BIFURCATION DIAGRAMS OF QUADRATIC DIFFERENTIAL SYSTEMS HMING ONE FOCUS AND ONE SADDLE

In[1]wehavestudiedthebifurcationdiagramsofthequadraticdifferentialsystem-.x=-y 6x lxz mxy y',i=x(1 ax by)inthe(b,m)parameterplanewhena,6andlsatisfythecondition:ao,l 16'(6 l),andwesawsaddleloopcanbifurcateatmosttwolimitcycles(LC,forabbre-viation).Ontheotherhand,inl2]L.A.CherkasandV.A.Gaikohaveprovedthatanyquadraticsystemhavingonlytwofinitecriticalpoints:onefocusandonesaddlepoint,canalwaysbetransformedbynon-degenratelineartransforma-tionsintotheform:whereP1=y(l 7y)-axy-by',Q1=x(…     

SOME CHARACTERIZATIONS OF α-BLOCH SPACES ON THE UNIT BALL OF C~n

1IntroductionLetH(B)denotetheclassofall11olomorphicfunctionsintheunitballB.WesaythatfEB",ifWhenn=1,replacethembyH(D)alldB"(D).HaxdyandLittlewoodprovedthat([3,2]):B"(D)=Lip(1--or).WeknowthatLipscanbeusedtodescribethedualspaceofHardyspaceHP(D)for0相似文献   

一类含平方数因子的伪素数

笔者曾构造出一类表示伪素数的公式 [1] ,张善立在文 [3]中指出这一类中存在含平方数因子 1 0 932 的伪素数 ,有没有含其它平方数因子的伪素数呢 ?本文将从文 [1 ]给出的公式中找出含平方数因子 1 0 932 和 351 1 2 的伪素数 (本文中字母为正整数 ,p为奇素数 ) .引理 1　设 A≥ 2 ,( p,A) =1 ,满足 ( p,2 A - 1 ) =1及 A 2 A( 2 A( p-1) - 1 )2 A - 1则　 n =2 Ap - 12 A - 1 是伪素数[1] .引理 2　 2 Q1- 1 | 2 Q1Q2 - 1 [1] .引理 3　设使同余式 :2 r ≡ 1 ( mod m)成立的最小正整数为 r,则 2 a≡ 1 ( mod m)成立的充要条件是 r| a[3…     

The Structure of Hopf Bigalois Extension over Noncommutative Rings

For a commutative ring k,the group G in a G-Galois extension A/ k is uniquely deter-mined by the ring extension A/ k.On the contrary,the Hopf algebra H in an H-Galoisextension A/ k is not unique(see[1 ] ) .In[2 ] ,for a commutative Hopf algebra and acommutative faithfully flat H-Galois extension A/ B(B=Aco H) ,the authors constructedanother commutative Hopf algebra Lsuch that A is also a faithfully flat L-Galois ex-tension.P.Schauenburg generalized H and A to noncommutative cases in[…     

THE CONVERGENCE ON A FAMILY OF ITERATIONS WITH CUBIC ORDER

1. IntroductionLet E and F be real or complex Banach spaces and f: D G E --+ F be a nonlinar twicediffereotiable operator. FOr solving the eqwtionconsider a one-parametered edly of iterationswhere Hj(x) = f'(x)--'f,,(x)j'(x)--'f(x) and 0 S A 5 1. This hal is cubically convergent(see [3, 7] ) and includes, as particular cases, Chebyshev method (^ = 0, see [5, 9l), Halleymethod (A = l, see [1, 4, 6, 10]) and suPer-Halley method (A = 1, see [7]).In [3], I.K. AIgyros et al analyze the con…     

关于《具有缺项和负系数的多叶函数》一文的注记

In this note we shall use the same notations and terms as ones in[1,3].Very recently,Sarangi and Patil[1]if considered the subclass P_k(p,A,B)of analyticp-valent functions with negative and missing coefficients and proved seven theorems con-cerning the class P_k(p,A,B).Theorems1,2,3,5,6 and7 of[l]generalize the correspondingresults obtained by Goel and Sohi[2]from k=1 to k≥2.However,it is easy to see thatsuch a generalization is trivial and simple.     

具有粗糙核的多线性算子在Herz型空间的有界性

§ 1　Introduction and main resultsLet b∈BMO(Rn) and T be a standard Calderon-Zygmund singular integral operator.Define the commutator[b,T] as follows.[b,T] f(x) =b(x) Tf(x) -T(bf) (x) .In [3 ] ,the boundedness ofthe commutator[b,T] wasestablished on Lp(Rn) .There are thesimilar results in [1 ,2 ] when the commutator was substituted with the multilinearoperators generated by the singular integral operator T and a Taylor series A(see thedefinition below) .Recently,many mathematicians h…     

The Gradient Theory of the Phase Transitions with Dirichlet Boundary Condition

§ 1.Introduction　 In this paper,we study the asymptotic behavior of minimizers{ uε} ε>0 (ε→ 0 ) of thevariational probleminf∫Ω[εp- 1 | Du| p + 1εW(x,u) ] dx:u∈ W1 ,p(Ω ) ,u =g,x∈ Ω (Pε)where p>1 ,N≥ 2 ,andΩ RN is a bounded open domain with Ω∈C2 .　 This kind of problems comes from the theory of phase transitions and are widelystudied in recent years.In [1 ]— [1 4] ,the authors investigated these problems underthe integral constraint∫Ωudx =constant,In [1 5]— [1 6] ,…     

C_(p,q)~(s+α)-forms Solution of _b-equ ation and Its L_(p,q)~s Estimates

§ 1.　Introduction　　TheinhomogeneoustangentialCauchy Riemannoperators,or b complex ,onthebound aryofadomainDinCn(n ≥ 2 ) ,hasplayedanimportantroleinthestudyofboundaryvaluesofholomorphicfunctionsandholomorphicextensioninCn(examplesseeKOHNandROSSI[3] ,AndrettiandHILL [4] ) .In 1 977,HENKIN [1 ]wasfirstlyobservedthat,if b equa tions bu =fon Dissolvable ,the (p ,1 ) form ,f ,0≤ p≤ 2 ,mustsatisfythecompatibilitycondition∫ Df∧Ψ =0forall b closed (2 p ,0 ) formsΨonDinwhich…     

ON INTERPOLATION BY S_3~1(△_mn~(1))(I)

§0 Introduction Let △_(mn)~(1) be a subdivision of D: [a,b]x[c,d] (Fig.1). l_1,_2be lengths of [a,b] and [c,d] respectively, and h_1=l_1/m, h_2=l_2/n,l=max(l_1,l_2). Suppose that S∈S1/3(△_(mn)~(1)). In §1-§5 we consider thefollowing interpolation problem:     

TO SOLVE MATRIX EQUATION sum (A~iXBi=c) BY THE SMITH NORMAL FORM

§ 1　IntroductionLet F be a field,F[λ] be the polynomial ring over F,Fm× n( or Fm× n[λ] ) be the setofall m×n matrices over F( or F[λ] ) .Let M(i) be the ith column of M∈Fm× m[λ] ,i=1 ,...,n.A g-inverse of M∈Fm× n will be denoted by M- and understood as a matrix for whichMM- M=M.In this paper,we discuss the linear matrix equation ki=0Ai XBi =C, ( 1 )where A∈Fm× m,Bi∈Fn× q,i=0 ,1 ,...,k,and C∈Fm× q.Equation( 1 ) is called universally solvable if ithas a solution f…     

Apollonius定理的一个空间形式

设△ A1A2 A3 的三边长分别为 a1,a2 ,a3 ,边 A1A2 上的中线长为 m3 ,则m23 =12 ( a21 a22 ) - 14 a23 . ( 1 )这就是著名的 Apollonius定理 ,由余弦定理 ,易将 ( 1 )变换成m23 =14 ( a21 a22 2 a1a2 cos A3 ) ( 2 )本文将给出关于 ( 2 )的一个十分有趣的空间形式 .定理　在四面体 A1A2 A3 A4 中 ,设 A5是棱 A1A2 的中点 ,S△ A2 A3A4=S1,S△ A1A3A4=S2 ,S△ A3A4A5=M3 4,二面角 A1A3 A4 A2 =3 4,则M23 4=14 ( S21 S22 2 S1S2 cos3 4) ( 3)先介绍两个基本公式 ,其证明参见文 [1] [2 ] .引理 1 [1]　设四面体的底…     

关于A+,A+MN的表达式及其应用

For A ∈ Cm×nr, let M and N be Hermitian positive definite matrices oforder m and n respectively. We derived the representation of the Moore-PenroseA+MN in terms of maximal nonsin-inverse A+ and weighted Moore-Penrose inverse +gular submatrices of A. In our notation,A+ = | detA[plq]|2A+pq1/vol2(A) (p,q)∈N(A)AM+N = 1/vol2(A)(p,q)∈N(A) |detA[p|q]|2N-1/2A+pqM1/2where A=M1/2 AN -1/2. From this, we propose a new method to calculate A+A+MN. The results generalize that of Moore-Penrose inverse in [2][3].     

Controllability of Linear Systems in Non Reflexive Banch Spaces

§ 1.Introduction　 Let Xand Ube Banch spaces,S(t) a C0 semi-group in X,A the infinitesimal gen-erator of S(t) ,and B∈L(U,X) .Consider the systemΣP:x(t) =Ax(t) + Bu(t) ,　 t≥ 0 ,(1 .1 )where u∈ Lploc([0 ,∞ ) ,U) ,1≤p<∞ .　 The problem of controllability of linear systems in Banch spaces has been studied bymany authors.In [1 ] ,the weak and strong reachability and controllability of infinite di-mensional linear systems were investigated.In[2 ,3 ,4 ] ,the controllability of infi…     

Optimum Modified Extrapolated Jacobi Method for Consistently Ordered Matrices

1.IlltroductionandPreliminariesWeconsiderthelineaxsystemAx=b,(1.1)wheteAERn,",bERnanddet(A)/O.WeaJ8oassumethatAhasthef0rmwhereA11,A22axesquarenonsingular(usuallydiagonal)matrices-Asisknow[61,AisaconSisentlyordered2-cyclicmatris.Forsolving(1.1)weintendtousethef0lfowngsimpleiterativemethod:In(1.4)and(1.6),w1,w2arenonzeroparameters(extraP0lati0nparameters)andI1,I2areidelltitymatricesofthesamesizesasA11anA22respectively.Theconstructionofmeth0d(1.3)isbasedonthesplittingA=M-N,whereM=Dfl-1…     

On the Diophantine Equation sum from k=1 to m((k~n)=(m+1)~n)

P. Erds has conjectured [1] that the Diophantine equation 1~n+2~n+…+m~n=(m+1)~n (1) has no positive integer solutions except that n=1, m=2. It is true when m≤10~(10) [3]. A generalized form of (1) has been investigated in [1] [2], and various     

具有无限时滞的二阶微分方程解的存在性

§ 1.　IntroductionRecently ,thedifferentialequationswithdeviatingargumentswereusuallydiscussed(see[1 ],[4],[5 ]) .In [1 ],AGARWALRPandO’REGANDconsideredequationy″(t) =f(t,y(t) ,y(σ(t) ) ) ,　a.e .t∈ [0 ,1 ]y(t) =ψ(t) ,　　　　　　　　t∈ [-r ,0 ]y( 1 ) =a ,( )andtheydiscussedtheexistenceofatleastonesolutionforequation ( ) .Inthispaper ,weconsideramoregeneralequation-x″(t) =f(t ,xt) ,　t∈ [0 ,1 ]x(t) =ψ(t) ,　　　　t∈ ( -∞ ,0 ]x( 0 ) =x( 1 ) =0 ,( 1 .1 )andsomeexistencetheor…     

高阶Hermite-Fejer型插值的渐近估计

1. Introduction Let f∈C[-1,1] and X_k=X_(kn)=COSθ_k=COS(2k-1)π/(2n)(k=1,…,n) be the zeros of the Chebyshev polynomial T_n(x)=cosnθ(x=cosθ). Let ω(t) be a given modulus of continuity and H_ω={f;ω(f,t)≤ω(t),for all.t≥0}. In this paper, c will always denote different constant independent of x, n and f and the sign"A～B" means that there exist two positive constants c_1相似文献   

一类二阶微分议程边值问题三解的存在性

§ 1　 IntroductionWe are interested in the existence ofthree-solutions ofthe following second-order dif-ferential equations with nonlinear boundary value conditionsx″=f( t,x,x′) ,　　 t∈ [a,b] ,( 1 .1 )g1 ( x( a) ,x′( a) ) =0 ,　　 g2 ( x( b) ,x′( b) ) =0 ,( 1 .2 )where f:[a,b]×R1 ×R1 →R1 ,gi:R1 ×R1 →R1 ( i=1 ,2 ) are continuous functions.The study ofthe existence of three-solutions ofboundary value prolems forsecond or-der differential equations was initiated by Amann[1 ] .In[1 …     

圆内接闭折线垂心的一个性质

众所周知 ,三角形的垂心有如下性质[1] :定理 1　设△ ABC的外接圆半径为 R,垂心为 H ,则 ( AB2 BC2 CA2 ) ( H A2 H B2 H C2 ) =1 2 R2 .将这个定理推广到一般圆内接闭折线中 ,可得定理 2　设闭折线 A1A2 A3 … An A1内接于⊙ ( O,R) ,其垂心为 H ,则　 ∑ni=1Ai A2i 1 ∑ni=1H A2i =n( n 1 ) R2 ,( * )其中 An 1为 A1.证明　以圆心 O为原点建立直角坐标系x Oy(图略 ) ,设顶点 Ai 的坐标为 ( xi,yi) ( i =1 ,2 ,… ,n) ,垂心 H的坐标为 ( x H,y H) ,则有[2 ]x H =∑ni=1xi,　 y H =∑ni=1yi. 1由两点间的距…