Exceptional orthogonal polynomials and the Darboux transformation

We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.Comment: corrected abstract, added references, minor correction

Increase in insulin-induced relaxation of consecutive arterial segments toward the periphery: role of vascular oxidative state

Rationale: The oxidative state has been implicated in the signaling of various vasomotor functions, yet its role is less known regarding the vasomotor action of insulin. Objective: We investigated the insulin-evoked relaxations of consecutive arterial segments of different oxidative state and the role of extracellular signal-regulated kinase (ERK) pathway. Methods and Results: The oxidative state, as assessed by ortho-tyrosine was higher in thoracic aorta of rats, followed by the abdominal aorta, and was the lowest in the femoral artery. Vasomotor function of vessels of same origin was studied using a small-vessel myograph. Insulin-induced relaxations increased toward the periphery (i.e. thoracic < abdominal < femoral). Aortic banding and hydrogen peroxide/aminotriazole increased oxidative state of the thoracic aorta that was accompanied by ERK activation and decreased relaxation to insulin, and vice versa, acutely lowered oxidative state by superoxide dismutase/catalase improved relaxation. In contrast, insulin-induced relaxation of the femoral artery could be enhanced with higher, and reduced with lower oxidative state. Conclusions: Oxidative state of vessels modulates the magnitude of vasomotor responses to insulin, which appears to be mediated via the ERK signaling pathway