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.     

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     

A new exclusion test for finding the global minimum

Exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find the global minimum of a function over a compact domain. These algorithms are based on a minimization condition that can be applied to each cell in the domain. In this paper, we consider Lipschitz functions of order α and give a new minimization condition for the exclusion algorithm. Furthermore, convergence and complexity results are presented for such algorithm.     

Synthesis of 6-acylpyrido[2,3-<Emphasis Type="Italic">d</Emphasis>]pyrimidine derivatives from 5-acetyl-4-aminopyrimidine hydrochlorides

Reaction of 2,6-disubstituted 5-acetyl-4-aminopyrimidine hydrochlorides with acetylacetone or benzoylacetone afforded new substituted 6-acylpyrido[2,3-d]pyrimidines. The reaction of these hydrochlorides with ethyl acetoacetate also proceeds according to the classical version of the Friedländer condensation to form the corresponding pyrido[2,3-d]pyrimidine-6-carboxylic acid esters.     

Recognition of alkyl ketone molecules based on thickness-shear-mode acoustic sensors with calixarene derivatives

The frequency response characteristics for twenty-two organic vapours by piezoelectric thickness-shear-mode (TSM) acoustic wave sensors coated with four supramolecule compounds—calixarenes have been investigated. Among them, 2,8,14,20-tetraethyl-4,6,10,12,16,18,22,24-octahydroxylcalix[4]arene (I) was the most efficient actively adsorptive material for host-guest recognizing alkyl ketone molecules such as 2-butanone and acetone. The supramolecule recognition mechanism has been discussed, that is based on the formation of C-H?π bond interaction between the methyl group of ketone molecule and the phenyl ring of the calixarene compound. The linear range of the TSM sensor upon exposure to 2-butanone vapour was 0-940.5 ppm with a detection limit of 2.67 ppm when the coating mass of the compound I was selected as 19 μg. The kinetics behaviours in the adsorption and desorption processes have been examined with polynomial curve fitting procedure. Furthermore, the proposed TSM sensor possessed good selectivity, reversibility, reproducibility and high stability. Compared with gas chromatography (GC) method, the proposed sensor can be used for on-line determination of 2-butanone vapour in air with a recovery of 94.8-106.5%, which was in consistent with those obtained by GC method.     

A quadratic programming approach to the Randić index

Let G(k, n) be the set of connected graphs without multiple edges or loops which have n vertices and the minimum degree of vertices is k. The Randi? index χ = χ(G) of a graph G   is defined by χ(G)=_∑(uv)(_δuδv)^-1/2$\chi (G)={\sum }_{(\mathit{uv})}({\delta }_u{\delta }_v{)}^{-1/2}$, where δ_u is the degree of vertex u and the summation extends over all edges (uv) of G. Caporossi et al. [G. Caporossi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs IV: Chemical trees with extremal connectivity index, Computers and Chemistry 23 (1999) 469–477] proposed the use of linear programming as one of the tools for finding the extremal graphs. In this paper we introduce a new approach based on quadratic programming for finding the extremal graphs in G(k, n) for this index. We found the extremal graphs or gave good bounds for this index when the number n_k of vertices of degree k is between n − k and n. We also tried to find the graphs for which the Randi? index attained its minimum value with given k (k ? n/2) and n. We have solved this problem partially, that is, we have showed that the extremal graphs must have the number n_k of vertices of degree k less or equal n − k and the number of vertices of degree n − 1 less or equal k.     

贝利相与非线性输运

电子输运现象包含一系列很重要的材料性质,并可以用来提供关于载体物理系统的很多信息。在 最近的二三十年,人们逐渐认识到除了电子能谱之外,电子在布里渊区的几何性质,比如贝利曲率和 轨道磁矩,会在电子输运中起到关键性的作用。在线性输运现象中的此种关联已引起广泛的兴趣并得 到深入的研究。然而,在非线性输运现象中的电子几何性质的作用的研究最近才逐渐起步。此种关联 可以大大加深对各种非线性输运现象的认识,并对如何从材料控制上调节非线性现象的强度提供有价 值的研究视角和指导原则。基于此背景,本文试图引入研究输运现象的半经典理论框架,并在几种输 运现象中举例说明贝利相的作用。