TRANSITIVITY AND ROTATION SETS WITH NONEMPTY INTERIOR FOR HOMEOMORPHISMS OF THE 2-TORUS

We show that if f is a homeomorphism of the 2-torus isotopic to the identity and its lift (f) over tilde is transitive, or even if it is transitive outside the lift of the elliptic islands, then (0,0) is in the interior of the rotation set of (f) over tilde. This proves a particular case of Boyland's conjecture.

ASSOCIATIVE SYMMETRY BY PIGEONS AFTER FEW-EXEMPLAR TRAINING

The present experiment investigated whether pigeons can show associative symmetry on a two-alternative matching to-sample procedure The procedure consisted of a within subject sequence of training and testing with reinforcement and It provided (a) exemplars of symmetrical responding and (b) all prerequisite discriminations among test samples and comparisons After pigeons had learned two arbitrary matching tasks (A B and C D) they were given a reinforced symmetry test for half of the baseline relations (B1-A1 and D1-C1) To control for the effects of reinforcement during testing two novel nonsymmetrical responses were concurrently reinforced using the other baseline stimuli (D2-A2 and B2-C2) Pigeons matched at chance on both types of relations thus indicating no evidence for symmetry These symmetrical and nonsymmetrical relations were then directly trained in order to provide exemplars of symmetry and all prerequisite discriminations for a second test The symmetrical test relations were now B2-A2 and D2-C2 and the nonsymmetrical relations were D1-A1 and B1-C1 On this test 1 pigeon showed clear evidence of symmetry 2 pigeons showed weak evidence and 1 pigeon showed no evidence The previous training of all prerequisite discriminations among stimuli and the within subject control for testing with reinforcement seem to have set favorable conditions for the emergence of symmetry in nonhumans However the variability across subjects shows that methodological variables still remain to be controlled

Evaluating links through spectral decomposition

Spectral decomposition has rarely been used to investigate complex networks. In this work we apply this concept in order to define two kinds of link-directed attacks while quantifying their respective effects on the topology. Several other kinds of more traditional attacks are also adopted and compared. These attacks had substantially diverse effects, depending on each specific network (models and real-world structures). It is also shown that the spectrally based attacks have special effects in affecting the transitivity of the networks.