Penalty finite element approximations of the stationary power- law Stokes problem

Finite element approximations of the stationary power-law Stokes problem using penalty formulation are considered. A priori error estimates under appropriate smoothness assumptions on the solutions are established without assuming a discrete version of the BB condition. Numerical solutions are presented by implementing a nonlinear conjugate gradient metho

Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions

We introduce an iterative method for computing the first eigenpair $(\lambda_{p},e_{p})$ for the $p$-Laplacian operator with homogeneous Dirichlet data as the limit of $(\mu_{q,}u_{q})$ as $q\rightarrow p^{-}$, where $u_{q}$ is the positive solution of the sublinear Lane-Emden equation $-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$ with same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of $u_{q}$ to $e_{p}$ is in the $C^{1}$-norm and the rate of convergence of $\mu_{q}$ to $\lambda_{p}$ is at least $O(p-q)$. Numerical evidence is presented.Comment: Section 5 was rewritten. Jed Brown was added as autho

FABP7: a glial integrator of sleep, circadian rhythms, plasticity, and metabolic function

Sleep and circadian rhythms are observed broadly throughout animal phyla and influence neural plasticity and cognitive function. However, the few phylogenetically conserved cellular and molecular pathways that are implicated in these processes are largely focused on neuronal cells. Research on these topics has traditionally segregated sleep homeostatic behavior from circadian rest-activity rhythms. Here we posit an alternative perspective, whereby mechanisms underlying the integration of sleep and circadian rhythms that affect behavioral state, plasticity, and cognition reside within glial cells. The brain-type fatty acid binding protein, FABP7, is part of a larger family of lipid chaperone proteins that regulate the subcellular trafficking of fatty acids for a wide range of cellular functions, including gene expression, growth, survival, inflammation, and metabolism. FABP7 is enriched in glial cells of the central nervous system and has been shown to be a clock-controlled gene implicated in sleep/wake regulation and cognitive processing. FABP7 is known to affect gene transcription, cellular outgrowth, and its subcellular localization in the fine perisynaptic astrocytic processes (PAPs) varies based on time-of-day. Future studies determining the effects of FABP7 on behavioral state- and circadian-dependent plasticity and cognitive processes, in addition to functional consequences on cellular and molecular mechanisms related to neural-glial interactions, lipid storage, and blood brain barrier integrity will be important for our knowledge of basic sleep function. Given the comorbidity of sleep disturbance with neurological disorders, these studies will also be important for our understanding of the etiology and pathophysiology of how these diseases affect or are affected by sleep

Clinical practice: Swallowing problems in cerebral palsy

Cerebral palsy (CP) is the most common physical disability in early childhood. The worldwide prevalence of CP is approximately 2–2.5 per 1,000 live births. It has been clinically defined as a group of motor, cognitive, and perceptive impairments secondary to a non-progressive defect or lesion of the developing brain. Children with CP can have swallowing problems with severe drooling as one of the consequences. Malnutrition and recurrent aspiration pneumonia can increase the risk of morbidity and mortality. Early attention should be given to dysphagia and excessive drooling and their substantial contribution to the burden of a child with CP and his/her family. This review displays the important functional and anatomical issues related to swallowing problems in children with CP based on relevant literature and expert opinion. Furthermore, based on our experience, we describe a plan for approach of investigation and treatment of swallowing problems in cerebral palsy