A characterization of (4,2)-choosable graphs

A graph $G$ is $\left(a,b\right)$-choosable if given any list assignment $L$ with $\text{◂=▸}\mid L\left(v\right)\mid =a$ for each $\text{◂+▸}v\in V\left(G\right)$ there exists a function $\phi$ such that $\text{◂⊆▸}\phi \left(v\right)\subseteq L\left(v\right)$ and $\text{◂=▸}\mid \phi \left(v\right)\mid =b$ for all $\text{◂+▸}v\in V\left(G\right)$, and whenever vertices $x$ and $y$ are adjacent $\text{◂+▸}\phi \left(x\right)\cap \phi \left(y\right)=\varnothing$. Meng, Puleo, and Zhu conjectured a characterization of (4,2)-choosable graphs. We prove their conjecture.     

Computability of graphs

We consider topological pairs $(A,B)$, $B\subseteq A$, which have computable type, which means that they have the following property: if X is a computable topological space and $f:A\to X$ a topological imbedding such that $f(A)$ and $f(B)$ are semicomputable sets in X, then $f(A)$ is a computable set in X. It is known, e.g., that $(M,\partial M)$ has computable type if M is a compact manifold with boundary. In this paper we examine topological spaces called graphs and we show that we can in a natural way associate to each graph G a discrete subspace E so that $(G,E)$ has computable type. Furthermore, we use this result to conclude that certain noncompact semicomputable graphs in computable metric spaces are computable.     

Control problems for energy harvester model and interpolation in Hardy space

Three control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler–Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3-component functions. The control is introduced as a separable forcing term $\text{◂⋅▸}\mathbf{g}(x)f(t)$ on the right-hand side of the operator equation. The first control problem deals with an explicit construction of $f(t)$ that steers an initial state to zero on a time interval [0, T]. The second control problem deals with the construction of $f(t)$ such that the voltage output is equal to some given function $\mathbf{v}(t)$ (with $\mathbf{g}(x)$ being given as well). The third control problem deals with an explicit construction of both the force profile, $\mathbf{g}(x)$, and the control, $f(t)$, which generate the desired voltage output $\mathbf{v}(t)$. Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems.