Hyperbolic entire functions and the Eremenko–Lyubich class: Class B or not class B?

Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire functions. There does not appear to be an agreed definition of the concept in this context, due to complications arising from the non-compactness of the phase space. In this article, we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity. We show that this definition is equivalent to another commonly used one: a transcendental entire function is hyperbolic if and only if its postsingular set is a compact subset of the Fatou set. This leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions, and, in particular, that speaking about hyperbolicity makes sense only within the Eremenko–Lyubich classB of transcendental entire functions with a bounded set of singular values. We also considerably strengthen a recent characterisation of the class B, by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially. In particular, this implies that no transcendental entire function can be expanding with respect to the spherical metric. Finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let f:Rd→Rd be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates fn tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f)

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

Functions of genus zero for which the fast escaping set has Hausdorff dimension two

We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside the Eremenko-Lyubich class. We show that for functions in this family the fast escaping set has Hausdorff dimension equal to two

Towards evolutionary ambient assisted living systems

Ambient Assisted Living (AAL) is advocated as technological solutions that will enable the elderly population maintain their independence for a longer time than would otherwise be the case. Though the facts motivating the need for AAL are indisputable, the inherently heterogeneous nature and requirements of the elderly population raise significant difficulties. One particular challenge is that of designing AAL systems that can evolve to meet the requirements of individuals as their needs and circumstances change. This demands the availability of an adaptive, open, scalable software platform that incorporates a select combination of autonomic and intelligent techniques. Given that the first generation of AAL systems will be deployed in the near future, it is incumbent on designers to factor this need for evolution and adaptivity in their designs and implementations. Thus this paper explores AAL from a number of prospective and considers an agent-based middleware approach to realising an architecture for evolutionary AAL.Science Foundation Irelan

Kids in action : participatory health research with children

Involving children in participatory health research (PHR) provides exciting opportunities to gain insights into their perspectives and capacities and encourages them to make a meaningful contribution to issues affecting their lives. It is underpinned by a rights-based approach, where children’s evolving expertise is valued. In PHR, children are not just research participants – they are co-researchers. This raises challenges and ethical issues on several levels, including the nature of child engagement, the role of adults, and methods to promote child agency, while also ensuring their safety is not compromised. This chapter uses real international exemplars to illuminate the complexity and debate on theory and practice in relation to PHR with children. These examples reflect the flexibility inherent in the approach to accommodate culture and context. They also highlight the need to avoid tokenism, which might undermine children’s agency, and encourage first attempts, reflexive practice, and progressive improvements in PHR competence. Children’s right to subvert or deny the participatory process is recognized as a challenge to the adults involved but is also a way to realize impacts and outcomes that are more relevant for those children. Finally, Kids in Action is presented as an international network of PHR projects involving children to promote best practice, develop and share resources, link similar projects to maximize impacts, and provide a platform for children’s voices in relation to global issues

Aging society and gerontechnology: A solution for an independent living?

International audienceRecent studies report that the majority of older adults wish to live in their own homes, for as long as possible. This creates a growing interest in technologies to enable older people to remain living independently at home. The purpose of this article is to provide a narrative review of current technology appropriate for older adults’ home use. The key research questions were as follow: 1- What is the evidence demonstrating that gerontechnologies are effective in enabling independent living? 2- What are devices designed specifically for frail elderly persons ? Several publications were identified about devices targeting social isolation (videophonic communication, affective orthotic devices or companion-type robots, personal emergency response systems [security]), autonomy loss (technologies for maintenance of autonomy in the activities of daily living) and cognitive disorders (cognitive orthotics, wandering management systems, telemonitoring). Very few articles dealt specifically with the frail older person. In particular, there was extremely limited evidence on use and efficacy of these devices within this population. There is a need to obtain a consensus on definition of the technologies, and also to revisit work strategies and develop innovative business models. To meet this goal, we need to create a network of technological companies, aging services organizations, end-users, academics, and government representatives to explore the real needs of the frail older population and to develop and validate new devices promoting aging at home