Abstract. We give conditions under which a homomorphism between two Zariski dense subgroups of connected semisimple Lie groups G and G without compact factors and with trivial center can be extended to a continuous isomorphism between G and G . In particular we prove the marked length rigidity and the marked translation vector rigidity. This last result was motivated by a Margulis's question.
<abstract><p>This paper aims to investigate the fourth-order boundary value problems with distributional potentials. We first prove that the operators associated with the problems are self-adjoint and the corresponding eigenvalues are real. Then we obtain that the eigenvalues of the problems depend not only continuously but also smoothly on the parameters of the problems: the boundary conditions, the coefficient functions and the endpoints. Moreover, we find the differential expressions for each parameter.</p></abstract>
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