具有抛物线解的一类三次系统的中心判定问题

对于一类具有抛物线解、直线解和中心-焦点型奇点的三次系统,证明它以原点为中心的充要条件是其前五阶焦点量全为零.此中心条件是通过不变代数曲线构造积分因子或对称原理得以证明.     

不变代数曲线与一类三次系统的中心判定问题

对于一类具有一条抛物线解、两条直线解和中心-焦点型奇点的三次系统,证明它以原点为中心的充要条件是它的第一阶焦点量为零.系统在原点的中心条件是通过不变代数曲线构造积分因子或利用Poincare对称原理得以证明.     

一类四次椭圆Hamilton向量场在三次多项式下的扰动

本文研究一类四次椭圆Hamilton向量场在所有三次多项式下的扰动,证明了如下结论: (1)除全局中心外,围绕一个中心定义的Abel积分的孤立零点的个数不超过12; (2)存在一个三次系统,它在扰动前属于一个鞍点环的情形,而在扰动后至少存在3个极限环.     

一类三次无理函数积分的求解方法

设Pn(x)为n次多项式,a0≠0,m≥2且m∈N,得到形如∫Pn(x)ma0x3+a1x2+a2x+a3dx的三次无理函数积分可解的充要条件,且其解的形式为∫Pn(x)ma0x3+a1x2+a2x+a3dx=Qn-2(x).m(a0x3+a1x2+a2x+a3)m-1+C,其中Qn-2(x)为各项系数待定的(n-2)次多项式.运用待定系数法可求出Qn-2(x)的各项系数.     

二次系统极限环的相对位置与个数

<正> 中的P_2(x,y)与Q_2(x,y)为x,y的二次多项式.文[1].曾指出,系统(1)最多有三个指标为+1的奇点,且极限环只可能在两个指标为+1的奇点附近同时出现.如果方程(1)的极限环只可能分布在一个奇点外围,我们就说此系统的极限环是集中分布的.本文主要研究具非粗焦点的方程(1)的极限环的集中分布问题,和极限环的最多个数问题.文[2]-[5]曾证明,当方程(1)有非粗焦点与直线解或有两个非粗焦点或有非粗焦点与具特征根模相等的鞍点时。方程(1)无极限环.本文给出方程(1)具非粗焦点时,极限环集     

Riccati方法与微分方程的渐近积分

考虑二阶常微分方程x″ f(t)x=0,t≥a,(1)假设应用Riccati方法得到方程(1)的主解(principal solution)的一个渐近积分并研究其副解(nonprincipal solutions)的三种不同的渐近性质.主要结果如下:定理1 若(Ⅰ)成立,则方程(1)有解x_1满足及另一解x_2满足x_2(t)=t[1 o(1)]. 反之,若方程(1)有解x(t)→1,t→∞,则(Ⅰ)成立. 定理2 设(Ⅰ)成立.(i)若(Ⅱ)成立,则方程(1)有解x_2使x_2’(t)=1 [tF(t) G(t)][1 o(1)] o(1). (ii) 反之,若方程(1)有解x使x’→1,t→∞,则(Ⅱ)成立. 定理3 若(Ⅲ)和(Ⅳ)成立,则方程(1)有解x_1满足(2)及解x_2满足     

斜率与离心率的一个有趣性质

本文介绍椭圆双曲线离心率与其有关斜率的一个有趣关系式 ,并说明它的应用 ,供读者参考 .定理　l1是过椭圆 x2a2 + y2b2 =1 (a >b >0 )焦点F且与x轴垂直的直线 ,A ,l2 是与F相对应的顶点和准线 ,经过椭圆中心O作斜率为k的直线l与l1,l2 分别交于P ,Q两点 ,则AP⊥AQ的充要条件是k2 + 2 =e +1e(e是离心率 ) .证明　由对称性 ,不妨设F是左焦点 ,则l1,l2 的方程分别是x =-c和x =- a2c.又知l的方程为y =kx ,分别与l1,l2 的方程联立解得点P( -c ,-kc)和Q( - a2c,ka2c) .又知点A( -a ,0 ) ,所以AP⊥AQ kAPkAQ=- 1 - kca -c·- ka2ca - a2…     

关于二次根式的化简

二次根式的化简是二次根式运算的基础 ,是本章教材的中心内容 .由于题型变化较多 ,化简中所涉及的知识面广 ,方法灵活多样 ,因此它又是本章学习的难点 .在学习过程中 ,善于积累和总结二次根式化简的方法显然十分必要 .下面归纳列举一些二次根式化简的方法和技巧供读者参考 .一、利用乘法公式与整式和分式的化简类似 ,二次根式的化简中如果注意观察题型 ,巧用乘法公式 ,可以使问题得以简化 .例 1　化简下列各式 :( 1) ｘ -ｙｘ +ｙ;( 2 ) ( 2 - 3+ 5) ( 2 + 3- 5) ;　 ( 3) 134 + 36 + 39.解 :( 1)原式 =(ｘ) 2 - (ｙ) 2ｘ + ｙ=(ｘ + ｙ) (…     

一类具有中心的三次系统

本文证明了一类缺二次项的三次系统当原点为中心时,中心与细焦点不能共存。同时给出了原点为中心的情况下系统(1)具有多中心的充要条件,并证明此时系统(1)不存在极限环、焦点和结点。     

一个有理分式Diophantine方程

本文研究了方程(x2-1)/(y2-1)=(z2-1)2.利用四次和二次方程的性质,获得了它的全部正整数解(x,y,z).     

Limit Boundary Value Problems of a class of Nonlinear Differential Equations of the Second order

In this paper, the existence and uniqueness of solution of the limit boundary value problem $\[\ddot x = f(t,x)g(\dot x)\]$(F) $\[a\dot x(0) + bx(0) = c\]$(A) $\[x( + \infty ) = 0\]$(B) is considered, where $\[f(t,x),g(\dot x)\]$ are continuous functions on $\[\{ t \ge 0, - \infty < x,\dot x < + \infty \} \]$ such that the uniqueness of solution together with thier continuous dependence on initial value are ensured, and assume: 1)$\[f(t,0) \equiv 0,f(t,x)/x > 0(x \ne 0);\]$; 2) f(t,x)/x is nondecreasing in x>0 for fixed t and non-increasing in x<0 for fixed t, 3)$\[g(\dot x) > 0\]$, In theorem 1, farther assume: 4) $\[\int\limits_0^{ \pm \infty } {dy/g(y) = \pm \infty } \]$ Condition (A) may be discussed in the following three cases $x(0)=p(p \neq 0)$(A_1) $\[x(0) = q(q \ne 0)\]$(A_2) $\[x(0) = kx(0) + r{\rm{ }}(k > 0,r \ne 0)\]$(A_3) The notation $\[f(t,x) \in {I_\infty }\]$ will refer to the function f(t,x) satisfying $\[\int_0^{ + \infty } {\alpha tf(t,\alpha )dt = + \infty } \]$ for each $\alpha \neq 0$, Theorem. 1. For each $p \neq 0$, the boundary value problem (F), (A_1), (B) has a solution if and only if $f(t,x) \in I_{\infty}$ Theorem 2. For each$q \neq 0$, the boundary value problem (F), (A_2), (B) has a solution if and only if $f(t, x) \in I_{\infty}$. Theorem 3. For each k>0 and $r \neq 0$, the boundary value problem (F), (A_3), (B) has a solution if and only if f(t, x) \in I_{\infty}, Theorem 4. The boundary value problem (F), (A_j), (B) has at most one solution for j=l, 2, 3. .     

Nonnegative Interpolation

The problem discussed in this paper is to determine a nonnegative interpolating polynomial which takes the prescribed nonegative values $y_0,y_1,\cdots,y_n$ at given distinct points $x_0,x_1,\cdots,x_n$: $p(x_i)=y_i),i=0,1,\cdots,n$. This paper shows：(1) $2n$ is the least number of $m$ such that there exists a polynomial $p\in P_m^{+}$, the set of all nonnegative polynomials of degree $\leq m$, satisfying the above equations for any choice of $y_i\geq 0$. (2) the above equations have a unique solution in $P_{2n}^{+}$ if and only if at most one of the $y_i's$ is nonzero.     

关于一类中心循环的有限P-群的自同构群

确定了一类中心循环的有限p-群G的自同构群.设G=X_3(p~m)~(*n)*Z_(p~(m+r)),其中m≥1,n≥1和r≥0,并且X_3(p~m)=x,y|x~(p~m)=y~(p~m)=1,[x,y]~(p~m)=1,[x,[x,y]]=[y,[x,y]]=1.Aut_nG表示Aut G中平凡地作用在N上的元素形成的正规子群,其中G'≤N≤ζG,|N|=p~(m+s),0≤s≤r,则(i)如果p是一个奇素数,那么AutG/Aut_nG≌Z_(p~((m+s-1)(p-1))),Aut_nG/InnG≌Sp(2n,Z_(p~m))×Z_(p~(r-s)).(ii)如果p=2,那么AutG/Aut_nG≌H,其中H=1(当m+s=1时)或者Z_(2~(m+s-2))×Z_2(当m+s≥2时).进一步地,Aut_nG/InnG≌K×L,其中K=Sp(2n,Z_(2~m))(当r0时)或者O(2n,Z_(2~m))(当r=0时),L=Z_(2~(r-1))×Z_2(当m=1,s=0,r≥1时)或者Z_(2~(r-s)).     

A NEW REGULARITY CLASS FOR THE NAVIER-STOKES EQUATIONS IN IR~n

ＡＮＥＷＲＥＧＵＬＡＲＩＴＹＣＬＡＳＳＦＯＲＴＨＥＮＡＶＩＥＲ－ＳＴＯＫＥＳＥＱＵＡＴＩＯＮＳＩＮＩＲ~ｎ￥Ｈ．ＢＥＩＲａＯＤＡＶＥＩＧＡ（ＤｅｐｏｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＰｉｓａＵｎｉｖｅｒｓｉｔｙ，Ｐｉｓａ，Ｉｔａｌｙ）Ａｂｓｔｒａ?..     

Asymptotic Behavior in a Quasilinear Fully Parabolic Chemotaxis System with Indirect Signal Production and Logistic Source

In this paper, we study the asymptotic behavior of solutions to a quasilinear fully parabolic chemotaxis system with indirect signal production and logistic sourceunder homogeneous Neumann boundary conditions in a smooth bounded domain $Ω⊂\mathbb{R}^n$ $(n ≥1)$, where $b ≥0$, $γ ≥1$, $a_i ≥1$, $µ$, $b_i >0$ $(i =1,2)$, $D$, $S∈ C^2([0,∞))$ fulfilling $D(s) ≥ a_0(s+1)^{−α}$, $0 ≤ S(s) ≤ b_0(s+1)^β$ for all $s ≥ 0,$ where $a_0,b_0 > 0$ and $α,β ∈ \mathbb{R}$ are constants. The purpose of this paper is to prove that if $b ≥ 0$ and $µ > 0$ sufficiently large, the globally bounded solution $(u,v,w)$ with nonnegative initial data $(u_0,v_0,w_0)$ satisfies $$\Big\| u(·,t)− \Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\|_{L^∞(Ω)}+\Big\| v(·,t)−\frac{b_1b_2}{a_1a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)} +\Big\| w(·,t)−\frac{b_2}{a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)}→0$$ as $t→∞$.     

Constructions of Metric $(n+1)$-Lie Algebras

Metric n-Lie algebras have wide applications in mathematics and mathematical physics. In this paper, the authors introduce two methods to construct metric (n+1)-Lie algebras from metric n-Lie algebras for n≥2. For a given m-dimensional metric n-Lie algebra(g, [, ···, ], B_g), via one and two dimensional extensions ￡=g+IFc and g0= g+IFx~(-1)+IFx~0 of the vector space g and a certain linear function f on g, we construct(m+1)-and (m+2)-dimensional (n+1)-Lie algebras(￡, [, ···, ]cf) and(g0, [, ···, ]1), respectively.Furthermore, if the center Z(g) is non-isotropic, then we obtain metric(n + 1)-Lie algebras(L, [, ···, ]cf, B) and(g0, [, ···, ]1, B) which satisfy B|g×g = Bg. Following this approach the extensions of all(n + 2)-dimensional metric n-Lie algebras are discussed.     

On the Serrin's Regularity Criterion for the β-Generalized Dissipative Surface Quasi-geostrophic Equation

The authors establish a Serrin's regularity criterion for the β-generalized dissipative surface quasi-geostrophic equation.More precisely,it is shown that if the smooth solution θ satisfies ▽θ∈L~q(0,T;L~P(R~2)) with α/q+2/p≤α+β-1,then the solution θcan be smoothly extended after time T.In particular,when α+β≥2,it is shown that if α_yθ∈L~q(0,T;L~P(R~2)) with α/q+2/p≤α+β-1,then the solution θ can also be smoothly extended after time T.This result extends the regularity result of Yamazaki in 2012.     

DISSIPATIVITY AND EXPONENTIAL STABILITY OF θ-METHODS FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH A BOUNDED LAG

This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small ε > 0. We will study the numerical solution defined by the linear θ-method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small ε > 0 if and only if θ = 1.     

Centers for the Kukles homogeneous systems with even degree

For the polynomial differential system $\dot{x}=-y$, $\dot{y}=x +Q_n(x,y)$, where $Q_n(x,y)$ is a homogeneous polynomial of degree $n$ there are the following two conjectures done in 1999. (1) Is it true that the previous system for $n \ge 2$ has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We give a step forward in the direction of proving both conjectures for all $n$ even. More precisely, we prove both conjectures in the case $n = 4$ and for $n\ge 6$ even under the assumption that if the system has a center or an isochronous center at the origin, then it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space. The case of $n$ odd was studied in [8].     

带有负指数 Kirchhoff 方程稳定解的Liouville型定理

In this paper, we consider the Liouville-type theorem for stable solutions of the following Kirchhoff equation ■,where M(t) = a + bt~θ, a 0, b, θ≥ 0, θ = 0 if and only if b = 0. N ≥ 2, q 0 and the nonnegative function g(x) ∈ L_(loc)~1(R~N). Under suitable conditions on g(x), θ and q, we investigate the nonexistence of positive stable solution for this problem.