Is perception of upper body orientation based on the inertia tensor? Normogravity versus microgravity conditions

During lateral leg raising, a synergistic inclination of the supporting leg and trunk in the opposite direction to the leg movement is performed in order to preserve equilibrium. As first hypothesized by Pagano and Turvey (J Exp Psychol Hum Percept Perform, 1995, 21:1070-1087), the perception of limb orientation could be based on the orientation of the limb's inertia tensor. The purpose of this study was thus to explore whether the final upper body orientation (trunk inclination relative to vertical) depends on changes in the trunk inertia tensor. We imposed a loading condition, with total mass of 4 kg added to the subject's trunk in either a symmetrical or asymmetrical configuration. This changed the orientation of the trunk inertia tensor while keeping the total trunk mass constant. In order to separate any effects of the inertia tensor from the effects of gravitational torque, the experiment was carried out in normo- and microgravity. The results indicated that in normogravity the same final upper body orientation was maintained irrespective of the loading condition. In microgravity, regardless of loading conditions the same (but different from the normogravity) orientation of the upper body was achieved through different joint organizations: two joints (the hip and ankle joints of the supporting leg) in the asymmetrical loading condition, and one (hip) in the symmetrical loading condition. In order to determine whether the different orientations of the inertia tensor were perceived during the movement, the interjoint coordination was quantified by performing a principal components analysis (PCA) on the supporting and moving hips and on the supporting ankle joints. It was expected that different loading conditions would modify the principal component of the PCA. In normogravity, asymmetrical loading decreased the coupling between joints, while in microgravity a strong coupling was preserved whatever the loading condition. It was concluded that the trunk inertia tensor did not play a role during the lateral leg raising task because in spite of the absence of gravitational torque the final upper body orientation and the interjoint coupling were not influenced.

On subspace lattices I. Closedness type properties and tensor products

We study properties of subspace lattices related to the continuity of the map Lat and the notion of reflexivity. We characterize various “closedness” properties in different ways and give the hierarchy between them. We investigate several properties related to tensor products of subspace lattices and show that the tensor product of the projection lattices of two von Neumann algebras, one of which is injective, is reflexive.

Tensor products of operator systems

The purpose of the present paper is to lay the foundations for a systematic study of tensor products of operator systems. After giving an axiomatic definition of tensor products in this category, we examine in detail several particular examples of tensor products, including a minimal, maximal, maximal commuting, maximal injective and some asymmetric tensor products. We characterize these tensor products in terms of their universal properties and give descriptions of their positive cones. We also characterize the corresponding tensor products of operator spaces induced by a certain canonical inclusion of an operator space into an operator system. We examine notions of nuclearity for our tensor products which, on the category of C*-algebras, reduce to the classical notion. We exhibit an operator system S which is not completely order isomorphic to a C*-algebra yet has the property that for every C*-algebra A, the minimal and maximal tensor product of S and A are equal.