[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

(Roberts) A

city is divided into 3 areas 1, 2, and 3. It is estimated that amounts [math]u_1[/math], [math]u_2[/math], and [math]u_3[/math] of pollution are emitted each day from these three areas. A fraction [math]q_{ij}[/math] of the pollution from region [math]i[/math] ends up the next day at region [math]j[/math]. A fraction [math]q_i = 1 - \sum_j q_{ij} \gt 0[/math] goes into the atmosphere and escapes. Let [math]w_i^{(n)}[/math] be the amount of pollution in area [math]i[/math] after [math]n[/math] days.

• Show that [math]\mat{w}^{(n)} = \mat{u} + \mat{u} \mat{Q} +\cdots + \mat{u}\mat{Q}^{n - 1}[/math].
• Show that [math]\mat{w}^{(n)} \to \mat{w}[/math], and show how to compute \mat{w} from \mat{u}.
• The government wants to limit pollution levels to a prescribed level by prescribing [math]\mat{w}.[/math] Show how to determine the levels of pollution [math]\mat{u}[/math] which would result in a prescribed limiting value [math]\mat{w}[/math].

Notes