Summary of Topics
Pascal's Triangle and Probability
If we flip n coins, there are 2^n possible outcomes. The position-r number in the number-n row of Pascal's Triangle tells us how many of these outcomes will have r Heads. This number is denoted C(n,r). Advice from Cooter & Bubba: list everything that can happen for 2 flips of a coin. Be lazy and use this list to git a new list for 3 flips. Then use this list to make you a list for 4 flips. What th' heck. Go one more step in create a list for 5 flips. Count the number of ways you could git 0 Heads, 1 Head, 2 Heads, 3 Heads, 4 Heads, 5 Heads, 6 Heads, 7 Heads .. Hold on there, Cooter. You went to far. Well y'all figure it out. Then compare your results to Pascal's three-sided what'cha'm'callit. Then use common sense and get some probabilities.
If we random-walk along a line, then whether a given step goes right or left effectively depends on a coin flip. If our random walk is n steps, then the number of possible random walks is 2^n, the same as the number of possible outcomes on n flips of a coin. The number of ways we can take exactly r steps to the right is the same as the number of point-flip outcomes that have r Heads--i.e., there are C(n,r) ways to take exactly r steps to the right. When we take exactly r out of n steps to the right, we take exactly n-r steps to the left. So our position is r - (n-r) = 2r - n steps to the right. Our distance from the starting point is therefore | 2 r - n | steps. Advice from Cooter & Bubba: Mess with this situation a little and put some sensible numbers in for r and n, and while you're at it, think about the random walk you did. Get a buddy and a coin and go random-walkin' in the hall. Figure out what this paragraph means.
Rate of Change
If a quantity Q changes by amount `dQ while the time on a running clock changes by `dt, then the average rate of change of Q with respect to clock time t between those to clock times is `dQ / `dt. If we know the average rate at which Q changes between two clock times, then if `dt is the change in the clock time, we can get the change in Q by multiplying the average rate by `dt. Q could be any quantity that changes with clock time--the amount of money in your interest-bearing account, the position of your car, the speed of your cat, the fish population of a pond, the energy falling on your head from the Sun. The rate could be the number of new bugs per transition, the number of newly demented people per day, the speed of your cat, or the rate at which your car's speed changes. And lots of other things. Cooter & Bubba say: These two sentences are sayin' the same thing, but it ain't double-talk. Sometimes you know `dQ and `dt, sometimes you know average rate and `dt, and the way you put what you know together depends on which way you gotta be lookin at it. Think it through in every situation you can think up.
Diffusion can be modeled equally well by transitions between neighboring 'cells' or lattice points, with the transition rules represented in a single tri-diagonal matrix, or by random walks. Diffusion can occur in one dimension along a line, or in 2 or 3 or even more dimensions. Geometrical configurations and the nature of the transition rules determine the behavior of the model, and are chosen to represent the behavior of such things as the diffusion of thermal energy through the interior or the Earth, species within an ecosystem, or nutrients across a cell wall. The introductory model is of bugs migrating at random from tree to tree. Cooter & Bubba say: Be sure to think about them trees plainted in circles. We did that once and you shoulda seen the way the bugs grouped up in the middle.
Sane and Demented
The standard sane/demented model, in which at each transition within a fixed population a given percentage of sane individuals become demented, and another given percentage of demented people become sane, leads to the first example of a stochastic process, with the transitions modeled by a stochastic matrix. The populations of sane and demented exponentially approach limits which can be easily calculated from the initial population and the transition percentages. Powers of the transition matrix remain stochastic and approach a limit in which all the numbers in a given row are identical, and the ratio of the numbers in one row to those in the other are identical to the limiting ratios of the corresponding populations. Cooter & Bubba say: We think we're sane but since they took our Wrigley's away we feel lak we've been demented.
A transition matrix is designed so that when the matrix is multiplied by a column vector representing a population configuration prior to a transition, the resulting column vector represents the population configuration after the transition. Row number i of the transition matrix multiplies the column vector to give the population of 'tree' number i (or cell number i, or lattice point number i), with the numbers in the row being chosen so that the numbers in the column vector are multiplied by the appropriate factors before being added to obtain the new population of 'tree' number i. Cooter & Bubba say: The matrix just organizes yer information so you kin tell quick what percent to take o' what when. Think about that and they's not that hard to figger out.
A trapezoidal graph of y vs. x starts with a finite number of points on the graph. y is represented by the vertical and x by the horizontal axis. The x coordinate of each point is indicated on the x axis. A line segment is drawn from each point to the horizontal axis, and the y value is written somewhere near the middle of the line and just to its right. Each point is connected to the next by a straight line segment and the slope of that segment is written in a rectangular box just above the segment. The segment connecting the two points, along with the two vertical lines extending to the x axis from the segment, form a trapezoid. The area of this trapezoid is equal to the product of its average altitude and its width. This area is calculated and placed with a circle around it, near the middle of the trapezoid. The accumulated area, which is the total of all areas up to and including the present trapezoid, is placed just below the area, in parentheses. For each point except the first and last we find the change in slope at that point, and divide by the x value between the midpoints of the two trapezoids bordering that point, which give us the rate at which the slope changes. This quantity is placed above the graph point between a < and a > symbol (e.g., <34.9> ). Cooter & Bubba say: It's just like a map. If you know how to read 'em they give you all the information you need.
Interpreting Trapezoidal Graphs
The rise of the line at the top of a trapezoid represents the change in the quantity y for the corresponding interval. The run represents the corresponding change in x. The slope of the graph for an interval is the rate at which the quantity y changes with respect to the quantity x for that interval. This interpretation is valid for any quantity y that depends on x. If y is itself the rate at which some quantity changes with respect to x, then the vertical lines bounding a trapezoid represent the rates at the beginning and the end of the corresponding interval, so the average altitude of the trapezoid represents the average rate at which the quantity changes and the area of the trapezoid represents the change in that quantity (not the change in y) for the corresponding x interval. The accumulated area represents the total change in the quantity since the very first x value. Cooter & Bubba say: The main thing is to understand what the rises, runs and average altitudes mean in terms of the situation you find yerself in.
Rate of change of a function
Given a function y = f(x), the average rate of change of y with respect to x over the interval from x1 to x2 is `dy / `dx = change in y / change in x. The change in y is obtained by evaluating y = f(x) at x1 and x2, obtaining y1 = f(x1) and y2 = f(x2). We subtract the first value from the second to obtain the change `dy = y2 - y1 = f(x2) - f(x1). The change in x is x2 - x1. We then find the average rate to be [ f(x2) - f(x1) ] / (x2 - x1). Cooter & Bubba say: Functions is one of the most useful ideas every to come outta mathematics. They lets you talk and thank about thangs in ways that you just couldn't without 'em. That rate thing is a powerful idea too--divide the change in one thing by the change in the thing it depends on.
Rate functions and what we can find from them
If y = r(x) represents the rate at which some quantity Q changes with respect to x, then we can use this function to find the approximate change in Q between x = x1 and x = x2. We first evaluate r at x = x1 and at x = x2, obtaining the rates r(x1) and r(x2) at which Q is changing with respect to x. We then need to multiply the average rate by the change in x. Since we don't know the average rate between x=x1 and x=x2, we estimate it by averaging the rates we found at x1 and x2. Then we can multiply the approximate average rate by the change in x. We note that this gives us the approximate change in Q, not the change in Q. Cooter & Bubba say: Thissum makes us thank about our ol' coon dawg at home, an' the way he'd lap up the water from the ol' water tank when we'd plug good it with a 30-06 round. Or the way we'd figure out the distance coastin' down a hill watchin the speedometer and our pocket watches 'cause our odometer didn't work. If we hadn't done this when we was young, we wouldn't have our Nobel Prizes now. We sure miss the ol' coon dawg, up here at the university whar they won't let us have one. Cain't far the rifles much neither. They say it's bad for our research, or somethin'. But we digress. Be sure you remember that if you got two rates you can average `em to get an approximate average rates, but it's only approximate--accuracy depends on how straight the rate graph is over the x interval. If it dips down you'll estimate too low, if it humps up you'll estimate too high.
Position changes with respect to clock time, strength changes with respect to the number of daily repetitions, grades change with respect to daily study time, velocity changes with respect to clock time, fish populations change with respect to clock time, the time it takes you to make a trip changes with respect to your average speed, the amount your daily $10 will be worth on your birthday in 2010 changes with respect to clock time (think of a slow-moving clock indicating the day and year), the reading on your speedometer changes with respect to clock time. In every case some quantity Q changes with respect to x. Cooter & Bubba say: Think about what Q means and what x means for each of these situations. Then think about why Q would change if x did. Then think about what the rate is telling you.
Velocity functions and what they tell us
Two important ways to analyze sequences of numbers are successive differences and ratio of differences. If the nth successive difference is a nonzero constant then the sequence can be exactly modeled by a polynomial of degree n. If the ratios of a sequence are constant then the sequence can be exactly modeled by an exponential function which is asymptotic to the x axis. If the ratios of the differences of a sequence are constant then the sequence can be exactly modeled by an exponential function which is asymptotic to a line parallel to the x axis.
If S denotes a sequence of numbers then S' denotes the sequence of differences of that sequence. S'' denotes the sequence of differences of S', or the 'second differences' of S.
A cellular automaton is collection of 2-state 'cells' (the two states can be denoted '1' and '0', or 'on' and 'off') which undergo transitions in their states, and a rule by which the states of the cells change at each transition. Many cellular automata fall quickly into a repeating sequence of configurations; such a sequence is called an 'orbit'. The orbit into which an initial configuration falls will always be the same for that specific configuration. The set of all initial configurations that fall into the same orbit is called the 'basin of attraction' for that orbit.
Exponential Population Growth
If the rate of change of a population P is proportional to the population, then we write the rate-of-change equation dP/dt = k P. If we know k then for any population P we can just plug P into this equation to find the rate dP/dt at which the population changes. If we know the initial value of P we can then approximate the change for some time interval `dt, which gives us an approximate value of P at a new clock time. Then using the new value of P we can approximate the change for another interval `dt. We can then continue the process starting from the new approximate value of P. The smaller the time interval `dt the longer it takes to do the calculations but the more accurate our approximations. We can make the approximations much more accurate using a 'predictor-corrector' procedure in which we average the rates at the beginning of the interval and at the end of the interval, the latter as predicted by the standard calculation, in order to make a better final estimate.
The precise population function corresponding to dP / dt = k P is P = Po e^(kt), where A is the population at t = 0. This follows from the nature of the rate function of an exponential.
Flow from a uniform cylinder
Given the diameter of a cylinder full of water and the diameter of a hole through which water escapes at the bottom of the cylinder, assuming that the velocity of the escaping water is v = `sqrt(1960 y) (where y is depth in cm and v is velocity in cm/s), we can calculate the change in depth over a series of short time intervals, obtaining a good approximation to the depth vs. clock time behavior of the system. Experiment indicates that depth is a quadratic function of clock time, so that rate of depth change should be a linear function of clock time.
Rate of change of a quadratic
A quadratic depth function y = a t^2 + b t + c implies a linear rate-of-depth-change function y ' = 2 a t + b. This expression is obtained relatively easily by calculating the average rate of depth change between t and t + `dt, then taking the limit as `dt -> 0. If we know a linear rate-of-change function y ' we can work this process backwards to obtain values of a and b for the depth function, but we won't know the value of c unless we have further information about actual depth. However we can use the depth function with unknown c to find the change in depth between two clock times. The result agrees with that we would obtain by finding the average rate of change and multiplying by the time interval; this is so because of the linearity of the rate function, which implies that the average value is equal to the average of initial and final values.
Rate of change of an exponential function:
The exponential function y = A e^(kt) has rate-of-change function y ' = k A e^(kt). If we approximate changes in y using the rate function y', the nonlinearity, or 'curvature' of the function causes our approximations to either underestimate (for a function which is concave upward) or overestimate (for a function which is concave downward) the change in y.
Rate-of-change relationships and rate-of-change equations
If y stands for population and t for clock time, the unrestricted population growth is characterized by the relationship y ' = dy / dt = k y, which says that the rate of change of y is proportional to the value of y. If y stands for depth and t for clock time, then for the flow-from-a-uniform-cylinder model we have y ' = dy / dt = k `sqrt(y), which is similar to y ' = k y except for the square root. If y ' = k y then y must be an exponential function of form y = A e^(kt) + c, where A and c can be any numbers. If y ' = k `sqrt(y) then y must be a quadratic function y = a t^2 + b t + c, where a and b have a fairly complicated relationship with k and c can be any number. There are other rate-of-change equations, like y ' = k P ( L - P) for restricted population growth, or y ' = k / y^(3/2) for flow from a cone, or y ' = g - k y for a body falling through the air. All the equations mentioned here can be solved using a little Calculus. Many rate-of-change equations can't be solve at all, except by approximations such as the predictor-corrector model.
Logistic population model
When a population has unrestricted access to food and space it tends to grow exponentially, with the rate of population growth proportional to the population: y ' = dy / dt = k y. When the population approaches the carrying capacity L of the environment the rate tends to decrease to 0birth and death numbers tend to balance. This situation can be modeled by the equation y ' = dy / dt = k y ( L y). A graph of y ' vs. y has y ' = 0 when y = 0 and also when y = L, which makes good sense: If there is no population (y = 0) then there will be no growth in population, and if population has reached the carrying capacity L (y = L) there is zero growth. The graph of y ' vs. y has a peak halfway between y = 0 and y = L. If the population is modeled, starting with an initial population P0 < < L, using a small time increment `dt, we see the emergence of a 'sigmoid' curve with asymptotes at the t axis and at the line y = L. The point of maximum slope shows the clock time t and the population y at which the population grows most rapidly.
If we iterate the rule "x is replaced by m x (1 x )" for increasing values of m between m = 0 and m = 4, we find that for m = 0 through m = 3 the iterations approach a constant number. The approach is more rapid for small values of m, and less and less rapid as m approaches 3. The number approached is 0 for small values of m, then at a certain point begins increasing as m continues to increase. A graph of this convergent value vs. m stays on the horizontal axis up to a certain value of m, then begins increasing but at a decreasing rate as m increases to 3. Past m = 3 the same behavior continues for a short ways, but the convergence gets slower and slower, with x values 'bouncing' back and forth between two values gradually which approach a common limit, but taking longer and longer to do so. Finally at a certain value of m the two values no longer come together but remain apart, and the x values approach two limiting values. At this point we see that our graph bifurcatessplits into two branches. As m continues to increase the graph bifurcates again and again, with the x values for increasing values of m eventually 'bouncing' around between 4 values, then 8, then 16, etc. until at a specific value of m the number of possible x values becomes infinite. At this point we say that the mapping has entered the realm of chaos.
The Normal Curve is defined by the function N(z) = 1 / `sqrt(2`pi) e^-(z^2/2), where z stands for the number of standard deviations from the mean. Trapezoidal approximation graphs can be used to show that approximately 34% of the area under the curve lies between z = 0 and z = 1i.e., between the mean and 1 standard deviation greater than the meanand about 14% between z = 1 and z = 2. The symmetry of the distribution shows that similar percentages apply between z = 0 and z = -1, and between z =-1 and z = -2. The Normal curve can be thought of as the limiting curve of the Binomial Distribution, which is characterized by the rows of Pascal's Triangle. As n gets very large, a normalized histogram of the binomial distribution approaches the Normal Curve. The standard deviation of the Binomial Distribution for n coin flips is std dev = `sqrt( .25 n ).
Flow from a cone
Flow from a cone is identical to flow from a cylinder, except that the water lost from the hole at the bottom vacates a slightly beveled cylinder whose radius depends on the depth of the water. At every step we need to calculate the amount of water lost during our time increment `dt and the cross-sectional area of the cone at the present depth. The change in depth during the time increment `dt is then easily calculated.
If interest is compounded continuously the value of an amount A at a time z years in the future is A e^(rz), where r is the annual interest rate. If you make money at a rate of Q dollars per year, and if your money comes in a constant stream, then in time increment `dt you make Q `dt dollars. If t is the clock time of the present instant, and if you want to find the value of your money at clock time Tfinal, then 'future value' at clock time Tfinal of the money you make in time interval `dt near the present instant is the value of Q `dt dollars at a time Tfinal t years in the future. That value is A e^(rz) = Q `dt * e^(r ( Tfinal t) ), which rearranges to Q e^(r (Tfinal-t) ) `dt. So the rate at which your money is accumulating is Q e^( r (Tfinal-t) ). If you can find the amount function corresponding to this rate function, you can evaluate it at t = 0 and at t = Tfinal to see how much money you are going to have at t = Tfinal.