Three constructive examples of separable pre-Hilbert spaces nonisomorphic to their closed hyperplanes

An example of a sigma -compact infinite-dimensional pre-Hilbert space H is constructed such that any continuous linear operator T: H --> H is of the form T = lambdaI + F for some lambda is an element of R and for a finite-dimensional continuous linear operator F. A class of simple examples of pre-Hilbert spaces nonisomorphic to their closed hyperplanes is given. A sigma -compact pre-Hilbert space H isomorphic to H x R x R and nonisomorphic to H x R is also constructed.

An infinite-dimensional separable pre-Hilbert space non-homeomorphic to any of its closed hyperplanes

An example is constructed of an infinite-dimensional separable pre-Hilbert space non-homeomorphic to any of its closed hyperplanes.

Fighting for shells: how private information about resource value changes hermit crab pre-fight displays and escalated fight behaviour

Pre-fight displays typically provide honest, but sometimes dishonest, information about resource holding potential and may be influenced by assessment of resource value and hence motivation to acquire the resource. These assessments of potential costs and benefits are also predicted to influence escalated fight behaviour. This is examined in shell exchange contests of hermit crabs in which we establish an information asymmetry about a particularly poor quality shell. The poor shell was created by gluing sand to the interior whereas control shells lacked sand and the low value of the poor shell could not be accurately assessed by the opponent. Crabs in the poor shell showed changes in the use of pre-fight displays, apparently to increase the chances of swapping shells. When the fights escalated, crabs in poor shells fought harder if they took the role of attacker but gave up quickly if in the defender role. These tactics appear to be adaptive but do not result in a major shift in the roles taken or outcome. We thus link resource assessment with pre-fight displays, the roles taken, tactics used during escalation and the outcome of these contests.