In 1988, Mayer proved the remarkable fact that ∞ is an explosion point for the set E( f a ) of endpoints of the Julia set of f a : C → C; e z + a with a < −1; that is, the set E( f a ) is totally separated (in particular, it does not have any non-trivial connected subsets), but E( f a )∪{∞} is connected. Answering a question of Schleicher, we extend this result to the setẼ( f a ) of escaping endpoints in the sense of Schleicher and Zimmer, for any parameter a ∈ C for which the singular value a 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 setẼ( f ) ∪ {∞} is connected for any transcendental entire function f of finite order with bounded singular set. We also discuss corresponding results for all endpoints in the case of exponential maps; to do so, we establish a version of Thurston's no wandering triangles theorem for exponential maps.

Aim: The aim of the study is to objectively assess the benefits that patients gain by switching from conventional insulin injections to insulin pump therapy. Introduction: An Insulin pump is a device used to treat diabetes and by infusing the insulin through a subcutaneous needle. It has become increasingly popular over the last several decades. Its mode of action is the closest mimicker to the pancreas and in theory it should properly manage blood sugar. However its impact on diabetes control was found to be modest and the remarkable improvement in glycemic control was demonstrated in observational studies and could not be confirmed by randomized controlled trails. Methods: Our study is an observational cross sectional study done in a single center. Forty patients with type I diabetes on insulin pump therapy were retrospectively assessed and compared with a matched control group that received multiple daily insulin injections. Biochemical data were collected including hemoglobin A1C (HbA1c), and all patients were interviewed and asked to complete a validated questionnaire to assess the quality of life. Results: Compared with the control group, patients on insulin pump therapy showed significantly lower HbA1c levels, with a mean of 1.8, had fewer hypoglycemic episodes, and had higher scores on the quality of life questionnaire. Conclusion: In our center, the insulin pump therapy was shown to remarkably improve the diabetes control and quality of life in type I diabetes to a greater extent than what has been reported in many observational studies.

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