Cartan-Calculus and its Generalizations via a Path-Integral Approach to Classical Mechanics

In this paper we review the recently proposed path-integral counterpart of the Koopman-von Neumann operatorial approach to classical Hamiltonian mechanics. We identify in particular the geometrical variables entering this formulation and show that they are essentially a basis of the cotangent bundle to the tangent bundle to phase-space. In this space we introduce an extended Poisson brackets structure which allows us to re-do all the usual Cartan calculus on symplectic manifolds via these brackets. We also briefly sketch how the Schouten-Nijenhuis, the Fr\"olicher- Nijenhuis and the Nijenhuis-Richardson brackets look in our formalism.Comment: 6 pages, amste

NEWTON's trajectories versus MOND's trajectories

MOND dynamics consists of a modification of the acceleration with respect to the one provided by Newtonian mechanics. In this paper we investigate whether it can be derived from a velocity-dependent deformation of the coordinates of the systems. The conclusion is that it cannot be derived this way because of the intrinsic non-local character in time of the MOND procedure. This is a feature pointed out some time ago already by Milgrom himself.Comment: Improved the abstract, the conclusions and inserted some further new reference

Classical and quantum mechanics via supermetrics in time

Koopman-von Neumann in the 30's gave an operatorial formululation of Classical Mechanics. It was shown later on that this formulation could also be written in a path-integral form. We will label this functional approach as CPI (for classical path-integral) to distinguish it from the quantum mechanical one, which we will indicate with QPI. In the CPI two Grassmannian partners of time make their natural appearance and in this manner time becomes something like a three dimensional supermanifold. Next we introduce a metric in this supermanifold and show that a particular choice of the supermetric reproduces the CPI while a different one gives the QPI.Comment: To appear in the proceedings of the conference held in Trieste in October 2008 with title: "Theoretical and Experimental aspects of the spin statistics connection and related symmetries

Dynamics of Attention in Depth: Evidence from Mutli-Element Tracking

The allocation of attention in depth is examined using a multi-element tracking paradigm. Observers are required to track a predefined subset of from two to eight elements in displays containing up to sixteen identical moving elements. We first show that depth cues, such as binocular disparity and occlusion through T-junctions, improve performance in a multi-element tracking task in the case where element boundaries are allowed to intersect in the depiction of motion in a single fronto-parallel plane. We also show that the allocation of attention across two perceptually distinguishable planar surfaces either fronto-parallel or receding at a slanting angle and defined by coplanar elements, is easier than allocation of attention within a single surface. The same result was not found when attention was required to be deployed across items of two color populations rather than of a single color. Our results suggest that, when surface information does not suffice to distinguish between targets and distractors that are embedded in these surfaces, division of attention across two surfaces aids in tracking moving targets.National Science Foundation (IRI-94-01659); Office of Naval Research (N00014-95-1-0409, N00014-95-1-0657

Attention in Depth: Disparity and Occlusion Cues Facilitate Multi-Element Visual Tracking

Human observers can track up to five moving targets in a display with ten identical elements (Pylyshyn and Storm, 1988; Yantis, 1992). Previous experiments manipulated element trajectories to prevent intersections of element boundaries, evidently in the belief that transient overlaps among homogeneous elements make the task too hard. We examine whether depth cues such as occlusion (T-junctions) and disparity affect performance in a tracking task when element boundaries, as projected onto the two-dimensional plane of the monitor screen, are allowed to intersect. Elements move smoothly in depth, as well as in horizontal and vertical position, throughout a 7-second tracking period. A probe is then flashed, and subjects report whether the flash occurred on a target or on a non-target. Overlapping circular objects form T-junctions when shaded to appear like spheres or figure eight regions when rendered as disks. Two factors, disparity and T-junctions, are considered. Results from eight naive observers show that performance improves for displays with depth information (T-junctions or disparity), suggesting that depth cues are useful for multi-element tracking.National Science Foundation (IRI-94-01659); Office of Naval Research (N00014-92-J-1309, N00014-95-1-0657, N00014-94-1-0597, N00014-95-1-0409