Merge branch 'master' into modernize-python2-code

File_Transfer_Protocol/ftp_client_server.py

@@ -0,0 +1,58 @@
+# server
+
+import socket                   # Import socket module
+
+port = 60000                    # Reserve a port for your service.
+s = socket.socket()             # Create a socket object
+host = socket.gethostname()     # Get local machine name
+s.bind((host, port))            # Bind to the port
+s.listen(5)                     # Now wait for client connection.
+
+print 'Server listening....'
+
+while True:
+    conn, addr = s.accept()     # Establish connection with client.
+    print 'Got connection from', addr
+    data = conn.recv(1024)
+    print('Server received', repr(data))
+
+    filename='mytext.txt'
+    f = open(filename,'rb')
+    l = f.read(1024)
+    while (l):
+       conn.send(l)
+       print('Sent ',repr(l))
+       l = f.read(1024)
+    f.close()
+
+    print('Done sending')
+    conn.send('Thank you for connecting')
+    conn.close()
+
+
+# client side server
+
+import socket                   # Import socket module
+
+s = socket.socket()             # Create a socket object
+host = socket.gethostname()     # Get local machine name
+port = 60000                    # Reserve a port for your service.
+
+s.connect((host, port))
+s.send("Hello server!")
+
+with open('received_file', 'wb') as f:
+    print 'file opened'
+    while True:
+        print('receiving data...')
+        data = s.recv(1024)
+        print('data=%s', (data))
+        if not data:
+            break
+        # write data to a file
+        f.write(data)
+
+f.close()
+print('Successfully get the file')
+s.close()
+print('connection closed')

Maths/ModularExponential.py

@@ -0,0 +1,20 @@
+def modularExponential(base, power, mod):
+	if power < 0:
+		return -1
+	base %= mod
+	result = 1
+
+	while power > 0:
+		if power & 1:
+			result = (result * base) % mod
+		power = power >> 1
+		base = (base * base) % mod
+	return result
+
+
+def main():
+	print(modularExponential(3, 200, 13))
+
+
+if __name__ == '__main__':
+	main()

Neural_Network/bpnn.py

@@ -0,0 +1,190 @@
+'''
+
+A Framework of Back Propagation Neural Network（BP） model
+
+Easy to use:
+    * add many layers as you want ！！！
+    * clearly see how the loss decreasing
+Easy to expand:
+    * more activation functions
+    * more loss functions
+    * more optimization method
+
+Author: Stephen Lee
+Github : https://github.com/RiptideBo
+Date: 2017.11.23
+
+'''
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+def sigmoid(x):
+    return 1 / (1 + np.exp(-1 * x))
+
+class DenseLayer():
+    '''
+    Layers of BP neural network
+    '''
+    def __init__(self,units,activation=None,learning_rate=None,is_input_layer=False):
+        '''
+        common connected layer of bp network
+        :param units: numbers of neural units
+        :param activation: activation function
+        :param learning_rate: learning rate for paras
+        :param is_input_layer: whether it is input layer or not
+        '''
+        self.units = units
+        self.weight = None
+        self.bias = None
+        self.activation = activation
+        if learning_rate is None:
+            learning_rate = 0.3
+        self.learn_rate = learning_rate
+        self.is_input_layer = is_input_layer
+
+    def initializer(self,back_units):
+        self.weight = np.asmatrix(np.random.normal(0,0.5,(self.units,back_units)))
+        self.bias = np.asmatrix(np.random.normal(0,0.5,self.units)).T
+        if self.activation is None:
+            self.activation = sigmoid
+
+    def cal_gradient(self):
+        if self.activation == sigmoid:
+            gradient_mat = np.dot(self.output ,(1- self.output).T)
+            gradient_activation = np.diag(np.diag(gradient_mat))
+        else:
+            gradient_activation = 1
+        return gradient_activation
+
+    def forward_propagation(self,xdata):
+        self.xdata = xdata
+        if self.is_input_layer:
+            # input layer
+            self.wx_plus_b = xdata
+            self.output = xdata
+            return xdata
+        else:
+            self.wx_plus_b = np.dot(self.weight,self.xdata) - self.bias
+            self.output = self.activation(self.wx_plus_b)
+            return self.output
+
+    def back_propagation(self,gradient):
+
+        gradient_activation = self.cal_gradient() # i * i 维
+        gradient = np.asmatrix(np.dot(gradient.T,gradient_activation))
+
+        self._gradient_weight = np.asmatrix(self.xdata)
+        self._gradient_bias = -1
+        self._gradient_x = self.weight
+
+        self.gradient_weight = np.dot(gradient.T,self._gradient_weight.T)
+        self.gradient_bias = gradient * self._gradient_bias
+        self.gradient = np.dot(gradient,self._gradient_x).T
+        # ----------------------upgrade
+        # -----------the Negative gradient direction --------
+        self.weight = self.weight - self.learn_rate * self.gradient_weight
+        self.bias = self.bias - self.learn_rate * self.gradient_bias.T
+
+        return self.gradient
+
+
+class BPNN():
+    '''
+    Back Propagation Neural Network model
+    '''
+    def __init__(self):
+        self.layers = []
+        self.train_mse = []
+        self.fig_loss = plt.figure()
+        self.ax_loss = self.fig_loss.add_subplot(1,1,1)
+
+    def add_layer(self,layer):
+        self.layers.append(layer)
+
+    def build(self):
+        for i,layer in enumerate(self.layers[:]):
+            if i < 1:
+                layer.is_input_layer = True
+            else:
+                layer.initializer(self.layers[i-1].units)
+
+    def summary(self):
+        for i,layer in enumerate(self.layers[:]):
+            print('------- layer %d -------'%i)
+            print('weight.shape ',np.shape(layer.weight))
+            print('bias.shape ',np.shape(layer.bias))
+
+    def train(self,xdata,ydata,train_round,accuracy):
+        self.train_round = train_round
+        self.accuracy = accuracy
+
+        self.ax_loss.hlines(self.accuracy, 0, self.train_round * 1.1)
+
+        x_shape = np.shape(xdata)
+        for round_i in range(train_round):
+            all_loss = 0
+            for row in range(x_shape[0]):
+                _xdata = np.asmatrix(xdata[row,:]).T
+                _ydata = np.asmatrix(ydata[row,:]).T
+
+                # forward propagation
+                for layer in self.layers:
+                    _xdata = layer.forward_propagation(_xdata)
+
+                loss, gradient = self.cal_loss(_ydata, _xdata)
+                all_loss = all_loss + loss
+
+                # back propagation
+                # the input_layer does not upgrade
+                for layer in self.layers[:0:-1]:
+                    gradient = layer.back_propagation(gradient)
+
+            mse = all_loss/x_shape[0]
+            self.train_mse.append(mse)
+
+            self.plot_loss()
+
+            if mse < self.accuracy:
+                print('----达到精度----')
+                return mse
+
+    def cal_loss(self,ydata,ydata_):
+        self.loss = np.sum(np.power((ydata - ydata_),2))
+        self.loss_gradient = 2 * (ydata_ - ydata)
+        # vector (shape is the same as _ydata.shape)
+        return self.loss,self.loss_gradient
+
+    def plot_loss(self):
+        if self.ax_loss.lines:
+            self.ax_loss.lines.remove(self.ax_loss.lines[0])
+        self.ax_loss.plot(self.train_mse, 'r-')
+        plt.ion()
+        plt.show()
+        plt.pause(0.1)
+
+
+
+
+def example():
+
+    x = np.random.randn(10,10)
+    y = np.asarray([[0.8,0.4],[0.4,0.3],[0.34,0.45],[0.67,0.32],
+                    [0.88,0.67],[0.78,0.77],[0.55,0.66],[0.55,0.43],[0.54,0.1],
+                    [0.1,0.5]])
+
+    model = BPNN()
+    model.add_layer(DenseLayer(10))
+    model.add_layer(DenseLayer(20))
+    model.add_layer(DenseLayer(30))
+    model.add_layer(DenseLayer(2))
+
+    model.build()
+
+    model.summary()
+
+    model.train(xdata=x,ydata=y,train_round=100,accuracy=0.01)
+
+if __name__ == '__main__':
+    example()

Neural_Network/neuralnetwork_bp3.py

@@ -1,152 +0,0 @@
-#-*- coding:utf-8 -*-
-'''
-Author: Stephen Lee
-Date: 2017.9.21
-
-BP neural network with three layers
-'''
-
-import numpy as np
-import matplotlib.pyplot as plt
-
-class Bpnn():
-
-    def __init__(self,n_layer1,n_layer2,n_layer3,rate_w=0.3,rate_t=0.3):
-        '''
-        :param n_layer1: number of input layer
-        :param n_layer2: number of hiden layer
-        :param n_layer3: number of output layer
-        :param rate_w: rate of weight learning
-        :param rate_t: rate of threshold learning
-        '''
-        self.num1 = n_layer1
-        self.num2 = n_layer2
-        self.num3 = n_layer3
-        self.rate_weight = rate_w
-        self.rate_thre = rate_t
-        self.thre2 = -2*np.random.rand(self.num2)+1
-        self.thre3 = -2*np.random.rand(self.num3)+1
-        self.vji = np.mat(-2*np.random.rand(self.num2, self.num1)+1)
-        self.wkj = np.mat(-2*np.random.rand(self.num3, self.num2)+1)
-
-    def sig(self,x):
-        return 1 / (1 + np.exp(-1*x))
-
-    def sig_plain(self,x):
-        return 1 / (1 + np.exp(-1*x))
-
-    def do_round(self,x):
-        return round(x, 3)
-
-    def trian(self,patterns,data_train, data_teach, n_repeat, error_accuracy, draw_e=False):
-        '''
-        :param patterns: the number of patterns
-        :param data_train: training data x; numpy.ndarray
-        :param data_teach: training data y; numpy.ndarray
-        :param n_repeat: echoes
-        :param error_accuracy: error accuracy
-        :return: None
-        '''
-        data_train = np.asarray(data_train)
-        data_teach = np.asarray(data_teach)
-        # print('-------------------Start Training-------------------------')
-        # print(' - - Shape: Train_Data  ',np.shape(data_train))
-        # print(' - - Shape: Teach_Data  ',np.shape(data_teach))
-        rp = 0
-        all_mse = []
-        mse  = 10000
-        while rp < n_repeat and mse >= error_accuracy:
-            alle = 0
-            final_out = []
-            for g in range(np.shape(data_train)[0]):
-                net_i = data_train[g]
-                out1 = net_i
-
-                net_j = out1 * self.vji.T  - self.thre2
-                out2=self.sig(net_j)
-
-                net_k = out2 * self.wkj.T  - self.thre3
-                out3 = self.sig(net_k)
-
-                # learning process
-                pd_k_all = np.multiply(np.multiply(out3,(1 - out3)),(data_teach[g]-out3))
-                pd_j_all = np.multiply(pd_k_all * self.wkj,np.multiply(out2,1-out2))
-                #upgrade weight
-                self.wkj = self.wkj + pd_k_all.T * out2 *self.rate_weight
-                self.vji = self.vji + pd_j_all.T * out1 * self.rate_weight
-                #upgrade threshold
-                self.thre3 = self.thre3 - pd_k_all * self.rate_thre
-                self.thre2 = self.thre2 - pd_j_all * self.rate_thre
-                #calculate sum of error
-                errors = np.sum(abs((data_teach[g] - out3)))
-
-                alle = alle + errors
-                final_out.extend(out3.getA().tolist())
-            final_out3 = [list(map(self.do_round,each)) for each in final_out]
-
-            rp = rp + 1
-            mse = alle/patterns
-            all_mse.append(mse)
-        def draw_error():
-            yplot = [error_accuracy for i in range(int(n_repeat * 1.2))]
-            plt.plot(all_mse, '+-')
-            plt.plot(yplot, 'r--')
-            plt.xlabel('Learning Times')
-            plt.ylabel('All_mse')
-            plt.grid(True,alpha = 0.7)
-            plt.show()
-        # print('------------------Training Complished---------------------')
-        # print(' - - Training epoch: ', rp, '     - - Mse: %.6f'%mse)
-        # print(' - - Last Output: ', final_out3)
-        if draw_e:
-            draw_error()
-
-    def predict(self,data_test):
-        '''
-        :param data_test: data test, numpy.ndarray
-        :return: predict output data
-        '''
-        data_test = np.asarray(data_test)
-        produce_out = []
-        # print('-------------------Start Testing-------------------------')
-        # print(' - - Shape: Test_Data  ',np.shape(data_test))
-        # print(np.shape(data_test))
-        for g in range(np.shape(data_test)[0]):
-
-            net_i = data_test[g]
-            out1 = net_i
-
-            net_j = out1 * self.vji.T - self.thre2
-            out2 = self.sig(net_j)
-
-            net_k = out2 * self.wkj.T - self.thre3
-            out3 = self.sig(net_k)
-            produce_out.extend(out3.getA().tolist())
-        res = [list(map(self.do_round,each)) for each in produce_out]
-        return np.asarray(res)
-
-
-def main():
-    #example data
-    data_x = [[1,2,3,4],
-              [5,6,7,8],
-              [2,2,3,4],
-              [7,7,8,8]]
-    data_y = [[1,0,0,0],
-              [0,1,0,0],
-              [0,0,1,0],
-              [0,0,0,1]]
-
-    test_x = [[1,2,3,4],
-              [3,2,3,4]]
-
-    #building network model
-    model = Bpnn(4,10,4)
-    #training the model
-    model.trian(patterns=4,data_train=data_x,data_teach=data_y,
-                n_repeat=100,error_accuracy=0.1,draw_e=True)
-    #predicting data
-    model.predict(test_x)
-
-if __name__ == '__main__':
-    main()

Project Euler/Problem 01/sol3.py

@@ -13,8 +13,8 @@ try:
 except NameError:
     raw_input = input  # Python 3
 n = int(raw_input().strip())
-sum=0;
+sum=0
-num=0;
+num=0
 while(1):
     num+=3
     if(num>=n):
@@ -44,4 +44,5 @@ while(1):
     if(num>=n):
         break
     sum+=num
+
 print(sum);

Project Euler/Problem 02/sol1.py

@@ -14,7 +14,9 @@ except NameError:
     raw_input = input  # Python 3
 
 n = int(raw_input().strip())
-i=1; j=2; sum=0
+i=1
+j=2 
+sum=0
 while(j<=n):
     if((j&1)==0): #can also use (j%2==0)
         sum+=j

Project Euler/Problem 03/sol1.py

@@ -18,7 +18,7 @@ def isprime(no):
             return False
     return True
 
-max=0
+maxNumber = 0
 n=int(input())
 if(isprime(n)):
     print(n)
@@ -32,8 +32,8 @@ else:
         for i in range(3,n1,2):
             if(n%i==0):
                 if(isprime(n/i)):
-                    max=n/i
+                    maxNumber = n/i
                     break
                 elif(isprime(i)):
-                    max=i
+                    maxNumber = i
-        print(max)
+        print(maxNumber)

Project Euler/Problem 04/sol1.py

@@ -4,13 +4,26 @@ A palindromic number reads the same both ways. The largest palindrome made from
 Find the largest palindrome made from the product of two 3-digit numbers which is less than N.
 '''
 from __future__ import print_function
-n=int(input())
+limit = int(input("limit? "))
-for i in range(n-1,10000,-1):
+
-    temp=str(i)
+# fetchs the next number
-    if(temp==temp[::-1]):
+for number in range(limit-1,10000,-1):
-        j=999
+
-        while(j!=99):
+    # converts number into string.
-            if((i%j==0) and (len(str(i/j))==3)):
+    strNumber = str(number)
-                print(i)
+
+    # checks whether 'strNumber' is a palindrome.
+    if(strNumber == strNumber[::-1]):
+
+        divisor = 999
+
+        # if 'number' is a product of two 3-digit numbers
+        # then number is the answer otherwise fetch next number.
+        while(divisor != 99): 
+            
+            if((number % divisor == 0) and (len(str(number / divisor)) == 3)):
+
+                print(number)
                 exit(0)
-            j-=1
+
+            divisor -=1

Project Euler/Problem 07/sol2.py

@@ -0,0 +1,16 @@
+# By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13. What is the Nth prime number?
+def isprime(number):
+	for i in range(2,int(number**0.5)+1):
+		if number%i==0:
+			return False
+	return True
+n = int(input('Enter The N\'th Prime Number You Want To Get: ')) # Ask For The N'th Prime Number Wanted
+primes = []
+num = 2
+while len(primes) < n:
+	if isprime(num):
+		primes.append(num)
+		num += 1
+	else:
+		num += 1
+print(primes[len(primes) - 1])

Project Euler/Problem 16/sol1.py

@@ -0,0 +1,15 @@
+power = int(input("Enter the power of 2: "))
+num = 2**power
+
+string_num = str(num)
+
+list_num = list(string_num)
+
+sum_of_num = 0
+
+print("2 ^",power,"=",num)
+
+for i in list_num:
+    sum_of_num += int(i)
+
+print("Sum of the digits are:",sum_of_num)

Project Euler/Problem 20/sol1.py

@@ -0,0 +1,27 @@
+# Finding the factorial.
+def factorial(n):
+    fact = 1
+    for i in range(1,n+1):
+        fact *= i
+    return fact
+
+# Spliting the digits and adding it.
+def split_and_add(number):
+    sum_of_digits = 0
+    while(number>0):
+        last_digit = number % 10
+        sum_of_digits += last_digit
+        number = int(number/10) # Removing the last_digit from the given number.
+    return sum_of_digits
+
+# Taking the user input.
+number = int(input("Enter the Number: "))
+
+# Assigning the factorial from the factorial function.
+factorial = factorial(number)
+
+# Spliting and adding the factorial into answer.
+answer = split_and_add(factorial)
+
+# Printing the answer.
+print(answer)

Project Euler/Problem 29/solution.py

@@ -0,0 +1,34 @@
+def main():
+    """ 
+    Consider all integer combinations of ab for 2 <= a <= 5 and 2 <= b <= 5:
+
+    22=4, 23=8, 24=16, 25=32
+    32=9, 33=27, 34=81, 35=243
+    42=16, 43=64, 44=256, 45=1024
+    52=25, 53=125, 54=625, 55=3125
+    If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:
+
+    4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125
+
+    How many distinct terms are in the sequence generated by ab for 2 <= a <= 100 and 2 <= b <= 100?
+    """
+
+    collectPowers = set()
+
+    currentPow = 0
+
+    N = 101     # maximum limit
+
+    for a in range(2,N):
+
+        for b in range (2,N):
+
+            currentPow = a**b   # calculates the current power
+            collectPowers.add(currentPow)   # adds the result to the set
+            
+
+    print "Number of terms ", len(collectPowers)
+
+
+if __name__ == '__main__':
+    main()

Project Euler/README.md

@@ -49,3 +49,10 @@ PROBLEMS:
     Using the rule above and starting with 13, we generate the following sequence:
     13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
     Which starting number, under one million, produces the longest chain?
+
+16. 2^15 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
+    What is the sum of the digits of the number 2^1000?
+20. n! means n × (n − 1) × ... × 3 × 2 × 1
+    For example, 10! = 10 × 9 × ... × 3 × 2 × 1 = 3628800,
+    and the sum of the digits in the number 10! is 3 + 6 + 2 + 8 + 8 + 0 + 0 = 27.
+    Find the sum of the digits in the number 100!

dynamic_programming/abbreviation.py

@@ -0,0 +1,29 @@
+"""
+https://www.hackerrank.com/challenges/abbr/problem
+You can perform the following operation on some string, :
+
+1. Capitalize zero or more of 's lowercase letters at some index i
+   (i.e., make them uppercase).
+2. Delete all of the remaining lowercase letters in .
+
+Example:
+a=daBcd and b="ABC"
+daBcd -> capitalize a and c(dABCd) -> remove d (ABC)
+"""
+def abbr(a, b):
+    n = len(a)
+    m = len(b)
+    dp = [[False for _ in range(m + 1)] for _ in range(n + 1)]
+    dp[0][0] = True
+    for i in range(n):
+        for j in range(m + 1):
+            if dp[i][j]:
+                if j < m and a[i].upper() == b[j]:
+                    dp[i + 1][j + 1] = True
+                if a[i].islower():
+                    dp[i + 1][j] = True
+    return dp[n][m]
+
+
+if __name__ == "__main__":
+    print abbr("daBcd", "ABC") # expect True

dynamic_programming/minimum_partition.py

@@ -20,7 +20,7 @@ def findMin(arr):
             if (arr[i-1] <= j):
                 dp[i][j] = dp[i][j] or dp[i-1][j-arr[i-1]]
 
-    for j in range(s/2, -1, -1):
+    for j in range(int(s/2), -1, -1):
         if dp[n][j] == True:
             diff = s-2*j
             break;

machine_learning/scoring_functions.py

@@ -61,3 +61,18 @@ def rmsle(predict, actual):
     score = np.sqrt(mean_square_diff)
 
     return score
+
+#Mean Bias Deviation
+def mbd(predict, actual):
+    predict = np.array(predict)
+    actual = np.array(actual)
+
+    difference = predict - actual
+    numerator = np.sum(difference) / len(predict) 
+    denumerator =  np.sum(actual) / len(predict)
+    print str(numerator)
+    print str(denumerator)
+
+    score = float(numerator) / denumerator * 100
+
+    return score

other/primelib.py

@@ -0,0 +1,605 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Thu Oct  5 16:44:23 2017
+
+@author: Christian Bender
+
+This python library contains some useful functions to deal with
+prime numbers and whole numbers.
+
+Overview:
+
+isPrime(number)
+sieveEr(N)
+getPrimeNumbers(N)
+primeFactorization(number)
+greatestPrimeFactor(number)
+smallestPrimeFactor(number)
+getPrime(n)
+getPrimesBetween(pNumber1, pNumber2) 
+
+----
+
+isEven(number)
+isOdd(number)
+gcd(number1, number2)  // greatest common divisor
+kgV(number1, number2)  // least common multiple
+getDivisors(number)    // all divisors of 'number' inclusive 1, number
+isPerfectNumber(number)
+
+NEW-FUNCTIONS
+
+simplifyFraction(numerator, denominator)
+factorial (n) // n!
+fib (n) // calculate the n-th fibonacci term.
+
+-----
+
+goldbach(number)  // Goldbach's assumption
+
+"""
+
+def isPrime(number):
+    """
+        input: positive integer 'number'
+        returns true if 'number' is prime otherwise false.
+    """
+    import math # for function sqrt
+    
+    # precondition
+    assert isinstance(number,int) and (number >= 0) , \
+    "'number' must been an int and positive"
+    
+    status = True
+    
+    # 0 and 1 are none primes. 
+    if number <= 1:
+        status = False
+    
+    for divisor in range(2,int(round(math.sqrt(number)))+1):
+        
+        # if 'number' divisible by 'divisor' then sets 'status'
+        # of false and break up the loop. 
+        if number % divisor == 0:
+            status = False
+            break
+    
+    # precondition
+    assert isinstance(status,bool), "'status' must been from type bool"    
+    
+    return status
+
+# ------------------------------------------
+
+def sieveEr(N):
+    """
+        input: positive integer 'N' > 2
+        returns a list of prime numbers from 2 up to N.
+        
+        This function implements the algorithm called
+        sieve of erathostenes.         
+        
+    """
+    
+    # precondition
+    assert isinstance(N,int) and (N > 2), "'N' must been an int and > 2"
+    
+    # beginList: conatins all natural numbers from 2 upt to N
+    beginList = [x for x in range(2,N+1)]
+
+    ans = [] # this list will be returns.     
+    
+    # actual sieve of erathostenes
+    for i in range(len(beginList)):
+        
+        for j in range(i+1,len(beginList)):
+            
+            if (beginList[i] != 0) and \
+            (beginList[j] % beginList[i] == 0):
+                beginList[j] = 0
+    
+    # filters actual prime numbers.           
+    ans = [x for x in beginList if x != 0]
+    
+    # precondition
+    assert isinstance(ans,list), "'ans' must been from type list"    
+    
+    return ans
+    
+
+# --------------------------------
+
+def getPrimeNumbers(N):
+    """
+        input: positive integer 'N' > 2
+        returns a list of prime numbers from 2 up to N (inclusive)
+        This function is more efficient as function 'sieveEr(...)'
+    """    
+    
+    # precondition
+    assert isinstance(N,int) and (N > 2), "'N' must been an int and > 2"
+    
+    ans = []    
+    
+    # iterates over all numbers between 2 up to N+1 
+    # if a number is prime then appends to list 'ans'
+    for number in range(2,N+1):
+        
+        if isPrime(number):
+            
+            ans.append(number)
+   
+    # precondition
+    assert isinstance(ans,list), "'ans' must been from type list"
+        
+    return ans
+
+
+# -----------------------------------------
+    
+def primeFactorization(number):
+    """
+        input: positive integer 'number' 
+        returns a list of the prime number factors of 'number'
+    """
+
+    import math    # for function sqrt
+    
+    # precondition
+    assert isinstance(number,int) and number >= 0, \
+    "'number' must been an int and >= 0"
+    
+    ans = [] # this list will be returns of the function.
+
+    # potential prime number factors.
+
+    factor = 2    
+
+    quotient = number
+    
+    
+    if number == 0 or number == 1:
+        
+        ans.append(number)
+        
+    # if 'number' not prime then builds the prime factorization of 'number'    
+    elif not isPrime(number):
+    
+        while (quotient != 1):
+            
+            if isPrime(factor) and (quotient % factor == 0):
+                    ans.append(factor)
+                    quotient /= factor
+            else:
+                    factor += 1
+    
+    else:
+        ans.append(number)
+    
+    # precondition
+    assert isinstance(ans,list), "'ans' must been from type list"    
+    
+    return ans
+    
+
+# -----------------------------------------
+    
+def greatestPrimeFactor(number):
+    """
+        input: positive integer 'number' >= 0
+        returns the greatest prime number factor of 'number'
+    """
+    
+    # precondition
+    assert isinstance(number,int) and (number >= 0), \
+    "'number' bust been an int and >= 0"
+    
+    ans = 0    
+    
+    # prime factorization of 'number'
+    primeFactors = primeFactorization(number)
+
+    ans = max(primeFactors)     
+    
+    # precondition
+    assert isinstance(ans,int), "'ans' must been from type int"    
+    
+    return ans
+    
+
+# ----------------------------------------------
+    
+    
+def smallestPrimeFactor(number):
+    """
+        input: integer 'number' >= 0
+        returns the smallest prime number factor of 'number'
+    """
+    
+    # precondition
+    assert isinstance(number,int) and (number >= 0), \
+    "'number' bust been an int and >= 0"
+    
+    ans = 0    
+    
+    # prime factorization of 'number'
+    primeFactors = primeFactorization(number)
+    
+    ans = min(primeFactors)
+
+    # precondition
+    assert isinstance(ans,int), "'ans' must been from type int"    
+    
+    return ans
+    
+    
+# ----------------------
+    
+def isEven(number):
+    """
+        input: integer 'number'
+        returns true if 'number' is even, otherwise false.
+    """   
+
+    # precondition
+    assert isinstance(number, int), "'number' must been an int" 
+    assert isinstance(number % 2 == 0, bool), "compare bust been from type bool"
+    
+    return number % 2 == 0
+    
+# ------------------------
+    
+def isOdd(number):
+    """
+        input: integer 'number'
+        returns true if 'number' is odd, otherwise false.
+    """   
+
+    # precondition
+    assert isinstance(number, int), "'number' must been an int" 
+    assert isinstance(number % 2 != 0, bool), "compare bust been from type bool"
+    
+    return number % 2 != 0
+    
+# ------------------------
+    
+    
+def goldbach(number):
+    """
+        Goldbach's assumption
+        input: a even positive integer 'number' > 2
+        returns a list of two prime numbers whose sum is equal to 'number'
+    """
+    
+    # precondition
+    assert isinstance(number,int) and (number > 2) and isEven(number), \
+    "'number' must been an int, even and > 2"
+    
+    ans = [] # this list will returned
+    
+    # creates a list of prime numbers between 2 up to 'number'
+    primeNumbers = getPrimeNumbers(number)
+    lenPN = len(primeNumbers)    
+
+    # run variable for while-loops.
+    i = 0
+    j = 1
+    
+    # exit variable. for break up the loops
+    loop = True
+    
+    while (i < lenPN and loop):
+        
+        j = i+1;
+        
+        
+        while (j < lenPN and loop):
+            
+            if primeNumbers[i] + primeNumbers[j] == number:
+                loop = False
+                ans.append(primeNumbers[i])
+                ans.append(primeNumbers[j])
+                
+            j += 1;
+            
+            
+        i += 1
+        
+    # precondition
+    assert isinstance(ans,list) and (len(ans) == 2) and \
+    (ans[0] + ans[1] == number) and isPrime(ans[0]) and isPrime(ans[1]), \
+    "'ans' must contains two primes. And sum of elements must been eq 'number'"
+        
+    return ans
+    
+# ----------------------------------------------
+
+def gcd(number1,number2):
+    """
+        Greatest common divisor
+        input: two positive integer 'number1' and 'number2'
+        returns the greatest common divisor of 'number1' and 'number2'
+    """
+    
+    # precondition
+    assert isinstance(number1,int) and isinstance(number2,int) \
+    and (number1 >= 0) and (number2 >= 0), \
+    "'number1' and 'number2' must been positive integer."
+
+    rest = 0    
+    
+    while number2 != 0:
+        
+        rest = number1 % number2
+        number1 = number2
+        number2 = rest
+
+    # precondition
+    assert isinstance(number1,int) and (number1 >= 0), \
+    "'number' must been from type int and positive"    
+    
+    return number1
+    
+# ----------------------------------------------------
+    
+def kgV(number1, number2):
+    """
+        Least common multiple
+        input: two positive integer 'number1' and 'number2'
+        returns the least common multiple of 'number1' and 'number2'
+    """
+    
+    # precondition
+    assert isinstance(number1,int) and isinstance(number2,int) \
+    and (number1 >= 1) and (number2 >= 1), \
+    "'number1' and 'number2' must been positive integer."
+    
+    ans = 1 # actual answer that will be return.
+     
+    # for kgV (x,1)
+    if number1 > 1 and number2 > 1:
+        
+        # builds the prime factorization of 'number1' and 'number2'
+        primeFac1 = primeFactorization(number1)
+        primeFac2 = primeFactorization(number2)
+        
+    elif number1 == 1 or number2 == 1:
+        
+        primeFac1 = []
+        primeFac2 = []
+        ans = max(number1,number2)
+    
+    count1 = 0
+    count2 = 0
+     
+    done = [] # captured numbers int both 'primeFac1' and 'primeFac2'
+    
+    # iterates through primeFac1
+    for n in primeFac1:
+        
+        if n not in done:
+        
+            if n in primeFac2:
+            
+                count1 = primeFac1.count(n)
+                count2 = primeFac2.count(n)
+            
+                for i in range(max(count1,count2)):
+                    ans *= n
+        
+            else:
+                
+                count1 = primeFac1.count(n)
+                
+                for i in range(count1):
+                    ans *= n
+                    
+            done.append(n)
+    
+    # iterates through primeFac2
+    for n in primeFac2:
+        
+        if n not in done:
+            
+            count2 = primeFac2.count(n)
+            
+            for i in range(count2):
+                ans *= n
+                    
+            done.append(n)
+    
+    # precondition
+    assert isinstance(ans,int) and (ans >= 0), \
+    "'ans' must been from type int and positive"    
+    
+    return ans
+    
+# ----------------------------------
+    
+def getPrime(n):
+    """
+        Gets the n-th prime number.
+        input: positive integer 'n' >= 0
+        returns the n-th prime number, beginning at index 0
+    """
+    
+    # precondition
+    assert isinstance(n,int) and (n >= 0), "'number' must been a positive int"
+    
+    index = 0
+    ans = 2 # this variable holds the answer
+    
+    while index < n:
+        
+        index += 1
+        
+        ans += 1   # counts to the next number     
+        
+        # if ans not prime then
+        # runs to the next prime number. 
+        while not isPrime(ans):
+            ans += 1
+    
+    # precondition
+    assert isinstance(ans,int) and isPrime(ans), \
+    "'ans' must been a prime number and from type int"    
+    
+    return ans
+    
+# ---------------------------------------------------
+    
+def getPrimesBetween(pNumber1, pNumber2):
+    """
+        input: prime numbers 'pNumber1' and 'pNumber2'
+                pNumber1 < pNumber2
+        returns a list of all prime numbers between 'pNumber1' (exclusiv)
+                and 'pNumber2' (exclusiv) 
+    """
+    
+    # precondition
+    assert isPrime(pNumber1) and isPrime(pNumber2) and (pNumber1 < pNumber2), \
+    "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"
+    
+    number = pNumber1 + 1 # jump to the next number
+    
+    ans = [] # this list will be returns.
+    
+    # if number is not prime then
+    # fetch the next prime number. 
+    while not isPrime(number):
+        number += 1
+    
+    while number < pNumber2:
+        
+        ans.append(number)
+        
+        number += 1
+        
+        # fetch the next prime number. 
+        while not isPrime(number):
+            number += 1
+            
+    # precondition
+    assert isinstance(ans,list) and ans[0] != pNumber1 \
+    and ans[len(ans)-1] != pNumber2, \
+    "'ans' must been a list without the arguments"
+            
+    # 'ans' contains not 'pNumber1' and 'pNumber2' !
+    return ans
+    
+# ----------------------------------------------------
+
+def getDivisors(n):
+    """
+        input: positive integer 'n' >= 1
+        returns all divisors of n (inclusive 1 and 'n')
+    """
+    
+    # precondition
+    assert isinstance(n,int) and (n >= 1), "'n' must been int and >= 1"
+
+    from math import sqrt        
+        
+    ans = [] # will be returned.
+    
+    for divisor in range(1,n+1):
+        
+        if n % divisor == 0:
+            ans.append(divisor)
+       
+       
+    #precondition
+    assert ans[0] == 1 and ans[len(ans)-1] == n, \
+    "Error in function getDivisiors(...)"
+    
+    
+    return ans
+
+
+# ----------------------------------------------------
+
+
+def isPerfectNumber(number):
+    """
+        input: positive integer 'number' > 1
+        returns true if 'number' is a perfect number otherwise false.
+    """
+    
+    # precondition
+    assert isinstance(number,int) and (number > 1), \
+    "'number' must been an int and >= 1"
+    
+    divisors = getDivisors(number)
+ 
+    # precondition
+    assert isinstance(divisors,list) and(divisors[0] == 1) and  \
+    (divisors[len(divisors)-1] == number), \
+    "Error in help-function getDivisiors(...)"
+    
+    # summed all divisors up to 'number' (exclusive), hence [:-1]
+    return sum(divisors[:-1]) == number
+
+# ------------------------------------------------------------
+
+def simplifyFraction(numerator, denominator):
+    """
+        input: two integer 'numerator' and 'denominator'
+        assumes: 'denominator' != 0
+        returns: a tuple with simplify numerator and denominator.
+    """      
+    
+    # precondition
+    assert isinstance(numerator, int) and isinstance(denominator,int) \
+    and (denominator != 0), \
+    "The arguments must been from type int and 'denominator' != 0"
+    
+    # build the greatest common divisor of numerator and denominator.
+    gcdOfFraction = gcd(abs(numerator), abs(denominator))
+
+    # precondition
+    assert isinstance(gcdOfFraction, int) and (numerator % gcdOfFraction == 0) \
+    and (denominator % gcdOfFraction == 0), \
+    "Error in function gcd(...,...)"
+    
+    return (numerator // gcdOfFraction, denominator // gcdOfFraction)
+    
+# -----------------------------------------------------------------
+    
+def factorial(n):
+    """
+        input: positive integer 'n'
+        returns the factorial of 'n' (n!)
+    """
+    
+    # precondition
+    assert isinstance(n,int) and (n >= 0), "'n' must been a int and >= 0"
+    
+    ans = 1 # this will be return.
+    
+    for factor in range(1,n+1):
+        ans *= factor
+    
+    return ans
+    
+# -------------------------------------------------------------------
+    
+def fib(n):
+    """
+        input: positive integer 'n'
+        returns the n-th fibonacci term , indexing by 0
+    """  
+    
+    # precondition
+    assert isinstance(n, int) and (n >= 0), "'n' must been an int and >= 0"
+    
+    tmp = 0
+    fib1 = 1
+    ans = 1 # this will be return
+    
+    for i in range(n-1):
+        
+        tmp = ans
+        ans += fib1
+        fib1 = tmp
+        
+    return ans

sorts/normal_distribution_QuickSort_README.md

@@ -1,15 +1,15 @@
-#Normal Distribution QuickSort
+# Normal Distribution QuickSort
 
 
 Algorithm implementing QuickSort Algorithm where the pivot element is chosen randomly between first and last elements of the array and the array elements are taken from a Standard Normal Distribution.
 This is different from the ordinary quicksort in the sense, that it applies more to real life problems , where elements usually follow a normal distribution. Also the pivot is randomized to make it a more generic one.
 
 
-##Array Elements
+## Array Elements
 
 The array elements are taken from a Standard Normal Distribution , having mean = 0 and standard deviation 1.
 
-####The code
+#### The code
 
 ```python
 
@@ -52,7 +52,7 @@ The array elements are taken from a Standard Normal Distribution , having mean =
 
 --
 
-##Plotting the function for Checking 'The Number of Comparisons' taking place between Normal Distribution QuickSort and Ordinary QuickSort 
+## Plotting the function for Checking 'The Number of Comparisons' taking place between Normal Distribution QuickSort and Ordinary QuickSort 
 
 ```python
 >>>import matplotlib.pyplot as plt