具有三对特殊方向的一类平面齐五次系统的全局拓扑结构

研究了一类平面齐五次系统dxdxdt=a50x5+a41x4y+a32x3y2+a23x2y3+a14xy4+a05y5,dydt=b50x5+b41x4y+b32x3y2+b23x2y3+b14xy4+b05y5当其只有唯一的有限远奇点且具有三对特殊方向时的全局拓扑结构及系数条件.假设系统只有唯一的有限远奇点(0,0),不妨设b50=0,其特殊方向由示性方程G(θ)=0给出,引进poincare变换研究无穷远奇点,再根据定理中的系数条件,列出系统所有可能的无穷远奇点和特殊方向,并判断其类型,由此画出系统具有三对特殊方向时的全局相图.     

考点40 函数的极限与连续

1.(辽宁卷,2)极限limx→x0f(x)存在是函数f(x)在点x=x0处连续的().(A)充分而不必要的条件(B)必要而不充分的条件(C)充要条件(D)既不充分也不必要的条件2.(广东卷,3)limx→-3x+3x2-9=().(A)-16(B)0(C)61(D)313.(全国卷,5)limx→1(x2-31x+2-2x2-4x+3)=().(A)-21(B)21(C)-61(D)614.(湖北卷,8)若limx→1(1-a x-1-b x2)=1,则常数a,b的值为().(A)a=-2,b=4(B)a=2,b=-4(C)a=-2,b=-4(D)a=2,b=45.(江西卷,8)若limx→1f(x-1)x-1=1,则limx→1x-1f(2-2x)=().(A)-1(B)1(C)-21(D)21考点40函数的极限与连续1.f(x)在x=x0处连续,必有limx→x0f(x)存在,…     

世界各地数学奥林匹克试题摘编

本文中|A|表示集合A的元素个数.1设P(x)=x~3-3x 1.求一个多项式Q(x),使得Q(x)的根是P(x)的根的5次幂.解设a,b,c是P(x)的根.由根与系数的关系,有依题意知,Q(x)=(x-a5)(x-b5)(x-c5)=x3-(a5 b5 c5)x2 (a5b5 a5c5 b5c5)x-a5b5c5=x3-S5x2 T5x 1.这里S5=a5 b5 c5,T5=a5b5 b5c5 c5a5.对于正整数n,令Sn=an bn cn,则有T5=21(S52-S10),所以要求Q(x),只需求出S5与S10.∵S1=a b c=0,S2=(a b c)2-2(ab bc ca)=6.又a,b,c是方程x3=3x-1的根,所以a3=3a-1,b3=3b-1,c3=3c-1,由此易得Sn 3=3Sn 1-Sn(n≥1),∴S3=3(a b c)-3=-3,S4=3×S2-S1=3×6-0=18…     

与一道USAMO试题有关的不等式链

第 2 6届美国数学奥林匹克有一道试题 :对 a、b、c∈ R ,有( a3 b3 abc) -1 ( b3 c3 abc) -1 　 ( c3 a3 abc) -1 ≤ ( abc) -1 . ( 1)本文将通过以下定理证得与 ( 1)有关的不等式链 .定理　设 x、y、z∈ R ,且 xyz =1,则3x y z≤ ∑ 1x y 1≤ ∑ 1x 2≤ 1, ( 2 )其中 ∑ 表示对 x、y、z的轮换求和 .证明　设 x y z =a,xy yz xz =b,由xyz =1,易知 a≥ 3,b≥ 3,a2 ≥ 3b.且x2 y2 z2 =a2 - 2 b,x2 y xy2 y2 z yz2 z2 x zx2 =ab - 3.经运算可得　 ∑ 1x 2= ( y 2 ) ( z 2 ) ( x 2 )…     

初二代数第一学期期末自测题

A组一、选择题 (每小题 3分 ,共 3 0分 )1 .下列由左边到右边的变形 ,属于因式分解的是(　　 ) .A .( 2a +b) ( 2a -b) =4a2 -b2B .x2 -2x -3 =(x +1 ) (x-3 )C .4x2 y3 -2xy2 +1 =2xy2 ( 2xy -1 ) +1D .a2 +1 =a(a+1a)2 .若 4x2 +2 (k-4)x+1是一个完全平方式 ,则k的值为 (　　 ) .A .4和 2　　　　B .5和 2C .6和 2　　　　D . -4和 23 .下列各式中 ,变形不正确的是A .c-ab=-cab　　　B .-b-3a=b3aC .3b-4a=-3b4a　　D .--7b3a =7b-3a4.若 4a =5b(b≠ 0 ) ,则 a2 -b2b2 的值等于 (　　 ) .A .-15 　　　B .14　　　C .91 6　　　D .…     

初二代数第一学期期末自测题

A组一、填空题 (每小题 3分 ,共 3 0分 )1 .当x时 ,分式 13x -2 的值为正 ,当x时 ,分式 x2 -9x-3 的值为零 .2 .a2 x2 -2a2 xy a2 y2 分解因式的结果是.3 .x2 mx 1 6是一个完全平方式 ,则m的值是.4.当m =时 ,方程2mx 1m -x =2的根为 12 .5 .化简 a b-1a -b 2b -1b-a=.6.当a ,b满足条件时 ,方程 (a -b)x =a2 -b2 的解是x =a b.7.已知 x3 =y4=z5 ,则2x y-3zx y z =.8.已知 xx -1 xx 1 =Ax2 Bxx2 -1 ,则A =,B =.9.如果ab≠ 0 ,a2 ab -2b2 =0 ,那么2a -b2a b的值为 .1 0 .解方程 2xx -1 -1 =ax -1 时 ,能使方程产生增根的a的值是 .二、选择题 (每小题 3分 ,共 3 0分 )1 .把多项式 4x -x2 -4分解因式 ,结果正确的是(　 ) .A . -x( 4 -x) -4　　　　B .4x -(x 2 ) (x-2 )C . -(x-2 ) ...     

向量不等式a·b≤|a|·|b|的应用

对于任意两个向量 a,b,有不等式 a.b≤|a|. |b|当且仅当向量 a与 b同向时为等式 .此不等式结构简单 ,形式隽永 ,内涵丰富 .运用它处理某些与不等式相关的代数问题简捷明快 ,颇具特色 .1　求函数的最值例 1　求函数 f(x) =3x +2 +44- x2 的最大值 .解　令 a =(3,4 ) ,b =(x,4 - x2 ) ,则 f(x) =a . b +2 ,|a|=5 ,|b|=2 .故 f(x)≤ |a|. |b|+2 =12 ,当且仅当 a与 b同向 ,即 3x=44 - x2 >0时取等式 .解之　 x =65 .故当　 x =65 时 ,f(x) m ax =12 .例 2　求实数 x,y的值 ,使得 f(x,y) =(1- y) 2 +(x +y - 3) 2 +(2 x +y - 6 ) 2取得最小值 . (…     

对一个不等式的简证及探究

文[1]给出并证明了如下不等式:若a,b,c是正数,且a b c=1,则有:(1/(b c)-a)(1/(c a)-b)(1/(a b)-c)≥(7/6)3(1)当且仅当a=b=c=13时,不等式(1)取等号.文[1]的证明方法虽然精妙,但过程繁琐且不宜推广,现给出不等式(1)的一种简单证法.证明由a b c=1可得a=1-(b c),b=1-(a c),c=1-(a b),故不等式(1)等价于(1b c b c-1)(1c a c a-1)(1a b a b-1)≥(76)3(2)令f(x)=ln(1x x-1),00,故f(x)为(0,1)上的下凸函数,从而由Jensen不等式,有f(b c)…     

∑(a/(b c))~(1/2)<1 2(3~(1/2)/3)的初等证明

本刊文 [1 ]中用导数方法证明了 :在△ ABC中 ,有　　 ∑ ab c<1 2 33 . (1 )本文给出一个初等的证明 .证明　由对称性 ,不妨设 a≥ b≥ c=1 ,易知 a b≥ 2 ,a 相似文献   

争鸣

问题问题98 a,b∈R,不等试acosx bcos3x≤1对任意实数x恒成立,求b的取值范围.解因为不等式acosx bcos3x≤1对任意实数x恒成立,所以令x=0得a b≤1;x=π得a b≥-1-1≤a b≤1(1)又当x=3π时,有2a-b≤1-2a b≥-1;x=23π时,-2a b≤1,故-1≤-2a b≤1-2≤-a 2b≤2(2)由(1) (2)得-3≤3b≤3,所以-1≤b≤1即为所求.1)以上解法是否正确?请给出判断结果及理由.2)若解法正确,其中x分别选取等于0,π,3π,2π3的依据是什么?若解法不正确,其正确解法又如何?3)若改为求a的取值范围,又当如何解决?佟成军提供(江苏省海州高级中学222023)评析问题84该问题共收稿…     

On a Żołądek theorem

We prove the existence of cubic systems of the form $$ \begin{gathered} \dot x = y[1 - 2r(5 + 3r^2 )x + \gamma \lambda ^2 x^2 ] + a_0 x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 , \hfill \\ \dot y = - x(1 - 8rx)(1 - 3r\gamma x) - 2x[2(1 - 3r^2 ) - r\gamma (7 - 15r^2 )x]y \hfill \\ - [r(11 + r^2 ) + \gamma (1 - 22r^2 - 3r^4 )x]y^2 \hfill \\ - 2r\gamma \delta y^3 + a_0 y + a_7 x^2 + a_8 xy + a_9 y^2 + a_{10} x^3 + a_{11} x^2 y, \hfill \\ \end{gathered} $$ where α = 3r ² + 17, γ = r ² + 3, δ = 1 ? r ², and λ = 3r ² + 1, that have at least eleven limit cycles in a neighborhood of the point O(0, 0).     

Алгебраические дроби Чебышева—Маркова на нескольких отрезках

The Chebyshev-Markov problem about real algebraic functions of the form $A_n (x) = \frac{{x^n + c_1 x^{n - 1} + ... + c_n }}{{\left( {\prod {_{i = 1}^{2n} } \left( {1 - a_{i,n} x} \right)} \right)^{1/2} }}$ deviated least from zero on a system of intervals $\left[ {b_1 ;b_2 } \right] \cup ... \cup \left[ {b_{2p - 1} ;b_{2p} } \right], - \infty< b_1 \leqslant b_2< ...< b_{2p - 1} \leqslant b_{2p}< + \infty $ is considered. The expression under the square root above is a real polynomial of degree less than 2n, which is positive on [b ₁;b _2p ]. The solution of this problem is given in a parametric form in terms of automorphic Schottky-Burside functions. Similar functions were first used by N. I. Akhyeser in the approximation theory.     

The Stability of Systems of Neutral Type with Small Time Lag

In this paper, we have obtained the equivalence theorems of stability between the system of differential equations $[{\dot x_i}(t) = \sum\limits_{j = 1}^n {{a_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{b_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{c_{ij}}{{\dot x}_j}(t)} (i = 1,2, \cdots ,n)\]$ and the system of differential-difference equations of neutral type $[{\dot x_i}(t) = \sum\limits_{j = 1}^n {{a_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{b_{ij}}{x_j}(t - {\Delta _{ij}})} + \sum\limits_{j = 1}^n {{c_{ij}}{{\dot x}_j}(t - {\Delta _{ij}})} (i = 1,2, \cdots ,n)\]$ where a_ij, b_ij, c_ij are given constants, and \Delta_ij are non-negative real constants.     

一类上三角矩阵环$W_{n}(p, q)$的斜Armendariz性质

设α是环R的一个自同态,称环R是α-斜Armendariz环,如果在R[x;α]中,(∑_(i=0)~ma_ix~i)(∑_(j=0)~nb_jx~j)=0,那么a_ia~i(b_j)=0,其中0≤i≤m,0≤j≤n.设R是α-rigid环,则R上的上三角矩阵环的子环W_n(p,q)是α~—-斜Armendariz环.     

On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

     

On the period function of a class of reversible quadratic centers

This paper is devoted to the study of the period function for a class of reversible quadratic system

Permanence for a delayed predator-prey model of prey dispersal in two-patch environments

In this paper, we study the following delayed predator-prey model of prey dispersal in two-patch environments $$\begin{array}{rcl}\dot{x}_1(t)&=&\displaystyle x_1(t)[r_1(t)-a_{11}(t)x_1(t)-a_{13}(t)y(t)]+D(t)(x_2(t)-x_1(t)),\\[3mm]\dot{x}_2(t)&=&\displaystyle x_2(t)[r_2(t)-a_{22}(t)x_2(t)-a_{23}(t)y(t)]+D(t)(x_1(t)-x_2(t)),\\[3mm]\dot{y}(t)&=&\displaystyle y(t)[-r_3(t)+a_{31}(t)x_1(t-\tau_1)+a_{32}(t)x_2(t-\tau _1)-a_{33}(t)y(t-\tau_2)].\end{array}$$ By giving the detail analyzing of the right-hand side functional of the system, sufficient and necessary condition which guarantee the predator and the prey species to be permanent are obtained. Numeric simulations show the feasibility of main results. In additional to the above, sufficient condition on the permanence of the above system with predator density-independence are established.     

Stability criteria for linear second order delay differential system

Consider second order delay differential system where r is a positive constant and all coefficients are real constants.Our main results are as follows:(1) The maximal length of the delay for which the stability of system (*) is maintained is given in the case where the zero solution of system (*) is asymptotically stable in the absence of delay.(2) The necessary and sufficient criteria for judging that asymptetical stability of system (*) is preserved for an arbitrary large delay are obtained.     

Existence and uniqueness of limit cycles on a cubic kolmogorov differential system in the Predator-Prey relation

In this paper, we analyse qualitatively a cubic Kolmogorov system: which is one of the mathematical models in ecology describing the interaction between Predator-Prey of two populations; and give the conditions of nonexistence, existence and uniqueness of limit cycles for three different cases.Fulfilled during engagement in advanced studies at the Institute of Mathematics, Academia Sinica.     

Uniqueness for non-harmonic trigonometric series

When , and

if

then

More generalized results are given.