Sharp Hardy inequalities in the half space with trace remainder term

In this paper we deal with a class of inequalities which interpolate the Kato's inequality and the Hardy's inequality in the half space. Starting from the classical Hardy's inequality in the half space \rnpiu =\R^{n-1}\times(0,\infty), we show that, if we replace the optimal constant $\frac{(n-2)^2}{4}$ with a smaller one $\frac{(\beta-2)^2}{4}$, $2\le \beta <n$, then we can add an extra trace-term equals to that one that appears in the Kato's inequality. The constant in the trace remainder term is optimal and it tends to zero when $\beta$ goes to $n$, while it is equal to the optimal constant in the Kato's inequality when $\beta=2$

NF2/merlin in hereditary neurofibromatosis 2 versus cancer: biologic mechanisms and clinical associations.

Inactivating germline mutations in the tumor suppressor gene NF2 cause the hereditary syndrome neurofibromatosis 2, which is characterized by the development of neoplasms of the nervous system, most notably bilateral vestibular schwannoma. Somatic NF2 mutations have also been reported in a variety of cancers, but interestingly these mutations do not cause the same tumors that are common in hereditary neurofibromatosis 2, even though the same gene is involved and there is overlap in the site of mutations. This review highlights cancers in which somatic NF2 mutations have been found, the cell signaling pathways involving NF2/merlin, current clinical trials treating neurofibromatosis 2 patients, and preclinical findings that promise to lead to new targeted therapies for both cancers harboring NF2 mutations and neurofibromatosis 2 patients

Exploiting classical nucleation theory for reverse self-assembly

In this paper we introduce a new method to design interparticle interactions to target arbitrary crystal structures via the process of self-assembly. We show that it is possible to exploit the curvature of the crystal nucleation free-energy barrier to sample and select optimal interparticle interactions for self-assembly into a desired structure. We apply this method to find interactions to target two simple crystal structures: a crystal with simple cubic symmetry and a two-dimensional plane with square symmetry embedded in a three-dimensional space. Finally, we discuss the potential and limits of our method and propose a general model by which a functionally infinite number of different interaction geometries may be constructed and to which our reverse self-assembly method could in principle be applied.Comment: 7 pages, 6 figures. Published in the Journal of Chemical Physic