A numerical approach to the approximate and the exact minimum rank of a covariance matrix

A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.The authors are obliged to Paul Bekker for stimulating and helpful comments.     

The Icarus flight of speculation: Philosophers' vices as perceived by nineteenth-century historians and physicists

Why did nineteenth-century German historians and physicists habitually warn against vices that they believed philosophers in particular embodied: speculation, absence of common sense, and excessive systematizing? Drawing on a rich array of sources, this article interprets this vice-charging as a rhetorical practice aimed at delineating empirical research from Naturphilosophie and Geschichtsphilosophie as practiced in the heyday of German Idealism. The strawman of “the philosopher” as invoked by historians and physicists served as a negative model for strongly empiricist scholars committed to virtues like precision, thoroughness, and conscientiousness. In their historical narratives, historians and physicists consistently depicted philosophy either as a relic from the past or as a phase that they had virtuously left behind. This boundary work was central to the self-positioning of both history and physics, which makes clear that the persona of the philosopher mattered not only to philosophers but also to scholars in adjacent disciplines.     

The typical rank of tall three-way arrays

The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank ofI × J × K arrays withK = 2 to a particular class of arrays withK ≥ 2. It is shown that the typical rank isI when the array is tall in the sense thatJK − J < I < JK. In addition, typical rank results are given for the case whereI equalsJK − J. The author is obliged to Henk Kiers, Tom Snijders, and Philip Thijsse for helpful comments.     

Switching on and off … : Does smartphone use obstruct the possibility to engage in recovery activities?

This study examines the impact of work-related smartphone use on daily recovery from work-related efforts. The literature shows that work–home interference (WHI) is an important inhibitor of the recovery process. We propose that the extensive use of smartphones with its implicit request of 24/7 availability inhibits the process of engaging in activities that are required for daily recovery. A total of 80 employees (40 smartphone users, 40 controls) completed a 6-day diary questionnaire over a time period of 2 weeks. Contrary to our hypothesis, smartphone users did not experience more overall WHI than nonusers. Furthermore, four activities aimed at recovery were examined. We predicted that daily WHI would increase employees’ engagement in recovery activities, but only if they did not use a smartphone. Results showed that, for the control group, WHI was positively related to psychological detachment, relaxation, mastery, and control activities, whereas smartphone users facing high WHI did not succeed in engaging in these recovery activities. This implies that being connected to work in the evening hours through smartphones has consequences for the extent to which employees succeed in undertaking recovery activities.     

Complete axiomatizations for XPath fragments

We provide complete axiomatizations for several fragments of Core XPath, the navigational core of XPath 1.0 introduced by Gottlob, Koch and Pichler. A complete axiomatization for a given fragment is a set of equivalences from which every other valid equivalence is derivable; equivalences can be thought of as (undirected) rewrite rules. Specifically, we axiomatize single axis fragments of Core XPath as well as full Core XPath. Our completeness proofs use results and techniques from modal logic.     

Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices

A key feature of the analysis of three-way arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled from a continuous distribution, and the third component matrix has full column rank. In this context, we obtain almost sure sufficient uniqueness conditions for the Candecomp/Parafac and Indscal models separately, involving only the order of the three-way array and the number of components in the decomposition. Both uniqueness conditions are closer to necessity than the classical uniqueness condition by Kruskal. Part of this research was supported by (1) the Flemish Government: (a) Research Council K.U. Leuven: GOA-MEFISTO-666, GOA-Ambiorics, (b) F.W.O. project G.0240.99, (c) F.W.O. Research Communities ICCoS and ANMMM, (d) Tournesol project T2004.13; and (2) the Belgian Federal Science Policy Office: IUAP P5/22. Lieven De Lathauwer holds a permanent research position with the French Centre National de la Recherche Scientifique (C.N.R.S.). He also holds an honorary research position with the K.U. Leuven, Leuven, Belgium.