Dunkl operators and a family of realizations of osp(1|2)

In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1|2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator.Comment: 28 pages, some small changes, accepted in Trans. Amer. Math. So

Bcl-xL gain of function and p19ARF loss of function cooperate oncogenically with Myc in vivo by distinct mechanisms

SummaryOverexpression of Bcl-xL, loss of p19ARF, and loss of p53 all accelerate Myc oncogenesis. All three lesions are implicated in suppressing Myc-induced apoptosis, suggesting that this is a common mechanism by which they synergize with Myc. However, using an acutely switchable model of Myc-induced tumorigenesis, we demonstrate that each lesion cooperates with Myc in vivo by a distinct mechanism. While Bcl-xL blocks Myc-induced apoptosis, inactivation of p19ARF enhances it. However, this increase in apoptosis is matched by increased Myc-induced proliferation. p53 inactivation shares features of both lesions, partially suppressing apoptosis while augmenting proliferation. Bcl-xL and p19ARF loss together synergize to further accelerate Myc oncogenesis. Thus, differing lesions cooperate oncogenically with Myc by discrete mechanisms that can themselves synergize with each other