A Refinement of the Discrete Wirtinger Inequality

AbstractIn this paper, we obtain an improved discrete Wirtinger inequality associated with a nonlinear second order differential equation. We apply this result to prove a Bonnesen-style isoperimetric inequality for plane polygons and reinterpret the main theorem as a weighted exponential inequality

Does Notch play a tumor suppressor role across diverse squamous cell carcinomas?

The role of Notch pathway in tumorigenesis is highly variable. It can be tumor suppressive or pro-oncogenic, typically depending on the cellular context. Squamous cell carcinoma (SCC) is a cancer of the squamous cell, which can occur in diverse human tissues. SCCs are one of the most frequent human malignancies for which the pathologic mechanisms remain elusive. Recent genomic analysis of diverse SCCs identified marked levels of mutations in NOTCH1, implicating Notch signaling pathways in the pathogenesis of SCCs. In this review, evidences highlighting NOTCH's role in different types of SCCs are summarized. Moreover, based on accumulating structural information of the NOTCH receptor, the functional consequences of NOTCH1 gene mutations identified from diverse SCCs are analyzed, emphasizing loss of function of Notch in these cancers. Finally, we discuss the convergent view on an intriguing possibility that Notch may function as tumor suppressor in SCCs across different tissues. These mechanistic insights into Notch signaling pathways will help to guide the research of SCCs and development of therapeutic strategies for these cancers