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

Designers initiating open innovation with multi-stakeholder through co-reflection sessions

This paper explores a designerly approach to open innovation initiation as start of the PhD research of the third author. More specifically, it presents the application of co-reflection sessions by designers in a healthcare open innovation project to initiate multi-stakeholder participation. Integrating co-reflection in open innovation initiation provides designers with the opportunity to a) negotiate with and function in multi-disciplinary environments consisting of stakeholder representatives and stakeholder customers (possible end-users); b) analyze complexity and structure of stakeholder ambitions, wishes, concerns and restrictions in order to frame a collaboration space; c) synthesize, visualize and materialize the value proposition to communicate the benefits to multi-stakeholder networks in order to define a design space and motivate their participation; and what is more important, keeping the balance between design thinking and design action. Lessons learned from this study a) can be used to provide a set of skills and practical guidance to designers when initiating open innovation b) define a spectrum for research on how designers can initiate innovation

Discovery of polarised emission from the long period intermediate polar RX J2133.7+5107

Aims. We intended to investigate the magnetic field properties of the recently identified intermediate polar RX J2133.7+5107. Methods. We carried out UBVRI photopolarimetric observations of the target using TURPOL on the Nordic Optical Telescope over 2 nights in July/August 2006. Results. We found that RX J2133.7+5107 emits circularly polarized light in all UBVRI bands (up to 3%). This is the first detection of circular polarization in this object. The circular polarization modulations and flux variations give hints of cyclotron beaming effects and suggest that the field strength in RX J2133.7+5107 is possibly one of the highest found amongst the IPs. Conclusions. The highly asynchronous rotation of RX J2133.7+5107 (the spin to orbital period ratio is ~0.022), suggests that it has only recently come into contact and although it is likely to evolve into a polar, it is currently a long way from doing so. We suggest a possible link between the detection of a soft X-ray blackbody component and polarized optical emission in intermediate polars

The Deficit of Distant Galaxy Clusters in the RIXOS X-ray Survey

Clusters of galaxies are the largest gravitationally bound systems and therefore provide an important way of studying the formation and evolution of the large scale structure of the Universe. Cluster evolution can be inferred from observations of the X-ray emission of the gas in distant clusters, but interpreting these data is not straightforward. In a simplified view, clusters grow from perturbations in the matter distribution: their intracluster gas is compressed and shock-heated by the gravitational collapse$^{1}$. The resulting X-ray emission is determined by the hydrostatic equilibrium of the gas in the changing gravitational potential. However, if processes such as radiative cooling or pre-collapse heating of the gas are important, then the X-ray evolution will be strongly influenced by the thermal history of the gas. Here we present the first results from a faint flux-limited sample of X-ray selected clusters compiled as part of the ROSAT International X-ray and Optical Survey (RIXOS). Very few distant clusters have been identified. Most importantly, their redshift distribution appears to be inconsistent with simple models based on the evolution of the gravitational potential. Our results suggest that radiative cooling or non-gravitational heating of the intracluster gas must play an important role in the evolution of clusters.Comment: uuencoded compressed postscript. The preprint is also available at http://www.ast.cam.ac.uk/preprint/PrePrint.htm

Impaired osteoclast homeostasis in the cystatin B-deficient mouse model of progressive myoclonus epilepsy

Peer reviewe

Conformal loop ensembles and the stress-energy tensor

We give a construction of the stress-energy tensor of conformal field theory (CFT) as a local "object" in conformal loop ensembles CLE_\kappa, for all values of \kappa in the dilute regime 8/3 < \kappa <= 4 (corresponding to the central charges 0 < c <= 1, and including all CFT minimal models). We provide a quick introduction to CLE, a mathematical theory for random loops in simply connected domains with properties of conformal invariance, developed by Sheffield and Werner (2006). We consider its extension to more general regions of definition, and make various hypotheses that are needed for our construction and expected to hold for CLE in the dilute regime. Using this, we identify the stress-energy tensor in the context of CLE. This is done by deriving its associated conformal Ward identities for single insertions in CLE probability functions, along with the appropriate boundary conditions on simply connected domains; its properties under conformal maps, involving the Schwarzian derivative; and its one-point average in terms of the "relative partition function." Part of the construction is in the same spirit as, but widely generalizes, that found in the context of SLE_{8/3} by the author, Riva and Cardy (2006), which only dealt with the case of zero central charge in simply connected hyperbolic regions. We do not use the explicit construction of the CLE probability measure, but only its defining and expected general properties.Comment: 49 pages, 3 figures. This is a concatenated, reduced and simplified version of arXiv:0903.0372 and (especially) arXiv:0908.151

Modular Equations and Distortion Functions

Modular equations occur in number theory, but it is less known that such equations also occur in the study of deformation properties of quasiconformal mappings. The authors study two important plane quasiconformal distortion functions, obtaining monotonicity and convexity properties, and finding sharp bounds for them. Applications are provided that relate to the quasiconformal Schwarz Lemma and to Schottky's Theorem. These results also yield new bounds for singular values of complete elliptic integrals.Comment: 23 page

The Polyakov action on the supertorus

A consistent method for obtaining a well-defined Polyakov action on the supertorus is presented. This method uses the covariantization of derivative operators and enables us to construct a Polyakov action which is globally defined.Comment: 15 pages LaTe

Tsuji functions with segments of Julia

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46297/1/209_2005_Article_BF01112579.pd

Conformal Mappings and Dispersionless Toda hierarchy

Let $\mathfrak{D}$ be the space consists of pairs $(f,g)$, where $f$ is a univalent function on the unit disc with $f(0)=0$, $g$ is a univalent function on the exterior of the unit disc with $g(\infty)=\infty$ and $f'(0)g'(\infty)=1$. In this article, we define the time variables $t_n, n\in \Z$, on $\mathfrak{D}$ which are holomorphic with respect to the natural complex structure on $\mathfrak{D}$ and can serve as local complex coordinates for $\mathfrak{D}$. We show that the evolutions of the pair $(f,g)$ with respect to these time coordinates are governed by the dispersionless Toda hierarchy flows. An explicit tau function is constructed for the dispersionless Toda hierarchy. By restricting $\mathfrak{D}$ to the subspace $\Sigma$ consists of pairs where $f(w)=1/\bar{g(1/\bar{w})}$, we obtain the integrable hierarchy of conformal mappings considered by Wiegmann and Zabrodin \cite{WZ}. Since every $C^1$ homeomorphism $\gamma$ of the unit circle corresponds uniquely to an element $(f,g)$ of $\mathfrak{D}$ under the conformal welding $\gamma=g^{-1}\circ f$, the space $\text{Homeo}_{C}(S^1)$ can be naturally identified as a subspace of $\mathfrak{D}$ characterized by $f(S^1)=g(S^1)$. We show that we can naturally define complexified vector fields \pa_n, n\in \Z on $\text{Homeo}_{C}(S^1)$ so that the evolutions of $(f,g)$ on $\text{Homeo}_{C}(S^1)$ with respect to \pa_n satisfy the dispersionless Toda hierarchy. Finally, we show that there is a similar integrable structure for the Riemann mappings $(f^{-1}, g^{-1})$. Moreover, in the latter case, the time variables are Fourier coefficients of $\gamma$ and $1/\gamma^{-1}$.Comment: 23 pages. This is to replace the previous preprint arXiv:0808.072