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New concepts related to approximating a Lipschitz function between Banach spaces by aane functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the(More)
BACKGROUND Trials of statin therapy have had conflicting findings on the risk of development of diabetes mellitus in patients given statins. We aimed to establish by a meta-analysis of published and unpublished data whether any relation exists between statin use and development of diabetes. METHODS We searched Medline, Embase, and the Cochrane Central(More)
Let f be a Lipschitz mapping of a separable Banach space X to a Banach space Y. We observe that the set of points at which f is diierentiable in a spanning set of directions but not G^ ateaux diierentiable is-directionally porous. Since Borel-directionally porous sets, in addition to being rst category sets, are null in Aronszajn's (or, equivalently, in(More)
OBJECT The ability to map cerebrovascular reactivity (CVR) at the tissue level in patients with moyamoya disease could have considerable impact on patient management, especially in guiding surgical intervention and assessing the effectiveness of surgical revascularization. This paper introduces a new noninvasive magnetic resonance (MR) imaging-based method(More)
BACKGROUND Whether intensive control of glucose reduces macrovascular events and all-cause mortality in individuals with type 2 diabetes mellitus is unclear. We undertook a meta-analysis of randomised controlled trials to determine whether intensive treatment is beneficial. METHODS We selected five prospective randomised controlled trials of 33 040(More)
BACKGROUND Statins increase the risk of new-onset type 2 diabetes mellitus. We aimed to assess whether this increase in risk is a consequence of inhibition of 3-hydroxy-3-methylglutaryl-CoA reductase (HMGCR), the intended drug target. METHODS We used single nucleotide polymorphisms in the HMGCR gene, rs17238484 (for the main analysis) and rs12916 (for a(More)
We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every ǫ > 0, a point of ǫ-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which appeared in the literature under a variety of names.(More)