EXISTENCE PROBLEMS OF ADDITIVE SELECTION MAPS FOR ANOTHER TYPE SUBADDITIVE SET-VALUED MAP

In this paper,we consider the following subadditive set-valued map F:X → P 0 (Y):F r ∑ i=1 x i + s ∑ j=1 x r+ j rF r ∑ i=1 x i r + sF s ∑ j=1 x r+ j s, x i ∈ X,i=1,2,,r + s,where r and s are two natural numbers.And we discuss the existence and unique problem of additive selection maps for the above set-valued map.     

柯西不等式的两个推论及应用

在中学数学中常遇到如下一个不等式:(n∑i=1xiyi)2≤(n∑i=1xi2)·(n∑i=1yi2),其中xi,yi为任意实数,且等号成立当且仅当xi=kyi(i=1,2,…,n),这就是著名的柯西不等式.推论1已知ai(i=1,2,…,n)是正数,xi∈R(i=1,2,…n)且n∑i=1ai=1,则n∑i=1aixi2≥(n∑i=1aixi)2.证∵ai∈R (i=1     

RANDOM APPROXIMATION OF AN ADDITIVE FUNCTIONAL EQUATION OF m-APPOLLONIUS TYPE

In this paper,using the fixed-point and direct methods,we prove the HyersUlam stability of the following m-Appolonius type functional equation:∑mi=1 f(z-xi)=mf(z-1/m2∑mi=1xi)-1/m∑1≤i相似文献   

SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS

We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem{x(3)(t) + f(t, x(t), x′(t)) = 0, 0 t 1,x(0)-m1∑i=1 αi x(ξi) = 0, x′(0)-m2∑i=1 βi x′(ηi) = 0, x′(1)=0,where 0 ≤ ai≤m1∑i=1 αi 1, i = 1, 2, ···, m1, 0 ξ1 ξ2 ··· ξm1 1, 0 ≤βj≤m2∑i=1βi1,J=1,2, ···, m2, 0 η1 η2 ··· ηm2 1. And we obtain some necessa βi =11, j = 1,ry and sufficient conditions for the existence of C1[0, 1] and C2[0, 1] positive solutions by constructing lower and upper solutions and by using the comparison theorem. Our nonlinearity f(t, x, y)may be singular at x, y, t = 0 and/or t = 1.     

对两个代数不等式的探索

文[1]给出了如下定理及猜想:定理1对于任意实数x,y,a,b有(x-a)2 (y-b)2≥(x2 y2-a2 b2)2.定理2已知x,y,xi,yi∈R(i=1,2,…,n),且x2 y2≥n∑i=1xi2 yi2,则(x-n∑i=1xi)2 (y-n∑i=1yi)2≥(x2 y2-n∑i=1xi2 yi2)2(1)猜想,已知x,y,xi,yi∈R(i=1,2,…,n),则(x-n∑i=1xi)2 (y-n∑i=1y     

关于正定厄米特矩阵的一个不等式的推广

本文推广了正定厄米特矩阵的一个不等式 ,得到以下结果 :设 A( i) ,B( i) ,… ,C( i) ( i=1 ,2 ,… ,m)都是 n阶正定厄米特矩阵 ,A( i)11,B( i)11,… ,C( i)11为其相应矩阵的 k阶顺序主子阵 ,1≤ k≤ n-1 ,α,β,… ,γ都是正实数 ,且 α+β+… +γ=p≥ 1 ,则有∑mi=1|A( i) |α|A( i)11|α,|B( i) |β|B( i)11|β… |C( i) |γ|C( i)11|γ) <∑mi=1A( i) α∑mi=1A( i)11α.∑mi=1B( i) β∑mi=1B( i)11β…∑mi=1C( i) γ∑mi=1C( i)11γ     

Some Notes on Submanifolds of an Euclidean Space with Conformal Gauss Map

Let (M,g) be an m-dimensional Riemannian manifold and i:M→E~n an isometricimmersion of (M,g) into an n-dimensional Euclidean space E~n. Let VM be anopen set in which the immersion i:M→E~n is given by x~h=x~h(y~a), (h=1,…,n; α=1,…,m). Here and in the sequel x~h(h=1,…,n) are rectangular coordinates of E~nand y~α(α=1,…,m) are local coordinates of a generic point in V. The tangent planeiM_(ip)=i(M_p), P∈V, of iM can be considered after a suitable parallel displacementas a point Γ_i(P) of the Grassmann manifold G(m,n-m).The mapping Γ:iM→G(m,n-m), ipΓ(iP) =Γ_i(P) is called the Gauss map. The mapping Γ_i:M→G(m,n-m),p_i(P) is called the Gauss map associated with the immersion i, and _i(M)=F(iM) the Gauss image of M.     

一个多元绝对值函数的最小值

对于函数F(x1，x2，…，xn)=|α1x1 α2x2 … αnxn A|，由绝对值的意义知F(x1，x2，…，xn)≥0，特别，当αi，xi，A∈Z(i=1，2，…，n)时，该函数有更精确的下界，本文将给出这个结论。     

The definite integral

<正>1.What does the definite integral mean?The definite integral of f(x)fromato b is defined the limit of the sum as n→∞.That is limn→∞∑n i=1f(ξi)·Δxi.We divide the interval[a,b]into n subintervals of equal widthΔx=(b-a)n.Let x0=a,x1,x2,…,xn=b be the endpoints of these subintervals and we chooseξiis any point in the ith subinterval,that is,xi-1≤ξi≤xi,then,the sum∑n f(ξi)·Δxiis called a Rie-     

APPROXIMATION OF A CAUCHY-JENSEN ADDITIVE FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN NORMED SPACES＊

Using the fixed point and direct methods,we prove the Hyers-Ulam stability of the following Cauchy-Jensen additive functional equation 2f p i=1 xi + q j=1 yj + 2 d k=1 zk 2 = p i=1 f(xi) + q j=1 f(yj) + 2 d k=1 f(zk),where p,q,d are integers greater than 1,in non-Archimedean normed spaces.     

一个不等式的推广及猜想

文[1]介绍了如下一个不等式:若xi>0,i=1,2,3,且∑3i=1xi=1,则11 x12 1 1x22 1 1x32≤1170(1)文[1]给出了(1)的一个初等证明·今推广(1)如下:若xi>0,i=1,2,3,4,且∑4i=1xi=1,则1x13 1 1x32 1 1x33 1 1x43≤26556(2)证明先证明:对任意0相似文献   

Vector subdivision schemes in (Lp(Rs))r(1 ≤ p ≤∞) spaces

The purpose of this paper is to investigate the refinement equations of the form ψ(x) = ∑α∈Zs a(α)ψ(Mx - α), x ∈ Rs,where the vector of functions ψ=(ψ1,…,ψr)T is in (Lp(Rs))r, 1≤p≤∞,a(α),α∈Zs,is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix suchthat lim n→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vectorof compactly supported functions ψ0 ∈ (Lp(Rs))r and use the iteration schemes fn := Qnaψ0,n = 1,2,…,where Qa is the linear operator defined on (Lp(Rs))r given by Qaψ:= ∑α∈Zs a(α)ψ(M·- α),ψ∈ (Lp(Rs))r. This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of somelinear operators determined by the sequence a and the set B restricted to a certain invariant subspace, wherethe set B is a complete set of representatives of the distinct cosets of the quotient group Zs/MZs containing 0.     

四面体的外p号心及其性质

本文拟用解析法建立四面体的外 p号心的概念 ,并探讨其相关性质 .设四面体A1A2 A3A4 的外接球球心为O ,以O为原点 ,建立空间直角坐标系Oxyz ,设Ai 的坐标为(xi,yi,zi) (i=1,2 ,3,4 ) ,令 xp=1p∑xi, yp=1p∑yi, zp=1p ∑zi(其中 p∈N ,∑为i=1,2 ,3,4的循环和 ) ,则称点Qp1p∑xi,1p∑yi,1p∑zi 为四面体A1A2 A3A4的外 p号心 ,于是H (∑xi,∑ yi,∑zi) , H 12 ∑xi,12 ∑yi,12 ∑zi ,F 13∑xi,13∑yi,13∑zi ,G14 ∑xi,14 ∑yi,14 ∑zi 分别是四面体A1A2 A3A4 的外 1,2 ,3,4号心 ;这里外 4号心G便是四面体A1A2 A3A4 的重心 ;如果…     

n元三次多项式不等式猜想的证明

猜想 [1]　设 x1,x2 ,… ,xn∈ R+ ,n为正整数 ,证明或否定 :n( n - 1 ) ∑ni=1x3 i + ( ∑ni=1xi) 3　≥ ( 2 n - 1 ) ∑ni=1xi∑ni=1x2i ( 1 )这是杨学枝老师近日提出的一个猜想 .经探讨发现 ,此猜想成立 .为证明 ( 1 )式成立 ,先给出如下引理 .引理 1　 x1,x2 ,… ,xn∈ R,n为正整数 ,则( ∑ni=1xi) 3 =∑ni=1x3 i + 3∑i≠ jx2ixj+ 6 ∑1≤ i相似文献   

MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Applying Nevanlinna theory of the value distribution of meromorphic functions,we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form∑nj=1aj(z)f1(λj1)(z+cj) = R2(z, f2(z)),∑nj=1βj(z)f2(λj2)(z+cj)=R1(Z,F1(z)).(*)where λij(j = 1, 2, ···, n; i = 1, 2) are finite non-negative integers, and cj(j = 1, 2, ···, n)are distinct, nonzero complex numbers, αj(z), βj(z)(j = 1, 2, ···, n) are small functions relative to fi(z)(i = 1, 2) respectively, Ri(z, f(z))(i = 1, 2) are rational in fi(z)(i = 1, 2)with coefficients which are small functions of fi(z)(i = 1, 2) respectively.     

也谈“改变观点”的创造性技巧

文 [1]例说了创造性思维方法中的“改变观点”的技巧 ,读罢很受启发 .由于概率统计将列入高中数学的教学内容[2 ] ,将传统的某些数学教学内容试改以概率观点来研究和处理将别有一番趣味 ,本文将对文 [1]所创造的 5个新问题给以概率证明 .新问题 1　若 cos2 α1 cos2 α2 … cos2 αn=1,则ctg2 α1 ctg2 α2 … ctg2 αn≥ nn-1( n≥ 2 ) ,这个问题的更一般的形式是新问题 1′　若 xi>0 ,i=1,… ,n,n≥ 2 ,且∑ni=1 xi=1,则∑ni=1xi1-xi≥ nn-1( 1)注意到此问题的条件的特点 :诸 xi 非负且其和为1,这使人联想到概率论中离散型随机变量…     

一道竞赛试题解法探究

例 1　 ( 1995年数学冬令营第五题 )设xi >0 ,∑ni =1xi=1(i =1,2 ,… ,n) ,求证 :∑ni =1xi1+ (x1+x2 +… +xi- 1)xi+xi+ 1+… +xn≤ π2 .证　令sinθi=∑ik =1xk ,θ0 =0 (i =1,2 ,… ,n) ( 0<θi≤ π2 ) ,则∑ni=1xi1+ (x1+x2 +… +xi - 1)xi+xi + 1+… +xn=∑ni =1sinθi-sinθi- 11+sinθi - 11-sinθi- 1=∑ni =12sin θi-θi - 12 cosθi+θi- 12cosθi - 1≤∑ni =12sinθi-θi - 12<∑ni =1(θi-θi - 1)=θn -θ0 =π2 .例 1的命制及解法均含有高等数学中的思想方法 ,为了说明问题 ,我们给出如下两个结论 .定理 1　设 f(x) 是区…     

一类二次方程组的一个定理及其运用

定理　在方程组∑ni=1xi=A∑ni=1x2i=B中 ,A、B是实数 ,记Δ=n B-A2 .若 xi∈ R( i=1,2 ,… ,n) ,则Δ≥ 0 ,当且仅当x1 =x2 =… =xn=An时 Δ=0 .证明　 ∑1≤ i相似文献   

SMOOTH AND PATH CONNECTED BANACH SUBMANIFOLD ∑r OF B(E,F) AND A DIMENSION FORMULA IN B(IRn,IRm)

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear oper-ators from E into F, ∑r the set of all operators of finite rank r in B(E, F), and ∑#r the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submani-fold in B(E, F) with given expression of its tangent space at each A ∈∑r. In this paper, the equality ∑#r = 1 is proved. Consequently, the following theorem is obtained: for any non-negative integer r, ∑r is a smooth and path connected Banach submanifold in B(E, F) with the tangent space TA∑r={B∈B(E,F): BN(A) R(A)} at each A∈∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = IRn and F = IRm, then ∑r is a smooth and path connected submanifold of B(IRn,IRm) and its dimension is dim ∑r = (m + n)r- r2 for each r, 0 ≤ r < min{n,m}.     

一个不等式猜想的证明

对于正数ai>0,i=1,2,…,n,k为给定的正整数,若∑ni=1ai=1,笔者在文[1]末提出了猜想:∏n-1i=1(1∑kj=1ai j-∑nj=k 1ai j)≥(nk kn-1)n(1)其中an i=ai(i=1,2,…,n-1),k为常数,且0相似文献