A generalization of Kristof's theorem on the trace of certain matrix products

Kristof has derived a theorem on the maximum and minimum of the trace of matrix products of the form \(X_1 \hat \Gamma _1 X_2 \hat \Gamma _2 \cdots X_n \hat \Gamma _n\) where the matrices \(\hat \Gamma _i\) are diagonal and fixed and theX _i vary unrestrictedly and independently over the set of orthonormal matrices. The theorem is a useful tool in deriving maxima and minima of matrix trace functions subject to orthogonality constraints. The present paper contains a generalization of Kristof's theorem to the case where theX _i are merely required to be submatrices of orthonormal matrices and to have a specified maximum rank. The generalized theorem contains the Schwarz inequality as a special case. Various examples from the psychometric literature, illustrating the practical use of the generalized theorem, are discussed.     

Some clarifications of the CANDECOMP algorithm applied to INDSCAL

Carroll and Chang have claimed that CANDECOMP applied to symmetric matrices yields equivalent coordinate matrices, as needed for INDSCAL. Although this claim has appeared to be valid for all practical purposes, it has gone without a rigorous mathematical footing. The purpose of the present paper is to clarify CANDECOMP in this respect. It is shown that equivalent coordinate matrices are not granted at global minima when the symmetric matrices are not Gramian, or when these matrices are Gramian but the solution not globally optimal.Part of this research has been supported by The Netherlands Organization for Scientific Research (NWO), PSYCHON-grant (560-267-011).     

Psychometric characteristics of Chicano and non-Hispanic White children's performance on the preschool inventory

Two studies are reported that independently examined the psychometric characteristics of children's performance on the Preschool Inventory. The sample in Study I consisted of 100 Chicano children tested longitudinally at ages $\text{3}\text{1}\text{2}$ and 4 years. The sample in Study II consisted of 200 Chicano and non-Hispanic White children matched on mothers' schooling level in a cross-sectional design involving two age levels—prekindergarten-age children and children enrolled in kindergarten. Comparisons were made by ethnic group, sex, and age level with regard to item characteristics, internal-consistency reliabilities, test—retest stability, standard errors of measurement, validity, and performance level. Comparisons also were made between the psychometric characteristics of the Chicano children's performance on the Preschool Inventory and the psychometric characteristics reportedly by other investigators for the same instrument administered to children of other ethnic, socioeconomic, and sociolinguistic backgrounds. The results provide impressive evidence of the Preschool Inventory's psychometric strengths and demonstrate that the examined psychometric characteristics of the Preschool Inventory are as exemplary when the test is administered to Chicano children as they are exemplary when it is administered to children of other ethnic backgrounds. Certain limitations of the Preschool Inventory are identified. Recommendations are made regarding possible avenues for future test development efforts.     

Does awareness of speech as a sequence of phones arise spontaneously?

It was found that illiterate adults could neither delete nor add a phone at the beginning of a non-word; but these tasks were rather easily performed by people with similar environment and childhood experiences, who learned to read rudimentarily as adults. Awareness of speech as a sequence of phones is thus not attained spontaneously in the course of general cognitive growth, but demands some specific training, which, for most persons, is probably provided by learning to read in the alphabetic system.     

Revisiting Reichenbach’s logic

Synthese - In this paper we show that, when analyzed with contemporary tools in logic—such as Dunn-style semantics, Reichenbach’s three-valued logic exhibits many interesting features,...