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

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

Familial Adenomatous Polyposis (FAP) and Other Polyposis Syndromes

There have been significant advances in our knowledge about the molecular changes that precede and accompany the development of inherited predispositions to colorectal cancer. In this review the clinical relationship to the molecular changes associated with the polyposis syndromes is presented. The aim is to put into context the diverse findings that have been shown to be associated with the development of colorectal cancer in persons who harbor a predisposition to develop disease at unusually early ages. The main focus will be on familial adenomatous polyposis as it serves as a model disease and is the most extensively studied of all of the polyposis syndromes. In addition some information is provided that explains the relationship between a germline change in one gene and what consequences that can have for a particular cell and the development of disease