New series of odd non-congruent numbers

We determine all square-free odd positive integers n such that the 2-Selmer groups S _n and Ŝ _n of the elliptic curve E _n : y ² = x(x − n)(x − 2n) and its dual curve ê _n : y ² = x ³ + 6nx ² + n ² x have the smallest size: S _n = {1}, Ŝ _n = {1, 2, n, 2n}. It is well known that for such integer n, the rank of group E _n (ℚ) of the rational points on E _n is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves E _n with rank zero and such series of integers n are non-congruent numbers. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

时间随机环境中一维紧邻随机游动的常返性质

设环境q={q(n)}∞0是取值于[0,1]上一列独立同分布的随机变量列,且Eq(0)=p;{Sn}∞0是随机环境q中取整数值随机游动,S0=0,且满足:对任意的整数xi(i≥0),x,y,P(Sn+1=y|S1=x1,…,Sn-1=xn-1,Sn=x,q)={q(n),y=x+1,1-q(n),y=x-1,0,其他.我们证明了:p>1/2时,Sn→+∞,a.e.,n→∞;p<1/2时,Sn→-∞,a.e.,n→∞;p=1/2时,-∞=(lim infSn)/(n→+∞)<(lim supSn)/(n→+∞)=+∞,a.e.,n→∞.     

常微分方程的區域分析理論(Ⅰ)

<正> 引言 常微分方程理論的一般性研究概括可分為兩個方面;即性質的方面和數量的方面.前一方面由於Poincare H.和的創造發展成為“定性理論”這一分枝.(例如参考[1]).它既研究個別的曲線和運動,也研究整族的曲線和運動;既研究在一特殊點附近的情况,也研究在大範圍內的情况.在數量方面,由於準確公式的難於求得,重點轉向數值積分.即由已給定之初值開始,逐步求出某一特定的解的近     

关于丢番图方程（195n）^x＋（2Sn）^y=（197n）^z

运用同余及元素阶的性质,证明了对任意的正整数n,丢番图方程（195n）^x＋（2Sn）^y=（197n）^z仅有正整数解（x,y,Z）=（2,2,2）．     

Integral Averaging Technique for Oscillation of Elliptic Equations of Second Order

The elliptic differential equations of second order \sum^n_{i,j=1}D_i[A_{ij}(x, y)D_jy] + P(x, y) + Q(x, y, ∇y) = e(x), x ∈ Ω . will be considered in an exterior domain Ω⊂ R^n, n ≥ 2. Some oscillation criteria are given by integral averaging technique.     

ISOGENOUS OF THE ELLIPTIC CURVES OVER THE RATIONALS

AbstractAn elliptic curve is a pair (E,O), where ?is a smooth projective curve of genus 1 and O is a point of E, called the point at infinity. Every elliptic curve can be given by a Weierstrass equationE:y2 a1xy a3y = x3 a2x2 a4x a6.Let Q be the set of rationals. E is said to be dinned over Q if the coefficients ai, i = 1,2,3,4,6 are rationals and O is defined over Q.Let E/Q be an elliptic curve and let E(Q)tors be the torsion group of points of E denned over Q. The theorem of Mazur asserts that E(Q)tors is one of the following 15 groupsE(Q)tors　Z/mZ, m = 1,2,..., 10,12,Z/2Z × Z/2mZ, m = 1,2,3,4.We say that an elliptic curve E'/Q is isogenous to the elliptic curve E if there is an isogeny, i.e. a morphism : E E' such that (O) = O, where O is the point at infinity.We give an explicit model of all elliptic curves for which E(Q)tors is in the form Z/mZ where m= 9,10,12 or Z/2Z × Z/2mZ where m = 4, according to Mazur's theorem. Morever, for every family of such elliptic curves, we give an explicit m     

关于数论函数$\sigma(n)$的一个注记

对于两个不相同的正整数$m$和$n$, 如果满足$\sigma(m)=\sigma(n)=m+n$, 则称之为一对亲和数, 这里$\sigma(n)=\sum_{d|n}d$.本文给出了$f(x,y)=x^{2^{x}}+y^{2^{x}}(x>y\geq{1},(x,y)=1)$不与任何正整数构成亲和数对的结论, 这里$x$,$y$具有不同的奇偶性, 即, 关于$z$的方程$\sigma(f(x,y))=\sigma(z)=f(x,y)+z$不存在正整数解.     

OSCILLATION CRITERIA FOR NONLINEAR SECOND ORDER ELLIPTIC DIFFERENTIAL EQUATIONS

1.IntroductionOscillationtheoryforellipticdifferentialequationswithvariablecoefficielltshasbeenextensivelydevelopedinrecentyearsbymanyauthors(seee-g-,Allegretto[1,2l,Bugir['],Fiedler[5]'KitamuraandKusano['],Kural4]KusanoandNaitol1o]'NaitoandYoshidal12]'NoussairandSwanson[l3'14]'Swansonl15'l6]andthereferencescitedtherein).Inthispaper,weareconcernedwiththeoscillatorybehaviorofsolutionsofsecondorderellipticdifferelltialequatiollswithalternqtingcoefficients.Asusual,pointsinn-dimensionalEucli…     

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Let p_M(t,x,y) be the minimal heat kernel of a d-dimenional compact Riemannian manifold M for any time t\in(0,1] and x,y\in M. Using the horizontal Brown bridge on M, we prove that, for any nonnegative integers n and m, there is a constant C depending on n,m and the manifold M, such that |\nabla^n_x\nabla^m_y\ln p_M(t,x,y)|\leq C[d(x,y)/t+1/\sqrt{t}\,]^{n+m}$, which generalizes the conclusion of the higher derivatives of the logarithmic heat kernel \ln p_M(t,x,y) about single variable in \ncite{1}.     

Algebraic Dilogarithm Identities

The Rogers L-function satisfies the functional equation .From this we derive several other such equations, including Euler's identity L(x)+L(1-x)=L(1) and various identities arising from summation and transformation formulas for basic hypergeometric series. We also obtain some new equations of the form where is algebraic and the c _k are integers.     

关于几类图的L(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 .Griggs和Yeh猜想对最大度为Δ的一般图G ,有λ(G) ≤Δ2 .本文给出了Kneser图 ,Mycieklski图 ,Descartes图 ,Halin图的λ值的上界 ,并证明了上述猜想对以上几类图成立     

ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR （I）

ＯＮＡＭＵＬＴＩＬＩＮＥＡＲＯＳＣＩＬＬＡＴＯＲＹＳＩＮＧＵＬＡＲＩＮＴＥＧＲＡＬＯＰＥＲＡＴＯＲ（Ｉ）ＣＨＥＮＷＥＮＧＵＨＵＧＵＯＥＮＬＵＳＨＡＮＺＨＥＮＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＯｃｔｏｂｅｒ１８，１９９４．ＲｅｖｉｓｅｄＤｅｃｅ...     

椭圆曲线y2=2px(x2+1)上正整数点的个数

设p是奇素数,N(p)是椭圆曲线E:y2=2px(x2+1)的正整数点(x,y)的个数.主要讨论了N(p)的性质,运用初等方法及四次Diophantine方程的性质,对某些特殊素数p,给出了N(p)的上界.证明了当p≡1(mod 8)且p=s2+32t,其中s,t是正整数时,N(p)≤3;当p≡1(mod 8)且p+s...     

孪生素数椭圆曲线E_的整数点

设p和q是孪生素数.本文讨论了椭圆曲线E_上的非平凡整数点.运用初等数论方法证明了该椭圆曲线至多有1组非平凡整数点(x,±y).     

Some estimates of solutions of degenerate (k,o)-elliptic equations

A class of nonlinear second-order equations of divergent form is distinguished, whose solutions have properties recalling the properties of solutions of ordinary elliptic equations. In the linear case these are equations of the form $$\sum\nolimits_{j = 1}^k \lambda _j (x)A_j^2 u + \sum\nolimits_{j = 1}^k {\mu _j (x)A_j u + c(x)u + f(x) = 0,} $$ where the \(A_j = \sum\nolimits_{\alpha = 1}^n {\alpha _j^\alpha (x)\frac{\partial }{{\partial x^\alpha }}(1 \leqslant j \leqslant k)} \) are linearly independent first-order differential operators whose Lie algebra is of rank n, 2 ? k ? n, and theλj(x) ? 0 are functions which can become0 zero or increase in a definite way. Harnack's inequality is proved for nonnegative solutions of these equations.     

方程xy+yz+zx=n的正整数解

本文用 Ｓｉｅｇｅｌ－Ｔａｔｕｚａｗａ定理证明了：当ｎ＞１．２×１０~１１时,至多有两个正 整数ｎ。使方程ｘｕ＋ｙｚ＋ｚｘ＝ｎ无适合（ｘ,ｙ,ｚ）＝１且０＜ｘ＜ｙ＜ｚ的解（ｘ,ｙ,ｚ）, 并给出类数为２的二次域与多项式表素数的一个结果．     

Uniqueness of Quasi-Regular Solutions for a Bi-Parabolic Elliptic Bi-Hyperbolic Tricomi Problem

The Tricomi equation $ yu_{xx} + u_{yy} = 0 $ was established in 1923 by Tricomi who is the pioneer of parabolic elliptic and hyperbolic boundary value problems and related problems of variable type. In 1945 Frankl established a generalization of these problems for the well-known Chaplygin equation $ K(\,y)u_{xx} + u_{yy} = 0 $ subject to the Frankl condition 1 + 2( K / K ')' > 0, y <0. In 1953 and 1955 Protter generalized these problems even further by improving the above Frankl condition. In 1977 we generalized these results in R n ( n > 2). In 1986 Kracht and Kreyszig discussed the Tricomi equation and transition problems. In 1993 Semerdjieva considered the hyperbolic equation $ K_1 (\,y)u_{xx} + (K_2 {\rm (\,}y{\rm )}u_y )_y + ru = f $ for y<0. In this paper we establish uniqueness of quasi-regular solutions for the Tricomi problem concerning the more general mixed type partial differential equation $ K_1 (\,y)(M_2 {\rm (}x{\rm )}u_x )_x + M_1 (x)(K_2 {\rm (\,}y{\rm )}u_y )_y + ru = f $ which is parabolic on both lines x = 0; y = 0, elliptic in the first quadrant x > 0, y > 0 and hyperbolic in both quadrants x< 0, y > 0; x > 0, y< 0. In 1999 we proved existence of weak solutions for a particular Tricomi problem. These results are interesting in fluid mechanics.     

再论无穷多个无穷小量的乘积

对文[6]提出的质疑给出回答,表明由于不同的无穷小量趋近于0的速度有快有慢,因此无穷多个无穷小量的乘积∏∞k=1{x_n~(k)}∞n=1,有可能不是无穷小量(其中对每个正整数k,{x_n~(k)}_(n=1)~∞表示极限为0的数列),而验证∏∞k=1{x_n~(k)}∞n=1是否是无穷多个无穷小量的乘积,只需验证对每个正整数k,当n→+∞时,{x_n~(k))_(n=1)~∞是否趋近于0,而无需考虑函数列{{x_n~(k)}_(n=1)~∞}_(k=1)~∞的极限limk→∞x_n~(k)是不是无穷小量.进而,对无穷多个无穷小量的乘积是无穷小量或不是无穷小量给出了一些充分条件,     

关于虚二次域类数的可除性

设α＞１，ｂ＞１，（α，ｂ）＝１，ｈ（－αｂ）表虚二次域的类数。如果有正整数ｘ，ｙ，ｎ，ｋ满足（１）αｘ￣２＋ｂｙ￣２＝４ｋ￣ｎ，ｂ且；或（２）αｘ￣２＋ｂｙ￣２＝ｋ￣ｎ，ｘ｜α，ｙ｜ｂ且αｂ≡２（ｍｏｄ４），则本文证明了关于ｈ（－αｂ）的可除性的两个定理（见定理１，２），其中符号ｘ｜α表示ｘ的每一个素因子整除α。     

条件经验随机过程的渐近性质

设(Xi,Yi)(i=1,2,…,n)是来自总体(X,Y)的样本(独立同分布),其中X∈R1,Y∈Rq.M(x y)是Y=y时X的条件分布,Mnkn(x y)为M(x y)的第kn个最近邻域的经验分布估计量,讨论条件经验过程Sn(t,x,y)=kn12(Mnkn(x y)-M(x y))的渐近性质,得出在适当条件下,对固定的y,Sn(t,x,y)(x,t为参数)弱收敛于某一G aussian过程S(.).