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The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure $mu$ describing the distribution of root hair cells, we seek to maximize a harvest functional $mathcal{H}$, computing the total amount of water and nutrients gathered by the roots, subject to a cost for transporting these nutrients from the roots to the trunk. Earlier papers had established the existence of an optimal measure, and a priori bounds. Here we derive necessary conditions for optimality. Moreover, in space dimension $d=2$, we prove that the support of an optimal measure is nowhere dense.
We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $Omega$ that varies over all subdomains of a given bounded domain $D$ of ${bf R}^d$. We show in a rather elementary way the ex
We provide high probability finite sample complexity guarantees for hidden non-parametric structure learning of tree-shaped graphical models, whose hidden and observable nodes are discrete random variables with either finite or countable alphabets. W
The performance of distributed and data-centric applications often critically depends on the interconnecting network. Applications are hence modeled as virtual networks, also accounting for resource demands on links. At the heart of provisioning such
We revisit the optimal control problem of maximizing biogas production in continuous bio-processes in two directions: 1. over an infinite horizon, 2. with sub-optimal controllers independent of the time horizon. For the first point, we identify a set
We consider a given region $Omega$ where the traffic flows according to two regimes: in a region $C$ we have a low congestion, where in the remaining part $Omegasetminus C$ the congestion is higher. The two congestion functions $H_1$ and $H_2$ are gi