Escaping endpoints explode

In 1988, Mayer proved the remarkable fact that infinity is an explosion point for the set of endpoints of the Julia set of an exponential map that has an attracting fixed point. That is, the set is totally separated (in particular, it does not have any nontrivial connected subsets), but its union with the point at infinity is connected. Answering a question of Schleicher, we extend this result to the set of "escaping endpoints" in the sense of Schleicher and Zimmer, for any exponential map for which the singular value belongs to an attracting or parabolic basin, has a finite orbit, or escapes to infinity under iteration (as well as many other classes of parameters). Furthermore, we extend one direction of the theorem to much greater generality, by proving that the set of escaping endpoints joined with infinity is connected for any transcendental entire function of finite order with bounded singular set. We also discuss corresponding results for *all* endpoints in the case of exponential maps; in order to do so, we establish a version of Thurston's "no wandering triangles" theorem.Comment: 35 pages. To appear in Comput. Methods Funct. Theory. V2: Authors' final accepted manuscript. Revisions and clarifications have been made throughout from V1. This includes improvements in the proof of Proposition 6.11 and Theorem 8.1, as well as corrections in Remarks 7.1 and 7.3 concerning differing definitions of escaping endpoints in greater generalit

Finite momentum at string endpoints

We argue that classical strings, both bosonic and supersymmetric, can have finite energy and momentum at their endpoints. We show that in a general curved background, string endpoints must propagate along null geodesics as long as their energy remains finite. Finite endpoint momentum allows strings with a fixed energy to travel a greater distance in an AdS5-Schwarzschild background than has been possible for classical solutions considered previously. We review the relevance to heavy ion phenomenology of the dependence of this distance on energy, and we propose a scheme for determining the instantaneous rate of energy loss.Comment: 28 pages, 4 figure

Role of patient-reported outcomes and other efficacy endpoints in the drug approval process in Europe (2008-2012)

The present study aimed at systematically reviewing the role and extent of patient-reported outcomes (PROs) usage within the package of scientific evidence considered for marketing authorization (MA). All regulatory information published by the European Medicines Agency (EMA) for products authorized between January 2008 and December 2012 and appearing in the European Public Assessment Report (EPAR) database was examined for efficacy endpoints. The endpoints here considered included: PROs, clinician reported outcomes (CROs), and laboratory reported outcomes (LROs). LROs were the most frequently reported endpoints. Out of the 180 products here selected, 99 (55%), 67 (37%), and 30 (17%), respectively, used LROs, CROs and PROs as primary endpoints (PEs). PROs as any endpoints were used in 82 (46%) products. Out of these, PROs were documented as PE in 30 (37%), with 27 (33%) products having used PROs both as primary and non-PEs. PRO usage was most frequently identified with nervous system and antineoplastic agents. During the study period, the use of all the three types of endpoints appeared to be static. Both the regulatory bodies and the industry should ensure complete and clear reporting of all endpoints used, including PROs, to improve transparency.Peer reviewedFinal Published versio

Integrating patients' views into health technology assessment: Analytic hierarchy process (AHP) as a method to elicit patient preferences

Background: Patient involvement is widely acknowledged to be a valuable component in health technology assessment (HTA) and healthcare decision making. However, quantitative approaches to ascertain patients' preferences for treatment endpoints are not yet established. The objective of this study is to introduce the analytic hierarchy process (AHP) as a preference elicitation method in HTA. Based on a systematic literature review on the use of AHP in health care in 2009, the German Institute for Quality and Efficiency in Health Care (IQWiG) initiated an AHP study related to its HTA work in 2010. - \ud Methods: The AHP study included two AHP workshops, one with twelve patients and one with seven healthcare professionals. In these workshops, both patients and professionals rated their preferences with respect to the importance of different endpoints of antidepressant treatment by a pairwise comparison of individual endpoints. These comparisons were performed and evaluated by the AHP method and relative weights were generated for each endpoint. - \ud Results: The AHP study indicates that AHP is a well-structured technique whose cognitive demands were well handled by patients and professionals. The two groups rated some of the included endpoints of antidepressant treatment differently. For both groups, however, the same six of the eleven endpoints analyzed accounted for more than 80 percent of the total weight. - \ud Conclusions: AHP can be used in HTA to give a quantitative dimension to patients' preferences for treatment endpoints. Preference elicitation could provide important information at various stages of HTA and challenge opinions on the importance of endpoints

Non-escaping endpoints do not explode

The family of exponential maps ƒα(z)=ez + α is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set J(ƒα). When α ∈ (−∞,−1), and more generally when α belongs to the Fatou set F(ƒα), it is known that J(ƒα) can be written as a union of hairs and endpoints of these hairs. In 1990, Mayer proved for α ∈ (−∞,−1) that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to the case where α ∈ F(ƒα), and showed that it holds even for the smaller set of all escaping endpoints. We show that, in contrast, the set of non-escaping endpoints together with infinity is totally separated. It turns out that this property is closely related to a topological structure known as a ‘spider’s web’; in particular we give a new topological characterisation of spiders’ webs that maybe of independent interest. We also show how our results can be applied to Fatou’s function, z ↦ z + 1 + e−z