一类奇异半线性热方程初值问题解 的唯一性结果

设ｕ（ｔ,ｘ）,ｕ（ｔ,ｘ）为初值问题在带形域ＳＴ＝（０,Ｔ）×Ｒｎ内的两个非负经曲解,ｆ（ｘ）连续有界非负的实函数,则有如下的结果：（１）若ｆ（ｘ）不恒为零,则在ＳＴ中ｕ（ｔ,ｘ）;（２）若γ＞１,则在ＳＴ中ｕ（ｔ,ｘ）ｕ（ｔ,ｘ）;（３）若０＞γ＞１,ｆ（ｘ）０,则问题（１．１）,（１．２）的解不唯一且它的所有非平凡解的集合为ｕ（ｔ,ｓ）＝这里ｓ≥０是参数,其中记号（γ）＋＝ｍａｘ｛γ,０｝．     

SUPER (SUB)-MIL AND UNIFORM INTEGRABILITY

In this paper, the following results have been proved: (i) If ( Xn,Fn ) is a uniformly integrable super(sub)-Mil, then (X.) lower (upper)-semiconverges (a. s. ) to an integrable random variable; (ii) If (Xn,Fn) is a uniformly integrable adapted sequence, then that (Xn) lower(upper)-semiconverges (a. s. ) is equivalent to that (Xn,Fn)is a super(sub)-Mil; and (iii) If (Xn,Fn) is a super(sub)-Mil of class (c-)((c+)), then (Xn) is lower(upper)-semiconvergent (a. s, ).     

关于图的L(3,2,1)-标号问题

图G的L( 2 ,1 )标号是一个从顶点集V(G)到非负整数集的函数f(x) .使得若d(x ,y) =1 .则｜f(x) -f(y) ｜≥ 2 ;若d(x ,y) =2 ,则｜f(x) -f(y)｜≥ 1 .图G的L( 2 ,1 )标号数λ(G)是使得G有max{f(v) ∶v∈V(G) }=k的L( 2 ,1 )标号中的最小数k .本文将L( 2 ,1 ) 标号问题推广到更一般的情形即L( 3,2 ,1 ) 标号问题 .我们首先定义了图G的顶点 3 着色及图的 3 色数 χ3 (G)等有关概念 ,并推导出 3 色数 χ3 (G)的上界 ;然后根据 χ3 (G)与λ3 (G)的关系 ,得出了对一般图G ,有λ3 (G) ≤ 3maxH Gδ(H) (Δ2 -Δ 1 )这一一般关系式 ;最后证明了对一般平面图G ,有λ3 (G)≤ 1 5(Δ2 -Δ 1 ) ,并得出了其它几类平面图的λ3 (G)的上界 .     

一类新型时滞积分不等式及其应用

主要研究了一种新型时滞积分不等式u(t)≤a(t)+∫0α(t)f(t,s)w(u(s))ds+∫0α(t)g(t,s∫)0sh(s,τ)φ(u(τ))dτds up(t)≤a(t)+p/p-q∫0α(t)(f(t,s)uq(s)w(u(s))+g(t,s)uq(s))dsup(t)≤a(t)+p/p-q∫0α(t)f(t,s)uq(s)w(u(s))ds+p/p-q∫0tg(t,s)uq(s)w(u(s))ds这里p>q≥0是常数且t∈[0,∞).并且用此结果研究了时滞微分积分方程解的全局存在性和有界性.     

直交多項式級数的求和

<正> 1.設φ_o(x),φ_1(x),…是区間(a,b)上之一系列的就范直交函数,孟孝夫証明:当級数∑(a_n log log_n)~2收斂时,直交函数級数 a_oφ_o(x)+a_1φ_1(x)+…+a_nφ_n(x)+…(1)在{φ_o(x)}的直交区間中,几乎到处可用正阶蔡查罗(Cesaro)求和法——(C,a)求和法,a>0——求和.当a=1时,这个定理还有波尔根(Borgen)和卡契馬尔茲(Karczmarz)     

具p-Laplacian算子型奇异方程组边值问题正解的存在性

本文讨论了一类具p-Laplacian算子型奇导方程组边值问题(φp(x'))'+α1(t),f(x(t),y(t))=0,(φp(y'))'+α2(t)g(x(t),y(t))=0,x(0)-β1x'(0)=0,x(1)+δ1x'(1)=0,y(0)-β2Y'(0)=0,y(1)+δ2y'(1)=0正解的存在性,其中φp(x)=|x|p-2x,p>1.通过使用不动点指数定理,在适当的条件下,建立了这类奇异方程组边值问题存在一个或者多个正解的充分条件.这些结果能用来研究椭圆型方程组边值问题径向对称解的存在性.     

拟线性系统的概周期解的存在性

<正> 考虑拟线性微分方程系 dX/dt=A(t)X十f(t)十μF(X,t,μ),(1)其中A(t)是t的n阶连续方阵,x是n向量,f(t),F(X,t,μ)是各变量的n连续向量,μ真是小参数. 当A(t)是常数方阵,f(t),F(X,t,μ)是t的一致概周期向量函数,Coddington,Levinson,等人建立了(1)的周期解的存在定理.此可参考[1]和[2].对A(t)为常数方阵,f(t),F(X,t,μ)是t的一致概周期向量函数,更进一步建立了(1)的概周期解的存在定理.     

关于Thue方程的解数

设ｍ是正整数，ｆ（Ｘ，Ｙ）＝ａ０Ｘｎ＋ａ１Ｘ（ｎ－１）Ｙ＋．．．＋ａｎＹｎ∈Ｚ［Ｘ，Ｙ］是Ｑ上不可约化的叫ｎ（ｎ≥３）次齐次多项式。本文证明了：当ｇｃｄ（ｍ，ａ０）＝１，ｎ≥４００且ｍ≥１０（３５）时，方程｜ｆ（ｘ，ｙ）｜＝ｍ，ｘ，ｙ∈ｚ，ｇｃｄ（ｘ，ｙ）＝１，至多有６ｎｖ（ｍ）组解（ｘ，ｙ），其中ｖ（ｍ）是同余式Ｆ（ｚ）＝ｆ（ｚ，１）≡０（ｍｏｄｍ）的解数。特别是当ｇｃｄ（ｍ，ＤＦ）＝１时，该方程至多有６ｎ（ω（ｍ）＋１）组解（ｘ，ｙ），其中ＤＦ是多项式Ｆ的判别式，ω（ｍ）是ｍ的不同素因数的个数．     

关于K_(2n)-E(C_m)的点可区别边色数

图G的一个k-正常边染色f被称为点可区别边染色是指任何两点的点及其关联边的色集合不同,所用最小的正整数k被称为G的点可区别边色数,记为X'_(vd)(G).用k_(2n)-E(C_m)表示2n阶完全图删去其中一条m阶路的边后得到的图,得到了K_(14)-E(C_4),K_(16)-E(C_4),K_(18)-E(C_5),K_(20)-E(C_5)的点可区别边色数分别为14,16,18,20.     

Toader型平均与几种经典平均的确界

给出了最佳参数α_1,α_2,α_3,β_1,β_2,β_3∈R,使得双向不等式α_1Q(a,b)+(1-α_1)G(a,b)0且a≠b成立.其中A(a,b)=(a+b)/2,H(a,b)=2ab/(a+b),G(a,b)=(ab)~(1/2),Q(a,b)=((a~2+b~2)/2)~(1/2),C(a,b)=(a~2+b~2)/(a+b),T(a,b)=2/π∫_0~(π/2)(a~2cos~2t+b~2sin~2)~(1/2)tdt分别是两个正数a和b的算术平均,调和平均,几何平均,二次平均,反调和平均和Toader平均.     

Mexican high school students' social representations of mathematics,its teaching and learning

This paper reports a qualitative research that identifies Mexican high school students’ social representations of mathematics. For this purpose, the social representations of ‘mathematics’, ‘learning mathematics’ and ‘teaching mathematics’ were identified in a group of 50 students. Focus group interviews were carried out in order to obtain the data. The constant comparative style was the strategy used for the data analysis because it allowed the categories to emerge from the data. The students’ social representations are: (A) Mathematics is…(1) important for daily life, (2) important for careers and for life, (3) important because it is in everything that surrounds us, (4) a way to solve problems of daily life, (5) calculations and operations with numbers, (6) complex and difficult, (7) exact and (6) a subject that develops thinking skills; (B) To learn mathematics is…(1) to possess knowledge to solve problems, (2) to be able to solve everyday problems, (3) to be able to make calculations and operations, and (4) to think logically to be able to solve problems; and (C) To teach mathematics is…(1) to transmit knowledge, (2) to know to share it, (3) to transmit the reasoning ability, and (4) to show how to solve problems.     

线性可移算子与可调和的随机过程

<正> §1.引言T.Onoyama 利用了 Weyl-Stone-Titchmarsh 的特征函数(Eigenfunction)展开公式,对具有二阶矩的实的连续机过程(本文中所用的极限,系指在均方意义下的极限)求得了下列隨机函数方程     

Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件

本文研究Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件,得到定理:设C0(U),C1(U,L,R,D,V),C2a(U),C2b(U,R),C3a(U),C3b(U,R)分别是Ⅱk空间上第0,Ⅰ,Ⅱa,Ⅱb,Ⅲa和Ⅲb类的算子代数,则(1)C0(U),C2a(U)或C3a(U)为一致闭(弱闭)的等价条件是U是Hibert空间G上的C*-代数(W*-代数;(2)C1(U,L,R,D,V)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间,V是闭算子,L对称闭的;(3)C2b(U,R)或C3b(U,R)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间．     

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. .     

三維黎曼空間中曲面的平均曲率及第二基本形式

<正> §1.T.Y.Thomas在三維歐氏空間的曲面論中將一般曲面的第二基本形式的係數用平均曲率及測度張量代數地表示出來.作者利用了高維歐氏空間超曲面的變形理論在高維歐氏空間的可變形超曲面上也得到同樣的結果,因此得知:對於歐氏空間的可變形超曲面一般地可以由測度張量及平均曲率完全予以確定.在該文的證明中曾經推廣T.Y.Thomas的結果到三維常曲率空間的曲面論.本文的目的是關     

测度链上非线性微分方程的三正解

运用文[1]中的Leggett—Williams不动点定理，我们给出了测度链上的非线性微分方程-x^△△(t)=f(t,x(σ(t))),t∈[a,b,]关于两点边值条件ax(a)-βx^△(a)=0，γx(σ(b)) δx^△(σ(b))=0三正解存在性准则。     

The Jump Conditions for Second Order Quasilinear Degenerate Parabolic Equations

ln this paper we consider the model problem for a second order quasilinear degenerate parabolic equation {D_xG(u) = t^{2N-1}D²_xK(u) + t^{N-1}D_x,F(u) \quad for \quad x ∈ R,t > 0 u(x,0) = A \quad for \quad x < 0, u(x,0) = B \quad for \quad x > 0 where A < B, and N > O are given constants; K(u) =^{def} ∫^u_Ak(s)ds, G(u)=^{def} ∫^u_Ag(s)ds, and F(u) =^{def} ∫^u_Af(s)ds are real-valued absolutely continuous functions defined on [A, B] such that K(u) is increasing, G(u) strictly increasing, and \frac{F(B)}{G(B)}G(u) - F(u) nonnegative on [A, B]. We show that the model problem has a unique discontinuous solution u_0 (x, t) when k(s) possesses at least one interval of degeneracy in [A, B] and that on each curve of discontinuity, x = z_j(t) =^{def} s_jt^N, where s_j= const., j=l,2, …, u_0(x, t) must satisfy the following jump conditions, 1°. u_0(z_j(t) - 0, t) = a_j, u_0 (z_j(t) + 0, t) = b_j, and u_0(z_j(t) - 0, t) = [a_j, b_j] where {[a_j, b_j]; j = 1, 2, …} is the collection of all intervals of degeneracy possessed by k (s) in [A, B], that is, k(s) = 0 a. e. on [a_j, b_j], j = 1, 2, …, and k(s) > 0 a. e. in [A, B] \U_j[a_j, b_j], and 2°. (z_j(t)G(u_0(x, t)) + t^{2N-1}D_xK(u_0(x, t)) + t^{N-1}F(u_0(x, t)))|\frac{s=s_j+0}{s=s_j-0} = 0     

Smarandache可求积因数对问题

对任意正整数n,设d(n)表示n的Dirichlet除数函数,即就是n的所有不同正因数的个数.Smarandache可求积因数对问题是:求所有正整数对m及n使得d(m)+d(n)=d(mn).主要目的是利用初等方法以及除数函数的性质研究这一问题,并给予彻底解决.具体地说也就是证明了正整数对m及n满足方程d(m)+d(n)=d(mn)当且仅当(m,n)=(pq~α,q)或者(m,n)=(p,p~αq),其中p及q为不同的素数,α为非负整数.     

GLOBAL EXISTENCE FOR A CLASS OF SYSTEMSOF NONLINEAR WAVE EQUATIONS INTHREE SPACE DIMENSIONS

Consider a system of nonlinear wave equationsfor i = 1, … , m, where F, (i = 1, … , m) are smooth functions of degree 2 near the origin of their arguments, and u = (u1, … ,um), while u and x u represent the first and second derivatives of u, respectively. In this paper, the author presents a new class of nonlineaxity for which the global existence of small solutions is ensured. For example, global existence of small solutions for arbitrary cubic terms,arbitrary cubic termswill be established, provided that c12 ≠ c22.     

The largest eigenvalue of a graph: A survey

This article is a survey of results concerning the largest eigenvalue (or index) of a grapn, catcgoiizeu as follows (1) inequalities lor the index, (2) graph with bounded index, (3) ordering graphs by their indices, (4) graph operations and modifications, (5) random graphs, (6) applications.