Remarks on common hypercyclic vectors

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set ? ? R+ × C which is not of zero three-dimensional Lebesgue measure, the family {a T + b I : (a, b) ? ?} has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if D = {z ? C : | z | < 1} and f ? H8 (D) is non-constant, then the family {z Mf{star operator} : b- 1 < | z | < a- 1} has a common hypercyclic vector, where Mf : H2 (D) ? H2 (D), Mf f = f f, a = inf {| f (z) | : z ? D} and b = sup {| f (z) | : | z | ? D}, providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family {a Tb : a, b ? C {set minus} {0}} has a common hypercyclic vector, where Tb f (z) = f (z - b) acts on the Fréchet space H (C) of entire functions on one complex variable.

Bevacizumab and rapamycin induce growth suppression in mouse models of hepatocellular carcinoma

Background/Aims: Hepatocellular carcinoma is a leading cause of global cancer mortality, with standard chemotherapy being minimally effective in prolonging survival. We investigated if combined targeting of vascular endothelial growth factor protein and expression might affect hepatocellular carcinoma growth and angiogenesis.