# Tagged Questions

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### Spectral radius of products of matrices of a certain form

Background Let $\mathcal{D}$ be the set of $S\times S$ diagonal matrices with elements in $[0,1)$. Let $\mathcal{G} = \{\Gamma \in \mathcal{D} : tr(\Gamma) = 1\}$, and let $B$ be an $S\times S$ ...
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### null controllability of linear wave equation

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 $(z,z_t)$ vanish ...
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### strong stability for the wave equation

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 $z(t) \to 0,$ as $t\to+\infty$ ...
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### Exponential stability in nonlinear differential equations

I have this nonlinear differential equation $d\textbf{x}/dt=f(\textbf{x})$, where $\textbf{x}\in \mathbb{R}^n$. There are results which guarantee the convergence of the dynamical system to ...
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### Avalanche Principle for higher dimensional unimodular matrices ?

Hello everyone, I have a quick question for people working on quasi-periodic Schrodinger operators, Lyapunov exponents for Schrodinger cocycles or in other fields that might make them aware of this ...
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### sum of Perron-Frobenius operators

My operator is the transfer operator $P$ on $L^1$ functions defined on compact $X$. It is the pre-dual of the operator $U:L^∞ \rightarrow L^∞$ defined by $U(ϕ)=ϕ\circ f$, for a fixed map f on X. I ...
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### Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...
Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional $$E(f) = \int_{\mathcal{M}} X_p(f)^2 dV$$ where $X_p(f)$ is the ...