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)

Predicting for activity of second-line trastuzumab-based therapy in her2-positive advanced breast cancer

<p>Abstract</p> <p>Background</p> <p>In Her2-positive advanced breast cancer, the upfront use of trastuzumab is well established. Upon progression on first-line therapy, patients may be switched to lapatinib. Others however remain candidates for continued antibody treatment (treatment beyond progression). Here, we aimed to identify factors predicting for activity of second-line trastuzumab-based therapy.</p> <p>Methods</p> <p>Ninety-seven patients treated with > 1 line of trastuzumab-containing therapy were available for this analysis. Her2-status was determined by immunohistochemistry and re-analyzed by FISH if a score of 2+ was gained. Time to progression (TTP) on second-line therapy was defined as primary study endpoint. TTP and overall survival (OS) were estimated using the Kaplan-Meier product limit method. Multivariate analyses (Cox proportional hazards model, multinomial logistic regression) were applied in order to identify factors associated with TTP, response, OS, and incidence of brain metastases. <it>p </it>values < 0.05 were considered to indicate statistical significance.</p> <p>Results</p> <p>Median TTP on second-line trastuzumab-based therapy was 7 months (95% CI 5.74-8.26), and 8 months (95% CI 6.25-9.74) on first-line, respectively (n.s.). In the multivariate models, none of the clinical or histopthological features could reliably predict for activity of second-line trastuzumab-based treatment. OS was 43 months suggesting improved survival in patients treated with trastuzumab in multiple-lines. A significant deterioration of cardiac function was observed in three patients; 40.2% developed brain metastases while on second-line trastuzumab or thereafter.</p> <p>Conclusion</p> <p>Trastuzumab beyond progression showed considerable activity. None of the variables investigated correlated with activity of second-line therapy. In order to predict for activity of second-line trastuzumab, it appears necessary to evaluate factors known to confer trastuzumab-resistance.</p

The iterated minimum modulus and conjectures of Baker and Eremenko

In transcendental dynamics significant progress has been made by studying points whose iterates escape to infinity at least as fast as iterates of the maximum modulus. Here we take the novel approach of studying points whose iterates escape at least as fast as iterates of the minimum modulus, and obtain new results related to Eremenko's conjecture and Baker's conjecture, and the rate of escape in Baker domains. To do this we prove a result of wider interest concerning the existence of points that escape to infinity under the iteration of a positive continuous function

A new characterisation of the Eremenko-Lyubich class

The Eremenko-Lyubich class of transcendental entire functions with a bounded set of singular values has been much studied. We give a new characterisation of this class of functions. We also give a new result regarding direct singularities which are not logarithmic