Nondivergence form degenerate linear parabolic equations on the upper half space

We study a class of nondivergence form second-order degenerate linear parabolic equations in $(-\infty, T) \times {\mathbb R}^d_+$ with the homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial {\mathbb R}^d_+$, where ${\mathbb R}^d_+ = \{x =(x_1,x_2,\ldots, x_d) \in {\mathbb R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $\mu(x_d)$ and bounded positive definite matrices, where $\mu(x_d)$ behaves like $x_d^\alpha$ for some given $\alpha \in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. The divergence form equations in this setting were studied in [14]. Under a partially weighted VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our research program is motivated by the regularity theory of solutions to degenerate viscous Hamilton-Jacobi equations.Comment: 40 pages, minor revision. Added a weighted parabolic embedding result and a local boundary estimat

Zbtb20 modulates the sequential generation of neuronal layers in developing cortex

BACKGROUND: During corticogenesis, genetic programs encoded in progenitor cells at different developmental stages and inherited in postmitotic neurons specify distinct layer and area identities. Transcription factor Zbtb20 has been shown to play a role for hippocampal development but whether it is implicated in mammalian neocortical morphogenesis remains unknown. RESULTS: Here, we report that during embyogenesis transcription factor Zbtb20 has a dynamic spatio-temporal expression pattern in mitotic cortical progenitors through which it modulates the sequential generation of cortical neuronal layer identities. Zbtb20 knock out mice exhibited enhanced populations of early born L6-L4 neuronal subtypes and a dramatic reduction of the late born L3/L2 neurons. This defect was due to a temporal misbalance in the production of earlier versus later born neurons, leading to a progressive diminishing of the progenitor pool for the generation of L3-L2 neurons. Zbtb20 implements these temporal effects in part by binding to promoter of the orphan nuclear receptor CoupTF1/Nr2f1. In addition to its effects exerted in cortical progenitors, the postmitotic expression of Zbtb20 in L3/L2 neurons starting at birth may contribute to their proper differentiation and migration. CONCLUSIONS: Our findings reveal Zbtb20 as a novel temporal regulator for the generation of layer-specific neuronal identities

A First Principles Study on Electronic and Magnetic Properties of Defects in ZnO/GaN Core-shell Nanowire Heterostructures

To date semiconductor nanowire (NW) heterostructures (HS) have attracted extensive attention as important components of electronic and optoelectronic nanodevices. Further NWs also show promising potency to enhance the solar energy harvesting, e.g. improving both light trapping, photo-carrier collection, and contacting surface area. In this work we show theoretically that the $d^{0}$-ferromagnetism and NW HS bandgap can be turned by engineering the HS interfaces in non-magnetic ZnO/GaN core/shell NW HS. In that NW HS the incorporation of one compound into the other leads to the bandgap narrowing in the nonisovalent alloy because of the type II band alignment betwwen ZnO and GaN. The $d^{0}$-ferromagnetic interface can be developed by creating $p$-type defect with $N$ and/or $n$-type defect with Zn in Ga--O interface bonds due to the defect-induced polar discontinuity. It's noted that the GaN/ZnO NW HS itself without defect or with same number defects of both types are not ferromagnetic. So that the induced magnetic moment is suggested to be related to the missing charge introduced at these defects. In our study we focused on the effects of GaN/ZnO interfaces on the electronic and magnetic properties, e.g. interface states within the bandgap and interface-induced ferromagnetism and impact of surface reconstruction and quantum confinement. The origin of this $d^{0}$-FM is revealed by analyses of spin-polarized bandstructure indicated by the asymmetrical spin-up and spin-down states near the Fermi level, the projected densities of states (PDOSs) and the spin-polarized mulliken charge differences, indicated that most spin-polarized states are dominated by the interface defect site N$p$ electrons. The calculated GaN/ZnO interface magnetism, have been compared with FM at the LaAlO-SrTiO\(_{3} interface which are theoretically predicted [30] and experimentally confirmed [31], where the magnetic moments also arise from the polar discontinuity

Structural brain anomalies in patients with FOXG1 syndrome and in Foxg1+/- mice

Objective FOXG1 syndrome is a rare neurodevelopmental disorder associated with heterozygous FOXG1 variants or chromosomal microaberrations in 14q12. The study aimed at assessing the scope of structural cerebral anomalies revealed by neuroimaging to delineate the genotype and neuroimaging phenotype associations. Methods We compiled 34 patients with a heterozygous (likely) pathogenic FOXG1 variant. Qualitative assessment of cerebral anomalies was performed by standardized re-analysis of all 34 MRI data sets. Statistical analysis of genetic, clinical and neuroimaging data were performed. We quantified clinical and neuroimaging phenotypes using severity scores. Telencephalic phenotypes of adult Foxg1+/- mice were examined using immunohistological stainings followed by quantitative evaluation of structural anomalies. Results Characteristic neuroimaging features included corpus callosum anomalies (82%), thickening of the fornix (74%), simplified gyral pattern (56%), enlargement of inner CSF spaces (44%), hypoplasia of basal ganglia (38%), and hypoplasia of frontal lobes (29%). We observed a marked, filiform thinning of the rostrum as recurrent highly typical pattern of corpus callosum anomaly in combination with distinct thickening of the fornix as a characteristic feature. Thickening of the fornices was not reported previously in FOXG1 syndrome. Simplified gyral pattern occurred significantly more frequently in patients with early truncating variants. Higher clinical severity scores were significantly associated with higher neuroimaging severity scores. Modeling of Foxg1 heterozygosity in mouse brain recapitulated the associated abnormal cerebral morphology phenotypes, including the striking enlargement of the fornix. Interpretation Combination of specific corpus callosum anomalies with simplified gyral pattern and hyperplasia of the fornices is highly characteristic for FOXG1 syndrome.Peer reviewe