关于Siegel-Tatuzawa定理

设0＜e＜l/(6log10),x为模k的实原特征且k大于e1/e.则除去至多一个可能的例外特征外,均有这改进了 Hoffstein等人的结果.     

关于Ozawa的一个定理

本文讨论了ＣＭ分享两个值的亚纯函数的唯一性问题，改进了一些结果．例子表明定理的条件是精确的．     

速率函数零点定理的一个新证明

本给出了速率函数零点定理的一种简捷证明方法。     

关于Darboux定理

本文细致地讨论了 Darboux定理及有关导函数的几种情形 ,并指出了文献 [1 ]中的一点不足之处 .     

关于微分多项式零点的注记

The value distribution of differential polynomials is studied.The re- suits in this paper improve and generalize some previous theorems given by Yang Chungchun(On deficiencies of differential polynomials,Math.Z.,116(1970),197- 204),H.S.Gopalakrishna and S.S.Bhoosnurmath(On distribution of values of differential polynomials,Indian J.Pure Appl.Math.,17(1986),367-372),I.Lahiri (A note on distribution of nonhomogeneous differential polynomials,Hokkaido Math. J.,31(2002),453-458)and Yi Hongxun(On zeros of differential polynomials,Adv. in Math.,18(1989),335-351)et al.Examples show that the results in this paper are sharp.     

关于亚纯函数的唯一性

设f（z）是非常数亚纯函数,n是正整数,F（z）=,其中aj（z）（j=0,1,2,…,n）均是f（z）的小函数．本文证明了：若f（z）满足N（r,f）=s（r,f）,且f（z）=b1（z）F（z）=b2（z）,这里b1（z）、b2（z）为f（z）的小函数,b1（z）0,b2（z）0,δ（0,f）＞,则或者f·Fb1·b2．     

具有亏值的亚纯函数的唯一性定理

本文主要就具有某些特殊亏值关系的亚纯函数及其ｎ阶导数具有公共值的唯一性得到了一个一般性定理，改进并推广了许多先前已知的结果。     

Airy函数的零点分布

作为二阶微分方程f＇＇-zf=0的解,Airy函数有可列个零点且均为负数,本借助Macdonal函数,证明这一重要结论,其证明过程不涉及整函数阶的问题,是一种较为初等的证明方法。     

Zeros of 2-adic -functions and congruences for class numbers and fundamental units

We study the imaginary quadratic fields such that the Iwasawa -invariant equals 1, obtaining information on zeros of -adic -functions and relating this to congruences for fundamental units and class numbers.

     

Algebraic limit cycles of degree 4 for quadratic systems

Yablonskii (Differential Equations 2 (1996) 335) and Filipstov (Differential Equations 9 (1973) 983) proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems. It was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems. Chavarriga (A new example of a quartic algebraic limit cycle for quadratic sytems, Universitat de Lleida, Preprint 1999) found a third family of this kind of algebraic limit cycles. Here, we prove that quadratic systems have exactly four different families of algebraic limit cycles. The proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields.     

实二次域上理想类的zeta-函数在-1处的值

用连分数给出了实二次域理想类的zeta-函数-1处值的一个具体的计算公式.     

ON RIESZ THEOREM

We prove a Riesz type criterion for a class of metric monoids: Local compactness implies finiteness of the Hausdorff dimension (and also of the topological dimension). We construct topological groups showing the necessity of some conditions. We finally prove that for some metric topological spaces finiteness of the algebraic dimension is equivalent to the finiteness of the Hausdorff dimension.     

ON THE VITALI-HAHN-SAKS-NIKODYM THEOREM

Abstract

The classical Vitali-Hahn-Saks-Nikodym Theorem [5, Thm. I.4.8] gives a limit criterion for when a sequence of strongly additive vector measures on a σ-field of sets having their range in a Banach space can be expected to be uniformly strongly additive. In [16, Cor. 8], Saeki proved that the limit condition on the sequence of vector measures could be substantially weakened as long as the Banach space in play is “good enough”. Saeki's result was based upon his work on a class of set functions too large to have Rosenthal's Lemma at his disposal. In Section 2, we prove Saeki's result with Rosenthal's Lemma at the basis of our work and then augment our characterization of Banach spaces enjoying Saeki's result in [1] with another natural equivalent condition. In Section 3 we extend Saeki's result to Boolean algebras having the Subsequential Interpolation property.     

VARIATIONS ON BELYI'S THEOREM

We establish a simple criterion for a variety to be definedover a number field, which allows us to extend to higher dimensionseveral known results relative to the field of definition ofcomplex curves as well as providing some new results in dimensionone.     

On class number relations in characteristic two

A theorem of F. Hirzebruch relates continued fractions to class numbers of quadratic number fields. A version for function fields of odd characteristic was established by D. R. Hayes and C. D. González. We present here a complete treatment of the even characteristic theory, in particular, two class number relations involving continued fractions are derived, one of which is an analogue of the Hirzebruch relation in characteristic 2. Research partially supported by National Science Council, Republic of China.     

Quadratic fields with noncyclic 5- or 7-class groups

We shall show that the number of quadratic fields with absolute discriminant ≤x and noncyclic 5- or 7-class group is ≫x ^1/4 improving the existing known bound for g=5 and for g=7 in Byeon (Ramanujan J. 11:159–163, 2006). This work was supported by KRF-2005-070-C00004.     

关于Wielandt-Hoffman定理

关于正规矩阵的任意扰动,有下述定理成立. 定理1.设A为n阶正规矩阵,C为n阶任一矩阵.A的特征值为λ_1,…,λ_n,C的特征值为μ_1…,μ_n.C~H表示C的转置共轭,||·||_2与||·||_F分别表示矩阵的谱范数与Frobenius范数.记