Sharp estimates for eigenvalues of integral operators generated by dot product kernels on the sphere

We obtain explicit formulas for the eigenvalues of integral operators generated by continuous dot product kernels defined on the sphere via the usual gamma function. Using them, we present both, a procedure to describe sharp bounds for the eigenvalues and their asymptotic behavior near 0. We illustrate our results with examples, among them the integral operator generated by a Gaussian kernel. Finally, we sketch complex versions of our results to cover the cases when the sphere sits in a Hermitian space.

Calorie restriction increases cerebral mitochondrial respiratory capacity in a NO center dot-mediated mechanism: Impact on neuronal survival

Calorie restriction (CR) enhances animal life span and prevents age-related diseases, including neurological decline. Recent evidence suggests that a mechanism involved in CR-induced life-span extension is NO-stimulated mitochondrial biogenesis. We examine here the effects of CR on brain mitochondrial content. CR increased eNOS and nNOS and the content of mitochondria] proteins (cytochrome c oxidase, citrate synthase, and mitofusin) in the brain. Furthermore, we established an in vitro system to study the neurological effects of CR using serum extracted from animals on this diet. In cultured neurons, CR serum enhanced nNOS expression and increased levels of nitrite (a NO product). CR serum also enhanced the levels of cytochrome c oxidase and increased citrate synthase activity and respiratory rates in neurons. CR serum effects were inhibited by L-NAME and mimicked by the NO donor SNAP. Furthermore, both CR sera and SNAP were capable of improving neuronal survival. Overall, our results indicate that CR increases mitochondrial biogenesis in a NO-mediated manner, resulting in enhanced reserve respiratory capacity and improved survival in neurons. (C) 2012 Elsevier Inc. All rights reserved.