Here is a computer algebra recipe t h a t might be used.

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For example, typing in Pen will produce Pendleton in any displayed o u t p u t. For example, in the next command line Seattle is connected to Portland, Oregon, and to Yakima, Washington. T h e remaining connections are entered in a similar manner. How many hours does it take to drive from Yakima to Bend? W h a t is the corresponding route number replace the colon with a semicolon? If the colon is replaced with a semicolon in the following edge-weight command, the various times in hours between cities in the graph will be displayed. The graph, which is produced with the hnear draw command, will not place the cities at their proper relative geographic positions, but instead will artistically group cities as follows.

The first city in the argument is Seattle S , which will be placed at the far left of the resulting picture. The next entry lists the cities of Portland Port and Yakima Yak. These cities will be placed to the right of Seattle, and lined up vertically, one above the other. Each successive list of cities is placed further to the right and the members of the list organized vertically, until finally Phoenix Ph is placed on the far right of the graph G. The cities are correctly joined according to the connections that were entered earlier.

The a l l p a i r s command is now used to calculate the shortest traveling time between any two cities in the graph. The output is suppressed for brevity. This produces Figure 1. The shortest route between Seattle and Phoenix is the only one in Figure 1. To draw this route, remove those cities that clearly do not connect Seattle and Phoenix in the shortest path tree. Then Bend and Boise become dead ends, so delete them too.

Finally, include Pendleton Pen and Yakima Yak in the following delete command. The cities of Portland, Reno, and Las Vegas lie along this route. This is the route that would be recommended to our friend Russell. Again, remember that the cities in the picture are not oriented in their proper geographic positions. Of course, for this example, the number of different routes was not so challenging that the calculation could not have been done fairly quickly with pen and paper.

But imagine doing this for a graph with all the cities and towns in North America or in Europe. Further, once one has created the graph, the shortest path between any two towns or cities is easily obtained. FermaVs principle in geometrical optics is based on a similar idea. In its simplest form [Tip91], it states that the path taken by light in traveling from one point to another is such that the time of travel is a minimum. Fermat's principle can be used to generate Snell's law in geometrical optics, as well as to account for the phenomena of mirages in a medium with a variable refractive index.

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Be sure to draw all the appropriate graphs. Problem Boise to Reno Use the provided recipe to determine the shortest travel time and the corresponding route to travel between Boise, Idaho, and Reno, Nevada. Show the appropriate graphs. Is this really the shortest route between these two cities? To answer this question, you might want to consult the appropriate road maps, add more connecting cities, and modify the recipe.

Problem Planning your route Choose two major American or European cities that are widely separated in distance and that have many possible connecting routes. Find the driving times and mileages between sizeable cities or towns along the various routes. To save on typing, do not include every village or hamlet that you would pass through.

Repeat the procedure outlined in the text to determine the route between the two major cities that minimizes the total driving time. Determine the route that minimizes the total distance. Here a and b are positive constants.

To confirm that this statement is true. Heather decides to use the log-log plotting procedure to determine the parameters a and b in the power law formula for the chimpanzee data. To make a log-log data plot, the plots library package must be entered. The d i s p l a y command was utilized by Heather because it allows more control of the plot options than loglogplot.

In particular, it allows use of the view command, which is often employed in this text to set the horizontal and vertical ranges in order to get a good picture. For complicated plot structures, it enables one to zoom in on any desired region without computing the graph again.

Heather notes that the data lie approximately along a straight line. From Table 1. Similarly, she finds that a? However, she is bothered by the fact that she doesn't know why this should be the case. Perhaps her older sister Jennifer, who is an MIT applied mathematician, can provide an explanation? In the next story, we shall hear what Jennifer has to say about this example. The age is in years and the volume in hundreds of board feet.

A board foot is the volume of a board 1 foot square and 1 inch thick. Does the curve suggest a power law? Is the curve approximately a straight line?

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Confirm that this is the case by plotting the log of the formula on the same graph as the log-log data. Plot the theoretical formula and data in the same log-log graph and also produce a normal graph. Problem Newton's law of cooling Russell, who has moved into his new lab in Phoenix, boils some water to make a cup of instant coffee.

While sipping cautiously on his hot drink, he has placed a mercury-in-glass thermometer in the remaining boiling water for a few minutes and then removed the thermometer. He records the readings in degrees Celsius on the thermometer t seconds after removal, the results being shown in Table 1.

The temperature of the room is a warm What might give rise to data points that deviate from the straight hne? That such a curve might apply to a given set of data can be ascertained by checking to see whether the observation points lie on a straight line when a log-log plot of the data is made. This is the procedure that Heather applied in the last recipe for the brain volumes V of adult chimpanzees. To understand how such a power law relation can arise.

## Computer Algebra Recipes: An Introductory Guide to the Mathematical Models of Science

I will try to keep it simple, but if you want a more complete treatment I would refer you to an interesting paper entitled "Fundamentals of zoological scaling" written by Herbert Lin and published in the American Journal of Physics [Lin82]. As you are undoubtedly becoming aware, introductory science courses tend to deal with highly idealized models of the real world that are set up to give unique, well-defined answers. In reality, experimentalists are often confronted with complex systems for which the properties could depend on many factors.

The appearance of power law 1. Let me give you a few simple examples. You're not in a hurry to get anywhere, are you? A log of about 10 cm diameter will take around an hour to burn, while 1 cm diameter kindling will burn in several minutes, and a fuse of diameter 1 mm will burn in several seconds. To understand this, note that burning takes place only at the surface, so the rate of combustion is limited by the surface area S with which oxygen, necessary for burning, must make contact. On the other hand, the rate must be inversely proportional to the amount of material present, i.

Thus, according to this scahng argument, since the time to burn is inversely proportional to the rate of combustion, the log should burn about 10 times slower than the kindling, which in turn will burn about 10 times slower than the fuse. This is in rough agreement with the observations. Although the precise burning times clearly would depend on other factors, the gross observed behavior is dominated by changes in characteristic size L. Can you also use scaling to explain it? Let's make the assumption that in order to maintain the same functional power, the adult chimpanzee brain volume V is proportional to the size L of the chimp.

That is to say, the 'bigger' an animal of a given species is, the bigger is its brain. But the body mass M is equal to the density times the body volume. The density of all mammals is fairly constant, especially within a given species such as the chimps. And, as you verified, this power law is in very good agreement with the experimental data.

Do you have any more simple examples of scaling? What got my mind going on it was all these movies that have appeared over the years featuring giant ants, apes, etc. These movies are often not very good, but more importantly in the context of our discussion, they are flawed from a scaling viewpoint. Can you give me a concrete example?

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Do you remember reading the novel Gulliver's Travels by Jonathan Swift? I am going to use scaling to punch some scientific holes in Swift's story. Recall that in the novel Gulliver travels to a number of strange lands. They are geometrically similar to humans, i. For calculation convenience, let's make the Lilliputians ten times as small and the Brobdingnagians Brobs, for short ten times as large. What effects might this have on their biological processes and on the scientific accuracy of Swift's tale?

So, the Lilliputians and Brobs would have surface areas times smaller and times larger than Gulliver. Now it is a well-known biological fact that warm-blooded animals lose heat through their skin.

Since heat is a form of energy, in equilibrium the heat energy loss must be balanced by energy intake in the form of food.