import timeit import random import matplotlib.pyplot as plt # Function to count crossings and nestings def count_crossings_and_nestings (matching): # Your implementation of count_crossings_and_nestings goes here pass # Function to generate random matching def random_matching (length): # Your implementation of random_matching goes here pass # Measure the average run times against the lengths of the matchings x = [n for n in range(1, 51)] # 50 arcs y_matching = [ ] for length in x: avg_runtime = timeit.timeit(lambda: count_crossings_and_nestings (random_matching (length)), number=1000) / 1000 y_matching.append(avg_runtime) # Calculate triangle numbers x_triangle = [j for j in range(1, 51)] y_triangle = [(m ** 2 + m) / 2 for m in range(1, 51)] # Plot both on the same plot plt.plot(x, y_matching, label="Average run times") plt.plot(x_triangle, y_triangle, label="Triangle numbers") plt.xlabel("Length of Matchings / Number of arcs") plt.ylabel("Time / Triangle numbers") plt.title("Average run times vs. Triangle numbers") plt. legend() plt.grid(True) plt.show() Time / Triangle numbers 1200 1000 800 600 400 200 Average run times vs. Triangle numbers Average run times Triangle numbers 0 10 20 30 40 Length of Matchings / Number of arcs 550

Using any programming language, implement the Leibniz series; Leibniz( ) as a recursive function and interactive functions using both a for and while loop.

Run my function on {(1,4),(0,6),(3,5),(2,9),(7,8)}. Is the output what you expect? Find the problem and fix it, writing a correct version of the function. Again, use the same function name, i.e., count_crossings_and_nestings. Demonstrate that the new version works by testing it on the example provided. Now write a well-documented version of your function count_crossings_and_nestings. Add a document string and plenty of comments.

There are three subtleties in the design of enumeration-controlled loops that are presented.

Give the steps to eliminate useless symbols?

12. Find the Laplace transform of the function t*e-2tcos(3t) using MATLAB and print it in the command window.

Answer the given question with a proper explanation and step-by-step solution.

Given a vector of real numbers r = (r1, r2, ..., rm). We can standardize the vector using the formulation: vi = "im, where m ri-m is the mean of the vector r, and s is the standard deviation of r. The vector v = (v1, v2, ..., Un) will be the scaled vector. Write a Python function scale_vec (r) that takes the vector r as input and returns the scaled vector v. Sample inputs and outputs: ● Input: np.array([1, 3, 5]), output: [-1.22474487 0. 1.22474487] • Input: np. array([3.3, 1.2, -2.7, -0.6]), output: [1.35457092 0.40637128 -1.35457092 -0.40637128] Hint: Use numpy.mean and numpy.std with default parameters. # Write your function here. Let's test your function. [ ] import numpy as np print (scale_vec (np.array([1, 3, 5]))) print (scale_vec (np.array([3.3, 1.2, -2.7, -0.6])))

Once you have a list of all the mathematical operations, you may put them in a sensible order.

in python with a print statment