Best Perplexity prompts for Mathematicians

Given a database schema with 15 tables, each containing between 10 and 100 columns, and a query that joins 7 of these tables on multiple columns, write a Python script using the SQL library to analyze the query execution plan, identify potential bottlenecks, and provide recommendations for index creation, statistics gathering, and query rewriting to improve performance by at least 30%. Assume the database is hosted on Snowflake and provide example code snippets for each optimization step.

Using a combination of Python libraries including Pandas, NumPy, and Statsmodels, analyze a time series dataset with 1000 data points, containing both trend and seasonal components, to identify the underlying patterns and relationships. Perform a decomposition of the time series into its constituent parts, estimate the parameters of an ARIMA model, and evaluate its performance using metrics such as mean absolute error and mean squared error. Provide a written summary of the results, including visualizations and recommendations for future analysis.

Design and implement an ETL pipeline using Python and the Apache Beam library to extract data from a REST API, transform the data into a suitable format for analysis, and load it into a data warehouse hosted on Snowflake. The pipeline should handle errors and exceptions, implement data validation and cleansing, and provide logging and monitoring capabilities. Provide example code snippets for each stage of the pipeline and discuss the trade-offs between different design choices.

Using a dataset with 100 features and 1000 samples, compare the performance of three different machine learning models (linear regression, decision tree, and random forest) on a regression task, using metrics such as mean squared error and R-squared. Implement a grid search hyperparameter tuning strategy using the Scikit-learn library to optimize the performance of each model, and provide a written summary of the results, including visualizations and recommendations for model selection and hyperparameter tuning.