Introductory Mathematical Analysis For Business Economics And The Life And Social Sciences – Ernest F Haeussler

But how do we model something complicated and irregular, like a bolt of lightning or the spread of a rumor? For this, instead of trying to write and solve a system of equations, we can use a different modeling technique: cellular automata. Cellular automata represent large, complex phe- nomena using collections of many small entities each following a few simple rules.

The best-known system of cellular automata is the game called LIFE devel- oped by John Conway in the late 1970s. It can be played by hand on a checkerboard, but using a com- puter program is faster and easier. Downloadable freeware can be found on the Internet. (Go to any search engine and search for “LIFE” and “Conway.”) Intriguing though LIFE is, it is not particularly good for modeling real-life processes. Better for such a task are cellular automata whose rules contain an ele- ment of randomness.

Here is an example. Let us model the seepage of an oil spill into the soil below. We will model the soil as a pattern of cells layered like bricks (Fig. 9.26). FIGURE 9.26 Cells for an oil spill model. Each cell represents a pore in the soil, a pocket of space between dirt particles. All cells start in the “empty” state. To simulate an oil spill, we switch the entire top layer of cells (the soil surface) from empty “Adapted from L. Charles Biehl, “Forest Fires, Oil Spills, and Fractal Geometry, Part 1: Cellular Automata and Modeling Natural Phenomena,” The Mathematics Teacher, 91 (November 1998), 682-87.

By permission of the National Council of Teachers of Mathematics. to “filled.” Depending on the microstructure of the pore arrangement, oil might flow from a pore either to both pores below, or just to the left one, or just to the right one, or to neither. We will model this by suppos- ing that at every junction between a filled cell and an empty one below it, the filled cell has a probability P of switching the empty cell to filled.

For a TI-83 graphics calculator, the following pro- gram models the process: Input “P?”,P ClrDraw AxesOff For(Y,0,46) Pxl-On(X,2Y+N) Pxl-On(X,2Y+N+l) End For(X,l,62) X-2iPart(X/2)^N For(Y,0,46) If ((pxl-Test(X-l, 2Y+N) and rand

a(b + c) = ab + ac a(b — c) — ab — ac (a + b)c = ac + be (a – b)c = ac – be a + 0 = a fl-0 = 0 a • 1 = a a + {-a) = 0 -(-a) = a ■ (“!)«= -a a – b = a + (-b) a – {-b) = a + b •(;) – ■ .

a 1 -b=a’b (-a)b = -(ab) = – a( -b) (-a)(-b) = ab -a a -b b -a a a ~b ” ‘~l ~ ^b a b a + b – + – = c c c a b a – b c c c a c ac ~b”d~ bd a/b ad c/d be a ac ~b~ Tc a” = 1 (a * 0) a~” = 1 {a* 0) ama” ■■ am+n (a'”)n = am” (ab)” = anb” (aV a” \b) b” y~a= a {rfa)n = a.^/a7, = a (fl>0) ^” = (^/a)m = a ^a~b= ^Ta^Tb r \n/~ .

n a Vfl v^” Sfo ml — P= mni—

(-v- a ,3 _ 3 x – 3 ax2 + 3a2x – a3 Factoring Formulas ab + ac = a(b + c) a~ – b2 = (a + b)(a- ~ b) a2 + lab + b2 — = (a + )2 d2- lab + b2 – = (a- a> + b3 = (a + b)(a2 – ab + b2) a’ – b3 = (a – b)(a2 + ab + b2) Quadratic Formula Straight Lines If a d.\: + bx 4 * 0.

then -b ± – c – 0, where Vb2 – 4oc x — 2a Inequalities [fa < 6, then a + c < b + c. If a < b and c > 0. then ac < be. If a < b and c > 0, then fl(-c) > b{-c). Counting m = ^2 – y\ y – y, = m(* – *,) y = mjr.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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