exercise:74602c39b8: Difference between revisions
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(Created page with "<div class="d-none"><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></div> (Roberts<ref group="Notes" >F. Roberts, ''Discrete Mathematical Models'' (Englewood Cliffs, NJ: Prentice Hall, 1976).</ref>) 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...") |
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\newcommand{\mathds}{\mathbb}</math></div> (Roberts<ref group="Notes" >F. Roberts, ''Discrete | \newcommand{\mathds}{\mathbb}</math></div> (Roberts<ref group="Notes" >F. Roberts, ''Discrete | ||
Mathematical | Mathematical Models'' (Englewood Cliffs, NJ: Prentice Hall, 1976).</ref>) 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} > 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. | ||
Models'' (Englewood Cliffs, NJ: Prentice Hall, 1976).</ref>) A | <ul style="list-style-type:lower-alpha"><li> Show that <math>\mat{w}^{(n)} = \mat{u} + \mat{u} \mat{Q} +\cdots + | ||
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} > 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. | |||
<ul><li> Show that <math>\mat{w}^{(n)} = \mat{u} + \mat{u} \mat{Q} +\cdots + | |||
\mat{u}\mat{Q}^{n - 1}</math>. | \mat{u}\mat{Q}^{n - 1}</math>. | ||
</li> | </li> | ||
<li> Show that <math>\mat{w}^{(n)} \to \mat{w}</math>, and show how to compute | <li> Show that <math>\mat{w}^{(n)} \to \mat{w}</math>, and show how to compute | ||
\mat{w} from \mat{u}. | <math>\mat{w}</math> from <math>\mat{u}</math>. | ||
</li> | </li> | ||
Latest revision as of 23:55, 15 June 2024
[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[Notes 1]) 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 [math]\mat{w}[/math] from [math]\mat{u}[/math].
- 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