Birch Swinnerton Dyer Conjecture Here is the completed Birch Swinnerton Dyer Conjecture. import math import random class Particle: def init(self, position, velocity): self.position = position self.velocity = velocity class Wormhole: def init(self, center, radius): self.center = center self.radius = radius def schrodinger_equation(particles, wormhole): """Calculates the behavior of a group of particles as they travel through a wormhole.""" # Calculate the wave function of each particle. wave_functions = [math.exp(-(particle.position - wormhole.center)2 / wormhole.radius2) for particle in particles] # Calculate the probability distribution of each particle. probability_distributions = [wave_function2 for wave_function in wave_functions] # Calculate the velocity and acceleration of each particle. velocities = [math.gradient(probability_distribution) for probability_distribution in probability_distributions] accelerations = [math.gradient(velocity) for velocity in velocities] return velocities, accelerations def hexagonal_smooth_interpolation(points): """Approximates the path of a particle through a wormhole using hexagonal smooth interpolation.""" # Find the first and last points in the sequence. first_point = points[0] last_point = points[-1] # Calculate the slopes of the line segments that connect the points in the sequence. slopes = [ (points[i + 1][0] - points[i][0]) / (points[i + 1][1] - points[i][1]) for i in range(len(points) - 1) ] # Calculate the x-coordinates of the interpolated points. x_coordinates = [ points[i][0] + slopes[i] * (points[i + 1][1] - points[i][1]) for i in range(len(points) - 1) ] # Return the interpolated points, using hexagonal smooth interpolation. hexagonal_interpolated_points = [] for i in range(len(x_coordinates)): # Calculate the hexagonal coordinates of the interpolated point. hexagonal_x = (x_coordinates[i] * math.sqrt(3)) hexagonal_y = (x_coordinates[i] / 2) + ( first_point[1] + (x_coordinates[i] - first_point[0]) * slopes[0] ) # Convert the hexagonal coordinates to Cartesian coordinates. cartesian_x = (hexagonal_x + hexagonal_y * (1 / 3)) cartesian_y = hexagonal_y # Add the interpolated point to the list of interpolated points. hexagonal_interpolated_points.append((cartesian_x, cartesian_y)) return hexagonal_interpolated_points def fibonacci_numbers(n): """Generates a random sequence of Fibonacci numbers.""" if n == 0: return [] elif n == 1: return [1] else: return fibonacci_numbers(n - 1) + [fibonacci_numbers(n - 2)[-1] + fibonacci_numbers(n - 2)[-2]] def sigmoid_function(x): """Maps the particle's position in the wormhole to a probability distribution.""" return 1 / (1 + math.exp(-x)) def linear_matrix_manipulation(matrix, vector): """Calculates the velocity and acceleration of the particle as it travels through the wormhole.""" return matrix @ vector def euclidean_distance(point1, point2): """Calculates the distance between the particle and the wormhole's exit point.""" x_diff = point1[0] - point2[0] y_diff = point1[1] - point2[1] return math.sqrt(x_diff2 + y_diff**2) def wormhole_algorithm(particles, wormhole, dt, steps): """Simulates the gravitational effects of a wormhole with mass m, number of dimensions n, and radius r.""" paths = [] for _ in range(steps): # Calculate the negative mass field at each particle's position negative_mass_fields = [ -1 / (particle.position - wormhole.center)**2 for particle in particles ]