In this paper, bifurcation and stability of two kinds of constant stationary solutions for non-reversible amplitude equations on a bounded domain with Neumann boundary conditions are investigated by using the perturbation theory and weak nonlinear analysis. The asymptotic behaviors and local properties of two explicit steady state solutions, and pitch-fork bifurcations are also obtained if the bounded domain is regarded as a parameter. In addition, the stability of a new increasing or decaying local steady state solution with oscillations are analyzed.     

竞争扩散时滞模型的稳定性和Hopf分歧

本文研究一类含扩散的竞争时滞模型的定常解的稳定性以及Ｈｏｐｆ分坡解的存在性,进而给出分歧周期解的稳定性和分歧方向。     

The local bifurcation and stability of nontrivial steady states of a logistic type of chemotaxis

Chemotaxis is a type of oriented movement of cells in response to the concentration gradient of chemical substances in their environment. We consider local existence and stability of nontrivial steady states of a logistic type of chemotaxis. We carry out the bifurcation theory to obtain the local existence of the steady state and apply the expansion method on the chemotaxis to investigate the bifurcation direction. Moreover, by applying the bifurcation direction, we obtain the bifurcating steady state is stable when the bifurcation curve turns to right under certain conditions.     

BIFURCATION OF TWO-DIMENSIONAL KURAMOTO-SIVASHINSKY EQUATION

This paper deals with the steady state bifurcation of the K-S equation in two spatial dimensions with periodic boundary value condition and of zero mean. With the increase of parameter a, the steady state bifurcation behaviour can be very complicated. For convenience, only the cases a=2 and a=5 witl be discussed. The asymptotic expressions of the steady state solutions bifurcated from the trivial solution near a=2 and a=5 are given. And the stability of thenontriviat sotutions bifurcated from a=2 is studied. Of course, the cases a=n^2 m^2,n,m∈N(a≠2,5) can be similarly discussed by the same method which is used to discussing the cases a=2 and a= 5.     

Spatial resonance and Turing–Hopf bifurcations in the Gierer–Meinhardt model

Gierer–Meinhardt system as a molecularly plausible model has been proposed to formalize the observation for pattern formation. In this paper, the Gierer–Meinhardt model without the saturating term is considered. By the linear stability analysis, we not only give out the conditions ensuring the stability and Turing instability of the positive equilibrium but also find the parameter values where possible Turing–Hopf and spatial resonance bifurcation can occur. Then we develop the general algorithm for the calculations of normal form associated with codimension-2 spatial resonance bifurcation to better understand the dynamics neighboring of the bifurcating point. The spatial resonance bifurcation reveals the interaction of two steady state solutions with different modes. Numerical simulations are employed to illustrate the theoretical results for both the Turing–Hopf bifurcation and spatial resonance bifurcation. Some expected solutions including stable spatially inhomogeneous periodic solutions and coexisting stable spatially steady state solutions evolve from Turing–Hopf bifurcation and spatial resonance bifurcation respectively.     

Stationary pattern of a diffusive prey–predator model with trophic intersections of three levels

The paper is concerned with a diffusive prey–predator model subject to the homogeneous Neumann boundary condition, which models the trophic intersections of three levels. We will prove that under certain assumptions, even though the unique positive constant steady state is globally asymptotically stable for the dynamics with diffusion, the non-constant positive steady state can exist due to the emergence of cross-diffusion. We demonstrate that the cross-diffusion can create stationary pattern. Moreover, we treat the cross-diffusion parameter as a bifurcation parameter and discuss the existence of non-constant positive solutions to the system with cross-diffusion.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.