A simplification of a result by zellini on the maximal rank of symmetric three-way arrays

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of ordern ×n as a linear combination of 1/2n(n+1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetricn ×n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In the present paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n+1) fixed binary matrices of rank one. The decomposition implies that ann ×n ×p array consisting ofp symmetricn ×n slabs has maximal rank 1/2n(n+1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n+1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n–1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large asp, the number of matrices analyzed.     

Authenticity as a foundational principle of medical ethics

Increasingly, contemporary medical ethicists have become aware of the need to explicate a foundation for their various models of applied ethics. Many of these theories are inspired by the apparent incompatibility of patient autonomy and provider beneficence. The principle of patient autonomy derives its current primacy to a large extent from its legal origins. However, this principle seems at odds with the clinical reality. In the bioethical literature, the notion of authenticity has been proposed as an alternative foundational principle to autonomy. This article examines this proposal in reference to various existentialist philosophers (Heidegger, Sartre, Camus and Marcel). It is concluded that the principle of autonomy fails to do what it is commonly supposed to do: provide a criterion of distinction that can be invoked to settle moral controversies between patients and providers. The existentialist concept of authenticity is more promising in at least one crucial respect: It acknowledges that the essence of human life disappears from sight if life's temporal character is reduced to a series of present decisions and actions. This also implies that the very quest for a criterion that allows physicians to distinguish between sudden, unexpected decisions of their patients to be or not to be respected, without recourse to the patient's past or future, is erroneous.     

Classical geometric illusion effects with nonclassical stimuli: Angular induction from decomposing lines into point arrays

Angular induction is the process by which one line segment can bias judgment of orientation and/or collinearity of another segment, and it has been established that the magnitude of error is a determinate function of the relative angle between the two. We examined how these known relationships are affected by decomposing the induction segment into an array of scattered points. The bias that was produced by such arrays was found to be consistent with a formal model of angular induction, with the strength of the effect decreasing as the scatter among the points was increased. This decline in strength was almost linear with a logarithmic transform of the dimensions of the stimulus array. We also evaluated the hypothesis that the induction stimulus is detected by one or more channels—for example, neurons—for which the sensitivity profiles are modeled as Gabor wavelets. The change in induction strength with increasing point scatter was not predicted by a single width of channel. However, the combined activity of an ensemble of channels that differed in width did match the perceptual effects if one also stipulated that each channel would respond maximally to a fine-line stimulus.     

Decompositions and biplots in three-way correspondence analysis

In this paper correspondence analysis for three-way contingency tables is presented using three-way generalisations of the singular value decomposition. It is shown that in combination with Lancaster's (1951) additive decomposition of interactions in three-way tables, a detailed analysis is possible of the deviations from independence. Finally, biplots are shown to produce powerful graphical representations of the results from three-way correspondence analyses. An example from child development is used to illustrate the theoretical developments.     

Some uniqueness results for PARAFAC2

Whereas the unique axes properties of PARAFAC1 have been examined extensively, little is known about uniqueness properties for the PARAFAC2 model for covariance matrices. This paper is concerned with uniqueness in the rank two case of PARAFAC2. For this case, Harshman and Lundy have recently shown, subject to mild assumptions, that PARAFAC2 is unique when five (covariance) matrices are analyzed. In the present paper, this result is sharpened. PARAFAC2 is shown to be usually unique with four matrices. With three matrices it is not unique unless a certain additional assumption is introduced. If, for instance, the diagonal matrices of weights are constrained to be non-negative, three matrices are enough to have uniqueness in the rank two case of PARAFAC2. The authors are obliged to Richard Harshman for stimulating this research, and to the Associate Editor and reviewers for suggesting major improvements in the presentation.     

Detachment and defeasibility in deontic logic

The purpose of the paper is to present a logical framework that allow to formalize a kind of prima facie duties, defeasible conditional duties, indefeasible conditional duties and actual (indefeasible) duties, as well as to show their logical interconnections.     

Spatial compatibility of signal-response position in pointing

Two experiments are reported in which, by means of a pointing task, we studied the stimulus-position effect, i.e. the inverted U-shape form of the reaction-time function in relation to stimulus position in tasks in which stimuli and/or responses are arranged in a horizontal array. The response consisted of aiming the index finger from a central starting point at a target area on a screen. Reaction time was the main dependent variable. The spatial relation between the position of the imperative signal and the position of the response was manipulated by varying the spatial S-R compatibility and physical distance that separated the positions of stimulus and response. The stimulus-position effect was shown to depend on the compatibility of the S-R relation (Exp. 1). In Exp. 2 it was found that the modulation of the stimulus-position effect by spatial compatibility disappeared completely when the distance between the positions of stimulus and response was reduced. None of the experiments revealed that the stimulus position effect depended on signal discriminability, which renders an interpretation of this effect in terms of perceptual processes unlikely. We argue that the attentional model of spatial coding provides the most reasonable explanation of the obtained reaction-time patterns.