Hello all, This is a question that might or might not be related to my previous one. Imagine you have two matrices: Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L …

Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries. The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has …

Define $\rho(A)$ to be the spectral radius of a square matrix $A$. Let $S$ and $T$ be two non-negative square matrices and $h$ a real number such that $\rho(S+T) < h$. Show that …

For congruence subgroups of $PSL(2,\mathbb{Z})$, the Weyl law for the eigenvalues of Maass cusp forms had been proven by Selberg. How is the status of such a Weyl law for eigenvalu …

Hi I have the following problem whose solution has lured me for some months now.... All matrices are complex $N\times N$. Let $A$ be a positive definite matrix with all eigenvalue …

Much has been said about bounds on Laplacian eigenvalues, and the literature can be tough to sort through! I am specifically interested in the case where the domain is a closed com …

Consider the linear wave equation : $$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$ Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which …

Consider the $n-$dimensional wave equation $$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$ where $\omega\subset \Omega.$ Can I have $z(t) \to 0 …

Hello! Let $(M,g)$ be a Riemannian manifold and $-\Delta$ the Laplacian on M (acting on smooth functions). Then the resolvent $R(\xi)$ of $-\Delta$ is a compact operator. Is it p …

I need a reference concerning a theorem that shows the following result, stated very roughly: Given a self-adjoint differential operator densely defined on a Hilbert space, then t …

For reasons which the margin of this page is too small to hold, I have been reading parts of a recent paper by O. Selim On submeasures on Boolean algebras, arXiv 1212.6822v3 and …

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

Isn't this question self-explanatory? There is a lot of literature about the Rado graph $R$ in various places. This graph is also known as the "Random Graph" because a countable …

Let $D$ be a real diagonal matrix $D=diag(a_1,a_2,\ldots,a_n)$ with $a_1\le a_2\le\ldots\le a_n$. Assume that at least one of the $a_i$ is positive. Let $P$ be an irreducible, real …

Say I have a positive semi-definite matrix with least positive eigenvalue x. Are there always principal minors of this matrix with eigenvalue less than x? (Here "semidefinite" can …