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Suppose that $A$ is an $n\times n$ nonnegative matrix whose eigenvalues are $\lambda =\rho \left(A\right),{\lambda }_2,...,{\lambda }_n$. Fiedler and others have shown that $det(\lambda I-A)\le {\lambda }^n-{\rho }^n$, for all $\lambda >\rho$, with equality for any such $\lambda$ if and only if $A$ is the simple cycle matrix. Let $a_i$ be the signed sum of the determinants of the principal submatrices of $A$ of order $i\times i$, $i=1,...,n-1$. We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: $det(\lambda I-A)+{\sum }_{i=1}^{n-1}{\rho }^{n-2i}\left|a_i\right|{(\lambda -\rho )}^i\le {\lambda }^n-{\rho }^n$, for all $\lambda \ge \rho$. We use this inequality to derive the inequality that: ${\prod }_2^n(\rho -{\lambda }_i)\le {\rho }^{n-2}{\sum }_{i=2}^n(\rho -{\lambda }_i)$. In the spirit of a celebrated conjecture due to Boyle-Handelman,...

Let $J_1$ be the Jacobian of the modular curve associated with ${\Gamma }_1\left(Np\right),(p,N)=1$ and $J_0$ the one associated with ${\Gamma }_1\left(N\right)\cap {\Gamma }_0\left(p\right)$. We study $J_1[p-1]$ as a Hecke and Galois-module. We relate a certain matrix of $p$-adic periods to the infinitesimal deformation of the $U_p$-operator.