In this note, we study the connection between the fractional Laplacian operator that appeared in the recent work of Caffarelli-Silvestre and a class of conformally covariant operators in conformal geometry.

A wide range of visual parameters used to evaluate binocular function were evaluated in a paediatric population (1056 subjects aged 6-12 years). Mean values are provided for these ages in optometric tests that directly assess the vergence system, horizontal phorias for near and far vision (measured by a modified version of the Thorington method), negative and positive vergence amplitude for near and far vision (step vergence testing), vergence facility (flippers 8 Delta BI/8 Delta BO), and near-point of convergence (penlight push-up technique and red-lens push-up technique), as well as stimulus accommodative convergence/accommodation ratio and stereoacuity (Randot test) which provide an overall evaluation of the vergence, accommodative and oculomotor systems. A statistical comparison (anova and Bonferroni post hoc test) of these values between ages was performed. The differences, although statistically significant, were not clinically meaningful, and therefore we identified two trends in the behaviour of these parameters. For all parameters, except for vergence facility, we established a single mean reference value for the age range studied. The difference between the means for vergence facility indicated the need to divide the population into two age ranges (6-8 and 8-12 years). This study establishes statistical normal values for these parameters in a paediatric population and their means are a valuable instrument for separating children with binocular anomalies from those with normal binocular vision.

In this paper we analyze the global existence of classical solutions to the initial boundaryvalue problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a source term given by a delta function. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Finally, we will also study the spectrum for the linear problem corresponding to uncoupled networks and its relation to Poincaré inequalities for studying their asymptotic behavior.

Determining the levels of human mitochondrial heteroplasmy is of utmost importance in several fields. In spite of this, there are currently few published works that have focused on this issue. In order to increase the knowledge of mitochondrial DNA (mtDNA) heteroplasmy, the main goal of this work is to investigate the frequency and the mutational spectrum of heteroplasmy in the human mtDNA genome. To address this, a set of nine primer pairs designed to avoid co-amplification of nuclear DNA (nDNA) sequences of mitochondrial origin (NUMTs) was used to amplify the mitochondrial genome in 101 individuals. The analysed individuals represent a collection with a balanced representation of genders and mtDNA haplogroup distribution, similar to that of a Western European population. The results show that the frequency of heteroplasmic individuals exceeds 61%. The frequency of point heteroplasmy is 28.7%, with a widespread distribution across the entire mtDNA. In addition, an excess of transitions in heteroplasmy were detected, suggesting that genetic drift and/or selection may be acting to reduce its frequency at population level. In fact, heteroplasmy at highly stable positions might have a greater impact on the viability of mitochondria, suggesting that purifying selection must be operating to prevent their fixation within individuals. This study analyses the frequency of heteroplasmy in a healthy population, carrying out an evolutionary analysis of the detected changes and providing a new perspective with important consequences in medical, evolutionary and forensic fields.

We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caffarelli-Silvestre. While this definition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at infinity and geometry plays a crucial role. First we give explicit calculations in the hyperbolic space, including a formula for the kernel and a trace Sobolev inequality. Then we consider more general noncompact manifolds, where the problem reduces to obtain suitable upper bounds for the heat kernel.