CTA200H  Problem Set 1

CTA200H  Problem Set 1
1. (Dan Green.) Write a script that finds and replaces a given word in all
the .txt files in the working directory, with the following properties:
a The script name is called with “name find replace” where find and replace are words the user chooses when running the program.
b A new directory named replace is created within the working directory.
c A copy of the .txt files with find is put in the replace directory, but with
all occurrences of ’find’ replaced by ’replace.’ If a file does not contain
find, it is ignored.
2. (Adapted from Newman.) The binomial coefficient is given by
(
n
k
)
=
n!
k! (n − k)! (1)
for k ≥ 1 and 1 for k = 0.
a Write your own routine to generate the binomial coefficients for a given
n and k. Make sure the result is an integer (not a float) and gives the
(correct) value 1 for k = 0. (Hint; you can do a cancellation to make it
more tractable.)
b Use your routine to write out the first 20 lines of Pascal’s triangle. This
gives the coefficients for expanding binomials of the form (x + y)
n.
c Consider a biased (e.g., bent) coin with probability p of coming up
heads, and probability 1 − p of coming up tails. The probability of
obtaining heads k times when flipping the coin n times is given by
(
n
k
)
p
k
(1 − p)
n−k
. (2)
Consider an experiment in which you flip the coin n times and hope
to obtain at least k values of heads. For values of p, n, and k of your
choice, use your routine to calculate the probability of obtaining heads
at least k times in n flips. (Example application: a baseball player with
batting average p = 0.250 has n = 4 attempts to obtain a hit, or at-bats,
during a given game. What is the chance that s/he will obtain at least
one hit during the game, assuming that all at-bats are statistically the
same and uncorrelated?)
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d Simulate the experiment N times for N ∈ {10, 100, 1000}, or a similar
range of your choice. For each value of N, which fraction of experiments
were successful? Consider showing the results in a plot.
3. (Adapted from Newman.) The Bessel function of the first kind of order
m is given by
Jm(x) = 1
π
∫ π
0
dθ cos(mθ − x sin θ). (3)
a Write a routine to calculate this integral directly, using the method
of integration of your choice, using only the library sine and cosine
functions (you may use the library Bessel functions to check the results
of your integral). Plot the results for a few values of m.
b Suppose that light passing through a circular telescope aperture of radius a is focussed on the telescope’s focal plane. If the source is effectively a point on the sky, then the pattern of intensity seen in the focal
plane at a distance q from the optical axis is given by
I(x) = I0
(
2J1(x)
x
)2
(4)
where I0 is the intensity at the center, and x =
2πaq
λR . Here λ is the
wavelength of the light and R is the distance from the aperture to the
focal plane. This is known as the “point spread function” in optical
astronomy, the “beam” in radio astronomy, and the “Airy disk” in general. (The geometrical ratio R/(2a) is known as the “f-number” of the
telescope and typically has values of several.) For a given set of parameters, make an image of the two-dimensional point spread function,
using a monochrome color scheme. Indicate the x and y axes with real
distance units. (It might be hard to see the detail away from the main
beam; if so, try increasing the contrast of your image in order to see the
features on the edge.)
c Download a high-resolution astronomical (or other) image of your choice.
If this image were viewed through a telescope with a finite aperture, each
pixel’s value would get smeared out to neighboring pixels, degrading the
image. This is known as convolution. To simulate how this image would
look when viewed through a telescope, use a canned convolution routine (e.g., scipy.ndimage.filters.convolve(), scipy.signal.convolve2d()) to
convolve the image by your point-spread function. Don’t worry about
being quantitative in this sub-section; rather play around to see what
happens to your image when you convolve it.
4. (George) Scene: Its 3pm and your supervisor has just handed you their
old Fortran77 code that they swear “solved this problem in the 80s” and
“is really smartly written and easy to follow”. After the meeting you take
one look at the code and realize it doesn’t. And it’s not.
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But, you said you’d have results by the group meeting the following day,
so lets see if we can understand what it is doing, fix it up, and get some
results! Also, this code has somehow already been converted into python 3. who
knows
The code can be found in the form of a jupyter notebook in problemset1/question6/question6.ipynb . Go through the examples near the top,
and fix up the code in the final cell.
5. For each point in the complex plane c = x + iy, with −2 < x < 2 and
−2 < y < 2, set z0 = 0 and iterate the equation zi+1 = z
2
i + c. Note
what happens to the zi
’s: some points will remain bounded in absolute
value |z|
2 = ℜ(z)
2 + ℑ(z)
2
, while others will run off to infinity. Make an
image in which your points c that diverge are given one color and those
that stay bounded are given another. (Once you have done this, you can
try coloring the points that diverge using a colorscale that indicates the
iteration number at which the given point diverged.) Try zooming in on
a portion of the image and trying again.
6. Write your own program simulating a space battle. It should have 3 types
of spaceships: standard spaceships, warships, and speeders. Each ship
has lasers, shields and hull strength (and most importantly, a name!).
Warships additionally have high powered missiles which they fire 30% of
the time, while speeders have a 50% chance to dodge incoming shots.
When a spaceship is shot by another ship, it first depletes its shields equal
to the strength of the shot. When it runs out of shields, it takes hull
damage at 50% of the shot strength. Once the hull is breached, the ship
is destroyed!
Your program should include a class Ship that defines the standard spaceship, and two classes which inherit from Ship called Warship and Speeder.
The classes should store the ship’s shield strength, hull strength, laser
power and name. There should be, at a minimum, methods to deal with
when a ship shoots, is shot at, whether the ship is destroyed or not, and
printing a diagnostic summary of the ship’s status.
Your program should instantiate 3 regular ships, 1 warship and 1 speeder.
The spaceships should shoot randomly at each other until only one remains
(targets cannot be themselves nor ships that are already destroyed). Print
a log of the battle as it progresses, and declare a final victor.
7. Submitting Your Assignment - Assignments must be submitted by
github.
(a) If you do not already have a github account, go to github.com and
create one
(b) Create your own public git directory named lastname_firstname_Assignment1,
where you use your firstname and lastname
(c) Make 6 subdirectories, called question1, question2, ..., question6
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(d) In each subdirectory include any python script or ipython notebook
required. An example of this directory setup can be seen at
https://github.com/CITA/CTA200H/tree/master/problemset1/Stein_George_Assignment1
(e) When you have the clean subdirectory system set up and all your
questions completed, email cta200@cita.utoronto.ca and include the
exact command for me to pull your directory system.
eg. git clone git@github.com:USERNAME/lastname_firstname_Assignment1.
(f) Check to make sure that it works (pull it yourself to a new
directory and make sure everything is there) before submitting
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