question:Design a function `combine_sorted_arrays` that takes in two sorted arrays of integers and returns a single sorted array that contains all elements from both input arrays. This function should maintain the sorted order. The function shall have the following signature: ```python def combine_sorted_arrays(arr1, arr2): # Your implementation goes here pass ``` # Parameters - `arr1` (list of int): The first sorted array. - `arr2` (list of int): The second sorted array. # Returns - (list of int): The merged sorted array containing all elements from `arr1` and `arr2`. # Constraints - You are not allowed to use any built-in sorting function (e.g., `sorted` in Python). - Both input arrays are already sorted in non-decreasing order. # Example Usage Given: ```python arr1 = [1, 3, 5] arr2 = [2, 4, 6] ``` When you call: ```python result = combine_sorted_arrays(arr1, arr2) ``` The function should return: ```python [1, 2, 3, 4, 5, 6] ``` Here's another example: Given: ```python arr1 = [-5, 0, 7] arr2 = [-10, -3, 8, 10] ``` When you call: ```python result = combine_sorted_arrays(arr1, arr2) ``` The function should return: ```python [-10, -5, -3, 0, 7, 8, 10] ``` # Implementation Notes: - Use a two-pointer approach to efficiently merge the arrays without additional sorting. - Consider edge cases where one of the arrays may be empty.

question:You are required to create a function called `find_factors` that will take an integer as a parameter and return a list of all its factors, in ascending order. Factors of a number are integers that divide the number without leaving any remainder. The primary purpose of this problem is to practice loop constructs and conditional statements in Python, while also getting familiar with basic mathematical operations. Key details: - Function name: `find_factors` - Parameter: `num` (an integer to find the factors of) - The function should return a list of factors of `num` in ascending order. Example: ```python print(find_factors(12)) # Output: [1, 2, 3, 4, 6, 12] print(find_factors(17)) # Output: [1, 17] print(find_factors(20)) # Output: [1, 2, 4, 5, 10, 20] print(find_factors(1)) # Output: [1] ``` #

question:You are provided with a dictionary where the keys are strings representing student names and the values are lists of numerical scores. Your task is to write a function named `calculate_student_averages` which calculates the average score for each student and returns a new dictionary with the student names as keys and their average scores as values. The function `calculate_student_averages` will take one argument: 1. `student_scores`: A dictionary where keys are student names (strings) and values are lists of numbers representing their scores. Your function should: 1. Calculate the average score for each student. 2. Return a new dictionary with the same keys (student names) and their respective average scores as values. For example, given the dictionary: ```python { 'Alice': [85, 90, 78], 'Bob': [92, 88, 79, 85], 'Charlie': [100, 87, 93] } ``` The resulting dictionary should look like: ```python { 'Alice': 84.33, 'Bob': 86.0, 'Charlie': 93.33 } ``` Note: Ensure the rounded average scores have two decimal places.

question:**Title: Implementing the PageRank Algorithm for a Small Web Graph** You need to design a function `pagerank` that implements the PageRank algorithm for a simplified web graph represented by an adjacency matrix. The function should compute the PageRank scores for all pages in the web graph. **Function: `pagerank(adj_matrix, damping_factor, num_iterations)`** **Parameters:** - `adj_matrix` (numpy.ndarray): A 2D NumPy array of shape (N, N) where `adj_matrix[i, j]` is 1 if there is a link from page `i` to page `j`, and 0 otherwise. - `damping_factor` (float): A value between 0 and 1 representing the probability that a user will continue clicking on links. - `num_iterations` (int): The number of iterations to perform in the PageRank algorithm. **Process:** 1. Initialize the PageRank scores for all pages to `1 / N`, where `N` is the number of pages (i.e., nodes). 2. For each iteration: - Update the PageRank score of each page using the formula: ``` PR(p) = (1 - damping_factor) / N + damping_factor * sum(PR(i) / L(i) for all i linking to p) ``` where `L(i)` is the number of outbound links from page `i`. - Ensure that dangling nodes (pages with no outbound links) are handled appropriately by redistributing their PageRank evenly across all pages. **Output:** - Return a NumPy array containing the final PageRank scores after all iterations. **Library to Use:** - `numpy`: For numerical operations and handling arrays. **Example:** ```python import numpy as np def pagerank(adj_matrix, damping_factor, num_iterations): N = adj_matrix.shape[0] pageranks = np.ones(N) / N for _ in range(num_iterations): new_pageranks = np.ones(N) * (1 - damping_factor) / N for i in range(N): for j in range(N): if adj_matrix[j, i] == 1: new_pageranks[i] += damping_factor * pageranks[j] / np.sum(adj_matrix[j]) pageranks = new_pageranks return pageranks # Sample usage adj_matrix = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]]) damping_factor = 0.85 num_iterations = 100 print(pagerank(adj_matrix, damping_factor, num_iterations)) ``` **Note:** This implementation should handle both cases where there are and aren't any outbound links from some pages. Ensure that the handling of dangling nodes (pages with no outbound links) is consistent with the PageRank algorithm.