The Quest for the Ultimate Anisotropic Banach Space

We present a new scale \(\mathcal {U}^{t,s}_p\) (\(s<-t<0\) and \(1\le p <\infty \)) of anisotropic Banach spaces, defined via Paley–Littlewood, on which the transfer operator \(\mathcal {L}_g \varphi = (g \cdot \varphi ) \circ T^{-1}\) associated to a hyperbolic dynamical system T has good spectral properties. When \(p=1\) and t is an integer, the spaces are analogous to the “geometric” spaces \(\mathcal {B}^{t,|s+t|}\) considered by Gouëzel and Liverani (Ergod Theory Dyn Syst 26:189–217, 2006). When \(p>1\) and \(-1+1/p<s<-t<0<t<1/p\), the spaces are somewhat analogous to the geometric spaces considered by Demers and Liverani (Trans Am Math Soc 360:4777–4814, 2008). In addition, just like for the “microlocal” spaces defined by Baladi and Tsujii (Ann Inst Fourier 57:127–154, 2007) (or Faure–Roy–Sjöstrand in Open Math J 1:35–81, 2008), the transfer operator acting on \(\mathcal {U}^{t,s}_p\) can be decomposed into \(\mathcal {L}_{g,b}+\mathcal {L}_{g,c}\), where \(\mathcal {L}_{g,b}\) has a controlled norm while a suitable power of \(\mathcal {L}_{g,c}\) is nuclear. This “nuclear power decomposition” enhances the Lasota–Yorke bounds and makes the spaces \(\mathcal {U}^{t,s}_p\) amenable to the kneading approach of Milnor–Thurson (Dynamical Systems (Maryland 1986–1987), Springer, Berlin, 1988) (as revisited by Baladi–Ruelle, Baladi in Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Monograph, 2016; Baladi and Ruelle in Ergod Theory Dyn Syst 14:621–632, 1994; Baladi and Ruelle in Invent Math 123:553–574, 1996) to study dynamical determinants and zeta functions.     

Discrimination between thermal quenching of the fluorescence and Auger upconversion processes using thermal lens technique

In this work we used the Thermal Lens (TL) technique to discriminate two important processes responsible to reduce the upper-state population and fluorescence quantum efficiency (η) in Cr³⁺ doped colquiriite crystals: the thermal quenching of the fluorescence (TQF) and the Auger upconversion (ETU). We observed a nonlinear increase of the TL signal with laser power due to the decrease of η by ETU and/or TQF. The analysis of these curves allowed the determination of the thermal load, the increase of the crystal temperature as a function of the pump excitation and the discrimination between ETU and TQF processes.     

Emergence of prime numbers as the result of evolutionary strategy

We investigate by means of a simple theoretical model the emergence of prime numbers as life cycles, as those seen for some species of cicadas. The cicadas, more precisely the Magicicadas, spend most of their lives below the ground and then emerge and die in a short period of time. The Magicicadas display an uncommon behavior: their emergence is synchronized and these periods are usually prime numbers. In the current work, we develop a spatially extended model at which preys and predators coexist and can change their evolutionary dynamics through the occurrence of mutations. We verified that prime numbers as life cycles emerge as a result of the evolution of the population. Our results seem to be a first step in order to prove that the development of such strategy is selectively advantageous, especially for those organisms that are highly vulnerable to attacks of predators.     

Fluorescence quantum efficiency measurements in the presence of Auger upconversion by the thermal lens method

Mode-mismatched thermal lens (TL) measurements in Cr3+-doped fluoride crystals (LiSrAIF6 and LiSrGaF6) are reported. A nonlinear increase of the TL signal, and decrease of quantum efficiency, with increasing excitation power was observed and attributed to energy-transfer upconversion (ETU). Assuming an upconversion rate that is proportional to the excited-state population, Wup = gammaN(e), the theoretical model developed fits the experimental data well. The ETU parameters (gamma) were determined with unprecedented accuracy for low Cr concentrations. Thermo-optical parameters (K, ds/dT) were also obtained.