An Optimal Control Derivation of Nonlinear Smoothing Equations

The purpose of this paper is to review and highlight some connections between the problem of nonlinear smoothing and optimal control of the Liouville equation. The latter has been an active area of recent research interest owing to work in mean-field games and optimal transportation theory. The nonlinear smoothing problem is considered here for continuous-time Markov processes. The observation process is modeled as a nonlinear function of a hidden state with an additive Gaussian measurement noise. A variational formulation is described based upon the relative entropy formula introduced by Newton and Mitter. The resulting optimal control problem is formulated on the space of probability distributions. The Hamilton's equation of the optimal control are related to the Zakai equation of nonlinear smoothing via the log transformation. The overall procedure is shown to generalize the classical Mortensen's minimum energy estimator for the linear Gaussian problem.Comment: 7 pages, 0 figures, under peer reviewin

Injective colorings of graphs with low average degree

Let \mad(G) denote the maximum average degree (over all subgraphs) of $G$ and let $\chi_i(G)$ denote the injective chromatic number of $G$. We prove that if $\Delta\geq 4$ and \mad(G)<\frac{14}5, then $\chi_i(G)\leq\Delta+2$. When $\Delta=3$, we show that \mad(G)<\frac{36}{13} implies $\chi_i(G)\le 5$. In contrast, we give a graph $G$ with $\Delta=3$, \mad(G)=\frac{36}{13}, and $\chi_i(G)=6$.Comment: 15 pages, 3 figure

Injective colorings of sparse graphs

Let $mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $\chi_i(G)$ denote the injective chromatic number of $G$. We prove that if $mad(G) \leq 5/2$, then $\chi_i(G)\leq\Delta(G) + 1$; and if $mad(G) < 42/19$, then $\chi_i(G)=\Delta(G)$. Suppose that $G$ is a planar graph with girth $g(G)$ and $\Delta(G)\geq 4$. We prove that if $g(G)\geq 9$, then $\chi_i(G)\leq\Delta(G)+1$; similarly, if $g(G)\geq 13$, then $\chi_i(G)=\Delta(G)$.Comment: 10 page

Hawking temperature from scattering off the charged 2D black hole

The charged 2D black hole is visualized as presenting an potential barrier $V^{OUT}(r^*)$ to on-coming tachyon wave. Since this takes the complicated form, an approximate form $V^{APP}(r^*)$ is used for scattering analysis. We calculate the reflection and transmission coefficients for scattering of tachyon off the charged 2D black hole. The Hawking temperature is also derived from the reflection coefficient by Bogoliubov transformation. In the limit of $Q \to 0$, we recover the Hawking temperature of the 2D dilaton black hole.Comment: 12 pages 3 figures, RevTeX, to obtain figures contact author ([email protected]