Existence and Spectral Theory for Weak Solutions of Neumann and Dirichlet Problems for Linear Degenerate Elliptic Operators with Rough Coefficients

In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated to second order linear degenerate elliptic partial differential operators $X$, with rough coefficients of the form $$X=-\text{div}(P\nabla )+{\bf HR}+{\bf S^\prime G} +F$$ in a geometric homogeneous space setting where the $n\times n$ matrix function $P=P(x)$ is allowed to degenerate. We give a maximum principle for weak solutions of $Xu\leq 0$ and follow this with a result describing a relationship between compact projection of the degenerate Sobolev space $QH^{1,p}$ into $L^q$ and a Poincar\'e inequality with gain adapted to $Q$

A systems perspective on the natural resources framework: comment on Hearnshaw et al

The Natural Resources Framework is a new approach to policy advice developed by the multi-agency natural resource sector in New Zealand. This framework has been implemented with some success, but also some teething problems. The framework is a ‘systems’ approach to understanding the interaction between the many actors in the natural resource management system, and as such could benefit from insights and lessons from the systems sciences. This article is a rejoinder to Hearnshaw et al. (2014), and presents three suggestions for how the framework could be improved based on literature from the fields of system dynamics and systems thinking

Harnack's inequality and H\"older continuity for weak solutions of degenerate quasilinear equations with rough coefficients

We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form \begin{eqnarray} \text{div}\big(A(x,u,\nabla u)\big) = B(x,u,\nabla u)\text{ for }x\in\Omega\nonumber \end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local H\"older continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin and N. Trudinger for quasilinear equations, as well as ones for subelliptic linear equations obtained by Sawyer and Wheeden in their 2006 AMS memoir article.Comment: 39 page

DNA Double Strand Break Repair and its Association with Inherited Predispositions to Breast Cancer

Mutations in BRCA1 account for the majority of familial aggregations of early onset breast and ovarian cancer (~70%) and about 1/5 of all early onset breast cancer families; in contrast, mutations in BRCA2 account for a smaller proportion of breast/ovarian cancer families and a similar proportion of early onset breast cancer families. BRCA2 has also been shown to be associated with a much more pleiotropic disease spectrum compared to BRCA1. Since the identification of both BRCA1 and BRCA2 investigations into the functions of these genes have revealed that both are associated with the maintenance of genomic integrity via their apparent roles in cellular response to DNA damage, especially their involvement in the process of double strand DNA break repair. This review will focus on the specific roles of both genes and how functional differences may account for the diverse clinical findings observed between families that harbour BRCA1 or BRCA2 mutations

Childhood status epilepticus: Structural consequences and assessment of a novel treatment

Status epilepticus (SE) is the commonest medical neurological emergency in childhood. In animal models of SE, the hippocampus is frequently damaged. The histological features resemble mesial temporal sclerosis (MTS), the commonest pathology in temporal lobe resections in adult humans. The commonest association with MTS is prolonged febrile convulsion (PFC). Hippocampal damage only occurs if seizures persist for at least 30 minutes. Early termination of seizures may decrease the incidence of MTS. Treatment with rectal diazepam is not always acceptable. An effective, convenient and acceptable method of treating SE would be advantageous. To address the question of whether MTS has different magnetic resonance (MR) characteristics dependent on antecedent, quantitative MR data from patients with histologically proven MTS was reviewed. Patients with a history of PFC have asymmetrical hippocampal volume (HCV) and T2 relaxation time (T2) when compared to patients with no history of PFC and controls. This may suggest that severity and extent of MTS may be, in part, determined by the cause. The assessment of whether SE results in acute brain abnormalities was carried out by prospectively investigating children using MR techniques. Within 48 hours of PFC there is an increase in HCV and T2 relaxation time when compared to controls. Patients with SE and no fever have an increased T2 relaxation time but normal hippocampal volume. PFC appears to result in acute hippocampal swelling, consistent with animal model data. The effect of non-febrile SE on limbic structures is less certain. Buccal midazolam was assessed as an effective, socially acceptable acute treatment for seizures. A pharmacokinetic/pharmacodynamic study confirmed rapid absorption into venous blood and brain. Buccal midazolam was shown to be an effective treatment for acute repetitive seizures and at least as effective as rectal diazepam in the treatment of seizures which have persisted for longer than 5 minutes