丢番图方程X~2-(a~2+1)Y~4=3-4a

设a≥2是正整数.本文证明了:当a=2时,方程X~2一(a~2+1)Y~4=3-4a仅有正整数解(X,Y)=(20,3);当a=3时,该方程仅有2组互素的正整数解(X,Y)=(1,1)和(79,5);当a≥4且4a+1非平方数时,该方程最多有4组互素的正整数解(X,Y);当a≥4且4a+1为平方数时,该方程最多有5组互素的正整数解(X,Y).     

界面移动法测定离子迁移数实验的优化

在基础物理化学实验界面移动法测定离子迁移数中,针对传统实验方案所用镉电极存在的不足之处,改用铜电极进行了平行对比实验,发现采用铜电极的实验方案在保证实验数据有良好准确度的前提下,既能降低电极材料的毒性,也能降低实验相关设备的配置要求。     

基于探测器模型的MPPC标定方法

探测器的光子数分辨能力是精确描述多光子态的突破口,近年来受到广泛的关注。光子数可分辨探测器有望被进一步应用于量子光学基础研究、量子成像、量子计算、量子通讯以及远距离激光测距等领域。基于空间复用原理的多像素光子计数器具有常温下稳定高效的光子数可分辨探测能力,加上尺寸和成本上的优势使得它成为光子数可分辨探测应用的首选。针对多像素光子计数器,我们通过建模给出了能描述探测器量子特性的标准矫正方法,系统地考虑了探测器量子效率、噪声、各像素点感光面的光子入射效率对量子特性的影响,给出了不同光子数态的正定算子估值测量和魏格纳函数。     

On the Ordering of a Class of Graphs with Respect to their Matching Numbers

Based on the number of k-matching m(G,k) of a graph G, Gutman and Zhang introduced an order relation of graphs: for graphs G ₁ and G ₂, if m(G ₁,k) ≥ m(G ₂,k) for all k. The order relation has important applications in comparing Hosaya indices and energies of molecular graphs and has been widely studied. Especially, Gutman and Zhang gave complete orders of six classes of graphs with respect to the relation . In this paper, we consider a class of graphs with special structure, which is a generalization of a class of graphs studied by Gutman and Zhang. Some order relations in the class of graphs are given, and the maximum and minimum elements with respect to the order relation are determined. The new results can be applied to order some classes of graphs by their matching numbers.     

Four distinctive 1-D lanthanide carboxylate coordination polymers: Synthesis,crystal structures and spectral properties

Four novel lanthanide coordination polymers [Pr(mal)(OH)(bipy) · 2H₂O]_n (1), {[Dy¹(SBA)₃(H₂O)₂][Dy²(SBA)₃(H₂O)₂] · 4H₂O}_n (2), {[Tb(OHnic)(Onic)(H₂O)₅ · (OHnicH)] · H₂O}_n (3) and {[Sm(OHnic)(Onic)(H₂O)₅ · (OHnicH)] · H₂O}_n (4) (Hmal = maleic acid, HSBA = 4-sulfobenzoic acid, OHnicH = 6-hydroxynicotinic acid and bipy = 2,2′-bipyridine) have been synthesized and determined by single crystal X-ray diffraction. Complex 1 is a 1-D helical chain with seven-coordinated praseodymium centers. Complex 2 forms 1-D chain-like molecular structure containing two crystallographically unique dysprosium centers, the Dy¹ center is seven-coordinated while Dy² is eight-coordinated. The isomorphous complexes 3 and 4 exhibit an unprecedented 1-D chain-like polymeric structure through hydroxyl oxygen atoms of bridging Onic²⁻ anions linking up the neighboring central ions, and there exist three types of 6-OHnicH ligands in the structural unit which is rare for lanthanide carboxylate complexes. The photophysical properties of these complexes were studied using ultraviolet absorption spectra, fluorescence excitation and emission spectra.     

Teiji Takagi, Founder of the Japanese School of Modern Mathematics

This article is a brief historical report on Teiji Takagi which was prepared at the commencement of ‘Takagi Lectures’ of The Mathematical Society of Japan. The first of its two purposes is to give some informations on the circumstances of education and research of mathematics in Japan surrounding Takagi who could finally established himself as the founder of the Japanese school of modern mathematics. The other is a brief overview on Takagi’s works of mathematics some of which are still attractive to and influential on especially ambitious students of mathematics. The author hopes that careful readers may find some hints for the questions how and why Takagi was able to establish his class field theory. At the end of this article the readers will find an English translation of the preface of his book Algebraic theory of numbers (in Japanese) which is the only thing that he left for us to see his total view over class field theory after the establishment of Artin’s reciprocity law.     

Maximum number of colorings of (2k,k2)‐graphs

Let consist of all simple graphs on 2k vertices and edges. For a simple graph G and a positive integer , let denote the number of proper vertex colorings of G in at most colors, and let . We prove that and is the only extremal graph. We also prove that as . © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 135–148, 2007     

Density of universal classes of series‐parallel graphs

A class of graphs ordered by the homomorphism relation is universal if every countable partial order can be embedded in . It is known (see [ 1 , 3 ]) that the class of k‐colorable graphs, for any fixed , induces a universal partial order. In 4 , a surprisingly small subclass of which is a proper subclass of the series‐parallel graphs (the K₄‐minor‐free graphs) is shown to be universal. On another side, Pan and Zhu in 7 proved a density result that for each rational number , there is a K₄‐minor‐free graph with circular chromatic number equal to a/b. In this note, we show for each rational number a/b within this interval the class of K₄‐minor‐free graphs with circular chromatic number a/b is universal if and only if , 5/2 or 3. This shows yet another surprising richness of the K₄‐minor‐free class that it contains universal classes as dense as the rational numbers. © 2006 Wiley Periodicals, Inc. J Graph Theory