


from __future__ import annotations

UpperCAmelCase    =					10






def  __lowerCAmelCase					(SCREAMING_SNAKE_CASE      )->						Tuple:




						"""simple docstring"""

						snake_case_ =   1
						snake_case_ =   max(__lowerCAmelCase      )
						while placement <= max_digit:
												# declare and initialize empty buckets
												snake_case_ =   [[] for _ in range(__lowerCAmelCase      )]
												# split list_of_ints between the buckets
												for i in list_of_ints:
																		snake_case_ =   int((i / placement) % RADIX      )
																		buckets[tmp].append(__lowerCAmelCase      )
												# put each buckets' contents into list_of_ints
												snake_case_ =   0
												for b in range(__lowerCAmelCase      ):
																		for i in buckets[b]:
																								snake_case_ =   i
																								a += 1
        # move to next
												placement *= RADIX
						return list_of_ints


if __name__ == "__main__":
			import doctest

			doctest.testmod()



from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
UpperCAmelCase    =					2_9979_2458

# Symbols
UpperCAmelCase,							UpperCAmelCase,							UpperCAmelCase,							UpperCAmelCase    =					symbols("""ct x y z""")






def  __lowerCAmelCase					(SCREAMING_SNAKE_CASE      )->						float:




      """simple docstring"""

      if velocity > c:
            raise ValueError('''Speed must not exceed light speed 299,792,458 [m/s]!'''      )
      elif velocity < 1:
            # Usually the speed should be much higher than 1 (c order of magnitude)
            raise ValueError('''Speed must be greater than or equal to 1!'''      )

      return velocity / c






def  __lowerCAmelCase					(SCREAMING_SNAKE_CASE      )->						float:




      """simple docstring"""

      return 1 / sqrt(1 - beta(SCREAMING_SNAKE_CASE      ) ** 2      )






def  __lowerCAmelCase					(SCREAMING_SNAKE_CASE      )->						np.ndarray:




      """simple docstring"""

      return np.array(
          [
              [gamma(SCREAMING_SNAKE_CASE      ), -gamma(SCREAMING_SNAKE_CASE      ) * beta(SCREAMING_SNAKE_CASE      ), 0, 0],
              [-gamma(SCREAMING_SNAKE_CASE      ) * beta(SCREAMING_SNAKE_CASE      ), gamma(SCREAMING_SNAKE_CASE      ), 0, 0],
              [0, 0, 1, 0],
              [0, 0, 0, 1],
          ]      )






def  __lowerCAmelCase					(SCREAMING_SNAKE_CASE      ,      SCREAMING_SNAKE_CASE = None      )->						np.ndarray:




      """simple docstring"""

      if event is None:
            snake_case_ =   np.array([ct, x, y, z]      )  # Symbolic four vector
      else:
            event[0] *= c  # x0 is ct (speed of light * time)

      return transformation_matrix(SCREAMING_SNAKE_CASE      ) @ event


if __name__ == "__main__":
   import doctest

   doctest.testmod()

   # Example of symbolic vector:
   UpperCAmelCase    =					transform(2997_9245)
   print("""Example of four vector: """)
   print(f'''ct\' = {four_vector[0]}''')
   print(f'''x\' = {four_vector[1]}''')
   print(f'''y\' = {four_vector[2]}''')
   print(f'''z\' = {four_vector[3]}''')

   # Substitute symbols with numerical values
   UpperCAmelCase    =					{ct: c, x: 1, y: 1, z: 1}
   UpperCAmelCase    =					[four_vector[i].subs(sub_dict) for i in range(4)]

   print(f'''\n{numerical_vector}''')