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We consider the problem of distributing two conducting materials in a ball with xed proportion in order to minimize the rst eigenvalue of a Dirichlet operator. It was conjectured that the optimal distribution consists of putting the material with the highest conductivity in a ball around the center. In this paper, we show that the conjecture is false for all dimensions greater than or equal to two  

In this paper the gain dynamics of an erbium-doped fiber laser (EDFL) with an inhomogeneous active medium in the presence of ion pairs is modeled. A two-level model for single ions and a three-level model for ion pairs are employed to write the propagation and rate equations of inhomogeneous laser medium. The governing equations are an uncountable system of partial differential equations (PDEs). By employing the moment method, the system of PDEs is converted to a finite system of ordinary differential equations (ODEs). The Solution of the system of ODEs is used to analyze the output power of a high concentration EDFL. As it expected, theoretical results show that the threshold pumping power...