Homework 01

Overall Info: Due date: 2024-03-25; Returns in terms of a *.py file with comments for the code; By group of 5-6 students

1. Datatypes, Control Flows, Functions

Note

For the following short exercises, no use of numpy.

1.1 Using conditional statements and loops, count the number of numbers between 0 and 10.000 which are divisible by 3 or 7 and print it.

1.2 Write a program that print the following pattern

ooooooooooooooooo                                                       
ooooooooooooooooo                                                       
ooooooooooooooooo                                                       
oooo                                                                    
oooo                                                                    
oooo                                                                    
ooooooooooooooooo                                                       
ooooooooooooooooo                                                       
ooooooooooooooooo                                                       
             oooo                                                       
             oooo                                                       
             oooo                                                       
ooooooooooooooooo                                                       
ooooooooooooooooo                                                       
ooooooooooooooooo

1.3 Given a list ["apple", "orange", "cabage", "lemon", "potato"] extract the subarray containing cabage and potato, and return their indices in the original list. Furthermore, write a program that inverse the order of the list.

1.5 Write a function that takes as input one of the name of the 12 month and return the number of days of this month (february is 28)

1.6 Write a function that takes a list of numbers and return a ordered list. i.e. [1, 4, 2] -> [1, 2, 4]. You can test this function by creating arbitraty list from numpy arrays x = list(np.random.rand(N).

2. Numpy

2.1 Using numpy and its functionalities redo the exercises 1.1 and 1.6

2.2 Create two arrays of numbers between -1 and 1. The first one has consecutive numbers equally spaced by 0.01 while the second one has exactly 100 elements.

2.3 Given natural numbers \(n\) and \(d\), create a \(n\times d\) matrix of the form

[
    [0, 1, ..., d-1],
    [d, d+1, ..., 2d-1]
    ...
    [(n-1)d,...,(nd) -1 ]
]

Note

There are several ways to do it, but remember that loops are not efficient in python. Have a look at the shape manipulations in numpy.

2.4 Create a 10x10 matrix where each line contains all the numbers 0 to 9 but shuffled uniformly randomly (check the random routines of numpy documentation)

2.5 Normalize a random 5x5 matrix such that the smallest element is 0 and largest is 1

2.6 Round a random array 5x5 with two digits after the coma

2.7 Given a random array 100 find the position and value of the closest element to 0.6

2.8 Write a function which given an array of size \(N\) of integers between 0 and 10 returns the histogram of the array by bins of 0.1 (that is number of elements between 0 and 0.1, number of elements between 0.1 and 0.2, ..., between 0.9 and 1

3. Numpy and Plotly: Collatz Conjecture

Collatz Conjecture

Consider the following sequence \(u = (u_i)\) of natural numbers given by

\[ \begin{equation} u_{i+1} = \begin{cases} u_i /2 & \text{if }u_i \text{ is even}\\ 3u_i + 1 & \text{if }u_i \text{ is odd} \end{cases} \end{equation} \]

starting with a number \(u_0 = n \in \mathbb{N}\).

The Collatz Conjecture states that whatever starting integer \(n \in \mathbb{N}\), the sequence will ultimately reach after some time \(1\). This Conjecture is still an open problem, some partial answer in this direction given recently by Terence Tao in 2019.

For \(n \in \mathbb{N}\), denote by

\[ \tau(n) : = \inf\{i \colon u_i = 1\} \]

which representes the first time where the sequence starting form \(n\) reaches \(1\). If this sequence never reaches \(1\) starting with \(n\), then \(\tau(n) = \infty\). Collatz conjecture states that \(\tau(n)<\infty\) for every \(n\).

There is strong confidence that this conjecture is true, and empirically, it holds for every natural number smaller than \(2^{68}\)

3.1 Using the control flow while, given an integer \(n\) write a function that returns a numpy array \((u_0, u_i, \ldots, u_{\tau(n)})\). Mathematically, also write what is the domain and codomain of this function.

3.2 Using plotly, provide a function where given a numpy array \(x = (n_0, \ldots, n_{d-1})\) it plots \(d\) paths \((u_0, \ldots, u_{\tau(n_k)})\) where \(u_0 = n_k\) for \(k=0, \ldots, d-1\).

3.4 Write a third function where given a numpy array \(x = (1, ldots, d)\) it returns the array \((\tau(1), \ldots, \tau(d))\).

3.5 Using bar ploting functionality (fig.add_bar(...)) of plotly, write a function that plot the histogram of the previous function, that is \((l_1, l_2, \ldots)\) where

\[ \begin{equation} l_k = \text{Cardinality}\{n\colon \tau(n) = k, \quad 0\leq n \leq N\} \end{equation} \]

for (depending on the performance of your computer) \(N=1000, 10000, 100000\).

4. Scipy

4.1 Consider the function \(x \mapsto f(x) = x ** 2 + 10 * sin(x)\) for \(x \in \mathbb{R}\)

plot the function
find the minimum

4.2 For the following distributions:

normal with std = 1, 2, 5
student with degree of freedom 2, 3, 4, 6

Plot the pdf and cdf of each class for different parameters and compare