The Young-Householder algorithm and the least squares multidimensional scaling of squared distances

It is shown that replacement of the zero diagonal elements of the symmetric data matrix of approximate squared distances by certain other quantities in the Young-Householder algorithm will yield a least squares fit to squared distances instead of to scalar products. Iterative algorithms for obtaining these replacement diagonal elements are described and relationships with the ELEGANT algorithm (de Leeuw 1975; Takane 1977) are discussed. In large residual situations a penalty function approach, motivated by the ELEGANT algorithm, is adopted. Empirical comparisons of the algorithms are given.An early version of this paper was presented at the Multidimensional Data Analysis Workshop, Pembroke College, Cambridge, July 1985. I want to thank Jan de Leeuw and Yoshio Takane for bringing the ELEGANT algorithm to my attention and for clarifying its rationale and notation. My thanks go also to Stephen du Toit for help with the ALSCAL computations reported in Section 7.     

用爆炸波起爆癸烷云雾的两种模式

用垂直激波管研究了400μm癸烷液滴与空气混合物的爆炸波起爆。每种混合物有两个极限起爆能值,即高值Ecu,和低值Ecl。当起爆能的值高于Ecu时,观察到100%发生爆轰;而当其低于Ecl时,就不会发生爆轰。有趣的是,当起爆能水平在Ecu和Ecl之间时观察到对应于不同的起爆能水平有不同的起爆行为。如实验结果所表明的,爆轰的激发不仅是由于直接起爆,而且也由于“爆燃”到爆轰的转变。     

Bolzano,Cauchy and the “new analysis” of the early nineteenth century

Summary In this paper I discuss the development of mathematical analysis during the second and third decades of the nineteenth century; and in particular I assert that the well-known correspondence of new ideas to be found in the writings of Bolzano and Cauchy is not a coincidence, but that Cauchy had read one particular paper of Bolzano and drew on its results without acknowledgement. The reasons for this conjecture involve not only the texts in question but also the state of development of mathematical analysis itself, Cauchy both as personality and as mathematician, and the rivalries which were prevalent in Paris at that time.