Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for x\in\RR^d, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold (\RR^d,{\bf g}), with a metric ${\bf g}$ which is conformal to the standard \RR^d metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker--Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of $m$.Comment: 37 page

The vestibular-related frontal cortex and its role in smooth-pursuit eye movements and vestibular-pursuit interactions.

In order to see clearly when a target is moving slowly, primates with high acuity foveae use smooth-pursuit and vergence eye movements. The former rotates both eyes in the same direction to track target motion in frontal planes, while the latter rotates left and right eyes in opposite directions to track target motion in depth. Together, these two systems pursue targets precisely and maintain their images on the foveae of both eyes. During head movements, both systems must interact with the vestibular system to minimize slip of the retinal images. The primate frontal cortex contains two pursuit-related areas; the caudal part of the frontal eye fields (FEF) and supplementary eye fields (SEF). Evoked potential studies have demonstrated vestibular projections to both areas and pursuit neurons in both areas respond to vestibular stimulation. The majority of FEF pursuit neurons code parameters of pursuit such as pursuit and vergence eye velocity, gaze velocity, and retinal image motion for target velocity in frontal and depth planes. Moreover, vestibular inputs contribute to the predictive pursuit responses of FEF neurons. In contrast, the majority of SEF pursuit neurons do not code pursuit metrics and many SEF neurons are reported to be active in more complex tasks. These results suggest that FEF- and SEF-pursuit neurons are involved in different aspects of vestibular-pursuit interactions and that eye velocity coding of SEF pursuit neurons is specialized for the task condition

Comparação entre dois métodos analíticos para determinação da lignina de algumas gramíneas forrageiras Comparison between two analytical methods for determining lignin concentration of some grass forages

Foram comparados dois métodos analíticos para a determinação da lignina (lignina em detergente ácido - LDA e lignina permanganato de potássio - LPer) bem como para averiguar a possível relação dos teores desse componente com a digestibilidade da fibra dos seguintes fenos: andropogon (Andropogon gayanus); aveia (Avena sativa); e dois tipos de coast-cross (Cynodon dactylon), um bem fenado e outro de baixa qualidade. Os valores de LDA e LPer foram diferentes (p <= 0,05) entre as gramíneas estudadas, com as concentrações de LPer consistentemente mais baixas do que as de LDA. Foram detectadas diferenças (p <= 0,05) nos valores de digestibilidade da fibra em detergente neutro (FDN) e da fibra em detergente ácido (FDA), porém, não foi detectada clara relação desses valores com teores de lignina. Os dados sugerem que a concentração de lignina não é o principal fator a explicar um dado valor de digestibilidade.<br>This work was carried out aiming to compare lignin concentration of some grass forages through two analytical methods (acid detergent lignin - ADL and permanganate lignin - PerL) as well to verify a possible relationship of lignin concentration with fiber digestion of the following grass hays: andropogon (Andropogon gayanus); oats (Avena sativa); a good quality and another of poor quality coast-cross (Cynodon dactylon). Acid detergent and permanganate lignin values were different (p <= 0.05) among the hays, however PerL concentrations were consistently lower than ADL values. There were differences (p <= 0.05) among the digestibility of neutral and acid detergent fiber fractions, however a clear relationship between these values with lignin concentration could not be assessed. The data suggested that lignin concentration, taken individually, is not the only factor to explain a given value of digestibility