Recurrence for discrete time unitary evolutions

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector \phi. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to \phi. We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.Comment: 27 pages, 5 figures, typos correcte

Fatou’s Associates

Suppose that f is a transcendental entire function, V⊊C is a simply connected domain, and U is a connected component of f-1(V). Using Riemann maps, we associate the map f : U→V to an inner function g : D→D. It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain

Fish Intelligence, Sentience and Ethics

Fish are one of the most highly utilised vertebrate taxa by humans; they are harvested from wild stocks as part of global fishing industries, grown under intensive aquaculture conditions, are the most common pet and are widely used for scientific research. But fish are seldom afforded the same level of compassion or welfare as warm-blooded vertebrates. Part of the problem is the large gap between people’s perception of fish intelligence and the scientific reality. This is an important issue because public perception guides government policy. The perception of an animal’s intelligence often drives our decision whether or not to include them in our moral circle. From a welfare perspective, most researchers would suggest that if an animal is sentient, then it can most likely suffer and should therefore be offered some form of formal protection. There has been a debate about fish welfare for decades which centres on the question of whether they are sentient or conscious. The implications for affording the same level of protection to fish as other vertebrates are great, not least because of fishing-related industries. Here, I review the current state of knowledge of fish cognition starting with their sensory perception and moving on to cognition. The review reveals that fish perception and cognitive abilities often match or exceed other vertebrates. A review of the evidence for pain perception strongly suggests that fish experience pain in a manner similar to the rest of the vertebrates. Although scientists cannot provide a definitive answer on the level of consciousness for any nonhuman vertebrate, the extensive evidence of fish behavioural and cognitive sophistication and pain perception suggests that best practice would be to lend fish the same level of protection as any other vertebrate

Social preferences and network structure in a population of reef manta rays

Understanding how individual behavior shapes the structure and ecology ofpopulations is key to species conservation and management. Like manyelasmobranchs, manta rays are highly mobile and wide ranging species threatened byanthropogenic impacts. In shallow-water environments these pelagic rays often formgroups, and perform several apparently socially-mediated behaviors. Group structuresmay result from active choices of individual rays to interact, or passive processes.Social behavior is known to affect spatial ecology in other elasmobranchs, but this isthe first study providing quantitative evidence for structured social relationships inmanta rays. To construct social networks, we collected data from more than 500groups of reef manta rays over five years, in the Raja Ampat Regency of West Papua.We used generalized affiliation indices to isolate social preferences from non-socialassociations, the first study on elasmobranchs to use this method. Longer lastingsocial preferences were detected mostly between female rays. We detectedassortment of social relations by phenotype and variation in social strategies, with theoverall social network divided into two main communities. Overall network structurewas characteristic of a dynamic fission-fusion society, with differentiated relationshipslinked to strong fidelity to cleaning station sites. Our results suggest that fine-scaleconservation measures will be useful in protecting social groups of M. alfredi in theirnatural habitats, and that a more complete understanding of the social nature of mantarays will help predict population response