A NOTE ON A TRANSFERENCE THEOREM OF THE SYSTEMS OF LINEAR CONGRUENCES

<正> Let p be a prime number. Let a_(ij)(1≤i≤t,1≤j≤s) be a set of st integers.Weuse the notations x=max(1, |x|), p_1=[(p-1)/2], p_2=[p/2], and (a)_p to denote the integer satisfying (a)_p=a(mod p),-p_1≤(a)_p≤p_2. Consider the dual of linear congruences     

BOUNDS OF EIGENVALUES OF A GRAPH

Let G be a simple graph with n vertices.We denote by λ_i(G) the i-th largest eigenvalue of G.In this paper,several results are presented concerning bounds on the eigenvalues of G.In particular,it is shown that -1≤λ_2(G)≤(n-2)/2,and the left hand equality holds if and only if G is a complete graph with at least two vertices;the right hand equality holds if and only if n is even and G?2K_(n/2).     

AN ELEMENTARY PROOF OF A THEOREM OF LEE

§ 1. Introduction Now we recall some basic notions and results in whitenoise analysis. Let S (R) (reop. S' (R)) denote the Sehwarz space of test functions (rosp. tempered distributions). Let A denoto tho self-adjoint operator-d~2/dt~2+1+t~2 in L~2(R).     

A GENERALIZATION OF SIMPSON''''S RULE

Let TA(f)=integral form n= to 1/2(P_~n(x) + P_b~n(x))dx and let TM(f)=integral form n= to P_((+b)/2)~(n+1)(x)dx, where P_c~n denotes theTaylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of theTrapezoidal rule and the midpoint rule, respectively. and are each exact for all polynomial of degree ≤n+1.We let L(f) = αTM(f) + (1-α)TA(f), where α =(2~(n+1)(n+1))/(2~(n+1)(n+1)+1), to obtain a numerical integrationrule L which is exact for all polynomials of degree≤n+3 (see Theorem l). The case n = 0 is just the classicolSimpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacingP_(+b)/2)~(n+1)(x) by the Hermite cabic interpolant at a and b. we obtain some known formulae by a different ap-proach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by takingpiecewise polynomials of odd degree, each piece being the Taylor polynomial off at a and b. respectively. Ofcourse all of our formulae can be compounded over subintervals of [a, b].     

A REMARK ON EXTINCTION OF A CLASS OF SUPERPROCESSES

1.IntroductionSuperprocesses,i.e.,measure-vauedMarkovprocesses,havebeenanattractivetopicintherecenty6ars,whichcanbedefinedasfollows.Let((,Px)..MbeaMarkovprocesswithstrongcontinuousMarkovsemigroupSt(denotebyAitsinfinitesimaloperator,andbyPt(x,y)theassociatedtransitiollfunction).DenotebyMPthesetofallRadonmeasurespsatisfyingandbypC(t)(resp.pCc(t))thesetofpositivelycontinuousfUnctions(resp.withcompactsupport)int-Thesthcalledsuperprocess(X,P#).6MPwithparameters(andopandtakingvalueinMrisdete…     

ON A THEOREM OF CHUI

Let f∈C_(2π) and let α_0/2+sum from k=1 to ∞ (a_k coskx+b_k sinkx) (1)be its Fourier series. Denote by S_n(f, x) the n-th partial sumof series (1) and by ω(f,δ) the modulus of continuity of functionf(x). For each q0, the Euler (E, q)-mean of series (1) is     

OSCILLATORY AND ASYMPTOTIC BEHMIOUR OF A CLASS OF NONLINEAR DIFFERENTIALEQUATIONS OF THIRD ORDER

1IntroductionInthispaperwedealwithoscillatoryandnonoscillatorybehaviourofsolutionsofnonlinearthirdorderdifferentialequationsoftheformwherea,b,cEC([er,co),R),rrER,fEC(R,R)suchthatac2P>0fory/0and7>0isaquotientofoddintegers.From"Theoryofequations"tsee[1],itfollowsthattheequationwithconstantcoefficientsMoreover,allnonoscillatorysolutionsof(1.3)tendtozeroifandonlyifequatioll(l.3)hasanoscillatorysolution.Theseobservationsarefoundtobetruetoalargeextentformoregeneralequationsoftheformwherealb,cE…     

ON THE SYMPLECTIC INVARIANTS OF A SUBSPACE OF A VECTOR SPACE~*

Let F be any commutative field. Let v be an integer≥1 and be a fixed 2v × 2v nonsingular alternate matrix over F. Define Sp_(zv)(F)={T: 2v×2v matrix over F|TKT~T=K}. It is well-known that Sp_(2v)(F) is a group with respect to the matrix multiplication and is called the symplectic group of degree 2v over F     

A BIVARIATE EXTENSION OF BLEIMANN-BUTZER-HAHN OPERATOR

Let C(R_+~2) be a class of continuous functions f on R+2. A bivariate extension L(?)(f,x,y) of Bleimann-Butzer-Hahn operator is defined and its standard convergence properties are given. Moreover, a local analogue of Voronovskaja theorem is also given for a subclass of C(R+2 ).     

APPROXIMATION BY A CLASS OF MODIFIED SZASZ-DURRMEYER OPERATORS

ThisprojectissupportedbyZhejiangProvincialFoundationofChina.1.IntroductionForjEC[0,1]ther-thBernsteinpolynomialisdefinedbyItwasshownbyH.BerensandG.G.Lorentz([2]in1972)thatif0相似文献   

LIMIT CYCLES OF A QUADRATIC HAMILTONIAN SYSTEM BY CUBIC PERTURBATIONS

In this paper,we are concerned with a cubic near-Hamiltonian system,whose unperturbed system is quadratic and has a symmetric homoclinic loop.By using the method developed in [12],we find that the system can have 4 limit cycles with 3 of them being near the homoclinic loop.Further,we give a condition under which there exist 4 limit cycles.     

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     

一个群体防卫捕-食系统的定性分析

该文对一个群体防卫捕一食系统进行了较全面的定性分析．讨论了分界线的相对位置，得到了极限环的存在性、唯一性以及分界线环的存在性，首次证明了群体防卫捕一食系统可以至少存在两个或三个极限环．     

生物化学反应中一类非线性系统极限环的存在惟一性

研究了生物化学反应中一类非线性系统，得到了该系统的环绕正奇点极限环的充分必要条件，并且证明了如果存在极限环，则必惟一。     

LIMIT CYCLES OF QUARTIC AND QUINTIC POLYNOMIAL DIFFERENTIAL SYSTEMS VIA AVERAGING THEORY

We study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging theory. More precisely,we prove that the perturbations of the period annulus of the center located at the origin of a cubic polynomial differential system,by arbitrary quartic and quintic polynomial differential systems,there respectively exist at least 8 and 9 limit cycles bifurcating from the periodic orbits of the period annu...     

THE LIMIT CYCLES AND HOPF BIFURCATION OF A CLASS OF SIMPLIFIED HOLLING TYPE-IV PREDATOR-PREY SYSTEM WITH LINEAR STATE FEEDBACK

In this paper, a class of simplified Type-IV predator-prey system with linear state feedback is investigated. We prove the boundedness of the positive solutions to this system, and analyze the quality of the equilibria and the existence of limit cycles of the system surrounding the positive equilibra. By Hopf bifurcation theory, the result of having two limit cycles to the system is obtained.     

THE PROBLEM OF LIMIT CYCLES FOR A LIENARD TYPE CUBIC SYSTEM

The purpose of this paper is to study a general Lienard type cubic system with one antisaddle and two saddles. We give some results of the existence and uniqueness of limit cycles as well as the evolution of limit cycles around the antisaddle for system (2) in the following when parameter a1 changes.     

A study on Zoladek's example

In this paper, we consider an example of third-order polynomial planar system, proposed by Zoladek who claimed that this example had eleven small-amplitude limit&nbsp;cycles around a center. We use focus value computation to show that for this example there may exist maximal nine small-amplitude limit cycles around the center due to Hopf bifurcation.     

Multiple limit cycles in a continuous culture vessel with variable yield

Determining the number of limit cycles for a continuous culture vessel system is always useful in analyzing the system. We prove the conditions that guarantee there exist three limit cycles for the chemostat with variable yield that was first proposed by Huang [Limit cycles in a continuous fermentation model, J. Math. Chem. 5 (1990) 287–296] and by Pilyugin and Waltman [Multiple limit cycles in the chemostat with variable yield, Math. Boisci. 182 (2003) 151–166].     

A CLASS OF DIFFERENTIAL SYSTEM WITH AT MOST FOUR ALGEBRAIC LIMIT CYCLES

A class of quartic and quintic differential system is introduced.We show that under suitable assumptions,one,two or four algebraic limit cycles can occur.These limit cycles are analytically given.