Dissolution of drug from its solid form to a dissolved form is an important consideration in the design and optimization of drug delivery devices, particularly owing to the abundance of emerging compounds that are extremely poorly soluble. When the solid dosage form is encapsulated, for example by the porous walls of an implant, the impact of the encapsulant drug transport properties is a further confounding issue. In such a case, dissolution and diffusion work in tandem to control the release of drug.

We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in ?H1- (R2) ? ?H s(R2) with s > 3 and for any 0 < < 1. Such result improves the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to H20(R2). More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in H1- (R2)?

We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ?. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an O(t-1/2) inviscid damping while the vorticity and density gradient grow as O(t1/2). The result holds at least until the natural, nonlinear timescale t??-2.

Given a random sample drawn from a Multivariate Bernoulli Variable (MBV), we consider the problem of estimating the structure of the undirected graph for which the distribution is pairwise Markov and the parameters' vector of its exponential form. We propose a simple method that provides a closed form estimator of the parameters' vector and through its support also provides an estimate of the undirected graph associated with the MBV distribution. The estimator is proved to be asymptotically consistent but it is feasible only in low-dimensional regimes.

We prove the strong ill-posedness of the two-dimensional Boussinesq system in vorticity form in L8pR2q without boundary, building upon the method that Shikh Khalil & Elgindi arXiv:2207.04556v1 developed for scalar equations. We provide examples of initial data with vorticity and density gradient of small L8pR2q size, for which the horizontal density gradient has a strong L8pR2q-norm inflation in infinitesimal time, while the vorticity and the vertical density gradient remain bounded.

We study the consistency and the oracle properties of the adaptive Lasso estimator for the coefficients of a linear AR(p) time series with a strictly stationary white noise (not necessarily described by i.i.d. r.v.'s). We apply the results to INAR(p) time series and to the non-parametric inference of the fertility function of a Hawkes point process. We present some numerical simulations to emphasize the advantages of the proposed procedure with respect to more classical ones and finally we apply it to a set of epidemiological data

A new technique for nonparametric regression of multichannel signals is presented. The technique is based on the use of the Rational-Dilation Wavelet Transform (RADWT), equipped with a tunable Q-factor able to provide sparse representations of functions with different oscillations persistence. In particular, two different frames are obtained by two RADWT with different Q-factors that give sparse representations of functions with low and high resonance.