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)

Escape rate and Hausdorff measure for entire functions

The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.Comment: 24 pages; some errors corrected, proof of Theorem 2 shortene

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

Rigidity of escaping dynamics for transcendental entire functions

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that this conjugacy is essentially unique. In particular, we show that an Eremenko-Lyubich class function f has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f and g which belong to the same parameter space are conjugate on their sets of escaping points.Comment: 28 pages; 2 figures. Final version (October 2008). Various modificiations were made, including the introduction of Proposition 3.6, which was not formally stated previously, and the inclusion of a new figure. No major changes otherwis

Reduced neural synchronization of gamma-band MEG oscillations in first-degree relatives of children with autism

<p>Abstract</p> <p>Background</p> <p>Gamma-band oscillations recorded from human electrophysiological recordings, which may be associated with perceptual binding and neuronal connectivity, have been shown to be altered in people with autism. Transient auditory gamma-band responses, however, have not yet been investigated in autism or in the first-degree relatives of persons with the autism.</p> <p>Methods</p> <p>We measured transient evoked and induced magnetic gamma-band power and inter-trial phase-locking consistency in the magnetoencephalographic recordings of 16 parents of children with autism, 11 adults with autism and 16 control participants. Source space projection was used to separate left and right hemisphere transient gamma-band measures of power and phase-locking.</p> <p>Results</p> <p>Induced gamma-power at 40 Hz was significantly higher in the parent and autism groups than in controls, while evoked gamma-band power was reduced compared to controls. The phase-locking factor, a measure of phase consistency of neuronal responses with external stimuli, was significantly lower in the subjects with autism and the autism parent group, potentially explaining the difference between the evoked and induced power results.</p> <p>Conclusion</p> <p>These findings, especially in first degree relatives, suggest that gamma-band phase consistency and changes in induced versus induced power may be potentially useful endophenotypes for autism, particularly given emerging molecular mechanisms concerning the generation of gamma-band signals.</p

Protein disulphide isomerase-assisted functionalization of proteinaceous substrates

Protein disulphide isomerase (PDI) is an enzyme that catalyzes thiol-disulphide exchange reactions among a broad spectrum of substrates, including proteins and low-molecular thiols and disulphides. As the first protein-folding catalyst reported, the study of PDI has mainly involved the correct folding of several cysteine-containing proteins. Its application on the functionalization of protein-based materials has not been extensively reported. Herein, we review the applications of PDI on the modification of proteinaceous substrates and discuss its future potential. The mechanism involved in PDI functionalization of fibrous protein substrates is discussed in detail. These approaches allow innovative applications in textile dyeing and finishing, medical textiles, controlled drug delivery systems and hair or skin care products.We thank to FCT 'Fundacao para a Ciencia e Tecnologia' (scholarship SFRH/BD/38363/2007) for providing Margarida Fernandes the grant for PhD studies