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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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