In this paper, we provide a framework for calculating expected utility in models with chaotic equilibria and consequently a framework for ranking chaos. Suppose that a dynamic economic model’s equilibria correspond to orbits generated by a chaotic dynamical system f : X → X where X is a compact metric space and f is continuous. The map f could represent the forward dynamics xt+1 = f(xt) or the ...

BACKGROUND AND AIMS Establishment of natural head position (NHP) in radiography or photography by estimation, which is called natural head orientation (NHO), has been recently advocated by various studies. This study aimed to determine the effect of chin position on the accuracy and reproducibility of NHO in a more realistic way. MATERIALS AND METHODS NHP-based digital photographs of 33 adult...

The forward di¤usion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene …xation, with the “weight”associated with the spikes corresponding to th...