Reinhardt域上一类推广的Roper-Suffridge算子

研究一类推广的Roper-Suffridge算子F(z)=(f(z_1)+f′(z_1)∑_(k=2)~nakz_k~pk,f′(z)1)(~1/p2)z_2,…,f′(z_1)~(1/pn)z_n)′,证明该算子在复欧氏空间中的Reinhardt域Ω_(n,p2,%…,pn)={z=(z_1,…,z_n)∈C~n:|z_|~2+∑_(k=2)~n|zk|~(pk)1,Pk∈N~+,k=2,…,n}上分别保持α次的殆β型螺形性,α次的β型螺形性及强β型螺形性.     

Cauchy多边形数定理的一个简短证明

本文对下述事实给出一个简单的证明:每个自然数是m+2个m+2边形数之和. 设m≥1,一个m+2边形数是形如 Pm(k)=m/2(k2-k)+k,(k=0,1,2,…)的数.Fermat[3]断言:每一个自然数是m+2个m+2边形数之和.对于m=2,Lagrange[5]证明了每一个自然数是4个平方数P2(k)=k2之和.对于m=1,Gauss [4]证明了每一个自然数是3个三角数P1(k)=1/2(k2+k)之和,或等价的,每一个满足n≡3(mod 8)的正整数n都是3个奇数平方之和,Cauchy[1]对所有的m≥3证明了Fermat的断言,Legendre[6]进一步细化和推广了这一结果.对于m≥3且n≤120m,Pepin [8]给出了将n写成m+2个m+2边形数之和的显示表达的表,其中至少有m-2个取值于0或1.     

一个不等式的逆向

文[1]证明了文[2]提出的一个猜想:说ai≥0,pi≥0,(i=1,2,…,n)且p1 p2 … pn=1,则n∑i=1piai≥n∏i=1aipi.本文将给出上述不等式的一个逆向不等式,从而得出一个不等式组.命题设ai>0,pi≥0(i=1,2,…,n),且p1 p2 … pn=1,则1∑ni=1piai≤n∏i=1aipi≤n∑i=1piai.证为了使命题证明     

广义Davey-Stewartson系统驻波的强不稳定性

本文研究方程驻波的强不稳定性iu_t+△u+a|u|~(p-1)u+E_1(|u|~2)u=0,t≥0,x∈R~n,其中a0,1p(n+2)/(n+2)~+,n∈{2,3}.当1+4/n≤pn+2/(n-2)~+)时,文[Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,Commun.Math.Phys.,2008,283:93-125]在驻波的频率满足一定假设条件下,证明了此方程驻波的强不稳定性.本文去掉这个假设,得到相同的结论.     

一个几何式不等的导数证法

设pn,qn分别为内接于、外切于半径为R的圆的正n边形的周界,证明:23pn 31qn>2πR.证由pn=2Rnsinπn,qn=2Rntanπn,得32pn 13qn=32Rn(2sinπn tgπn).要证32pn 13qn>2πR,只须证2sinπn tanπn>3nπ即可.令F(x)=2sinx tanx-3x,00,故有F(x)>F(0)=0即2sinx tanx-3x>0,从而有sinπn tanπn>3πn,于是有32pn 13qn>2πR.一个几何式不等的导数证法@谢先武$江西师大数信学院!江西…     

综合题新编选登

题197设a1=1,an 1=2an n 1.(Ⅰ)是否存在常数p,q使{an pn q}为等比数列?若存在,求出p,q的值,若不存在,说明理由:(Ⅱ)求{an}的通项公式;(Ⅲ)当n≥5时,证明:an>(n 2)2.解(Ⅰ)由an 1 p(n 1) q=2(an pn q)得an 1=2an pn (q-p).可见应有p=1,q-p=1,p=1,q=2,∴an 1 (n 1) 2=2(an n 2).     

一个几何不等式的推广

文[1]用导数方法证明了：设pn,qn分别为内接于、外切于半径为R的圆的正n边形的周界，则2/3pn＋1/3qn〉2πR。本文笔者拟采用其证明方法推广所证不等式。     

广义Ramanujan-Nagell方程x~2+D~m=p~n的解数

设a是正整数,D=3a2+1,P=4a2+1,其中p是素数.本文证明了:如果a不是4的倍数,则除了当(D,p)=(4,5)时方程x2+Dm=pn恰有3组正整数解(x,m,n)=(1,1,1),(3,2,2),(11,1,3)以外,该方程恰有2组正整数解(x,m,n)=(a,1,1)和(8a3+3a,1,3).     

涉及凸n边形内部一点的一个不等式

设 P是凸 n边形 A1A2 … An 内一点 ,ri 为P至边 Ai Ai+ 1的距离 ,wi是∠ Ai PAi+ 1=2αi的角平分线 ,Ri=PAi,ti =Ri Ri+ 1cosαi,i=1 ,2 ,… ,n,An+ 1=A1.文献 [1 ]中 ,H.C.L enhard证明了不等式 :　　　　∑ni=1Ri ≥ secπn .∑ni=1ti ( 1 )文献 [2 ]中 ,笔者建立了 (其中 s为凸 n边形的半周长 )∑ni=1Ri2 - ∑ni=1ti2 ≥ s2 ( 2 )并且根据不等式 ( 1 ) ,( 2 )证明了 ,当 secπn ≥ k≥ cosπn 时 ,有∑ni=1Ri - k∑ni=1ti ≥1 - kcosπnsin πns ( 3)本文应用不等式 ( 1 ) ,( 2 )建立类似于不等式( 3)的一个结论 .定理　设 P…     

几个不等式的简证

《数学通讯》(教师版)2006年上年度刊登了一组关于不等式研究的专题文章,笔者拜读之后受益匪浅,笔者探究发现其中的几个不等式更加简捷的证明方法,现写出来,供读者参考.例1[1]设ai>0,pi≥0(i=1,2,…,n),且p1 p2 … pn=1,则1n∑i=1piai≤n∏i=1aipi≤n∑i=1piai(1)这是文[1]对文[2]证明的如下一个不等式的逆向不等式:设ai>0,pi≥0,(i=1,2,…,n)且p1 p2 … pn=1.则n∑i=1piai≥n∏i=1aipi(2)文[1]通过构造函数,考察函数的凸性,然后用数学归纳法证明了(1)式.其实,有了(2)式,(1)式的证明便唾手可得,不必绕道而行.事实上,由(2)式∑ni=1piai=∑ni=…     

單葉函數的係數與開始多項式

<正> 以fk(z)表單位圓內的K次對稱單葉全純函數,亦即fk(z)=z+a_I~((k))z~(k+1)+a_2~((k))z~(2k+1)+…,|z|<1.以S_k表此種函數之全體.特別,書S以代S_1.     

Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

Let \(\bar{p}(n)\) denote the number of overpartitions of n. Fortin et al. and Hirschhorn and Sellers established some congruences modulo powers of 2 for \(\bar{p}(n)\). Recently, Xia and Yao found several congruences modulo powers of 2 and 3. In particular, they proved that \(\bar{p}(96n+12)\equiv 0 \ (\mathrm{mod}\ 9)\) and \(\bar{p}(24n+19)\equiv 0\ (\mathrm{mod\ }27)\). In this paper, we generalize the two congruences and establish several new infinite families of congruences modulo 9 and 27 for \(\bar{p}(n)\). Furthermore, we prove some strange congruences modulo 9 and 27 for \(\bar{p}(n)\) by employing some results due to Cooper et al. For example, we prove that for \(k\ge 0\), \(\bar{p}(4^{k+1})\equiv 2^{k+3}+6(-1)^k\ (\mathrm{mod} \ 27) \) and \(\bar{p}\left( 7^{2k}\right) \equiv 2-2k\ (\mathrm{mod}\ 9)\). We also present two conjectures on congruences for \(\bar{p}(n)\).     

Subclasses of regular and schlicht functions with fixed coefficient

In the open unit disk E={z, ¦z¦<1} we consider the class of regular functions q(z) = 1 + 2(1 = )e¹zⁿ + 2p_n+kz^n+k + ..., Re q(z)>, , [0, 1), [0, 2] and fixed, n, kN; fixing zE, we construct the range of the functional I₀=q(z). We find an analog of V. A. Zmorovich's variational formula for the expression zp'(z) and point out some applications.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 813–820, June, 1991.     

关于一个素数和一个素数的k次方和问题

设k≥2,Hk表示一个正整数n的集合,使对任意的正整数q,同余方程a+b2三n(modq)在模q的既约剩余系中有解a,b.Dk(N)表示n≤N,n∈Hk,但不能表成p1+p22=n的数的个数,其中p1,p2表示素数.则在GRH下,Dk(N)<<N1-1/k(h(k)+1)+ε,这里k=2,3;h(2)=2,h(3)=8.     

What can we do with only a pair of rusty compasses?

Compasses are called rusty, if one can draw only the unit circle with them. We prove that from two points A and B, with only rusty compasses, one can draw the points of k-section of AB, and all the vertices of a regular n-gon which has a side AB, where k is any integer greater than 1, and n=3, 4, 5, 6, 8, 12, 17, 257, ..., etc. Generally, let A be (0, 0) and let B be (, 0), then one can draw all the points (x, y) where x and y are any elements in some regular 2 ^m -extension of the rational field, for m=1, 2, 3, ...     

Coupling for τ-Dependent Sequences and Applications

Let X be a real-valued random variable and a -algebra. We show that the minimum -distance between X and a random variable distributed as X and independant of can be viewed as a dependence coefficient ( ,X) whose definition is comparable (but different) to that of the usual -mixing coefficient between and (X). We compare this new coefficient to other well known measures of dependence, and we show that it can be easily computed in various situations, such as causal Bernoulli shifts or stable Markov chains defined via iterative random maps. Next, we use coupling techniques to obtain Bennett and Rosenthal-type inequalities for partial sums of -dependent sequences. The former is used to prove a strong invariance principle for partial sums.     

The lotto numbers L(n,3,p,2)

An (n,k,p,t)‐lotto design is an n‐set N and a set of k‐subsets of N (called blocks) such that for each p‐subset P of N, there is a block for which . The lotto number L(n,k,p,t) is the smallest number of blocks in an (n,k,p,t)‐lotto design. The numbers C(n,k,t) = L(n,k,t,t) are called covering numbers. It is easy to show that, for n ≥ k(p ? 1), For k = 3, we prove that equality holds if one of the following holds:

• (i) n is large, in particular
• (ii)
• (iii) 2 ≤ p ≤ 6.

© 2006 Wiley Periodicals, Inc. J Combin Designs 14: 333–350, 2006     

Spectral properties and asymptotic periodicity of flows in networks

We combine functional analytical and graph theoretical methods in order to study flows in networks. We show that these flows can be described by a strongly continuous operator semigroup on a Banach space. Using Perron-Frobenius spectral theory we then prove that this semigroup behaves asymptotically periodic.This paper was written during the authors stay at the Mathematisches Institut der Universität Tübingen. The first author was supported by Virtugrade Baden-Württenberg and the second by the Marie Curie Host Fellowship Spectral theory for evolution equations contract number HPMT-CT-2001-00315. Both authors thank Rainer Nagel for many helpful discussions.     

Realizations for Schur upper triangular operators centered at an arbitrary point

Reproducing kernel spaces introduced by L. de Branges and J. Rovnyak provide isometric, coisometric and unitary realizations for Schur functions, i.e. for matrix-valued functions analytic and contractive in the open unit disk. In our previous paper [12] we showed that similar realizations exist in the nonstationary setting, i.e. when one considers upper triangular contractions (which appear in time-variant system theory as transfer functions of dissipative systems) rather than Schur functions and diagonal operators rather than complex numbers. We considered in [12] realizations centered at the origin. In the present paper we study realizations of a more general kind, centered at an arbitrary diagonal operator. Analogous realizations (centered at a point of the open unit disk) for Schur functions were introduced and studied in [3] and [4].     

新同伦不变数量 Ⅰ.一个短正合序列

<正> 自从Poincaré引入同调概念以来,人们一直想用依附于空间的各种代数结构来区分空间本身.这种代数结构,经过许多年和许多人的工作,越来越精细,越来越富有成果.如从同调论引伸出来的广义同调论,特别是其中的K理论,显示了很大的威力.但由于新代数结构的深入细致,不免也愈益复杂.