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Coral Quackenbush MAE 6127 February 14, 2007 1. Sites Relative to a Manhattan Distance r Find a formula for N(r), the total number of sites (i,j) within a distance d ≤ r, and B(r), the number of boundary sites with distance d = r. The formula (explicit?) to determine the number of sites (i,j) within an area of 2r2 units is N(r) = 2r2 +2r+1+4r. Which is the number of “inside” sites (2r2 + (2r+1)), plus the boundary sites (4r). The recursive formula that can be used is: sum of previous “r total sites” + 4r. I then used a slider to assist with observing when “r” becomes very large. # of Sites # of Sites TOTAL Inside Boundary SITES r 2r² 2r²+2r+1 4r 2r²+2r+1+ 4r 1 2 5 4 9 2 8 13 8 21 3 18 25 12 37 4 32 41 16 57 5 50 61 20 81 6 72 85 24 109 7 98 113 28 141 93 17298 17485 372 17857 What happens as r becomes very large? As r increases, the number of “inside” locations increases rapidly, (about 200+ for each value of r), while the number of boundary sites is still increasing at a steady rate (4 for each value of r). Pick’s Theorem… I am not sure what exactly is being asked, so my answer may not be clear. r 2r² 2r²+2r+1 4r 2r²+2r+1+ 4r 1000 2000000 2002001 4000 2006001 Coral Quackenbush MAE 6127 February 14, 2007 2. More One Way Manhattan Patterns What is the number of shortest paths… If i and j are even: N(i,j) = COMBIN((i-1)/2+(j+1)/2,(i+1)/2) If i is even and j is odd, or i is odd and j is even: N(i,j)= COMBIN((odd-1)/2+even/2,even/2) If i and j are both odd: Only half the streets and avenues can be used for the shortest paths, therefore: N(i,j) =COMBIN( i/2+j/2,i/2) 5 1 1 3 4 6 10 5 =COMBIN((C$9-1)/2+C7/2,C9/2) =COMBIN(D$9/2+$A2/2,D$9/2) 4 1 1 3 3 6 6 4 =COMBIN((C$9-1)/2+($A3+1)/2,(C$9+1)/2) =COMBIN((D$9-1)/2+A3/2, D9/2) 3 1 1 2 3 3 6 3 =COMBIN((C$9-1)/2+A4/2, C9/2) =COMBIN(D$9/2+A4/2,D$9/2) 2 1 1 2 2 3 3 1 1 1 1 2 1 3 2 =COMBIN((C$9-1)/2+($A5+1)/2,(C$9+1)/2) =COMBIN((D$9-1)/2+A5/2, D9/2) 0 1 1 1 1 1 1 1 =COMBIN((C9-1)/2+A6/2,C9/2) =COMBIN(D$9/2+A6/2,D$9/2) -1 -1 0 1 2 3 4 5 0 =COMBIN((C$9-1)/2+($A7+1)/2,(C$9+1)/2) =COMBIN((D$9-1)/2+A7/2, D9/2) Number of Shortest Paths to Origin Coral Quackenbush MAE 6127 February 14, 2007 3. Probability of 2-die sums Given two die, the possible sums Die 1 will range from 2 – 12, which is 1 2 3 4 5 6 10 possible sums. Probability is 1 2 3 4 5 6 7 ratio of the number of favorable Die 2 2 3 4 5 6 7 8 outcomes to the possible 3 4 5 6 7 8 9 outcomes. If there are two-fair 4 5 6 7 8 9 10 die, then there are a total of 36 5 6 7 8 9 10 11 possible outcomes. For instance, if the desired sum is “4”, the 6 7 8 9 10 11 12 possible values of the die are 1+3, 3+1, and 2+2. There are 3 ways to have a sum of 4 out of 36 possibilities. Therefore, the probability of a sum of 4 will be 3/36 = 1/12. The most likely to occur is 7, because there are 6 possible ways for that sum to occur. Sum 2 3 4 5 6 7 8 9 10 11 12 I Probability 1 2 3 4 5 6 5 4 3 2 1 36 36 36 36 36 36 36 36 36 36 36 used Excel to illustrate the possible sums. I then used the “countif” function to determine the probability. For example: =COUNTIF($C$3:$H$8,B$10) was used to determine the probability of a sum of 2. Coral Quackenbush MAE 6127 February 14, 2007 4. Short history of Brownian motion and diffusion Brownian motion, or movement, is either the random movement of particles suspended in a fluid or the mathematical model used to describe such random movements, often called a Wiener process. An example of where we see this model in everyday life is the fluctuations in the stock market. Itwas named for theScottish botanist Robert Brown. While examining the fertilization process of a flower, Brown noticed small, random fluctiations fo the pollen grains floating in the water. While other researchers had seen this movement, he was the first to study it in 1827, not only with particles of other organic material, but also chips of glass, granite, and smoke particles. Any nubmer of particles subject ot Brownian motion will tend to spread evenly throughout the medium. One early explanation incorrectly attributed the movement to thermal currents in the fluid. Further investigations from the mid to late 1800’s revealed that small particle size, low viscosity of surrounding fluid, and heat led to faster Brownian Motion. Consequently, a new explanation in 1877 suggested that it was due to motion in the molecules of the liquid or gas, fundamental to the kinetic theory of matter. This led Albert Einstein to develop his theory of Bownian Motion. Diffustion is a physical process by shich a substance spreads steadily from regions of higher concentration to regions of lower concentration. It replicates Brownian Motion on the microscopic level. Some relative diffusion processes that can be studied in terms of Brownian Motion include calcium through bone tissue and pollutants through the atmosphere. DRUNKEN SAILOR WALK: A random walk, sometimes called a "drunkard's walk," is a formalisation of the intuitive idea of taking successive steps, each in a random direction. A one-dimensional random walk can also be looked at as a Markov chain whose state space is given by the integers , for some probability , . We can call it a random walk because we may think of it as being a model for an individual walking on a straight line who at each point of time either takes one step to the right with probability p or one step to the left with probability 1 − p. Coral Quackenbush MAE 6127 February 14, 2007 5. Complete random walk spreadsheets as shown in class: 2D with variable step sizes as shown on slide 12 The path our “drunken sailor” will take is random, hence we use the “rand()” function in Excel. The values we want are not integer values, but rather vales that range from -.5 to +.5. Because of that, we will use the sum of the previous step plus the random value minus 0.5 as noted below in the snapshot. The picture to the left illustrates what the sailor’s path would look like if we were watching him from above as he stumbles around after leaving the bar. Coral Quackenbush MAE 6127 February 14, 2007 1D Discrete: Complete the spreadsheet and document the construction and discuss what it demonstrates in your narrative. To show the 1D path for ten steps, I used the random function. First, each step 0 has a size 0. The distance from each successive step is dependent upon the previous step. Using the RAND function, I used the previous step, plus 2 times the integer value of the product of 2 and the random value, and then subtracted one. This is done so each step will be greater than or equal to 1. I then dragged the function for each of the ten steps for 10 different walks. Using the F9 key, I can recalculate the random values. The resulting graph represents the absolute distance traveled after ten steps

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