Contents and Objective

Describing several commands and methods that will be used throughout the simulations
Note: Basic knowledge of programming languages and concepts is assumed. Only specific concepts that are different from, e.g., C++ or Matlab, are provided.
NOTE 2: The following summary is by no means complete or exhaustive, but only provides a short and simplified overview of the commands used throughout the simulations in the lecture. For a detailed introduction please have a look at one of the numerous web-tutorials or books on Python, e.g.,



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Cell Types

There are two types of cells:

Text cells (called 'Markdown'): containing text, allowing use of LaTeX
Math/code cells: where code is being executed

As long as you are just reading the simulations, there is no need to be concerned about this fact.



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Data Structures

In the following sections the basic data structures used in upcoming simulations will be introduced.
Basic types as int, float, string are supposed to be well-known.

Lists

Container-type structure for collecting entities (which may even be of different type)
Defined by key word list( ) or by square brackets with entities being separated by comma
Referenced by index in square brackets; Note: indexing starting at 0
Entries may be changed, appended, sliced,...



In [1]:

    
# defining lists
sport_list = [ 'cycling', 'football', 'fitness' ]
first_prime_numbers = [ 2, 3, 5, 7, 11, 13, 17, 19 ]

# getting contents
sport = sport_list[ 2 ]
third_prime = first_prime_numbers[ 2 ]

# printing
print( 'All sports:', sport_list )
print( 'Sport to be done:', sport )

print( '\nFirst primes:', first_prime_numbers )
print( 'Third prime number:', third_prime )









    



All sports: ['cycling', 'football', 'fitness']
Sport to be done: fitness

First primes: [2, 3, 5, 7, 11, 13, 17, 19]
Third prime number: 5



In [2]:

    
# adapt entries and append new entries
sport_list[ 1 ] = 'swimming'
sport_list.append( 'running' )
first_prime_numbers.append( 23 ) 

# printing
print( 'All sports:', sport_list )
print( 'First primes:', first_prime_numbers )









    



All sports: ['cycling', 'swimming', 'fitness', 'running']
First primes: [2, 3, 5, 7, 11, 13, 17, 19, 23]

Tuples

Similar to lists but "immutable", i.e., entries can be appended, but not be changed
Defined by tuple( ) or by brackets with entities being separated by comma
Referenced by index in square brackets; Note: indexing starting at 0



In [3]:

    
# defining tuple
sport_tuple = ( 'cycling', 'football', 'fitness' )

# getting contents
sport = sport_tuple[ 2 ]

# printing
print( 'All sports:', sport_tuple )
print( 'Sport to be done:', sport )









    



All sports: ('cycling', 'football', 'fitness')
Sport to be done: fitness



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# append new entries
sport_tuple += ( 'running', )

# printing
print( 'All sports:', sport_tuple )
print()

# changing entries will fail
# --> ERROR is being generated on purpose
# --> NOTE: Error is handled by 'try: ... except: ...' statement
try:
    sport_tuple[ 1 ] = 'swimming'
except:
    print('ERROR: Entries within tuples cannot be adapted!')









    



All sports: ('cycling', 'football', 'fitness', 'running')

ERROR: Entries within tuples cannot be adapted!

Dictionaries

Container in which entries are of type: ( key : value )
Defined by key word "dict" or by curly brackets with entities of shape "key : value" being separated by comma
Referenced by key in square brackets --> Note: Indexing by keys instead of indices might be a major advantage (at least sometimes)



In [5]:

    
# defining dictionaries
sports_days = { 'Monday': 'pause', 'Tuesday': 'fitness', 'Wednesday' : 'running', 
            'Thursday' : 'fitness', 'Friday' : 'swimming', 'Saturday' : 'cycling', 
            'Sunday' : 'cycling' }

print( 'Sport by day:', sports_days )
print( '\nOn Tuesday:', sports_days[ 'Tuesday' ])









    



Sport by day: {'Monday': 'pause', 'Tuesday': 'fitness', 'Wednesday': 'running', 'Thursday': 'fitness', 'Friday': 'swimming', 'Saturday': 'cycling', 'Sunday': 'cycling'}

On Tuesday: fitness



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# Changes are made by using the key as identifier
sports_days[ 'Tuesday' ] = 'running'
print( 'Sport by day:', sports_days )









    



Sport by day: {'Monday': 'pause', 'Tuesday': 'running', 'Wednesday': 'running', 'Thursday': 'fitness', 'Friday': 'swimming', 'Saturday': 'cycling', 'Sunday': 'cycling'}

Sets

As characterized by the naming, sets are representing mathematical sets; no double occurences of elements
Defined by keyword "set" of by curly brackets with entities being separated by comma
Note: As in maths, sets don't possess ordering, so there is no indexing of sets!



In [7]:

    
# defining sets
sports_set = { 'fitness', 'running', 'swimming', 'cycling'}
print( sports_set )
print()

# indexing will fail
# --> ERROR is being generated on purpose
try:
    print( sports_set[0] )
except:
    print('ERROR: No indexing of sets!')









    



{'running', 'fitness', 'swimming', 'cycling'}

ERROR: No indexing of sets!



In [8]:

    
# adding elements (or not)
sports_set.add( 'pause' )
print(sports_set)

sports_set.add( 'fitness' )
print(sports_set)









    



{'fitness', 'pause', 'running', 'cycling', 'swimming'}
{'fitness', 'pause', 'running', 'cycling', 'swimming'}



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# union of sets (also: intersection, complement, ...)
all_stuff_set = set( sports_set )
union_of_sets = all_stuff_set.union( first_prime_numbers)

print( union_of_sets )









    



{2, 3, 5, 7, 11, 13, 'pause', 17, 19, 23, 'cycling', 'swimming', 'fitness', 'running'}



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Flow Control

Standards commands as for, while, ...
Functions for specific purposes
Note: Since commands and their concept are quite self-explaining, only short description of syntax is provided

For Loops

for loops in Python allow looping along every so-called iterable as, e.g., list, tuple, dicts.... Note: Not necessarily int
Syntax: for i in iterable:
Note: Blocks are structured by indentation; sub-command (as, e.g., in a loop) are indented



In [10]:

    
# looping in lists simply parsing along the list
for s in sport_list:
    print( s )

print()

# looping in dictionaries happens along keys
for s in sports_days:
    print( '{}: \t{}'.format( s, sports_days[ s ] ) )









    



cycling
swimming
fitness
running

Monday: 	pause
Tuesday: 	running
Wednesday: 	running
Thursday: 	fitness
Friday: 	swimming
Saturday: 	cycling
Sunday: 	cycling

While Loops

while loops in Python are (as usual) constructed by checking condition and exiting loop if condition becomes False
Note: Blocks are structured by indentation; sub-command (as, e.g., in a loop) are indented



In [11]:

    
# initialize variables
sum_primes = 0
_n = 0

# sum primes up to sum-value of 20
while sum_primes < 20:
    
    # add prime of according index
    sum_primes += first_prime_numbers[ _n ]
    
    # increase index
    _n += 1
    
print( 'Sum of first {} primes is {}.'.format( _n, sum_primes ) )









    



Sum of first 5 primes is 28.

Functions

Defined by key-word "def" followed by list of arguments in brackets
Doc string defined directly after "def" by ''' TEXT '''
Values returned by key word "return"; Note: return "value" can be scalar, list, dict, vector, maxtrix,...



In [12]:

    
def get_n_th_prime( n, first_prime_numbers ):
    '''
    DOC String
    
    IN: index of prime number, list of prime numbers
    OUT: n-th prime number
    '''
    
    # do something smart as, e.g., checking that according index really exists
    # "assert" does the job by checking first arg and--if not being TRUE--providing text given as second arg
    try: 
        val = first_prime_numbers[ n - 1 ]
    except:
        return '"ERROR: Index not feasible!"'
    
    # NOTE: since counting starts at 0, (n-1)st number is returned
    # Furthermore, there is no need for a function here; a simple reference would have done the job!
    
    return first_prime_numbers[ n - 1 ]


# show doc string
print( help( get_n_th_prime ) )

# apply functions
N = 3
print( '{}. prime number is {}.'.format( N, get_n_th_prime( N, first_prime_numbers ) ) )

print()

N = 30
print( '{}. prime number is {}.'.format( N, get_n_th_prime( N, first_prime_numbers ) ) )









    



Help on function get_n_th_prime in module __main__:

get_n_th_prime(n, first_prime_numbers)
    DOC String
    
    IN: index of prime number, list of prime numbers
    OUT: n-th prime number

None
3. prime number is 5.

30. prime number is "ERROR: Index not feasible!".



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Several Usefull Commands and Wording

import: importing modules to be used as, e.g., modules for
- plotting (matplotlib),
- numerical calculations (numpy),
  - np.linalg: linear algebra
  - np.random: everything dealing with randomness, as, e.g., realization of different distributions
- scientific math (scipy),
- iterating along an iterable (itertools...

break: quitting current loop
continue: processing to next iteration of current loop

enumerate( x ): generates enumeration tuples of shape ( index, value ) parsing all entries of iterable x
len( x ): length of list/tuple x
range( N ): generates iterable with integers 0...N-1

iterable: something that can be iterated, i.e., " walked through"', as, e.g., lists, tuples, dicts

Module Numpy

In simulations imported as "import numpy as np" such that all functions or submodules within the module are being used as 'np.sub_func_mod'
Optimized for vector operations

np.arange( a, b, step ): generating a numpy array starting at a (included) up to b (not included) with step size 'step' (optional)
np.argmax( a ): returns first index at which a attains its maximum value
np.array( ): standard data type of numpy, essentially correponding to a vector as used in maths
np.average( x ): determines average value of list/array x
np.concatenate( x, y ): generates new vector/tuple ( x, y )
x.copy( ): creates a copy of x
np.corrcoeff( x, y): determines correlation coefficient of x and y
np.correlate( x, y, 'full'): determines correlation function (!) of vectors (signals) x and y
np.cumsum( x ): cumulative sum of x
np.histogram( values, bins ): generates histogram (relative frequency) of 'values' with respect to classes listed in 'bins'
np.isclose( a, b): returns Boolean describing if $a \approx b$ 'within tolerance'
np.linspace( a, b, num_points ): generates vector of 'num_points' equidistant points in $[a, b)$
np.ones( shape ): generates array of ones of shape 'shape'
np.ones_like( x ): generates array of ones of same shape and type as x
np.prod( x ): multiplying all elements of x
np.random.choice( A, size = S ): sampling S entities out of A; replacing elements and probability may be characterized as parameters
np.random.rand( size ): generates 'size' random values being uniformly distributed in $[0,1)$
np.random.randint( a, b, size ): generates 'size' random integers between a (included) and b (not included)
np.random.randn( size ): generates a vector of standard gaussian ($E(X)=0, D^2(X)=1$) random variables of size 'size'
np.sum( x ): summing up all elements of x
np.zeros( shape ): generates array of zeros of shape 'shape'
np.zeros_like( x ): generates array of zeros of same shape and type as x

Module Scipy

Contains several usefull scientific modules

special: several special functions, e.g., Bessel, binomial coefficients,...



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