From an evolutionary perspective, deiodinases may be considered pivotal players in the emergence and functional diversification of both thyroidal systems (TS) and their iodinated messengers. To better understand the evolutionary pathway and the concomitant functional diversification of vertebrate deiodinases, in the present review we summarized the highlights of the available information regarding this ubiquitous enzymatic component that represents the final, common physiological link of TS. The information reviewed here suggests that deiodination of tyrosine metabolites is an ancient feature of all chordates studied to date and consequently, that it precedes the integration of the TS that characterize vertebrates. Phylogenetic analysis presented here points to D1 as the oldest vertebrate deiodinase and to D2 as the most recent deiodinase gene, a hypothesis that agrees with the notion that D2 is the most specialized and finely regulated member of the family and plays a key role in vertebrate neurogenesis. Thus, deiodinases seem to be major participants in the evolution and functional expansion of the complex regulatory network of TS found in vertebrates.
This paper formulates some conjectures about the amplitude of 1. Introduction resonance in the General Standard Map. The main idea is to ex-2. Resonant Normal Forms pand the periodic perturbation function in Fourier series. Given 3. Rules to Determine the Resonant Normal Form any rational rotation number, we choose a finite number of har-4. Numerical computation monies in the Fourier expansion and we compute the amplitude 5. Estimate of A p/q for Values of e of the Order of 1 of resonance of the reduced perturbation function of the map, r ~ " £n using a suitable normal form around the resonance, which is 6. Collapse of Resonance _° M M r t order of 1.We find that some perturbation functions give rise "..,.. a phenomenon that we call collapse of resonance; this means Bibliography / K ' that the amplitude of resonance goes to zero for some value of the perturbation parameter. We find an empirical procedure to estimate this value of the parameter related to the collapse of resonance.
We accurately compute the golden and silver critical invariant circles of several area-preserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for example, the conjugacy between the circle map giving the dynamics on the invariant circle and a rigid rotation on the circle). The global Hölder regularities of these functions are low (some of them are not even once differentiable). We present several conjectures about the universality of the regularity properties of the critical circles and the related functions. Using a Fourier analysis method developed by de la Llave and one of the authors, we compute numerically the Hölder regularities of these functions. Our computations show thatwithin their numerical accuracy-these regularities are the same for the different maps studied. We discuss how our findings are related to some previous results: (a) to the constants giving the scaling behavior of the iterates on the critical invariant circle (discovered by Kadanoff and Shenker) and (b) to some characteristics of the singular invariant measures connected with the distribution of iterates. Some of the functions studied have pointwise Hölder regularity that has different values at different points. Our results give convincing numerical support to the fact that the points with different Hölder exponents of these functions are interspersed in the same way for different maps, which is a strong indication that the underlying twist maps belong to the same universality class. In particular, the numerical results on the regularity of the so-called big conjugacies imply that the Hölder spectra of the functions conjugating the dynamics on the critical invariant circle to a rigid rotation are the same. This, in turn, shows that the invariant measures on the critical circles have the same singularity spectra.
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