General Math Applications

Welcome to the General_Math_Applications repository! This space is designed to offer users a detailed look into a variety of mathematical methodologies, paired with their practical Python implementations. With modules ranging from Convex Optimization to Wavelets, the repository stands as a resource-rich platform for those interested in the crossroads of mathematics and computational applications.

Contents:

• Convex Optimization: This module employs the CVXPY Python package for optimization tasks, covering linear and quadratic programming. The project showcases CVXPY's capability to address challenges, such as optimizing dietary choices based on constraints.

• Differentiation: Focusing on the variations of functions using derivatives, this module provides insights into symbolic derivatives and numerical approximations. Supported by tools like numpy, sympy, and matplotlib, it demonstrates the implementation and visualization of differentiation techniques.

• Fourier Transform: This project delves into the Fourier transform, emphasizing its role in signal decomposition. With a focus on sound waves, it introduces the PeriodicSampling and SoundWave classes, aiding in the application and visualization of Fourier transformations.

• Gaussian Quadrature: An in-depth look at the Gaussian Quadrature method for approximating definite integrals, this module uses orthogonal polynomials, such as Legendre and Chebyshev. By harnessing the Jacobi Matrix, it provides a comparison of the precision of Gaussian Quadrature with actual integral values.

• Monte Carlo Integration: Adopting the Monte Carlo methods, this module is designed to estimate multi-dimensional integrals where traditional techniques fall short. The module covers areas such as volume estimation of n-dimensional unit balls and emphasizes the adaptability of Monte Carlo methods in high-dimensional integration tasks.

• Polynomial Interpolation: Exploring numerical analysis, this project centers on the Lagrange and Barycentric interpolation methods. Highlighting the construction of the Lagrange interpolating polynomial and the efficiency of the Barycentric method, the module also demonstrates real-world applications, such as air quality data interpolation.

• Wavelets: Presented within a Jupyter Notebook, this module offers an exploration of wavelets for signal and image analysis. With classes like "WaveletTransform", "Signal", and "Image", the project details applications in signal denoising and image compression, ensuring user-friendly interaction with the content.

I invite users to dive into each module and benefit from the blend of mathematical concepts and their corresponding computational applications.