Sleep-disordered breathing in acute ischemic stroke and transient ischemic attack: effects on short- and long-term outcome and efficacy of treatment with continuous positive airways pressure - rationale and design of the SAS CARE study

Sleep-disordered breathing represents a risk factor for cardiovascular morbidity and mortality and negatively affects short-term and long-term outcome after an ischemic stroke or transient ischemic attack. The effect of continuous positive airways pressure in patients with sleep-disordered breathing and acute cerebrovascular event is poorly known. The SAS CARE 1 study assesses the effects of sleep-disordered breathing on clinical evolution, vascular functions, and markers within the first three-months after an acute cerebrovascular event. The SAS CARE 2 assesses the effect of continuous positive airways pressure on clinical evolution, cardiovascular events, and mortality as well as vascular functions and markers at 12 and 24 months after acute cerebrovascular event.

Convolution roots and differentiability of isotropic positive definite functions on spheres

We prove that any isotropic positive definite function on the sphere can be written as the spherical self-convolution of an isotropic real-valued function. It is known that isotropic positive definite functions on d-dimensional Euclidean space admit a continuous derivative of order [(d − 1)/2]. We show that the same holds true for isotropic positive definite functions on spheres and prove that this result is optimal for all odd dimensions.