On the evolutionary origins of obesity: a new hypothesis

Obesity is an escalating threat of pandemic proportions, currently affecting billions of people worldwide and exerting a devastating socioeconomic influence in industrialized countries. Despite intensive efforts to curtail obesity, results have proved disappointing. Although it is well recognized that obesity is a result of gene-environment interactions and that predisposition to obesity lies predominantly in our evolutionary past, there is much debate as to the precise nature of how our evolutionary past contributed to obesity. The “thrifty genotype” hypothesis suggests that obesity in industrialized countries is a throwback to our ancestors having undergone positive selection for genes that favored energy storage as a consequence of the cyclical episodes of famine and surplus after the advent of farming 10 000 years ago. Conversely, the “drifty genotype” hypothesis contends that the prevalence of thrifty genes is not a result of positive selection for energy-storage genes but attributable to genetic drift resulting from the removal of predative selection pressures. Both theories, however, assume that selection pressures the ancestors of modern humans living in western societies faced were the same. Moreover, neither theory adequately explains the impact of globalization and changing population demographics on the genetic basis for obesity in developed countries, despite clear evidence for ethnic variation in obesity susceptibility and related metabolic disorders. In this article, we propose that the modern obesity pandemic in industrialized countries is a result of the differential exposure of the ancestors of modern humans to environmental factors that began when modern humans left Africa around 70 000 years ago and migrated through the globe, reaching the Americas around 20 000 years ago. This article serves to elucidate how an understanding of ethnic differences in genetic susceptibility to obesity and the metabolic syndrome, in the context of historic human population redistribution, could be used in the treatment of obesity in industrialized countries

On universal and periodic β-expansions, and the Hausdorff dimension of the set of all expansions

We study the topology of a set naturally arising from the study of β-expansions. After proving several elementary results for this set we study the case when our base is Pisot. In this case we give necessary and sufficient conditions for this set to be finite. This finiteness property will allow us to generalise a theorem due to Schmidt and will provide the motivation for sufficient conditions under which the growth rate and Hausdorff dimension of the set of β-expansions are equal and explicitly calculable.

Toeplitz operators on Dirichlet-Besov spaces

We study Toeplitz operators on the Besov spaces in the case of the open unit disk. We prove that a symbol satisfying a weak Lipschitz type condition induces a bounded Toeplitz operator. Such symbols do not need to be bounded functions or have continuous extensions to the boundary of the open unit disk. We discuss the problem of the existence of nontrivial compact Toeplitz operators, and also consider Fredholm properties and prove an index formula.