

In [1]:

    
%matplotlib inline



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import matplotlib.pyplot as plt

Functions, loops and branching

The following exercises let you practice Python syntax. Do not use any packages not in the standard library except for matplotlib.pyplot which has been imported for you.

If you have not done much programming, these exercises will be challenging. Don't give up!

1. Grading (20 points)

Write a function to assign grades to a student such that
```
A = [90 - 100]
B = [80 - 90)
C = [65 - 80)
D = [0, 65]
```

where square brackets indicate inclusive boundaries and parentheses indicate exclusive boundaries. However, studens whose attendance is 12 days or fewer get their grade reduced by one (A to B, B to C, C to D, and D stays D). The function should take a score and an attendance as an argument and return A, B, C or D as appropriate.(10 points)

Count the number of students with each grade from the given scores. (10 points)



In [3]:

    
scores = [ 84,  76,  67,  23,  83,  23,  50, 100,  32,  84,  22,  41,  27,
        29,  71,  85,  47,  77,  39,  25,  85,  69,  22,  66, 100,  92,
        97,  46,  81,  88,  67,  20,  52,  62,  39,  36,  79,  54,  74,
        64,  33,  68,  85,  69,  84,  30,  68, 100,  71,  33,  21,  95,
        92,  72,  53,  50,  31,  82,  53,  68,  49,  37,  40,  21,  94,
        30,  54,  58,  92,  95,  73,  80,  81,  56,  44,  22,  69,  70,
        25,  50,  59,  32,  65,  79,  27,  62,  27,  31,  78,  88,  68,
        53,  79,  69,  89,  38,  80,  55,  92,  55]

attendances = [17, 19, 21, 14, 10, 20, 14,  9,  6, 21,  5, 23, 21,  4,  5, 21, 20,
        2, 14, 14, 21, 22,  3,  0, 11,  0,  0,  4, 20, 14, 23, 16, 24,  5,
       12, 11, 22, 20, 15, 23,  0, 20, 20,  6,  4, 14,  6, 18, 17,  0, 18,
        6,  3, 19, 24,  7,  9, 15, 18, 10,  2, 15, 21,  2,  9, 21, 20, 11,
       24, 23, 14, 22,  4, 12,  7, 19,  6, 18, 23,  6, 14,  6,  1, 12,  7,
       11, 22, 21,  7, 22, 24,  4, 10, 17, 21, 15,  0, 20,  3, 20]



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# Your answer here

2. The period doubling route to chaos. (25 points)

Write a function $f(x, r) = rx(1-x)$ where $r$ is some real number. What is the value of $f(0.3, 2)$? (5 points)
Using a for loop that increments the value of $r$ from 0.1 to 4.0 in steps of 0.1, save the last 50 terms in the iterated sequence $x_{n+1} = f(x_n, r)$ stopping at $x_{1000}$ for each value of $r$. Use $x_0 = 0.13$ for each value of $r$(10 points)
Make a scatter plot of each $(r, x)$ value with $r$ on the horizontal axis and $x$ on the vertical axis. Use the plt.scatter function with s=1 to make the plot. (10 points)



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# Your answer here



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def f(x, r):
    """The discrete logistic function."""
    
    return r*x*(1-x)



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f(0.3, 2)









    Out[7]:





0.42



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xs = []
rs = []
for i in range(1, 41):
    r = i/10.0
    x = 0.13
    for j in range(1001 - 50):
        x = f(x, r)
    for j in range(50):
        x = f(x, r)
        xs.append(x)
        rs.append(r)

plt.scatter(rs, xs, s=1)
pass

3. Digit product sequences. (25 points)

A digit product sequence starts with a number $n_1$, then the ith number, $n_i$ in the sequence is obtained by multiplying the non-zero digits of $n_{i-1}$ and adding that product to $n_{i-1}$. For example, if $n_1$ is 1, we have

1, 2, 4, 8, 16, 22, 26, 38, ...

Find the first 100 numbers in the sequence above. Print out the first 10. (10 points)
Choose a starting $n_1$ different from 1 and generate its sequence. At some point, this new sequence will become identical with the sequence starting from 1 (this is known as joining). For $n_1 = 2$, joining occurs trivially at position 1 (counting from 0) of the 1-sequence. For $n_1 = 3$, the sequence is

3, 6, 12, 14, 18, 26, 38, 62, 74, 102, ...

and we see that the number 26 joins at position 6. Find the join value and position for sequences starting with 19. (15 points)



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# Your answer here



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def digits(n):
    """Return the elements of n in a list."""
    q, r = divmod(n, 10)
    ds = [r]
    while q != 0:
        q, r = divmod(q, 10)
        ds.append(r)
    return ds



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def dps(n, k):
    """Return digit product sequence of length k starting with n."""
    seq = [n]
    for i in range(k):
        prod = 1
        for d in [i for i in digits(n) if i != 0]:
            prod *= d
        n = n + prod
        seq.append(n)
    return seq



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s1 = dps(1, 10000)
print(s1[:10])









    



[1, 2, 4, 8, 16, 22, 26, 38, 62, 74]



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s3 = dps(3, 100)
for s in s3:
    if s in s1:
        print(s, s1.index(s))
        break



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s11 = dps(19, 100)
for s in s11:
    if s in s1:
        print(s, s1.index(s))
        break

4. The E. coli genome. (30 points)

Write a program to download the E. coli genomic sequence in FASTA format (5 points)
Open the downloaded file and read the entire sequence into a single string variable called seq, discarding the header information. Also remove any newline characters \n from seq. Print the first 100 letters in seq (10 points)
Find and report the starting index (counting from zero) and length of the longest contiguous sequence of any nucleotide in seq. If there are ties, report the starting index and length of the last contiguous sequence found. For example, in ATGACCCCCG the longest contiguous sequence is CCCC which starts at index 4 and has length 5. (15 points)



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# Your answer here



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import urllib.request

url = "https://www.genome.wisc.edu/pub/sequence/U00096.2.fas"

filename = 'ecoli.fas'
urllib.request.urlretrieve(url, filename)









    Out[16]:





('ecoli.fas', <http.client.HTTPMessage at 0x7fcb207ca208>)



In [17]:

    
with open(filename) as f:
    lines = f.readlines()
seq = ''.join([line.strip() for line in lines[1:]])
seq[:100]









    Out[17]:





'agcttttcattctgactgcaacgggcaatatgtctctgtgtggattaaaaaaagagtgtctgatagcagcttctgaactggttacctgccgtgagtaaat'



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def running(seq):
    """Find starting location and length of the last longest continuous sequence of elements in seq."""
    ref = seq[0]
    count = 0
    best = 0
    idx = 0
    for i in range(1, len(seq)):
        current = seq[i]
        if current == ref:
            count += 1
            if count > best:
                best = count
                idx = i
        else:
            ref = current
            count = 0
    return (idx - best, best + 1)



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s = 'ATGACCCCCG'
running(s)









    Out[19]:





(4, 5)



In [20]:

    
running(seq)









    Out[20]:





(379236, 10)



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