4 problems

Let $[n]=\{1,\ldots,n\}$, let $\mathfrak{S}_n$ be the set of permutations of $[n]$, and let $\mathfrak{C}_n$ be the set of $n$-cycles in $\mathfrak{S}_n$. A join/union expression i…

Let $P_0,P_1,P_2,P_{11},P_{12},P_{21},P_{22}$ be the transition matrices defining the active pieces of $\hat{L}_{\pi}^{\dagger,0}$, let $x_0,x_1,x_2,x_{11},x_{12},x_{21},x_{22}$ be…

Let $\hat{U}^{\dagger,0}:\mathbb{R}^N\to\mathbb{R}^N$ be the operator defined for the unknown stochastic shortest path problem, and let…

Consider sets $Xt subseteqmathbb{R}^{ntimes T}$ for $0\leq tleq T$, a control set $Usubseteqmathbb{R}^m$, and cost functions … Let $\mathcal{F}$ denote the set of pairs $(f,x_0)$ su…