EVERY LONGEST HAMILTONIAN PATH IN EVEN n-GONS

We single out every longest path of n-1 order that solves each of the [Formula: see text] Longest Euclidean Hamiltonian Path Problems on the even nth root of the unity, by means of a geometric and arithmetic procedure. This identification is done regardless of planar rotations and orientation. In addition, the uniqueness of the Euclidean Hamiltonian cycle that resolves the Maximum Traveling Salesman Problem is shown.

The Role of Edges in Object Recognition by Pigeons

In three experiments, we explored how pigeons use edges, corresponding to orientation and depth discontinuities, in visual recognition tasks. In experiment 1, we compared the pigeon's ability to recognize line drawings of four different geons when trained with shaded images. The birds were trained with either a single view or five different views of each object. Because the five training views had markedly different appearances and locations of shaded surfaces, reflectance edges, etc, the pigeons might have been expected to rely more on the orientation and depth discontinuities that were preserved over rotation and in the line drawings. In neither condition, however, was there any transfer from the rendered images to the outline drawings. In experiment 2, some pigeons were trained with line drawings and shaded images of the same objects associated with the same response (consistent condition), whereas other pigeons were trained with a line drawing and a shaded image of two different objects associated with the same response (inconsistent condition). If the pigeons perceived any correspondence between the stimulus types, then birds in the consistent condition should have learned the discrimination more quickly than birds in the inconsistent condition. But, there was no difference in performance between birds in the consistent and inconsistent conditions. In experiment 3, we explored pigeons' processing of edges by comparing their discrimination of shaded images or line drawings of four objects. Once trained, the pigeons were tested with planar rotations of those objects. The pigeons exhibited different patterns of generalization depending on whether they were trained with line drawings or shaded images. The results of these three experiments suggest that pigeons may place greater importance on surface features indicating materials, such as food or water. Such substances do not have definite boundaries—cued by edges—which are thought to be central to human recognition.